Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser .
Enter the email address you signed up with and we'll email you a reset link.
- We're Hiring!
- Help Center


A Douglas–Rachford splitting method for solving equilibrium problems

2012, Nonlinear Analysis: Theory, Methods & Applications
Related Papers
arXiv: Optimization and Control
xuan thanh le
It is well known that the projection method is not convergent for monotone equilibrium problems. Recently Sosa \textit{et al.} in \cite{SS2011} proposed a projection algorithm ensuring convergence for paramonotone equilibrium problems. In this paper we modify this algorithm to obtain a splitting convergent one for the case when the bifunction is the sum of the two ones. At each iteration, two strongly convex subprograms are required to solve separately, one for each component bifunction. We show that the algorithm is convergent for paramonotone bifunction without any Lipschitz type condition as well as H\"older continuity of the involved bifunctions. Furthermore, we show that the ergodic sequence defined by the algorithm's iterates converges to a solution without paramonotonicity property. We use the proposed algorithm to solve a jointly constrained Cournot-Nash model. The computational results show that this algorithm is efficient for the model with a restart strategy.

Optimization
We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.
Quoc Tran Dinh
We propose a new primal-dual splitting method for solving composite inclusions involving Lipschitzian, and parallel-sum-type monotone operators. Our approach extends the method proposed in \cite{Siopt4} to a more general class of monotone inclusions. The main idea is to represent the solution set of both the primal and dual problems using their associated Kuhn-Tucker set, and then develop an iterative projected method to successively approximate a feasible point of the Kuhn-Tucker set. We propose a primal-dual splitting algorithm that features the resolvent of each operator separately. We then prove the weak convergence of this algorithm to a solution of both the primal and dual problems. Applications to systems of monotone inclusions as well as composite convex minimization problems are also investigated.
Advances in Computational Mathematics
SIAM Journal on Optimization
Matthew Tam
In this paper, we introduce a new algorithm for finding a common fixed point of a finite family of continuous pseudocontractive mappings which is a unique solution of some variational inequality problem and whose image under some bounded linear operator is a common solution of some system of equilibrium problems in a real Hilbert space. Our result generalize and improve some well-known results.
Demonstratio Mathematica
ibrahim karahan
In this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.
Fixed Point Theory and Algorithms for Sciences and Engineering
Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.
Computational Optimization and Applications
Renato Monteiro
Journal of Global Optimization
RELATED PAPERS
Advances in Difference Equations
yasir arfat
Yeong-cheng Liou , Rudong Chen
Applied Mathematics & Optimization
Akindele Adebayo Mebawondu
Narin Petrot
Journal of Industrial & Management Optimization
Computers & Mathematics with Applications
muhammad fakhruddin noor
Matrix Methods: Theory, Algorithms and Applications
Oleg Zadvornov
Naseer Shahzad
Journal of Nonlinear Sciences and Applications
Abdul Latif
Mauro Passacantando
Journal of the Nigerian Mathematical Society
shuja rizvi
Annali Dell'universita' Di Ferrara
Computational and Applied Mathematics
rahul shukla
Optimization Letters
Cornell University - arXiv
ADEOLU TAIWO
Mathematical Problems in Engineering
International Journal of Mathematics and Mathematical Sciences
ayed e. Hashoosh
Journal of Applied Mathematics
Journal of Inequalities and Applications
Fixed Point Theory and Applications
Qamrul Hasan Ansari
Van Hien Nguyen
Afrah Abdou
Journal of Mathematics
mohammad farid
Mathematical Methods in The Applied Sciences
Yeong-cheng Liou
Carpathian Journal of Mathematics
Yasir Arfat
RELATED TOPICS
- We're Hiring!
- Help Center
- Find new research papers in:
- Health Sciences
- Earth Sciences
- Cognitive Science
- Mathematics
- Computer Science
- Academia ©2023
A Douglas–rachford Splitting Method for Solving Equilibrium Problems 1 1 1 This work was supported by CONICYT under grant FONDECYT N o 3120054.
We propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition. Our algorithm is a consequence of the Douglas–Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.
url] http://www.dim.uchile.cl/ ∼ lbriceno
1 Introduction
In the past years, several works have been devoted to the equilibrium problem
where C is a nonempty closed convex subset of the real Hilbert space H , and H : C × C → R satisfies the following assumption.
Assumption 1.1
The bifunction H : C × C → R satisfies
( ∀ x ∈ C ) H ( x , x ) = 0 .
( ∀ ( x , y ) ∈ C × C ) H ( x , y ) + H ( y , x ) ≤ 0 .
For every x in C , H ( x , ⋅ ) : C → R is lower semicontinuous and convex.
( ∀ ( x , y , z ) ∈ C 3 ) ¯ ¯¯¯¯¯¯ ¯ lim ε → 0 + H ( ( 1 − ε ) x + ε z , y ) ≤ H ( x , y ) .
Throughout this paper, the solution set of ( 1.1 ) will be denoted by S H .
Problem ( 1.1 ) models a wide variety of problems including complementarity problems, optimization problems, feasibility problems, Nash equilibrium problems, variational inequalities, and fixed point problems Alle77 ; Bian96 ; Blum94 ; ComH05 ; Flam97 ; Iuse03 ; Iuse03b ; Konn03 ; Moud02 ; Oett97 . Sometimes the bifunction H is difficult to manipulate but it can be considered as the sum of two simpler bifunctions F and G satisfying Assumption 1.1 (see, for example, Moud10 ). This is the context in which we aim to solve problem ( 1.1 ). Our problem is formulated as follows.
Problem 1.2
Let C be a nonempty closed convex subset of the real Hilbert space H . Suppose that F : C × C → R and G : C × C → R are two bifunctions satisfying Assumption 1.1 . The problem is to
under the assumption that such a solution exists, or equivalently, S F + G ≠ ∅ .
In the particular instance when G ≡ 0 , Problem 1.2 becomes ( 1.1 ) with H = F , which can be solved by the methods proposed in Flam97 ; Iuse03 ; Konn03 ; Moud03 ; MouT99 . These methods are mostly inspired from the proximal fixed point algorithm Mart70 ; Rock76 . The method proposed in Reic11 can be applied to this case when F : ( x , y ) ↦ ⟨ B x ∣ y − x ⟩ and B is maximally monotone. On the other hand, when G : ( x , y ) ↦ ⟨ B x ∣ y − x ⟩ , where B : H → H is a cocoercive operator, weakly convergent splitting methods for solving Problem 1.2 are proposed in ComH05 ; Moud02 . Several methods for solving Problem 1.2 in particular instances of the bifunction G can be found in Nfao1 ; Ceng08 ; Ceng10 ; Konn05 ; Peng10 ; Peng09 ; Peng09b ; Yao09 and the references therein. In the general case, sequential and parallel splitting methods are proposed in Moud09 with guaranteed ergodic convergence. A disadvantage of these methods is the involvement of vanishing parameters that leads to numerical instabilities, which make them of limited use in applications. The purpose of this paper is to address the general case by providing a non-ergodic weakly convergent algorithm which solves Problem 1.2 . The proposed method is a consequence of the Douglas-Rachford splitting method Lion79 ; Svai11 applied to an auxiliary monotone inclusion involving an appropriate choice of maximally monotone operators. This choice of monotone operators allows us to deduce interesting relations between monotone equilibrium problems and monotone inclusions in Hilbert spaces. Some of these relations are deduced from related results in Banach spaces Aoya08 ; Saba11 .
The paper is organized as follows. In Section 2 , we define an auxiliary monotone inclusion which is equivalent to Problem 1.2 under a suitable qualification condition, and some relations between monotone inclusions and equilibrium problems are examined. In Section 3 , we propose a variant of the Douglas–Rachford splitting studied in Livre1 ; Svai11 and we derive our method whose iterates converge weakly to a solution of Problem 1.2 . We start with some notation and useful properties.
Notation and preliminaries Throughout this paper, H denotes a real Hilbert space, ⟨ ⋅ ∣ ⋅ ⟩ denotes its inner product, and ∥ ⋅ ∥ denotes its induced norm. Let A : H → 2 H be a set-valued operator. Then dom A = { x ∈ H ∣ ∣ A x ≠ ∅ } is the domain of A and gra A = { ( x , u ) ∈ H × H ∣ ∣ u ∈ A x } is its graph. The operator A is monotone if
and it is called maximally monotone if its graph is not properly contained in the graph of any other monotone operator in H . In this case, the resolvent of A , J A = ( Id + A ) − 1 , is well defined, single valued, and dom J A = H . The reflection operator R A = 2 J A − Id is nonexpansive.
For a single-valued operator T : dom T ⊂ H → H , the set of fixed points is
We say that T is nonexpansive if
and that T is firmly nonexpansive if
Lemma 1.3 ( cf. ( Opti04 , , Lemma 5.1) )
Let T : dom T = H → H be a nonexpansive operator such that Fix T ≠ ∅ . Let ( μ n ) n ∈ N be a sequence in ] 0 , 1 [ and ( c n ) n ∈ N be a sequence in H such that ∑ n ∈ N μ n ( 1 − μ n ) = + ∞ and ∑ n ∈ N μ n ∥ c n ∥ < + ∞ . Let x 0 ∈ H and set
Then ( x n ) n ∈ N converges weakly to x ∈ Fix T and ( x n − T x n ) n ∈ N converges strongly to 0 .
Now let F : C × C → R be a bifunction satisfying Assumption 1.1 . The resolvent of F is the operator
which is single valued and firmly nonexpansive ( ComH05 , , Lemma 2.12) , and the reflection operator
is nonexpansive.
Let C ⊂ H be nonempty, closed, and convex. We say that 0 lies in the strong relative interior of C , in symbol, 0 ∈ sri C , if ⋃ λ > 0 λ C = ¯ ¯¯¯¯¯¯¯¯¯ ¯ span C . The normal cone of C is the maximally monotone operator
We denote by Γ 0 ( H ) the family of lower semicontinuous convex functions f from H to ] − ∞ , + ∞ ] which are proper in the sense that dom f = { x ∈ H ∣ ∣ f ( x ) < + ∞ } is nonempty. The subdifferential of f ∈ Γ 0 ( H ) is the maximally monotone operator ∂ f : H → 2 H : x ↦ { u ∈ H ∣ ∣ ( ∀ y ∈ H ) ⟨ y − x ∣ u ⟩ + f ( x ) ≤ f ( y ) } . For background on convex analysis, monotone operator theory, and equilibrium problems, the reader is referred to Livre1 ; Blum94 ; ComH05 .
2 Monotone inclusions and equilibrium problems
The basis of the method proposed in this paper for solving Problem 1.2 is that it can be formulated as finding a zero of the sum of two appropriate maximally monotone operators. In this section, we define this auxiliary monotone inclusion and, additionally, we study a class of monotone inclusions which can be formulated as an equilibrium problem.
2.1 Monotone inclusion associated to equilibrium problems
We first recall the maximal monotone operator associated to problem ( 1.1 ) and some related properties. The following result can be deduced from ( Aoya08 , , Theorem 3.5) and ( Saba11 , , Proposition 4.2) , which have been proved in Banach spaces.
Proposition 2.1
Let F : C × C → R be such that Assumption 1.1 holds and set
Then the following hold:
A F is maximally monotone.
S F = zer A F .
For every γ ∈ ] 0 , + ∞ [ , J γ A F = J γ F .
The following proposition allows us to formulate Problem 1.2 as an auxiliary monotone inclusion involving two maximally monotone operators obtained from Proposition 2.1 .
Theorem 2.2
Let C , F , and G be as in Problem 1.2 . Then the following hold.
zer ( A F + A G ) ⊂ S F + G .
Suppose that span ( C − C ) is closed. Then, zer ( A F + A G ) = S F + G .
Proof . (i) . Let x ∈ zer ( A F + A G ) . Thus, x ∈ C and there exists u ∈ A F x ∩ − A G x , which yield, by ( 2.1 ),
Hence, by adding both inequalities we obtain
and, therefore, x ∈ S F + G .
(ii) . Let x ∈ S F + G and define
Assumption 1.1 asserts that f and g are in Γ 0 ( H ) , dom f = dom g = C ≠ ∅ , and since x ∈ S F + G , ( 1.2 ) yields f + g ≥ 0 . Hence, it follows from Assumption 1.1 (i) and ( 2.4 ) that
Thus, Fermat’s rule ( Livre1 , , Theorem 16.2) yields 0 ∈ ∂ ( f + g ) ( x ) . Since span ( C − C ) is closed, we have 0 ∈ sri ( C − C ) = sri ( dom f − dom g ) . Therefore, it follows from ( Livre1 , , Corollary 16.38) that 0 ∈ ∂ f ( x ) + ∂ g ( x ) which implies that there exists u 0 ∈ H such that u 0 ∈ ∂ f ( x ) and − u 0 ∈ ∂ g ( x ) . This is equivalent to
Since Assumption 1.1 (i) and ( 2.4 ) yield f ( x ) = g ( x ) = 0 , we have that ( 2.6 ) is equivalent to
Hence, we conclude from ( 2.1 ) that u 0 ∈ A F x ∩ − A G x , which yields x ∈ zer ( A F + A G ) .
2.2 Equilibrium problems associated to monotone inclusions
We formulate some monotone inclusions as equilibrium problems by defining a bifunction associated to a class of maximally monotone operators. In the following proposition we present this bifunction and its properties.
Proposition 2.3
( cf. ( ComH05 , , Lemma 2.15) ) Let A : H → 2 H be a maximally monotone operator and suppose that C ⊂ int dom A . Set
F A satisfy Assumption 1.1 .
J F A = J A + N C .
Note that the condition C ⊂ int dom A allows us to take the maximum in ( 2.8 ) instead of the supremum. This is a consequence of the weakly compactness of the sets ( A x ) x ∈ C (see ( ComH05 , , Lemma 2.15) for details).
Proposition 2.5
Let A : H → 2 H be a maximally monotone operator and suppose that C ⊂ int dom A . Then zer ( A + N C ) = S F A .
Proof . Indeed, it follows from ( Livre1 , , Proposition 23.38) , Proposition 2.3 , and ( ComH05 , , Lemma 2.15(i)) that
which yields the result.
Note that, in the particular case when dom A = int dom A = C = H , Proposition 2.5 asserts that zer A = S F A , which is a well known result (e.g., see ( Konn01 , , Section 2.1.3) ).
In Banach spaces, the case when C = dom A ⊂ H is studied in ( Aoya08 , , Theorem 3.8) .
The following propositions provide a relation between the operators defined in Propositions 2.1 and 2.3 .
Proposition 2.7
Let B : H → 2 H be maximally monotone and suppose that C ⊂ int dom B . Then, A F B = B + N C .
Proof . Let ( x , u ) ∈ H 2 . It follows from ( 2.8 ) and ( Blum94 , , Lemma 1) (see also ( ComH05 , , Lemma 2.14) ) that
Proposition 2.8
Let G be such that Assumption 1.1 holds, and suppose that C = dom A G = H . Then, F A G ≤ G .
Proof . Let ( x , y ) ∈ C × C and let u ∈ A G x . It follows from ( 2.1 ) that G ( x , y ) + ⟨ x − y ∣ u ⟩ ≥ 0 , which yields
Since C = int dom A G = H , the result follows by taking the maximum in the left side of the inequality.
Note that the equality in Proposition 2.8 does not hold in general. Indeed, let H = R , C = H , and G : ( x , y ) ↦ y 2 − x 2 . It follows from ( ComH05 , , Lemma 2.15(v)) that G satisfy Assumption 1.1 . We have u ∈ A G x ⇔ ⇔ u = 2 x and, hence, for every ( x , y ) ∈ H × H , F A G ( x , y ) = ( y − x ) 2 x = 2 x y − 2 x 2 . In particular, for every y ∈ R ∖ { 0 } , F A G ( 0 , y ) = 0 < y 2 = G ( 0 , y ) .
In the general case when C = dom A ⊂ H , necessary and sufficient conditions for the equality in Proposition 2.8 are provided in ( Aoya08 , , Theorem 4.5) .
3 Algorithm and convergence
Theorem 2.2 (ii) characterizes the solutions to Problem 1.2 as the zeros of the sum of two maximally monotone operators. Our algorithm is derived from the Douglas-Rachford splitting method for solving this auxiliary monotone inclusion. This algorithm was first proposed in Doug56 in finite dimensional spaces when the operators are linear and the generalization to general maximally monotone operators in Hilbert spaces was first developed in Lion79 . Other versions involving computational errors of the resolvents can be found in Opti04 ; Ecks92 . The convergence of these methods needs the maximal monotonicity of the sum of the operators involved, which is not evident to verify ( Livre1 , , Section 24.1) . Furthermore, the iterates in these cases do not converge to a solution but to a point from which we can calculate a solution. These problems were overcame in Svai11 and, later, in ( Livre1 , , Theorem 25.6) , where the convergence of the sequences generated by the proposed methods to a zero of the sum of two set-valued operators is guaranteed by only assuming the maximal monotonicity of each operator. However, in Svai11 the errors considered do not come from inaccuracies on the computation of the resolvent but only from imprecisions in a monotone inclusion, which sometimes could be not manipulable. On the other hand, in ( Livre1 , , Theorem 25.6) the method includes an additional relaxation step but it does not consider inaccuracies in its implementation.
We present a variant of the methods presented in Svai11 and ( Livre1 , , Theorem 25.6) , which has interest in its own right. The same convergence results are obtained by considering a relaxation step as in Ecks92 ; Ecks09 and errors in the computation of the resolvents as in Opti04 ; Ecks92 .
Theorem 3.1
Let A and B be two maximally monotone operators from H to 2 H such that zer ( A + B ) ≠ ∅ . Let γ ∈ ] 0 , + ∞ [ , let ( λ n ) n ∈ N be a sequence in ] 0 , 2 [ , and let ( a n ) n ∈ N and ( b n ) n ∈ N be sequences in H such that b n ⇀ 0 ,
Let x 0 ∈ H and set
Then there exists x ∈ Fix ( R γ A R γ B ) such that the following hold:
J γ B x ∈ zer ( A + B ) .
( R γ A ( R γ B x n ) − x n ) n ∈ N converges strongly to 0 .
( x n ) n ∈ N converges weakly to x .
( y n ) n ∈ N converges weakly to J γ B x .
Proof . Denote T = R γ A R γ B . Since R γ A and R γ B are nonexpansive operators, T is nonexpansive as well. Moreover, since ( Livre1 , , Proposition 25.1(ii)) states that J γ B ( Fix T ) = z e r ( A + B ) , we deduce that Fix T ≠ ∅ . Note that ( 3.2 ) can be rewritten as
where, for every n ∈ N ,
Hence, it follows from the nonexpansivity of J γ A that, for every n ∈ N ,
and, therefore, from ( 3.1 ) and ( 3.4 ) we obtain
Moreover, since the sequence ( λ n ) n ∈ N is in ] 0 , 2 [ , it follows from ( 3.4 ) that ( μ n ) n ∈ N is a sequence in ] 0 , 1 [ and, from ( 3.1 ) we obtain
(i) . This follows from ( Livre1 , , Proposition 25.1(ii)) .
(ii) and (iii) . These follow from Lemma 1.3 .
(iv) . From the nonexpansivity of J γ B we obtain
It follows from (iii) and b n ⇀ 0 that ( x n ) n ∈ N and ( b n ) n ∈ N are bounded, respectively. Hence, ( 3.8 ) implies that ( y n ) n ∈ N is bounded as well. Let y ∈ H be a weak sequential cluster point of ( y n ) n ∈ N , say y k n ⇀ y , and set
It follows from ( 3.2 ) that
For every n ∈ N , we obtain from ( 3.9 )
Hence, (ii) yields ˜ z k n − ˜ y k n → 0 , and, therefore, from ( 3.10 ) we obtain that ˜ u k n + ˜ v k n → 0 . Moreover, it follows from b k n ⇀ 0 , y k n ⇀ y , and ( 3.2 ) that ˜ y k n ⇀ y , and, hence, ˜ z k n ⇀ y . Thus, from (iii) and ( 3.9 ), we obtain ˜ u k n ⇀ y − x and ˜ v k n ⇀ x − y . Altogether, from ( Livre1 , , Corollary 25.5) we deduce that y ∈ zer ( γ A + γ B ) = zer ( A + B ) , ( y , y − x ) ∈ gra γ A , and ( y , x − y ) ∈ gra γ B . Hence, y = J γ B x and y ∈ dom A . Therefore, we conclude that J γ B x is the unique weak sequential cluster point of ( y n ) n ∈ N and then y n ⇀ J γ B x .
Now we present our method for solving Problem 1.2 , which is an application of Theorem 3.1 to the auxiliary monotone inclusion obtained in Theorem 2.2 .
Theorem 3.2
Let C , F , and G be as in Problem 1.2 and suppose that span ( C − C ) is closed. Let γ ∈ ] 0 , + ∞ [ , let ( λ n ) n ∈ N be a sequence in ] 0 , 2 [ , and let ( a n ) n ∈ N and ( b n ) n ∈ N be sequences in H such that b n ⇀ 0 ,
Then there exists x ∈ Fix ( R γ F R γ G ) such that the following hold:
J γ G x ∈ S F + G .
( R γ F ( R γ G x n ) − x n ) n ∈ N converges strongly to 0 .
( y n ) n ∈ N converges weakly to J γ G x .
Proof . Note that, from Theorem 2.2 (ii) , we have that
where A F and A G are defined in ( 2.1 ) and maximally monotone by Proposition 2.1 (i) . In addition, it follows from Proposition 2.1 (iii) that ( 3.13 ) can be written equivalently as ( 3.2 ) with A = A F and B = A G . Hence, the results are derived from Theorem 3.1 , Proposition 2.1 , and Theorem 2.2 .
Note that the closeness of span ( C − C ) and Theorem 2.2 (ii) yields ( 3.14 ), which allows us to apply Theorem 3.1 for obtaining our result. However, it is well known that this qualification condition does not always hold in infinite dimensional spaces. In such cases, it follows from Theorem 2.2 (i) that Theorem 3.2 still holds if zer ( A F + A G ) ≠ ∅ . Conditions for assuring existence of solutions to monotone inclusions can be found in ( Nash , , Proposition 3.2) and Livre1 .
Finally, let us show an application of Theorem 3.2 for solving mixed equilibrium problems. Let f ∈ Γ 0 ( H ) . For every x ∈ H , prox f x is the unique minimizer of the strongly convex function y ↦ f ( y ) + ∥ y − x ∥ 2 / 2 . The operator prox f : H → H thus defined is called the proximity operator.
Example 3.4
In Problem 1.2 , suppose that G : ( x , y ) ↦ f ( y ) − f ( x ) , where f ∈ Γ 0 ( H ) is such that C ⊂ dom f . Then Problem 1.2 becomes
which is known as a mixed equilibrium problem. This problem arises in several applied problems and it can be solved by using some methods developed in Ceng08 ; Peng09b ; Peng10 ; Yao09 . However, all this methods consider implicit steps involving simultaneously F and f , which is not easy to compute in general. On the other hand, it follows from ( ComH05 , , Lemma 2.15(v)) that ( 3.13 ) becomes
which computes separately the resolvent of F and the proximity operator of f . If span ( C − C ) is closed, Theorem 3.2 ensures the weak convergence of the iterates of this method to a solution to ( 3.15 ). Examples of computable proximity operators and resolvents of bifunctions can be found in Smms05 and ComH05 , respectively.
4 Acknowledgement
I thank Professor Patrick L. Combettes for bringing this problem to my attention and for helpful discussions. In addition, I would like to thank the anonymous reviewers for their comments that help improve the manuscript.
- (1) G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Anal. Appl. 58 (1977) 1–10.
- (2) K. Aoyama, Y. Kimura, W. Takahashi, Maximal monotone operators and maximal monotone functions for equilibrium problems, J. Convex Anal. 15 (2008) 395–409.
- (3) H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
- (4) M. Bianchi, S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl. 90 (1996) 31–43.
- (5) E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994) 123–145.
- (6) L. M. Briceño-Arias, Outer approximation method for constrained composite fixed point problems involving Lipschitz pseudo contractive operators, Numer. Funct. Anal. Optim. 32 (2011) 1099–1115.
- (7) L. M. Briceño-Arias, P. L. Combettes, Monotone operator methods for Nash equilibria in non-potential games, in: D. Bailey, H.H. Bauschke, P. Borwein, F. Garvan, M. Théra, J. Vanderwerff, H. Wolkowicz (Eds.) Computational and Analytical Mathematics, Springer, New York, 2013, in press.
- (8) L.-C. Ceng, J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201.
- (9) L.-C. Ceng, J.-C. Yao, A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Anal. 72 (2010) 1922–1937.
- (10) P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization 53 (2004) 475–504.
- (11) P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136.
- (12) P. L. Combettes, V. R. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul. 4 (2005) 1168–1200.
- (13) J. Douglas, H. H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82 (1956) 421–439.
- (14) J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming 55 (1992) 293–318.
- (15) J. Eckstein, B. F. Svaiter, General projective splitting methods for sums of maximal monotone operators, SIAM J. Control Optim. 48 (2009) 787–811.
- (16) S. D. Flåm, A. Antipin, Equilibrium programming using proximal-like algorithms, Math. Programming 78 (1997) 29–41.
- (17) A. N. Iusem, W. Sosa, New existence results for equilibrium problems, Nonlinear Anal. 52 (2003) 621–635.
- (18) A. N. Iusem, W. Sosa, Iterative algorithms for equilibrium problems, Optimization 52 (2003) 301–316.
- (19) I. V. Konnov, Combined relaxation methods for variational inequalities, Springer-Verlag, Berlin, 2001.
- (20) I. V. Konnov, Application of the proximal point method to nonmonotone equilibrium problems, J. Optim. Theory Appl. 119 (2003) 317–333.
- (21) I. V. Konnov, S. Schaible, J. C. Yao, Combined relaxation method for mixed equilibrium problems, J. Optim. Theory Appl. 126 (2005) 309–322.
- (22) P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979) 964–979.
- (23) B. Martinet, Régularisation d’inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle 4 (1970) 154–158.
- (24) A. Moudafi, Mixed equilibrium problems: sensitivity analysis and algorithmic aspect, Comput. Math. Appl. 44 (2002) 1099–1108.
- (25) A. Moudafi, Second-order differential proximal methods for equilibrium problems, J. Inequal. Pure Appl. Math. 4 art. 18 (2003) 7 pp.
- (26) A. Moudafi, On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces, J. Math. Anal. Appl. 359 (2009) 508–513.
- (27) A. Moudafi, Proximal methods for a class of bilevel monotone equilibrium problems, J. Global Optim. 47 (2010) 287–292.
- (28) A. Moudafi, M. Théra, Proximal and dynamical approaches to equilibrium problems, in: M. Théra, R. Tichatschke (Eds.), Ill-Posed Variational Problems and Regularization Techniques, Lecture Notes in Economics and Mathematical Systems, 477, Springer-Verlag, New York, 1999, pp. 187–201.
- (29) W. Oettli, A remark on vector-valued equilibria and generalized monotonicity, Acta Math. Vietnamica 22 (1997) 215–221.
- (30) J. W. Peng, Iterative algorithms for mixed equilibrium problems, strict pseudocontractions and monotone mappings, J. Optim. Theory Appl. 144 (2010) 107–119.
- (31) J.-W. Peng, J.-C. Yao, Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings, Comput. Math. Appl. 58 (2009) 1287–1301.
- (32) J.-W. Peng, J.-C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Math. Comput. Modelling 49 (2009) 1816–1828.
- (33) G. Kassay, S. Reich, S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim. 21 (2011) 1319–1344.
- (34) R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976) 877–898.
- (35) S. Sabach, Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces, SIAM J. Optim. 21 (2011) 1289–1308.
- (36) B. F. Svaiter, On weak convergence of the Douglas-Rachford method, SIAM J. Control Optim. 49 (2011) 280–287.
- (37) Y. Yao, Y. J. Cho, R. Chen, An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems, Nonlinear Anal. 71 (2009) 3363–3373.
Want to hear about new tools we're making? Sign up to our mailing list for occasional updates.
If you find a rendering bug, file an issue on GitHub . Or, have a go at fixing it yourself – the renderer is open source !
For everything else, email us at [email protected] .

- 0">AIMS Journals

- 0">AIMS Press Math Journals
- 0">Book Series
- {{book.nameEn}}
- {{subColumn.name}}
- AIMS Press Math Journals
- Book Series
Journal of Industrial and Management Optimization

Projection methods for solving split equilibrium problems
- Dang Van Hieu ,
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
* Corresponding author: [email protected]
The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.
- Split equilibrium problem ,
- split inverse problem ,
- projection method ,
- diagonal subgradient method .

Figure 1. Algorithm 1 for $ (m, k) = (30, 20) $ and different sequences of $ \beta_n $. The number of iterations is 360,353,339,360,355,376, respectively

Figure 2. Algorithm 1 for ( m ; k ) = (60; 40) and different sequences of β n . The number of iterations is 258,333,336,326,291,293, respectively

Figure 3. Algorithm 1 for ( m ; k ) = (100; 50) and different sequences of β n . The number of iterations is 215,236,283,280,321,290, respectively

Figure 4. Algorithm 1 for $ (m, k) = (150,100) $ and different sequences of $ \beta_n $. The number of iterations is 161,188,219,209,245,264, respectively

Figure 5. Experiment for the algorithms with $ (m, k) = (30, 20) $. The number of iterations is 334,240,379,168,130, respectively

Figure 6. Experiment for the algorithms with ( m ; k ) = (60; 40). The number of iterations is 326,221,292,129,108, respectively

Figure 7. Experiment for the algorithms with ( m ; k ) = (100; 50). The number of iterations is 308,250,356,114, 89, respectively

Figure 8. Experiment for the algorithms with $ (m, k) = (150,100) $. The number of iterations is 254,192,271, 87, 69, respectively
Access History
Figures ( 8 )
Article Metrics
HTML views( 2119 ) PDF downloads( 413 ) Cited by( 0 )

Other Articles By Authors
- Dang Van Hieu
Export File

Algorithm 1 for $ (m, k) = (30, 20) $ and different sequences of $ \beta_n $ . The number of iterations is 360,353,339,360,355,376, respectively
Algorithm 1 for ( m ; k ) = (60; 40) and different sequences of β n . The number of iterations is 258,333,336,326,291,293, respectively
Algorithm 1 for ( m ; k ) = (100; 50) and different sequences of β n . The number of iterations is 215,236,283,280,321,290, respectively
Algorithm 1 for $ (m, k) = (150,100) $ and different sequences of $ \beta_n $ . The number of iterations is 161,188,219,209,245,264, respectively
Experiment for the algorithms with $ (m, k) = (30, 20) $ . The number of iterations is 334,240,379,168,130, respectively
Experiment for the algorithms with ( m ; k ) = (60; 40). The number of iterations is 326,221,292,129,108, respectively
Experiment for the algorithms with ( m ; k ) = (100; 50). The number of iterations is 308,250,356,114, 89, respectively
Experiment for the algorithms with $ (m, k) = (150,100) $ . The number of iterations is 254,192,271, 87, 69, respectively
- Open access
- Published: 04 November 2015
Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces
- Uamporn Witthayarat 1 ,
- Afrah A N Abdou 2 &
- Yeol Je Cho 2 , 3
Fixed Point Theory and Applications volume 2015 , Article number: 200 ( 2015 ) Cite this article
1890 Accesses
7 Citations
Metrics details
In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence \(\{x_{n}\}\) by the shrinking projection method. Our results improve and extend the previous results given in the literature.
1 Introduction
Throughout this paper, let \(\mathbb{R}\) and \(\mathbb{N}\) denote the set of all real numbers and the set of all positive integers, respectively. Let H be a real Hilbert space and C be a nonempty closed convex subset of H .
A mapping \(T:C\times C\to\mathbb{R}\) is said to be asymptotically nonexpansive if there exists a sequence \(\{k_{n}\}\subset[1,\infty)\) with \(\lim_{n\to\infty}k_{n}=1\) such that
for all \(x, y\in C\) . It is easy to see that, if \(k_{n}\equiv1\) , then T is said to be nonexpansive . We denote the set of fixed point of T by \(F(T)\) , that is, \(F(T)=\{x\in C:Tx=x\}\) . There are many iterative methods for solving a fixed point problem corresponding to an asymptotically nonexpansive mapping (see also [ 1 – 3 ]).
Recall that a Hilbert space H satisfies Opial’s condition [ 4 ], that is, for any subsequence \(\{x_{n}\}\subset H\) with \(x_{n}\rightharpoonup x\) , the following inequality
holds for all \(y\in H\) with \(y\neq x\) . Furthermore, a Hilbert space H has a Kadec-Klee property, i.e. , \(x_{n}\rightharpoonup x\) and \(\Vert x_{n}\Vert \to \Vert x\Vert \) imply \(x_{n}\to x\) . In fact, from
we can conclude that a Hilbert space has a Kadec-Klee property.
In 1994, Blum and Oettli [ 5 ] introduced the equilibrium problem which is to find \(x\in C\) such that
They denoted the solution set of problem ( 1.1 ) as \(EP(F)\) . Since the well-known problems were variational problems, complementary problems, fixed point problems, saddle point problems and other problems proposed from the equilibrium problem, it has become the most attractive topic for many mathematicians [ 6 – 8 ]. They have widely spread its applications to other applied disciplines including physics, chemistry, economics and engineering (see, for example, [ 9 – 12 ]).
In 1997, Combettes and Hirstoaga [ 13 ] proposed an iterative method for solving problem ( 1.1 ) by the assumption that \(EP(F)\neq \emptyset\) . Moreover, there are many new iteratively generated sequences for solving this problem together with fixed point problems (see [ 14 – 17 ]).
Later, the so-called split equilibrium problem was introduced (shortly, \(SEP\) ). Let \(H_{1}\) , \(H_{2}\) be two real Hilbert spaces. Let C , Q be closed convex subsets of \(H_{1}\) and \(H_{2}\) , respectively, and let \(A:H_{1}\to H_{2}\) be a bounded linear operator. Further, let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions. The \(SEP\) is to find the element \(x^{*}\in C\) such that
and such that
The solution sets of problems ( 1.2 ) and ( 1.3 ) are symbolized by \(EP(F_{1})\) and \(EP(F_{2})\) , respectively. Therefore, we denote \(\Omega=\{v\in C: v\in EP(F_{1}) \mbox{ such that } Av\in EP(F_{2})\}\) as the solution set of \(SEP\) .
Clearly, the \(SEP\) contains two equilibrium problems, that is, we find out the solution of one equilibrium problem, i.e. , its image under a given bounded linear operator, must be the solution of another equilibrium problem. In order to find a common solution of equilibrium problems, it has been mostly considered in the same spaces. However, we normally found that, in the real-life problems, it may be considered in different spaces. That is how the \(SEP\) works very well for this case (see, for example, [ 18 ]). Moreover, the split variational inequality problem (shortly, \(SVIP\) ) is its special case, which is to find \(x^{*}\in C\) such that
and corresponding to
where \(f:H_{1}\to H_{1}\) and \(g:H_{1}\to H_{2}\) are nonlinear mappings and \(A:H_{1}\to H_{2}\) is a bounded linear operator (see [ 19 ]).
In 2012, He [ 18 ] proposed the new algorithm for solving a split equilibrium problem and investigated the convergence behavior in several ways including both weak and strong convergence. Moreover, they gave some examples and mentioned that there exist many SEPs, and the new methods for solving it further need to be explored in the future. Later, in 2013, Kazmi and Rizvi [ 20 ] considered the iterative method to compute the common approximate solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in the framework of real Hilbert spaces. They generated the sequence iteratively as follows:
for each \(n\geq0\) , where \(A:H_{1}\to H_{2}\) is a bounded linear operator, \(D:C\to H_{1}\) is a τ -inverse strongly monotone mapping, \(F_{1}:C\times C\to\mathbb{R}\) , \(F_{2}:Q\times Q\to\mathbb{R}\) are two bifunctions. They found that, under the sufficient conditions of \(r_{n}\) , \(\lambda_{n}\) , γ , \(\beta_{n}\) and \(\gamma_{n}\) , the generated sequence \(\{ x_{n}\}\) converges strongly to a common solution of all mentioned problems.
Recently, in 2014, Bnouhachem [ 21 ] introduced a new iterative method for solving split equilibrium problem and hierarchical fixed point problems by defining the sequence \(\{x_{n}\}\) as follows:
for each \(n\geq0\) , where S , T are nonexpansive mappings, \(F:C\to C\) is a k -Lipschitz mapping and η -strongly monotone, \(U:C\to C\) is a τ -Lipschitz mapping. Also, they proved some strong convergence theorems for the proposed iteration under some appropriate conditions.
In this paper, motivated and inspired by the results [ 18 , 20 , 21 ] and the recent works in this field, we introduce the shrinking projection method for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems for the proposed new iterative method. In fact, our results improve and extend the results given by some authors.
2 Preliminaries
In this section, we recall some concepts including the assumption which will be needed for the proof of our main result.
Let H be a Hilbert space and C be a nonempty closed convex subset of H . For each \(x\in H\) , there exists a unique nearest point of C , denoted by \(P_{C}x\) , such that
for all \(y\in C\) . \(P_{C}\) is called the metric projection from H onto C . It is well known that \(P_{C}\) is a firmly nonexpansive mapping from H onto C , that is,
for all \(x,y\in H\) . Furthermore, for any \(x\in H\) and \(z\in C\) , \(z=P_{C}x\) if and only if
for all \(y\in C\) . A mapping \(A:C\to H\) is called α - inverse strongly monotone if there exists \(\alpha>0\) such that
for all \(x,y\in H\) . Moreover, we can investigate that, for each \(\lambda \in(0,2\alpha]\) , \(I-\lambda A\) is a nonexpansive mapping of C into H (see [ 22 ]).
In a Hilbert space H , the following identity holds :
for all \(x,y\in H\) and \(\lambda\in[0,1]\) .
Let T be an asymptotically nonexpansive mapping defined on a bounded closed convex subset C of a Hilbert space H . Assume that \(\{x_{n}\}\) is a sequence in C with the following properties :
\(x_{n}\rightharpoonup z\) ;
\(Tx_{n}-x_{n}\rightarrow0\) .
Then \(z\in F(T)\) .
Assumption 2.3
Let \(F:C\times C\to\mathbb{R}\) be a bifunction satisfying the following conditions:
\(F(x,x)=0\) for all \(x\in C\) ;
F is monotone, i.e. , \(F(x,y)+F(y,x)\leq0\) for all \(x,y\in C\) ;
for each \(x,y,z\in C\) , \(\lim_{t\downarrow 0}F(tz+(1-t)x,y)\leq F(x,y)\) ;
for each \(x\in C\) , \(y\mapsto F(x,y)\) is convex and lower semi-continuous.
Let C be a nonempty closed convex subset of a Hilbert space H and \(F:C\times C\to\mathbb{R}\) be a bifunction which satisfies conditions (A1)-(A4). For any \(x\in H\) and \(r>0\) , define a mapping \(T_{r}^{F}:H\to C\) by
Then \(T_{r}^{F}\) is well defined and the following hold :
\(T_{r}^{F}\) is single - valued ;
\(T_{r}^{F}\) is firmly nonexpansive , i . e ., for any \(x,y\in H\) ,
\(F(T_{r}^{F})=EP(F)\) ;
\(EP(F)\) is closed and convex .
3 Main results
In this section, we prove some strong convergence theorems of an iterative algorithm for solving a split equilibrium together with a fixed point problem revolving an asymptotically nonexpansive mapping in the framework of Hilbert spaces.
Theorem 3.1
Let \(H_{1}\) , \(H_{2}\) be two real Hilbert spaces and C , Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\) , respectively . Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)-(A4) and \(F_{2}\) be upper semi - continuous in the first argument . Let \(T:C\to C\) be an asymptotically nonexpansive mapping and \(A:H_{1}\to H_{2}\) be a bounded linear operator . Suppose that \(F(T)\cap\Omega\neq\emptyset\) , where \(\Omega=\{v\in C: v\in EP(F_{1})\textit{ such that }Av\in EP(F_{2})\}\) , and let \(x_{0}\in C\) . For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\) , define a sequence \(\{x_{n}\}\) iteratively as follows :
for each \(n\geq1\) , where \(\theta_{n}=(1-\alpha_{n})(k_{n}^{2}-1)\sup\{\Vert x_{n}-z\Vert ^{2}:z\in\Omega\}\) , \(0\leq\alpha_{n}\leq a<1\) for all \(n\in\mathbb{N}\) , \(0< b\leq r_{n}<\infty\) , \(\gamma\in(0,1/L)\) , L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A . Then the sequence \(\{x_{n}\}\) generated by ( 3.1 ) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\) .
First of all, we investigate that, for each \(n\in\mathbb{N}\) , \(A^{*}(I-T_{r_{n}}^{F_{2}})A\) is a \(\frac{1}{2L}\) -inverse strongly monotone mapping. Since \(T_{r_{n}}^{F_{2}}\) is firmly nonexpansive and \((I-T_{r_{n}}^{F_{2}})\) is \(\frac{1}{2}\) -inverse strongly monotone, it follows that
for all \(x,y\in H\) , from which it can be concluded that \(A^{*}(I-T_{r_{n}}^{F_{2}})A\) is a \(\frac{1}{2L}\) -inverse strongly monotone mapping. Moreover, we claim that since \(\gamma\in(0,\frac{1}{L})\) , \(I-\gamma A^{*}(I-T_{r_{n}}^{F_{2}})A\) is nonexpansive.
Next, we show that \(F(T)\cap\Omega\subset C_{n+1}\) for all \(n\in\mathbb {N}\) . Let \(p\in F(T)\cap\Omega\) , i.e. , \(T_{r_{n}}^{F_{1}}p=p\) and \((I-\gamma A^{*}(I-T_{r_{n}}^{F_{2}})A)p=p\) . By mathematical induction, we have \(p\in C=C_{1}\) and hence \(F(T)\cap\Omega\subset C_{1}\) . Let \(F(T)\cap \Omega\subset C_{k}\) for some \(k\in\mathbb{N}\) . It follows that
where \(M_{k}=\sup\{\Vert x_{k}-z\Vert :z\in\Omega\}\) and \(\theta_{k}=(1-\alpha _{k})(k_{k}^{2}-1)M_{k}^{2}\) . It can be concluded that \(p\in C_{k+1}\) and \(F(T)\cap\Omega\subset C_{k+1}\) and, further, \(F(T)\cap\Omega\subset C_{n}\) for all \(n\in\mathbb{N}\) .
Next, we show that \(C_{n}\) is closed and convex for all \(n\in\mathbb{N}\) . First, it is obvious that \(C_{1}=C\) is closed and convex. By induction, we suppose that \(C_{k}\) is closed and convex for some \(k\in\mathbb{N}\) . Let \(z_{m}\in C_{k+1}\subset C_{k}\) with \(z_{m}\to z\) . Since \(C_{k}\) is closed, it follows that \(x\in C_{k}\) and \(\Vert y_{k}-z_{m}\Vert ^{2} \leq \Vert z_{m}-x_{k}\Vert ^{2}+\theta _{k}\) . Then we have
Letting \(m\to\infty\) , we have
which means that \(z\in C_{k+1}\) . Let \(x,y\in C_{k+1}\subset C_{k}\) and \(z=\alpha x+(1-\alpha)y\) for any \(\alpha\in[0,1]\) . Since \(C_{k}\) is convex, \(z\in C_{k}\) , \(\Vert y_{k}-x\Vert ^{2}\leq \Vert x-x_{k}\Vert ^{2}+\theta_{k}\) and \(\Vert y_{k}-y\Vert ^{2}\leq \Vert x-x_{k}\Vert ^{2}+\theta_{k}\) and so
Therefore, \(z\in C_{k+1}\) and hence \(C_{k+1}\) is closed and convex. It is immediately concluded that \(C_{n}\) is closed and convex for all \(n\in \mathbb{N}\) , which implies that \(\{x_{n}\}\) is well defined.
Next, from \(x_{n}=P_{C_{n}}x_{0}\) , we have
for all \(y\in C_{n}\) . Since \(p\in F(T)\cap\Omega\) , we have
for all \(p\in F(T)\cap\Omega\) , that is, we have
This implies that
for all \(n\in\mathbb{N}\) . From \(x_{n}=P_{C_{n}}x_{0}\) and \(x_{n+1}=P_{C_{n+1}}x_{0}\in C_{n+1}\subset C_{n}\) , we also have
for all \(n\in\mathbb{N}\) , and so we have
Hence we have
that is, \(\Vert x_{n}-x_{0}\Vert \leq \Vert x_{0}-x_{n+1}\Vert \) for all \(n\in\mathbb{N}\) . From ( 3.4 ), it follows that \(\{x_{n}\}\) is bounded and \(\lim_{n\to \infty} \Vert x_{n}-x_{0}\Vert \) exists.
Next, we show that \(\Vert x_{n}-x_{n+1}\Vert \to0\) . From ( 3.5 ), we have
Since the limit of \(\{\Vert x_{n}-x_{0}\Vert \}\) exists, we have
Thus, by ( 3.7 ) and ( 3.14 ), we have
as \(n\to\infty\) . Furthermore, since \(T_{r_{n}}^{F_{1}}\) is firmly nonexpansive, we have
where \(z_{n}=(I-\gamma A^{*}(I-T_{r_{n}}^{F_{2}})A)x_{n}\) . Moreover,
which leads to
Letting \(\rho_{n}=k_{n}-1\) . Then it is clear that \(\rho_{n}\to0\) as \(n\to \infty\) and, by ( 3.9 ), we exactly have
By ( 3.8 ) and \(\rho_{n}\to0\) as \(n\to\infty\) , we have
as \(n\to\infty\) . Furthermore, since A is linear bounded and so is \(A^{*}\) , we can conclude that
Next, we show that \(\|u_{n}-x_{n}\|\to0\) . We investigate the following:
Consequently, by ( 3.12 ), we can conclude that
Next, we show that \(\|T^{n}x_{n}-x_{n}\|\to0\) . We first consider
and since \(x_{n+1}\in C_{n+1}\subset C_{n}\) , we have
which means that
and so \(\|T^{n}u_{n}-x_{n}\|\to0\) . Consider
Therefore, we have \(\|T^{n}x_{n}-x_{n}\|\to0\) as \(n\to\infty\) . Putting \(k_{\infty}=\sup\{k_{n}:n\geq1\}<\infty\) , we deduce that
Hence we have \(\|Tx_{n}-x_{n}\|\to0\) as \(n\to\infty\) . Without loss of generality, since \(\{x_{n}\}\) is bounded, we may assume that \(x_{n}\rightharpoonup x^{*}\) . It is easy to see that \(x^{*}\in C_{n}\) for all \(n\geq1\) . On the other hand, we have
It follows that
Hence \(\|x_{n}\|\to\|x^{*}\|\) . Since every Hilbert space has the Kadec-Klee property, we immediately have \(x_{n}\to x^{*}\) .
Finally, we prove that \(x^{*}\in F(T)\cap\Omega\) . Since \(x_{n}\to x^{*}\) and \(x_{n}-Tx_{n}\to0\) as \(n\to\infty\) , consider
We can see that \(\|x^{*}-Tx^{*}\|=0\) and, further, \(x^{*}\in F(T)\) . Therefore, we have \(x^{*}\in F(T)\) .
Next, we show that \(x^{*}\in\Omega\) . By ( 3.1 ), \(u_{n}=T_{r_{n}}^{F_{1}}(I-\gamma A^{*}(I-T_{r_{n}}^{F_{2}}))\) , that is,
for all \(y\in C\) . From (A2), it follows that
for all \(y\in C\) . Since \(\|A^{*}(T_{r_{n}}^{F_{2}}-I)Ax_{n}\|\to0\) , \(\| u_{n}-x_{n}\|\to0\) and \(\|x_{n}-x^{*}\|\to0\) as \(n\to\infty\) , we have
for all \(y\in C\) . Let \(y_{t}=ty+(1-t)x^{*}\) for any \(0< t\leq1\) and \(y\in C\) . It means that \(y_{t}\in C\) and hence
and then \(F_{1}(y_{t},y)\geq0\) . Letting \(t\to0\) , we immediately have \(F_{1}(x^{*},y)\geq0\) , i.e. , \(x^{*}\in EP(F_{1})\) .
Next, we show that \(Ax^{*}\in EP(F_{2})\) . Since A is a bounded linear operator and ( 3.11 ), we have
as \(n\to\infty\) , which yields that \(T_{r_{n}}^{F_{2}}Ax_{n}\to Ax^{*}\) . By the definition of \(T_{r_{n}}^{F_{2}}\) , we have
for all \(y\in C\) . Since \(F_{2}\) is upper semi-continuous in the first argument, taking lim sup in ( 3.15 ), it follows that
for all \(x,y\in C\) , from which it can be concluded that \(Ax^{*}\in EP(F_{2})\) . Consequently, \(x^{*}\in\Omega\) . This completes the proof. □
In Theorem 3.1 , if the mapping T is a nonexpansive mapping, then we immediately have the following.
Corollary 3.2
Let \(H_{1}\) , \(H_{2}\) be two real Hilbert spaces and C , Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\) , respectively . Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)-(A4) and \(F_{2}\) be upper semi - continuous in the first argument . Let \(T:C\to C\) be a nonexpansive mapping and \(A:H_{1}\to H_{2}\) be a bounded linear operator . Suppose that \(F(T)\cap\Omega\neq\emptyset\) , where \(\Omega=\{v\in C: v\in EP(F_{1})\textit{ such that }Av\in EP(F_{2})\}\) , and let \(x_{0}\in C\) . For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\) , define a sequence \(\{x_{n}\} \) iteratively as follows :
for each \(n\in\mathbb{N}\) , where \(M_{n}=\sup\{\|x_{n}-z\|:x\in\Omega\}\) and \(\theta_{n}=(1-\alpha_{n})(k_{n}^{2}-1)M_{n}^{2}\) , \(0\leq\alpha_{n}\leq a<1\) for all \(n\in\mathbb{N}\) , \(0< b\leq r_{n}<\infty\) , \(\gamma\in(0,1/L)\) , L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A . Then the sequence \(\{x_{n}\}\) generated by ( 3.16 ) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\) .
If \(H_{1}=H_{2}\) , \(C=Q\) and \(A=I\) in Theorem 3.1 , then we have the following.
Corollary 3.3
Let H be a real Hilbert space and C be a nonempty closed convex subset of a Hilbert space H . Let \(F_{1},F_{2}:C\times C\to\mathbb{R}\) be bifunctions satisfying conditions (A1)-(A4) and \(F_{2}\) be upper semi - continuous in the first argument . Let \(T:C\to C\) be an asymptotically nonexpansive mapping . Suppose that \(F(T)\cap\Omega\neq \emptyset\) , where \(\Omega=\{v\in C: v\in EP(F_{1})\cap EP(F_{2})\}\) , and let \(x_{0}\in C\) . For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\) , define a sequence \(\{ x_{n}\}\) iteratively as follows :
for each \(n\in\mathbb{N}\) , where \(M_{n}=\sup\{\|x_{n}-z\|:z\in\Omega\}\) and \(\theta_{n}=(1-\alpha_{n})(k_{n}^{2}-1)(M_{n})^{2}\) , \(0\leq\alpha_{n}\leq a<1\) and \(0< b\leq r_{n}<\infty\) for all \(n\in\mathbb{N}\) . Then the sequence \(\{x_{n}\}\) generated by ( 3.17 ) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\) .
4 Applications
4.1 applications to split variational inequality problems.
Firstly, we point out the so-called variational inequality problem (shortly, \(VIP\) ), which is to find a point \(x^{*}\in C\) which satisfies the following inequality:
for all \(z\in C\) . Its solution set is symbolized by \(VI(A,C)\) .
In 2012, Censor et al. [ 19 ] proposed the split variational inequality problem (shortly, \(SVIP\) ) which is formulated as follows:
where \(A:C\to C\) is a bounded linear operator. The solution set of split variational inequality problem is denoted by the \(SVIP\) .
Setting \(F_{1}(x,y)=\langle f(x),y-x\rangle\) and \(F_{2}(x,y)=\langle g(x),y-x\rangle\) , it is clear that \(F_{1}\) , \(F_{2}\) satisfy conditions (A1)-(A4), where f and g are \(\eta_{1}\) - and \(\eta_{2}\) -inverse strongly monotone mappings, respectively. Then, by Theorem 3.1 , we get the following.
Theorem 4.1
Let \(H_{1}\) , \(H_{2}\) be two real Hilbert spaces and C , Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\) , respectively . Let f and g be \(\eta_{1}\) - and \(\eta_{2}\) - inverse strongly monotone mappings , respectively . Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)-(A4), which are defined by f and g , and \(F_{2}\) be upper semi - continuous in the first argument . Let \(T:C\to C\) be an asymptotically nonexpansive mapping and \(A:H_{1}\to H_{2}\) be a bounded linear operator . Suppose that \(F(T)\cap\Omega\neq\emptyset\) , where \(\Omega=\{v\in C: v\in EP(F_{1})\textit{ such that }Av\in EP(F_{2})\}\) , and let \(x_{0}\in C\) . For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\) , define a sequence \(\{x_{n}\}\) iteratively as follows :
for each \(n\in\mathbb{N}\) , where \(M_{n}=\sup\{\|x_{n}-z\|:z\in\Omega\}\) and \(\theta_{n}=(1-\alpha_{n})(k_{n}^{2}-1)M_{n}^{2}\) , \(0\leq\alpha_{n}\leq a<1\) for all \(n\in\mathbb{N}\) , \(0< b\leq r_{n}<\infty\) , \(\gamma\in(0,1/L)\) , L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A . Then the sequence \(\{x_{n}\}\) generated by ( 4.1 ) strongly converges to a point \(z_{0}\in F(T)\cap\Omega\) .
The desired result can be proved directly through Theorem 3.1 . □
4.2 Applications to split optimization problems
In this section, we mention applications to the split optimization problem , which is to find \(x^{*}\in C\) such that
for all \(y\in Q\) . We symbolize Γ for the solution set of the split optimization problem.
Let \(f:C\to\mathbb{R}\) and \(g:Q\to\mathbb{R}\) be two functions satisfying the following assumption:
for each \(x,y\in C\) , \(f(tx+(1-t)y)\leq f(y)\) , and for each \(u,v\in Q\) , \(g(tu+(1-t)v)\leq g(v)\) ;
\(f(x)\) is concave and upper semi-continuous for all \(x\in C\) and \(g(u)\) is concave and upper semi-continuous for all \(u\in Q\) .
Let \(F_{1}(x,y)=f(x)-f(y)\) for all \(x,y\in C\) and \(F_{2}(u,v)=g(u)-g(v)\) for all \(u,v\in Q\) . If f and g satisfy conditions (1) and (2), then it is clear that \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to \mathbb{R}\) are two bifunctions satisfying conditions (A1)-(A4). Therefore, by Theorem 3.1 , we have the following.
Theorem 4.2
Let \(H_{1}\) , \(H_{2}\) be two real Hilbert spaces and C , Q be nonempty closed convex subsets of Hilbert spaces \(H_{1}\) and \(H_{2}\) , respectively . Let \(f:C\to\mathbb{R}\) and \(g:Q\to\mathbb{R}\) be two functions satisfying conditions (1) and (2). Let \(F_{1}:C\times C\to\mathbb{R}\) and \(F_{2}:Q\times Q\to\mathbb{R}\) be two bifunctions satisfying conditions (A1)-(A4) and \(F_{2}\) be upper semi - continuous in the first argument . Let \(A:H_{1}\to H_{2}\) be a bounded linear operator . Suppose that \(F(T)\cap\Gamma\neq\emptyset\) and let \(x_{0}\in C\) . For \(C_{1}=C\) and \(x_{1}=P_{C_{1}}x_{0}\) , define a sequence \(\{x_{n}\}\) iteratively as follows :
for each \(n\in\mathbb{N}\) , where \(0\leq\alpha_{n}\leq a<1\) , \(0< b\leq r_{n}<\infty\) , and \(\gamma\in(0,1/L)\) , L is the spectral radius of the operator \(A^{*}A\) and \(A^{*}\) is the adjoint of A . Then the sequence \(\{x_{n}\}\) generated by ( 4.3 ) strongly converges to a point \(z_{0}\in F(T)\cap\Gamma\) .
Inchan, I: Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces. Int. J. Math. Anal. 2 , 1135-1145 (2008)
MATH MathSciNet Google Scholar
Kim, JK, Nam, YM, Sim, JY: Convergence theorem of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal. 71 , 2839-2848 (2009)
Article MathSciNet Google Scholar
Kim, TH, Xu, HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mapping and semigroups. Nonlinear Anal. 64 , 1140-1152 (2006)
Article MATH MathSciNet Google Scholar
Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73 , 591-597 (1967)
Blum, E, Oettli, W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63 , 123-145 (1994)
Choudhury, BS, Kundu, S: A viscosity type iteration by weak contraction for approximating solutions of generalized equilibrium problem. J. Nonlinear Sci. Appl. 5 , 243-251 (2012)
Kang, SM, Cho, SY, Qin, X: Hybrid projection algorithms for approximating fixed points of asymptotically quasi-pseudocontractive mappings. J. Nonlinear Sci. Appl. 5 , 466-474 (2012)
Witthayarat, U, Cho, YJ, Kumam, P: Approximation algorithm for fixed points of nonlinear operators and solutions of mixed equilibrium problems and variational inclusion problems with applications. J. Nonlinear Sci. Appl. 5 , 475-494 (2012)
Chang, SS, Lee, HWJ, Chan, CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70 , 3307-3319 (2009)
Katchang, P, Kumam, P: A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space. J. Appl. Math. Comput. 32 , 19-38 (2010)
Plubtieng, S, Punpaeng, R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336 , 455-469 (2007)
Qin, X, Shang, M, Su, Y: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal. 69 , 3897-3909 (2008)
Combettes, PL, Hirstoaga, SA: Equilibrium programming using proximal like algorithms. Math. Program. 78 , 29-41 (1997)
Article Google Scholar
Agarwal, RP, Chen, JW, Cho, YJ: Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces. J. Inequal. Appl. 2013 , Article ID 119 (2013)
Tada, A, Takahashi, W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 , 359-370 (2007)
Takahashi, S, Takahashi, W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69 , 1025-1033 (2008)
Takahashi, S, Takahashi, W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331 , 506-515 (2007)
He, Z: The split equilibrium problem and its convergence algorithms. J. Inequal. Appl. 2012 , Article ID 162 (2012)
Censor, Y, Gibali, A, Reich, S: Algorithm for split variational inequality problems. Numer. Algorithms 59 , 301-323 (2012)
Kazmi, KR, Rizvi, SH: Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem. J. Egypt. Math. Soc. 21 , 44-51 (2013)
Bnouhachem, A: Strong convergence algorithm for split equilibrium problems and hierarchical fixed point problems. Sci. World J. 2014 , Article ID 390956 (2014)
Iiduka, H, Takahashi, W: Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings. Nonlinear Anal. 61 , 341-350 (2005)
Lin, PK, Tan, KK, Xu, HK: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. Nonlinear Anal. 24 , 929-946 (1995)
Combette, PL, Hirstoaga, SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6 , 117-136 (2005)
MathSciNet Google Scholar
Download references
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant No. (18-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2014R1A2A2A01002100).
Author information
Authors and affiliations.
Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand
Uamporn Witthayarat
Department of Mathematics, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
Afrah A N Abdou & Yeol Je Cho
Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, 660-701, Korea
Yeol Je Cho
You can also search for this author in PubMed Google Scholar
Corresponding authors
Correspondence to Afrah A N Abdou or Yeol Je Cho .
Additional information
Competing interests.
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.

Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Reprints and Permissions
About this article
Cite this article.
Witthayarat, U., Abdou, A.A.N. & Cho, Y.J. Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl 2015 , 200 (2015). https://doi.org/10.1186/s13663-015-0448-5
Download citation
Received : 10 August 2015
Accepted : 27 October 2015
Published : 04 November 2015
DOI : https://doi.org/10.1186/s13663-015-0448-5
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative
- split equilibrium problem
- asymptotically nonexpansive mapping
- fixed point problem
- Hilbert space

An official website of the United States government
The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
- Publications
- Account settings
- Advanced Search
- Journal List
- Springer Open Choice
The modified split generalized equilibrium problem for quasi-nonexpansive mappings and applications
Kanyarat cheawchan.
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand
Atid Kangtunyakarn
In this paper, we introduce a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. We introduce a new method of an iterative scheme { x n } for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without assuming a demicloseness condition and T ω : = (1 − ω ) I + ω T , where T is a quasi-nonexpansive mapping and ω ∈ (0, ½) ; a difficult proof in the framework of Hilbert space. In addition, we give a numerical example to support our main result.
Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H . The set of fixed points of T is denoted by F ( T ) . The mapping T : C → C is said to be quasi-nonexpansive if
for all x ∈ C and p ∈ F ( T ) .
Definition 1.1
Let T : H → H . Then the following are equivalent:
- T is firmly nonexpansive,
- ∥ T x − T y ∥ 2 ≤ 〈 x − y , T x − T y 〉 , ∀ x , y ∈ H ,
- 〈 T x − T y , ( I − T ) x − ( I − T ) y 〉 ≥ 0 , ∀ x , y ∈ H .
Let A : C → H be a mapping. The variational inequality is to find a point u ∈ C such that
for all v ∈ C . The set of solutions of ( 1.1 ) is denoted by V I ( C , A ) . A mapping A : C → H is called α-inverse strongly monotone if there exists a positive real number α > 0 such that
for all x , y ∈ C . They have been investigated in the literature; see, for example, [ 2 , 3 ]. Let F be a bifunction of C × C into ℝ , where ℝ is the set of real numbers. The equilibrium problem for F : C × C → ℝ is to find x ∈ C such that
The set of solutions of ( 1.2 ) is denoted by E P ( F ) . Equilibrium problems were introduced by [ 4 ] in 1994 and included many well-known problems such as variational inequality, optimization problem, nonexpansive mapping and fixed point problem; see, for example, [ 5 – 8 ].
Let F be a function of C × C into ℝ and let f : H → H be a mapping. The generalized equilibrium problem is to find x ∈ C such that
for all y ∈ C . The set of solutions of ( 1.3 ) is denoted by E P ( F , f ) . When f ≡ 0 , E P ( F , f ) is denoted by E P ( F ) and F ≡ 0 , E P ( F , f ) is denoted by V I ( C , f ) .
Throughout this section, let H 1 , H 2 be real Hilbert spaces and let C , Q be nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively. Let A : H 1 → H 2 be a bounded linear operator.
In 1994, Censor and Elfving [ 9 ] introduced the split feasibility problem (in short, SFP) which is to find a point x ∈ C such that A x ∈ Q . The set of all solutions of split feasibility problem is denoted by φ = { x ∈ C : A x ∈ Q } .
To solve the SFP, Byrne [ 10 ] introduced CQ algorithm whose sequence { x n } is generated by
where the initial x 0 ∈ H 1 and γ ∈ (0, 2/ L ) , L is the spectral radius of the operator A ∗ A and A ∗ is the adjoint of A . Then the CQ algorithm converges to a solution of the SFP, whenever solutions exist. If there are no solutions of the SFP, the CQ algorithm converges to a minimizer of the function
whenever such minimizers exist.
Let U : H 1 → H 1 and T : H 2 → H 2 be two nonlinear operators. The split common fixed points problem (SCFPP) [ 11 , 12 ] is to find a point x ∗ such that
The solution set of SCFPP is denoted by Φ = { p ∗ ∈ F ( U ): A p ∗ ∈ F ( T )} . The split common fixed point problem is a generalization of the split feasibility problem.
In 2017, Wang [ 13 ] introduced a new method for solving SCFPP as follows:
where ρ n ⊂ (0, ∞) is chosen such that
and U and T are firmly quasi-nonexpansive mappings. Then the sequence { x n } converges weakly to z , where z = lim n →∞ P Φ x n .
Censor et al. [ 11 , 14 ] introduced the prototypical split inverse problem (SIP) which is a generalization of the split common fixed points problem. In this, there are given two vector spaces X and Y and a linear operator A : X → Y . In addition, two inverse problems are involved. The first one, denoted IP 1 , is formulated in the space X and the second one, denoted IP 2 , is formulated in the space Y . Given these data, the split inverse problem is formulated as follows:
and such that
This problem is used in many modeling arising in sensor networks, radiation therapy treatment planning, color imaging, etc.
The split equilibrium problem (SEP) [ 12 ] is to find x ˆ ∈ C such that
where F 1 : C × C → ℝ and F 2 : Q × Q → ℝ be nonlinear bifunctions. If we consider only problem ( 1.7 ), it is the equilibrium problem and we denoted its solution set by E P ( F 1 ) . The solution set of SEP is denoted by Γ = { p ˆ ∈ E P ( F 1 ) : A p ˆ ∈ E P ( F 2 ) } . SEP is reduced to E P ( F ) , where H 1 ≡ H 2 , F 1 ≡ F 2 and A ≡ I . E P ( F ) is an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc.
The split variational inequality problems (in short, SVIP) were introduced and studied by Cencor et al. [ 11 ]: find x ‾ ∈ C such that
where f 1 : C → H 1 and f 2 : Q → H 2 are nonlinear mappings. The solution set of SVIP is denoted by Ψ = { p ‾ ∈ V I ( C , f 1 ) : A p ‾ ∈ V I ( Q , f 2 ) } . The split variational inequality problems have already been studied and used in practice as a model in intensity-modulated radiation therapy (IMRT) treatment planning; see, for example, [ 15 ] and the modeling of many inverse problems arising for phase retrieval and other real-world problems; for instance, in sensor networks in computerized tomography and data compression; see, for example, [ 16 , 17 ].
By investigating SEP and SVIP, we introduce the modified split generalized equilibrium problem (MSGEP) which is to find x ∗ ∈ C such that
where F 1 : C × C → ℝ and F 2 : Q × Q → ℝ are nonlinear bifunctions and f 1 : C → H 1 and f 2 : Q → H 2 are nonlinear mappings. The solution set of MSGEP is denoted by Ω = { p ∗ ∈ E P ( F 1 , f 1 ): A p ∗ ∈ E P ( F 2 , f 2 )} .
- If we put f 1 ≡ f 2 ≡ 0 in MSGEP then the MSGEP is reduced to SEP.
- If we put F 1 ≡ F 2 ≡ 0 in MSGEP then the MSGEP is reduced to SVIP.
- In the case of bifunctions F 1 and F 2 are according to (A1)–(A4). From ( 1.11 ), ( 1.12 ) and Lemma 2.2 , we have x ∗ ∈ F ( T r F 1 ( I − r f 1 ) ) and A x ∗ ∈ F ( T s F 2 ( I − s f 2 ) ) , for all r , s > 0 . So, MSGEP can be viewed as SCFPP.
MSGEP is a generalization of the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. So, this problem can be used in sensor networks, data compression, practice as a model in intensity-modulated radiation therapy (IMRT) treatment planning, robustness to marginal changes and equilibrium stability etc.
Example 1.2
Let H 1 = [0, 6] , H 2 = [0, 18] , C = [2, 5] and Q = [6, 10] . Let A : H 1 → H 2 be defined by A x = 3 x for all x ∈ H 1 . Let the mapping F 1 : C × C → ℝ be defined by
and F 2 : Q × Q → ℝ be defined by
Let the mapping f 1 : C → H 1 be defined by f 1 x = x − 2 9 , ∀ x ∈ C and the mapping f 2 : Q → H 2 be defined by f 2 x = x − 6 7 , ∀ x ∈ Q .
Then 2 ∈ Ω . Therefore 2 is a solution of MSGEP.
In 2012, Tain and Jin [ 18 ] introduced iterative algorithms involving a quasi-nonexpansive mapping. They generated the iterative as follows:
where A is a bounded linear operator on H , T is a quasi-nonexpansive mapping on H , f is a contraction with coefficient a under suitable conditions of the parameters α n , γ and ω . By assuming ω ∈ (0, ½) , T ω : = (1 − ω ) I + ω T and T is demiclosed on H .
Motivated by SFP and SVIP, we introduced a new problem, the modified split generalized equilibrium problem, which extends the generalized equilibrium problem, the split equilibrium problem and the split variational inequality problem. Many authors proved strong convergence theorem involving a quasi-nonexpansive mapping T by assuming T ω : = (1 − ω ) I + ω T and T is demiclosed on H ; a difficult proof. Motivated by [ 19 ], we introduced Remark 2.5 and [ 11 , 12 ] and [ 18 ], we introduce a new method of iterative scheme { x n } for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without the condition above in the framework of a Hilbert space.
Preliminaries
Let H be a real Hilbert space with inner product 〈 ⋅ , ⋅ 〉 and norm ∥ ⋅ ∥ . Throughout this paper, we use the notations of weak and strong convergence by “⇀” and “→” Opial’s condition [ 20 ], i.e., for any sequence { x n } with x n ⇀ x , the inequality lim n →∞ inf ∥ x n − x ∥ < lim n →∞ inf ∥ x n − y ∥ , holds for every y ∈ H with y ≠ x .
For solving the equilibrium problem, we assume that the bifunction F : C × C → ℝ satisfy the following conditions:
- F ( x , x ) = 0 for all x ∈ C ,
- F is monotone, i.e., F ( x , y ) + F ( y , x ) ≤ 0 for all x , y ∈ C ,
- for each x , y , z ∈ C , lim t ↓0 F ( t z + (1 − t ) x , y ) ≤ F ( x , y ) ,
- for each x ∈ C , y ↦ F ( x , y ) is convex and lower semicontinuous.
Let C be a nonempty closed convex subset of H and let F be a bifunction of C × C into ℝ satisfying (A1) – (A4). Let r > 0 and x ∈ H . Then there exists z ∈ C such that
Assume that F : C × C → ℝ satisfies (A1) – (A4). For r > 0 , define a mapping T r : H → C as follows :
for all x ∈ H . Then the following hold :
- T r is single - valued ,
- T r is firmly nonexpansive , i . e ., for any x , y ∈ H , ∥ T r ( x )− T r ( y )∥ 2 ≤ 〈 T r ( x )− T r ( y ), x − y 〉,
- F ( T r ) = E P ( F ) ,
- E P ( F ) is closed and convex .
Let H be a real Hilbert space , let C be a nonempty closed convex subset of H and let A be a mapping of C into H . Let u ∈ C . Then , for λ > 0 ,
where P C is the metric projection of H onto C .
Let C be a nonempty closed convex subset of a real Hilbert space H . Let { T i } i = 1 N be a finite family of quasi - nonexpansive mappings of C into H with ⋂ i = 1 N F ( T i ) ≠ ∅ and let 0 < a i < 1 with ∑ i = 1 N a i = 1 . Then
In this lemma, we show that ⋂ i = 1 N F ( T i ) = ⋂ i = 1 N V I ( C , I − T i ) and ⋂ i = 1 N V I ( C , I − T i ) = V I ( C , ∑ i = 1 N a i ( I − T i ) ) . Lastly, we have
To start with, it is easy to see that ⋂ i = 1 N F ( T i ) ⊆ ⋂ i = 1 N V I ( C , I − T i ) . Next, we show that ⋂ i = 1 N V I ( C , I − T i ) ⊆ ⋂ i = 1 N F ( T i ) . Let u ∈ ⋂ i = 1 N V I ( C , I − T i ) and ⋂ i = 1 N F ( T i ) ≠ ∅ . So, we get u ∈ V I ( C , I − T i ) , ∀ i = 1, 2, …, N . We may write
There exists v ∗ ∈ C such that v ∗ = T i v ∗ , ∀ i = 1, 2, …, N . Since T i is a quasi-nonexpansive mapping, ∀ i = 1, 2, …, N , it follows that
By using ( 2.1 ) and ( 2.2 ), we conclude that
It implies that u ∈ ⋂ i = 1 N F ( T i ) . Therefore ⋂ i = 1 N V I ( C , I − T i ) ⊆ ⋂ i = 1 N F ( T i ) . Hence
After that, we show ⋂ i = 1 N V I ( C , I − T i ) = V I ( C , ∑ i = 1 N a i ( I − T i ) ) where 0 < a i < 1 and ∑ i = 1 N a i = 1 . Observe that
Therefore ⋂ i = 1 N V I ( C , I − T i ) = V I ( C , ∑ i = 1 N a i ( I − T i ) ) . Hence ⋂ i = 1 N F ( T i ) = V I ( C , ∑ i = 1 N a i ( I − T i ) ) . □
From Lemma 2.3 and Lemma 2.4 , we have
for all λ > 0 and 0 < a i < 1 with ∑ i = 1 N a i = 1 .
Let { s n } be a sequence of nonnegative real numbers satisfying
where { α n } is a sequence in (0, 1) and { δ n } is a sequence such that
Then lim n →∞ s n = 0 .
Main results
Let C and Q be nonempty closed convex subsets of a real Hilbert spaces H 1 and H 2 , respectively . Let A : H 1 → H 2 be a bounded linear operator . Let F 1 : C × C → ℝ and F 2 : Q × Q → ℝ be the bifunctions satisfying (A1) – (A4). Let f 1 : H 1 → H 1 be a ρ - inverse strongly monotone mapping and f 2 : H 2 → H 2 be a firmly nonexpansive mapping . Then
- T r F 1 ( I − r f 1 ) and T s F 2 ( I − s f 2 ) are nonexpansive mapping ,
- ∥ T r F 1 ( I − r f 1 ) ( p + γ A ∗ ( T s F 2 ( I − s f 2 ) − I ) A p ) − T r F 1 ( I − r f 1 ) ( q + γ A ∗ ( T s F 2 ( I − s f 2 ) − I ) A q ) ∥ 2 ≤ ∥ p − q ∥ 2 + γ ( γ L − 1 ) ∥ ( T s F 2 ( I − s f 2 ) − I ) A p − ( T s F 2 ( I − s f 2 ) − I ) A q ∥ 2 ,
for all p , q ∈ C , where r ∈ (0, 2 ρ ) , s ∈ (0, 1) , γ ∈ (0, 1/ L ) , L is the spectral radius of the operator A ∗ A and A ∗ is the adjoint of A , T r F 1 : H 1 → C defined by
for all x ∈ H 1 and T s F 2 : H 2 → Q defined by
for all x ‾ ∈ H 2 .
Let p , q ∈ C . First, we show 1 is true. Since f 1 is a ρ -inverse strongly monotone mapping and r ∈ (0, 2 ρ ) , we obtain
Thus T r F 1 ( I − r f 1 ) is a nonexpansive mapping. Since f 2 is a firmly nonexpansive mapping and s ∈ (0, 1) , we get
for all p ‾ , q ‾ ∈ Q . Therefore T s F 2 ( I − s f 2 ) is a nonexpansive mapping.
Next, we show 2 is true. From Lemma 3.1 (1), we have
From the property of T s F 2 , we get
From ( 3.2 ), ( 3.3 ) and the property of firmly nonexpansive mapping, we get
Substituting ( 3.4 ) in ( 3.1 ), we obtain
□
Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C → C be a quasi - nonexpansive mapping with F ( T ) ≠ ∅ . Then
Let x ∈ C and z ∈ F ( T ) . Since T is a quasi-nonexpansive mapping, we get
We can conclude that
Let C be a nonempty closed convex subset of a real Hilbert space H . Let { T i } i = 1 N be a finite family of quasi - nonexpansive mappings of C into itself with ⋂ i = 1 N F ( T i ) ≠ ∅ . Then
for all x ∈ C , where 0 < k i < 1 with ∑ i = 1 N k i = 1 and 0 < λ ‾ < 1 .
Let x ∈ C and z ∈ ⋂ i = 1 N F ( T i ) . From Remark 2.5 and z ∈ ⋂ i = 1 N F ( T i ) , we have z ∈ F ( P C ( I − λ ‾ ( ∑ i = 1 N k i ( I − T i ) ) ) ) and z = T i z , ∀ i = 1, 2, …, N . Since P C is nonexpansive mapping, 0 < λ ‾ < 1 and Lemma 3.2 , we have
Next, we prove a strong convergence theorem for solving the modified split generalized equilibrium problem (MSGEP).
Theorem 3.4
Let C and Q be nonempty closed convex subsets of a real Hilbert spaces H 1 and H 2 , respectively . Let A : H 1 → H 2 be a bounded linear operator . Let D 1 , D 2 : C → H 1 be α , β - inverse strongly monotone mappings , respectively . Let F 1 : C × C → ℝ and F 2 : Q × Q → ℝ be the bifunctions satisfying (A1) – (A4). Let { T i } i = 1 N be a finite family of quasi - nonexpansive mappings of C into itself with ⋂ i = 1 N F ( T i ) ≠ ∅ . Let f 1 : H 1 → H 1 be a ρ - inverse strongly monotone mapping and f 2 : H 2 → H 2 be a firmly nonexpansive mapping . Assume F = V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ ⋂ i = 1 N F ( T i ) ∩ Ω ≠ ∅ . For given x 1 , u ∈ C and let { x n } , { u n } and { y n } be sequences generated by
where d 1 ∈ (0, 2 α ) , d 2 ∈ (0, 2 β ) , r ∈ (0, 2 ρ ) , s ∈ (0, 1) , a ∈ [0, 1] , 0 < k i < 1 with ∑ i = 1 N k i = 1 , γ ∈ (0, 1/ L ) , L is the spectral radius of the operator A ∗ A and A ∗ is the adjoint of A . Also { α n } , { β n } , { γ n } are sequences in [0, 1] with α n + β n + γ n = 1 for all n ∈ ℕ . Suppose the following conditions hold :
- (i) lim n →∞ α n = 0 and ∑ n = 1 ∞ α n = ∞ ,
- (ii) 0 < c ≤ β n , γ n ≤ d < 1 for some c , d > 0 for all n ≥ 1 ,
- (iii) ∑ n = 1 ∞ λ n < ∞ and 0 < λ n < 1 ,
- (iv) ∑ n = 1 ∞ | α n + 1 − α n | < ∞ , ∑ n = 1 ∞ | β n + 1 − β n | < ∞ .
Then { x n } , { u n } and { y n } converge strongly to z = P ℱ u .
Let x , y ∈ C and z ∈ ℱ . First, we show that ( I − d 1 D 1 ) is a nonexpansive mapping. Since D 1 is an α -inverse strongly monotone mapping, we obtain
Thus ( I − d 1 D 1 ) is a nonexpansive mapping. By using the same method as above, we see that ( I − d 2 D 2 ) is a nonexpansive mapping. Since f 1 is a ρ -inverse strongly monotone mapping and f 2 is a firmly nonexpansive mapping. From Lemma 3.1 (1), we have ( T r F 1 ( I − r f 1 ) ) and ( T s F 2 ( I − s f 2 ) ) are nonexpansive mappings. Since z ∈ ⋂ i = 1 N F ( T i ) and Lemma 3.3 , we have
Since z ∈ V I ( C , D 1 ) and z ∈ V I ( C , D 2 ) and using the property of ( I − d 1 D 1 ) and ( I − d 2 D 2 ) , we get
Since z ∈ Ω , we have z = T r F 1 ( I − r f 1 ) z and A z = T s F 2 ( I − s f 2 ) A z . From Lemma 3.1 (2) and γ ∈ (0, 1/ L ) , we obtain
Using the definition of x n , ( 3.7 ), ( 3.9 ) and ( 3.11 ), we get
Using induction, we can conclude that
for all n ≥ 1 . This implies that the sequence { x n } is bounded and so are { y n } and { u n } . From Lemma 3.1 (2) and γ ∈ (0, 1/ L ) , we obtain
Next, we show that lim n →∞ ∥ x n +1 − x n ∥ = 0 . According to Eq. ( 3.12 ), we have
From condition (i), (iii), (iv) and Lemma 2.6 , we have
According to Eqs. ( 3.7 ), ( 3.9 ) and ( 3.10 ), we have
This implies that
By using condition (i) and ( 3.13 ), we have
By using the same method as ( 3.16 ), we have
Let M n = x n + γ A ∗ ( T s F 2 ( I − s f 2 ) − I ) A x n . Applying the inequality ( 3.11 ), we have
Using the property of inverse strongly monotone operators and ( 3.18 ), we have
Substituting ( 3.19 ) in ( 3.15 ), we have
According to condition (i) and ( 3.13 ), we get
By the property of firmly nonexpansive mappings, we have
Substituting ( 3.22 ) in ( 3.15 ), we get
It follows that
From condition (i), ( 3.13 ) and ( 3.20 ), we ensure that
From ( 3.16 ) and ( 3.23 ), we also have
Then we have
By using the same method as ( 3.19 ), we have
Substituting ( 3.8 ) and ( 3.25 ) in ( 3.14 ), we have
Since P C is a firmly nonexpansive mapping and using the same method as ( 3.21 ), we get
Substituting ( 3.8 ) and ( 3.27 ) in ( 3.14 ), we have
From condition (i), ( 3.13 ) and ( 3.26 ), we get
Let k n = a u n + (1 − a ) P C ( I − d 2 D 2 ) u n . By using the same method as ( 3.19 ), we have
Substituting ( 3.29 ) in ( 3.14 ), we have
According to condition (i) and ( 3.13 ), we have
By using the same method as ( 3.21 ), we have
Substituting ( 3.31 ) in ( 3.14 ), we have
According to condition (i), ( 3.13 ) and ( 3.30 ), we get
From ( 3.28 ) and ( 3.33 )
we conclude that
By ( 3.24 ) and ( 3.34 ), we also conclude that
Afterward, we show that lim sup n →∞ 〈 u − z , x n − z 〉 ≤ 0 , where z = P ℱ u .
To show this, choose a subsequence { x n j } of { x n } such that
Without loss of generality, we may assume that x n j ⇀ ω as j → ∞ . From ( 3.35 ), we obtain y n j ⇀ ω as j → ∞ . From Lemma 2.3 , we have V I ( C , D 1 ) = F ( P C ( I − d 1 D 1 )) . Assume that ω ∉ V I ( C , D 1 ) , we have ω ≠ P C ( I − d 1 D 1 ) ω . Using Opial’s condition, ( 3.33 ), we obtain
This is a contradiction, so we have
From ( 3.24 ), we have u n j ⇀ ω as j → ∞ . By ( 3.28 ) and using the same method as ( 3.37 ), we obtain
Next, we show that ω ∈ ⋂ i = 1 N F ( T i ) . From Lemma 2.5 , we have
Assume that ω ∉ ⋂ i = 1 N F ( T i ) , and that ω ≠ P C ( I − λ n j ( ∑ i = 1 N k i ( I − T i ) ) ) ω . Using Opial’s condition, ( 3.17 ) and ( 3.35 ), we obtain
After that, we show that ω ∈ Ω . Assume ω ∉ E P ( F 1 , f 1 ) . Since E P ( F 1 , f 1 ) = F ( T r F 1 ( I − r f 1 ) ) , we obtain ω ≠ T r F 1 ( I − r f 1 ) ω . Using Opial’s condition and ( 3.23 ), we get
Next, we show that A ω ∈ E P ( F 2 , f 2 ) . Since A is bounded linear operator so that A x n j ⇀ A ω as j → ∞ . Assume A ω ∉ E P ( F 2 , f 2 ) . Since E P ( F 2 , f 2 ) = F ( T s F 2 ( I − s f 2 ) ) , we obtain A ω ≠ T s F 2 ( I − s f s ) A ω . Using Opial’s condition and ( 3.16 ), we have
We can conclude that ω ∈ Ω . Therefore ω ∈ ℱ . Since x n j ⇀ ω as j → ∞ , we have
Finally, we show that the sequence { x n } converges strongly to z = P ℱ u . By ( 3.7 ), ( 3.9 ) and ( 3.11 ), we get
According to condition (i), ( 3.42 ) and Lemma 2.6 , we can conclude that { x n } converges strongly to z = P ℱ u . By ( 3.24 ) and ( 3.35 ), we have { u n } and { y n } converge strongly to z = P ℱ u . This completes the proof. □
These results are directly proved from Theorem 3.4 . Therefore, we omit the proof.
Corollary 3.5
Let C and Q be nonempty closed convex subsets of a real Hilbert space H 1 and H 2 , respectively . Let A : H 1 → H 2 be a bounded linear operator . Let D 1 , D 2 : C → H 1 be α , β - inverse strongly monotone mappings , respectively . Let F 1 : C × C → ℝ and F 2 : Q × Q → ℝ be the bifunctions satisfying (A1) – (A4). Let T be a quasi - nonexpansive mapping of C into itself . Let f 1 : H 1 → H 1 be a ρ - inverse strongly monotone mapping and f 2 : H 2 → H 2 be a firmly nonexpansive mapping . Assume ℱ = V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ F ( T ) ∩ Ω ≠ ∅ . For given x 1 , u ∈ C , and let { x n } , { u n } and { y n } be sequences generated by
where d 1 ∈ (0, 2 α ) , d 2 ∈ (0, 2 β ) , r ∈ (0, 2 ρ ) , s ∈ (0, 1) , a ∈ [0, 1] , γ ∈ (0, 1/ L ) , L is the spectral radius of the operator A ∗ A and A ∗ is the adjoint of A . Also { α n } , { β n } , { γ n } are sequences in [0, 1] with α n + β n + γ n = 1 for all n ∈ ℕ . Suppose the conditions (i) – (iv) of Theorem 3.4 hold . Then { x n } , { u n } and { y n } converge strongly to z = P ℱ u .
Corollary 3.6
Let C be nonempty closed convex subset of a real Hilbert space H 1 . Let D 1 , D 2 : C → H 1 be α , β - inverse strongly monotone mappings , respectively . Let F 1 : C × C → ℝ be the bifunction satisfying (A1) – (A4). Let { T i } i = 1 N be a finite family of quasi - nonexpansive mappings of C into itself with ⋂ i = 1 N F ( T i ) ≠ ∅ . Let f 1 : H 1 → H 1 be a ρ - inverse strongly monotone mapping . Assume F = V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ ⋂ i = 1 N F ( T i ) ∩ E P ( F 1 , f 1 ) ≠ ∅ . For given x 1 , u ∈ C and let { x n }, { u n } and { y n } be sequences generated by
where d 1 ∈ (0, 2 α ) , d 2 ∈ (0, 2 β ) , r ∈ (0, 2 ρ ) , a ∈ [0, 1] , 0 < k i < 1 with ∑ i = 1 N k i = 1 . Also { α n } , { β n } , { γ n } are sequences in [0, 1] with α n + β n + γ n = 1 for all n ∈ ℕ . Suppose the conditions (i) – (iv) of Theorem 3.4 hold . Then { x n } , { u n } and { y n } converge strongly to z = P ℱ u .
Corollary 3.7
Let C and Q be nonempty closed convex subsets of a real Hilbert space H 1 and H 2 , respectively . Let A : H 1 → H 2 be a bounded linear operator . Let D 1 , D 2 : C → H 1 be α , β - inverse strongly monotone mappings , respectively . Let F 1 : C × C → ℝ and F 2 : Q × Q → ℝ be the bifunctions satisfying (A1) – (A4). Let { T i } i = 1 N be a finite family of quasi - nonexpansive mappings of C into itself with ⋂ i = 1 N F ( T i ) ≠ ∅ . Assume F = V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ ⋂ i = 1 N F ( T i ) ∩ Γ ≠ ∅ . For given x 1 , u ∈ C and let { x n } , { u n } and { y n } be sequences generated by
where d 1 ∈ (0, 2 α ) , d 2 ∈ (0, 2 β ) , a ∈ [0, 1] , 0 < k i < 1 with ∑ i = 1 N k i = 1 , γ ∈ (0, 1/ L ) , L is the spectral radius of the operator A ∗ A and A ∗ is the adjoint of A . Also { α n } , { β n } , { γ n } are sequences in [0, 1] with α n + β n + γ n = 1 for all n ∈ ℕ . Suppose the conditions (i) – (iv) of Theorem 3.4 hold . Then { x n } , { u n } and { y n } converge strongly to z = P ℱ u .
If we take N = 1 in Theorem 3.4 , we have a strong convergence for finding a common element of the set of solutions of variational inequality problems and the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the modified split generalized equilibrium problem. From previous result, we can apply by using the same method as Theorem 4.5 in [ 24 ]. We have a strong convergence for finding a common element of the set of solutions of variational inequality problems and the set of fixed points of a finite family of nonspreading mappings and the set of solutions of the modified split generalized equilibrium problem. By using our main result, Theorem 3.4 reduces to the Corollary 3.6 , the solution of the generalized equilibrium problem and Corollary 3.7 , the split equilibrium problem. All theorems are found as regards the solution of common fixed points of a finite family of quasi-nonexpansive mappings without assuming T ω : = (1 − ω ) I + ω T and T is demiclosed; a difficult proof in a framework of Hilbert space.
Application
The following knowledge is used to prove Theorem 4.4 . A mapping T : C → C is called nonspreading if
Such a mapping is defined by Kohsaka and Takahashi [ 25 ].
In 2009, Iemoto and Takahashi [ 26 ] proved that ( 4.1 ) is equivalent to
A nonspreading mapping T with F ( T ) ≠ ∅ is quasi-nonexpansive mapping T .
Let H be a Hilbert space , let C be a nonempty closed convex subset of H , and let S be a nonspreading mapping of C into itself . Then F ( S ) is closed and convex .
In 2009, Kangtunyakarn and Suantai[ 27 ] introduced the S -mapping generated by T 1 , T 2 , T 3 , …, T N and λ 1 , λ 2 , …, λ N as follows.
Definition 4.1
Let C be a nonempty convex subset of a real Banach space. Let { T i } i = 1 N be a finite family of (nonexpansive) mappings of C into itself. For each j = 1, 2, …, N , let α j = ( α 1 j , α 2 j , α 3 j ) ∈ I × I × I , where I ∈ [0, 1] and α 1 j + α 2 j + α 3 j = 1 . Define the mapping S : C → C as follows:
This mapping is called an S-mapping generated by T 1 , T 2 , …, T N and α 1 , α 2 , …, α N .
Let C be a nonempty closed convex subset of a real Hilbert space . Let { T i } i = 1 N be a finite family of nonspreading mappings of C into C with ⋂ i = 1 N F ( T i ) ≠ ∅ , and let α j = ( α 1 j , α 2 j , α 3 j ) ∈ I × I × I , j = 1, 2, …, N , where I = [0, 1] , α 1 j + α 2 j + α 3 j = 1 , α 1 j , α 3 j ∈ ( 0 , 1 ) for all j = 1, 2, …, N − 1 and α 1 N ∈ ( 0 , 1 ] , α 3 N ∈ [ 0 , 1 ) , α 2 j ∈ [ 0 , 1 ) for all j = 1, 2, …, N . Let S be the mapping generated by T 1 , T 2 , …, T N and α 1 , α 2 , …, α N . Then F ( S ) = ⋂ i = 1 N F ( T i ) and S is a quasi - nonexpansive mapping .
By using these results, we obtain the following theorems.
Theorem 4.4
Let C and Q be nonempty closed convex subsets of a real Hilbert space H 1 and H 2 , respectively . Let A : H 1 → H 2 be a bounded linear operator . Let D 1 , D 2 : C → H 1 be α , β - inverse strongly monotone mappings , respectively . Let F 1 : C × C → ℝ and F 2 : Q × Q → ℝ be the bifunctions satisfying (A1) – (A4). Let { T i } i = 1 N be a finite family of nonspreading mappings of C into C with ⋂ i = 1 N F ( T i ) ≠ ∅ , and let α j = ( α 1 j , α 2 j , α 3 j ) ∈ I × I × I , j = 1, 2, …, N , where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j , α 3 j ∈ ( 0 , 1 ) for all j = 1, 2, …, N − 1 and α 1 N ∈ ( 0 , 1 ] , α 3 N ∈ [ 0 , 1 ) , α 2 j ∈ [ 0 , 1 ) for all j = 1, 2, …, N . Let S be the mapping generated by T 1 , T 2 , …, T N and α 1 , α 2 , …, α N . Let f 1 : H 1 → H 1 be a ρ - inverse strongly monotone mapping and f 2 : H 2 → H 2 be a firmly nonexpansive mapping . Assume F = V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ ⋂ i = 1 N F ( T i ) ∩ Ω ≠ ∅ . For given x 1 , u ∈ C and let { x n } , { u n } and { y n } be sequences generated by
By using Corollary 3.5 and Lemma 4.3 , we obtain the conclusion. □
Example and numerical results
In this section, an example is given for supporting Theorem 3.4 . In Example 5.1 , we only instance an example in infinite dimensional Hilbert space for supporting Theorem 3.4 . We omit the computer programming.
Example 5.1
Let H 1 = H 2 = C = Q = ℓ 2 be the linear space whose elements consist of all 2-summable sequences ( x 1 , x 2 , …, x j , …) of scalars, i.e.,
with an inner product 〈 ⋅ , ⋅ 〉: ℓ 2 × ℓ 2 → ℝ defined by 〈 x , y 〉 = ∑ j = 1 ∞ x j y j where x = { x j } j = 1 ∞ , y = { y j } j = 1 ∞ ∈ ℓ 2 and a norm ∥ ⋅ ∥ : ℓ 2 → ℝ defined by ∥ x ∥ 2 = ( ∑ j = 1 ∞ | x j | 2 ) 1 2 where x = { x j } j = 1 ∞ ∈ ℓ 2 . Let the mapping A : ℓ 2 → ℓ 2 be defined by A x = ( x 1 3 , x 2 3 , … , x j 3 , … ) for all x = { x j } j = 1 ∞ ∈ ℓ 2 and A ∗ : ℓ 2 → ℓ 2 be defined by A ∗ z = ( z 1 3 , z 2 3 , … , z j 3 , … ) for all z = { z j } j = 1 ∞ ∈ ℓ 2 . Let D 1 , D 2 : ℓ 2 → ℓ 2 be defined by D 1 x = ( x 1 6 , x 2 6 , … , x j 6 , … ) and D 2 x = ( x 1 5 , x 2 5 , … , x j 5 , … ) , ∀ x = { x j } j = 1 ∞ ∈ ℓ 2 , respectively. Let the mapping T i : ℓ 2 → ℓ 2 be defined by T i x = ( 3 i x 1 5 i + 1 , 3 i x 2 5 i + 1 , … , 3 i x j 5 i + 1 , … ) , ∀ x = { x j } j = 1 ∞ ∈ ℓ 2 and k i = 6 7 i + 1 N 7 N for every i = 1, 2, …, N . Let the mapping F 1 , F 2 :ℝ 2 × ℝ 2 → ℝ be defined by
Let the mapping f 1 : ℓ 2 → ℓ 2 be defined by f 1 x = ( x 1 5 , x 2 5 , … , x j 5 , … ) , ∀ x = { x j } j = 1 ∞ ∈ ℓ 2 and the mapping f 2 : ℓ 2 → ℓ 2 be defined by f 2 x = ( x 1 7 , x 2 7 , … , x j 7 , … ) , ∀ x = { x j } j = 1 ∞ ∈ ℓ 2 . Let r = 1 and s = 0.5 . Since L = 1 9 , we choose γ = 0.5 . Let x 1 = ( x 1 1 , x 1 2 , … , x 1 j , … ) and u = ( u 1 , u 2 , …, u j , …) ∈ ℓ 2 and let the sequences { x n } , { y n } and { u n } be generated by ( 3.6 ) as follows:
for all n ≥ 1 , where x n = ( x n 1 , x n 2 , … , x n j , … ) , y n = ( y n 1 , y n 2 , … , y n j , … ) and u n = ( u n 1 , u n 2 , … , u n j , … ) . It easy to see that D 1 , D 2 , T i , F 1 , F 2 , f 1 and f 2 satisfy Theorem 3.4 . Moreover, we have V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ ⋂ i = 1 N F ( T i ) ∩ Ω = { 0 } , where ρ = d 1 = d 2 = 1 . From Theorem 3.4 , we can conclude that the sequences { x n } , { y n } and { u n } converge strongly to 0.
In Example 5.2 , we give computer programming to support our main result.
Example 5.2
Let H 1 = H 2 = C = Q = ℝ 2 be the two-dimensional Euclidean space of the real number with an inner product 〈 ⋅ , ⋅ 〉:ℝ 2 × ℝ 2 → ℝ be defined by 〈 x , y 〉 = x ⋅ y = x 1 y 1 + x 2 y 2 where x = ( x 1 , x 2 ) ∈ ℝ 2 and y = ( y 1 , y 2 ) ∈ ℝ 2 and a usual norm ∥ ⋅ ∥ :ℝ 2 → ℝ be defined by ∥ x ∥ = x 1 2 + x 2 2 where x = ( x 1 , x 2 ) ∈ ℝ 2 . Let the mapping A :ℝ 2 → ℝ 2 be defined by A x = (2 x 1 − x 2 , x 1 + 2 x 2 ) for all x = ( x 1 , x 2 ) ∈ ℝ 2 and A ∗ :ℝ 2 → ℝ 2 be defined by A ∗ z = (2 z 1 − z 2 , 2 z 2 − z 1 ) for all z = ( z 1 , z 2 ) ∈ ℝ 2 . Let D 1 , D 2 :ℝ 2 → ℝ 2 be defined by D 1 x = ( x 1 6 , x 2 6 ) and D 2 x = ( x 1 2 , x 2 3 ) , ∀ x = ( x 1 , x 2 ) ∈ ℝ 2 , respectively. Let the mapping T i :ℝ 2 → ℝ 2 be defined by T i x = ( 3 i x 1 3 i + 1 , 3 i x 2 3 i + 2 ) , ∀ x = ( x 1 , x 2 ) ∈ ℝ 2 and k i = 6 7 i + 1 N 7 N for every i = 1, 2, …, N . Let the mapping F 1 , F 2 :ℝ 2 × ℝ 2 → ℝ be defined by
Let the mapping f 1 :ℝ 2 → ℝ 2 be defined by f 1 x = ( x 1 5 , x 2 5 ) , ∀ x = ( x 1 , x 2 ) ∈ ℝ 2 and the mapping f 2 :ℝ 2 → ℝ 2 be defined by f 2 x = ( x 1 7 , x 2 7 ) , ∀ x = ( x 1 , x 2 ) ∈ ℝ 2 . Let r = 1 and s = 0.5 , the sequences z n = ( z n 1 , z n 2 ) , x n = ( x n 1 , x n 2 ) , u n = ( u n 1 , u n 2 ) , y = ( y 1 , y 2 ) ∈ ℝ 2 . By the definition of f 1 and f 2 , we get
Let G 1 ( y 1 ) = ( y 1 ) 2 + ( − x n 1 + 6 5 z n 1 ) y 1 + x n 1 z n 1 − 11 5 ( z n 1 ) 2 and G 2 ( y 2 ) = ( y 2 ) 2 + ( − x n 2 + 6 5 z n 2 ) y 2 + x n 2 z n 2 − 11 5 ( z n 2 ) 2 . G 1 ( y 1 ) and G 2 ( y 2 ) are quadratic functions with coefficients a 1 = 1 , b 1 = − x n 1 + 6 5 z n 1 , and c 1 = x n 1 z n 1 − 11 5 ( z n 1 ) 2 of G 1 ( y 1 ) and coefficients a 2 = 1 , b 2 = − x n 2 + 6 5 z n 2 , and c 2 = x n 2 z n 2 − 11 5 ( z n 2 ) 2 of G 2 ( y 2 ) , respectively. Determine the discriminant Δ 1 of G 1 as follows:
We know that G 1 ( y 1 ) ≥ 0 , ∀ y ∈ ℝ . If it has most one solution in ℝ , then Δ 1 ≤ 0 , so we obtain z n 1 = 5 x n 1 16 . Next, we determine the discriminant Δ 2 of G 2 by using the same method as above, we obtain z n 2 = 5 x n 2 16 . That is T r F 1 ( I − r f 1 ) z n = ( 5 x n 1 16 , 5 x n 2 16 ) . After that, we find the solution of u n = ( u n 1 , u n 2 ) in this inequality 0 ≤ F 2 ( u n , y ) + 〈 f 2 ( u n ) , y − u n 〉 + 1 s 〈 y − u n , u n − x n 〉 . By using the same method as z n = ( z n 1 , z n 2 ) , we obtain
That is, T s F 2 ( I − s f 2 ) u n = ( 7 x n 1 51 , 7 x n 2 51 ) .
Let x 1 = ( x 1 1 , x 1 2 ) and u = ( u 1 , u 2 ) ∈ ℝ 2 . The sequences { x n } , { y n } and { u n } are generated by ( 3.6 ), where k i = 6 7 i + 1 N 7 N , d 1 = 1 , d 2 = 1 , a = 0.5 , α n = 1 2 n , β n = 7 n − 4 12 n , γ n = 5 n − 2 12 n and λ n = 1 2 n 2 for all n ∈ ℕ . Since L = 5 , we choose γ = 0.1 . From the definition of D 1 , D 2 , T i , F 1 , F 2 , f 1 and f 2 , we have V I ( C , D 1 ) ∩ V I ( C , D 2 ) ∩ ⋂ i = 1 N F ( T i ) ∩ Ω = { 0 } . From Theorem 3.4 , we can conclude that the sequences { x n } , { y n } and { u n } converge strongly to 0. We can rewrite ( 3.6 ) as follows:
for all n ≥ 1 , where x n = ( x n 1 , x n 2 ) , y n = ( y n 1 , y n 2 ) and u n = ( u n 1 , u n 2 ) .
Table 1 shows the values of sequences { x n } , { y n } and { u n } where u = (5, −5) , x 1 = (5, −5) and n = 30 .
Table 1
The values of { x n } , { y n } and { u n } where u = (5, −5) , x 1 = (5, −5) and n = 30
- Example 5.1 is an example in infinite dimensional Hilbert space for supporting Theorem 3.4

The convergence comparison with different values N
- Theorem 3.4 guarantees the convergence of { x n } , { y n } and { u n } in Example 5.1 and Example 5.2 .
- By using the concept of Picard iteration, Wang [ 13 ] defined the iterative scheme { x n } for solving SCFPP as follows: x n + 1 = x n − ρ n ( ( I − U ) x n + A ∗ ( I − T ) A x n ) = ( I − ρ n ( ( I − U ) + A ∗ ( I − T ) A ) ) x n , 6.1 where ρ n is according to ( 1.4 ) and U and T are firmly quasi-nonexpansive mappings. Then the sequence { x n } converges weakly to z , where z = lim n →∞ P Φ x n . In Theorem 3.4 , we use the concept of Halpern iteration and suitable conditions of the parameters d 1 , d 2 , r , s , a , γ , L , { α n } , { β n } and { γ n } , the sequence { x n } defined by ( 3.6 ) converges strongly to z = P ℱ u , which is a different method from ( 6.1 ).
Acknowledgements
This paper was supported by the Royal Golden Jubilee (RGJ) Ph.D. Programme, the Thailand Research Fund (TRF), under Grant No. PHD/0082/2558 and the Research and Innovation Services of King Mongkut’s Institute of Technology Ladkrabang.
Authors’ contributions
The two authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Kanyarat Cheawchan, Email: [email protected] .
Atid Kangtunyakarn, Email: moc.liamtoh@kcorwaeb .
- Published: 19 June 2022
Linear approximation method for solving split inverse problems and its applications
- Guash Haile Taddele 1 ,
- Yuan Li 2 ,
- Aviv Gibali 3 ,
- Poom Kumam ORCID: orcid.org/0000-0002-5463-4581 4 &
- Jing Zhao 2
Advances in Computational Mathematics volume 48 , Article number: 39 ( 2022 ) Cite this article
151 Accesses
Metrics details
We study the problem of finding a common element that solves the multiple-sets feasibility and equilibrium problems in real Hilbert spaces. We consider a general setting in which the involved sets are represented as level sets of given convex functions, and propose a constructible linear approximation scheme that involves the subgradient of the associated convex functions. Strong convergence of the proposed scheme is established under mild assumptions and several synthetic and practical numerical illustrations demonstrate the validity and advantages of our method compared with related schemes in the literature.
This is a preview of subscription content, access via your institution .
Access options
Buy single article.
Instant access to the full article PDF.
Price includes VAT (Russian Federation)
Rent this article via DeepDyve.
Availability of data and material
Not applicable
Code availability
Censor, Y, Elfving, T, Kopf, N, Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21 (6), 2071–2084 (2005). Available from: https://doi.org/10.1088/0266-5611/21/6/017
Article MathSciNet MATH Google Scholar
Censor, Y, Bortfeld, T, Martin, B, Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol.. 51 (10), 2353–2365 (2006). Available from: https://doi.org/10.1088/0031-9155/51/10/001
Article Google Scholar
López, G, Martin, V, Xu, H, et al.: Iterative algorithms for the multiple-sets split feasibility problem. Biomed. Math.: Promising Directions in Imaging, Therapy Planning and Inverse Problems 243–279 (2009)
López, G, Martín-Márquez, V, Xu, HK: Perturbation techniques for nonexpansive mappings with applications. Nonlinear Anal. Real World Appl. 10 (4), 2369–2383 (2009). Available from: https://doi.org/10.1016/j.nonrwa.2008.04.020
López, G, Martín-Márquez, V, Wang, F, Xu, HK: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28 (8), 085004 (2012). Available from: https://doi.org/10.1088/0266-5611/28/8/085004
Censor, Y, Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8 (2), 221–239 (1994). Available from: https://doi.org/10.1007/bf02142692
Censor, Y, Segal, A.: Iterative projection methods in biomedical inverse problems. Mathematical methods in biomedical imaging and intensity-modulated radiation therapy (IMRT), vol. 10, pp. 65–96 (2008)
Wang, J, Hu, Y, Li, C, Yao, JC: Linear convergence of CQ algorithms and applications in gene regulatory network inference. Inverse Probl. 33 (5), 055017 (2017). Available from: https://doi.org/10.1088/1361-6420/aa6699
Ansari, QH, Rehan, A: Split feasibility and fixed point problems. In: Nonlinear Analysis. Available from: https://doi.org/10.1007/978-81-322-1883-8∖_9 , pp 281–322. Springer, India (2014)
Xu, HK: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26 (10), 105018 (2010). Available from: https://doi.org/10.1088/0266-5611/26/10/105018 https://doi.org/10.1088/0266-5611/26/10/105018
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20 (1), 103–120 (2003). Available from: https://doi.org/10.1088/0266-5611/20/1/006 https://doi.org/10.1088/0266-5611/20/1/006
Takahashi, W.: The split feasibility problem and the shrinking projection method in Banach spaces. J. Nonlinear Convex Anal. 16 (7), 1449–1459 (2015)
MathSciNet MATH Google Scholar
Xu, H. K.: A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22 (6), 2021–2034 (2006). Available from: https://doi.org/10.1088/0266-5611/22/6/007
Article MATH Google Scholar
Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Probl. 20 (4), 1261–1266 (2004). Available from: https://doi.org/10.1088/0266-5611/20/4/014
Gibali, A, Mai, D T, Vinh, N T: A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications. J. Ind. Manag. Optim.. 15 (2), 963–984 (2019). Available from: https://doi.org/10.3934/jimo.2018080
Sahu, D R, Cho, Y J, Dong, Q L, Kashyap, M R, Li, X. H.: Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces. Numer. Algorithms. Available from: https://doi.org/10.1007/s11075-020-00999-2 (2020)
Shehu, Y, Gibali, A.: New inertial relaxed method for solving split feasibilities. Optim. Lett. Available from: https://doi.org/10.1007/s11590-020-01603-1 (2020)
Liou, Y C, Zhu, L J, Yao, Y, Chyu, C C: Algorithmic and analytical approaches to the split feasibility problems and fixed point roblems. Taiwan. J. Math. 17 (5). Available from: https://doi.org/10.11650/tjm.17.2013.3175 (2013)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18 (2), 441–453 (2002). Available from: https://doi.org/10.1088/0266-5611/18/2/310
Dong, Q L, Yao, Y, He, S.: Weak convergence theorems of the modified relaxed projection algorithms for the split feasibility problem in Hilbert spaces. Optim. Lett. 8 (3), 1031–1046 (2013). Available from: https://doi.org/10.1007/s11590-013-0619-4
He, S, Zhao, Z.: Strong convergence of a relaxed CQ algorithm for the split feasibility problem. J. Inequal. Appl. 2013 (1). Available from: https://doi.org/10.1186/1029-242x-2013-197 (2013)
Yao, Y, Postolache, M, Liou, Y. C.: Strong convergence of a self-adaptive method for the split feasibility problem. Fixed Point Theory Appl. 2013 (1), 201 (2013). Available from: https://doi.org/10.1186/1687-1812-2013-201
Dang, Y, Sun, J, Xu, H: Inertial accelerated algorithms for solving a split feasibility problem. J. Ind. Manag. Optim. 13 (3), 1383–1394 (2017). Available from: https://doi.org/10.3934/jimo.2016078
Gibali, A, Liu, L W, Tang, Y. C.: Note on the modified relaxation CQ algorithm for the split feasibility problem. Optim. Lett. 12 (4), 817–830 (2017). Available from: https://doi.org/10.1007/s11590-017-1148-3 https://doi.org/10.1007/s11590-017-1148-3
Shehu, Y, Vuong, P T, Cholamjiak, P.: A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restoration problems. J. Fixed Point Theory Appl. 21 (2). Available from: https://doi.org/10.1007/s11784-019-0684-0 (2019)
Blum, E.: From optimization and variational inequalities to equilibrium problems. Mathematics Student 63 , 123–145 (1994)
Facchinei, F, Pang, J. S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer Series in Operations Research. Springer, New York (2003)
MATH Google Scholar
Konnov, I. V.: Equilibrium Models and Variational Inequalities, vol. 210 of Mathematics in Science and Engineering. Elsevier B. V., Amsterdam (2007)
Google Scholar
Muu, L D, Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18 (12), 1159–1166 (1992). Available from: https://doi.org/10.1016/0362-546X(92)90159-C https://doi.org/10.1016/0362-546X(92)90159-C
Chang, X, Liu, S, Deng, Z, Li, S.: An inertial subgradient extragradient algorithm with adaptive stepsizes for variational inequality problems. Optim. Methods Softw. 1–20 (2021)
Ceng, L C, Petruşel, A, Qin, X, Yao, J. C.: Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization 70 (5–6), 1337–1358 (2021). Available from: https://doi.org/10.1080/02331934.2020.1858832
Bnouhachem, A, Al-Homidan, S, Ansari, Q.: An iterative method for common solutions of equilibrium problems and hierarchical fixed point problems. Fixed Point Theory Appl. 2014 (1), 194 (2014). Available from: https://doi.org/10.1186/1687-1812-2014-194
Flåm, SD, Antipin, AS: Equilibrium programming using proximal-like algorithms. Math. Program. 78 (1), 29–41 (1996). Available from: https://doi.org/10.1007/bf02614504
Moudafi, A.: Second-order differential proximal methods for equilibrium problems. J. Inequalities Appl. Math. 4 (1), 1–7 (2003)
Shehu, Y, Iyiola, O S, Thong, D V, Van, N. T. C.: An inertial subgradient extragradient algorithm extended to pseudomonotone equilibrium problems. Math. Methods Oper. Res. 93 (2), 213–242 (2021). Available from: https://doi.org/10.1007/s00186-020-00730-w
Sombut, K, Plubtieng, S.: Weak convergence theorem for finding fixed points and solution of split feasibility and systems of equilibrium problems. Abstr. Appl. Anal. 2013 , 1–8 (2013). Available from: https://doi.org/10.1155/2013/430409
Qin, X, Wang, L.: A fixed point method for solving a split feasibility problem in Hilbert spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (1), 315–325 (2017). Available from: https://doi.org/10.1007/s13398-017-0476-6
Shehu, Y, Ogbuisi, FU: An iterative method for solving split monotone variational inclusion and fixed point problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 110 (2), 503–518 (2015). Available from: https://doi.org/10.1007/s13398-015-0245-3
Tang, Y, Gibali, A.: Several inertial methods for solving split convex feasibilities and related problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114 (3). Available from: https://doi.org/10.1007/s13398-020-00857-9 (2020)
Tang, Y, Zhu, C, Yu, H.: Iterative methods for solving the multiple-sets split feasibility problem with splitting self-adaptive step size. Fixed Point Theory Appl. 2015 (1). Available from: https://doi.org/10.1186/s13663-015-0430-2 (2015)
Bauschke, H H, Combettes, P. L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011). Available from: https://doi.org/10.1007/978-1-4419-9467-7
Book MATH Google Scholar
Goebel, K, Simeon, R.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984)
Goebel, K, Kirk, W. A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990). Available from: https://doi.org/10.1017/cbo9780511526152
Aubin, J. P.: Optima and Equilibria: an Introduction to Nonlinear Analysis, vol. 140. Springer Science & Business Media (2013)
Takahashi, S, Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331 (1), 506–515 (2007). Available from: https://doi.org/10.1016/j.jmaa.2006.08.036
Suzuki, T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005 (1), 685918 (2005). Available from: https://doi.org/10.1155/fpta.2005.103
Article MathSciNet Google Scholar
Xu, H. K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66 (1), 240–256 (2002). Available from: https://doi.org/10.1112/s0024610702003332
Maingé, P. E.: New approach to solving a system of variational inequalities and hierarchical problems. J. Optim. Theory Appl. 138 (3), 459–477 (2008). Available from: https://doi.org/10.1007/s10957-008-9433-z https://doi.org/10.1007/s10957-008-9433-z
Dai, Y. H.: Fast algorithms for projection on an ellipsoid. SIAM J. Optim. 16 (4), 986–1006 (2006). Available from: https://doi.org/10.1137/040613305
Yu, H, Zhan, W, Wang, F.: The ball-relaxed CQ algorithms for the split feasibility problem. Optimization 67 (10), 1687–1699 (2018). Available from: https://doi.org/10.1080/02331934.2018.1485677
He, S, Zhao, Z, Luo, B.: A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem. Optimization 64 (9), 1907–1918 (2015). Available from: https://doi.org/10.1080/02331934.2014.895898 https://doi.org/10.1080/02331934.2014.895898
Suantai, S, Pholasa, N, Cholamjiak, P.: Relaxed CQ algorithms involving the inertial technique for multiple-sets split feasibility problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2), 1081–1099 (2019). Available from: https://doi.org/10.1007/s13398-018-0535-7
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces, vol. 2057 of Lecture Notes in Mathematics. Springer, Heidelberg (2012)
Kesornprom, S, Pholasa, N, Cholamjiak, P.: On the convergence analysis of the gradient-CQ algorithms for the split feasibility problem. Numer. Algorithms 84 (3), 997–1017 (2020). Available from: https://doi.org/10.1007/s11075-019-00790-y
Reich, S, Truong, M T, Mai, T. N. H.: The split feasibility problem with multiple output sets in Hilbert spaces. Optim Lett. 14 (8), 2335–2353 (2020). Available from: https://doi.org/10.1007/s11590-020-01555-6 https://doi.org/10.1007/s11590-020-01555-6
Suantai, S, Eiamniran, N, Pholasa, N, Cholamjiak, P.: Three-step projective methods for solving the split feasibility problems. Mathematics 7 (8), 712 (2019). Available from: https://doi.org/10.3390/math7080712 https://doi.org/10.3390/math7080712
Wang, F.: Polyak’s gradient method for split feasibility problem constrained by level sets. Numer. Algorithms 77 (3), 925–938 (2018). Available from: https://doi.org/10.1007/s11075-017-0347-4
Download references
Acknowledgements
The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Guash Haile Taddele was supported by the ”Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut’s University of Technology Thonburi” (Grant No.37/2561). Moreover, this project is funded by National Research Council of Thailand (NRCT) under Research Grants for Talented Mid-Career Researchers (Contract no. N41A640089). We are also very grateful to the Editor and Reviewers for taking the time and effort necessary to review the manuscript. We sincerely appreciate all valuable comments and suggestions, which helped us to improve the quality of the manuscript.
This research was funded by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, KMUTT. The first author was supported by the ”Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut’s University of Technology Thonburi” with Grant No. 37/2561. Moreover, this project is funded by National Research Council of Thailand (NRCT) under Research Grants for Talented Mid-Career Researchers (Contract no. N41A640089).
Author information
Authors and affiliations.
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand
Guash Haile Taddele
College of Science, Civil Aviation University of China, Tianjin, China
Yuan Li & Jing Zhao
Department of Mathematics, ORT Braude College, 2161002, Karmiel, Israel
Aviv Gibali
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok, 10140, Thailand
You can also search for this author in PubMed Google Scholar
Contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Corresponding author
Correspondence to Poom Kumam .
Ethics declarations
Competing interests.
The authors declare no competing interests.
Additional information
Communicated by: Stefan Volkwein
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Reprints and Permissions
About this article
Cite this article.
Taddele, G.H., Li, Y., Gibali, A. et al. Linear approximation method for solving split inverse problems and its applications. Adv Comput Math 48 , 39 (2022). https://doi.org/10.1007/s10444-022-09959-x
Download citation
Received : 18 July 2021
Accepted : 13 May 2022
Published : 19 June 2022
DOI : https://doi.org/10.1007/s10444-022-09959-x
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative
- Split feasibility problem
- Equilibrium problem
- Self-adaptive technique
Mathematics Subject Classification (2010)
Advertisement
- Find a journal
- Publish with us
Help | Advanced Search
Mathematics > Optimization and Control
Title: projection methods for solving split equilibrium problems.
Abstract: The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP). It is known that almost solution methods for solving problem (SEP) are designed from two fundamental methods as the proximal point method and the extended extragradient method (or the two-step proximal-like method). Unlike previous results, in this paper we introduce a new algorithm, which is only based on the projection method, for finding solution approximations of problem (SEP), and then establish that the resulting algorithm is weakly convergent under mild conditions. Several of numerical results are reported to illustrate the convergence of the proposed algorithm and also to compare with others.
Submission history
Access paper:.
- Download PDF
- Other Formats
References & Citations
- Google Scholar
- Semantic Scholar
BibTeX formatted citation

Bibliographic and Citation Tools
Code, data and media associated with this article, recommenders and search tools.
- Institution
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

IMAGES
VIDEO
COMMENTS
. Throughout this paper, the solution set of (1.1) will be denoted by . Problem (1.1) models a wide variety of problems including complementarity problems, optimization problems, feasibility problems, Nash equilibrium problems, variational inequalities, and fixed point problems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10].
We propose a splitting method for solving an equilibrium problem involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of two appropriate maximally monotone operators. Our algorithm is a consequence of the Douglas--Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and ...
We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces.
We propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of...
Semantic Scholar extracted view of "A Douglas-Rachford splitting method for solving equilibrium problems" by L. Briceño-Arias
In this paper, we introduce a splitting algorithm for solving equilibrium problems given by the difference of two bifunctions in a real Hilbert space. Under suitable assumptions on component bifunctions, we prove strong convergence of the proposed algorithm.
Abstract In this paper, sequential and parallel splitting algorithms are proposed for solving equilibrium problems given by a sum of two functions. The convergence of the sequences generated by the proposed methods is guaranteed by assuming the Hölder continuity of each function.
We propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition. Our algorithm is a consequence of the Douglas-Rachford splitting applied to this auxiliary monotone ...
Some iterative methods for solving equilibrium problems are suggested and analyzed by using the technique of the auxiliary principle, see, for example [ 2, 3, 10, 17, 39, 45] and references therein. In the papers [ 17, 39, 45 ], the bifunction f is assumed to satisfy the following conditions (A1) for all ; (A2) f is monotone on C, i.e., for all ;
A Douglas-Rachford Splitting Method for Solving Equilibrium Problems Authors: Luis M. Briceño-Arias Universidad Técnica Federico Santa María Abstract We propose a splitting method for...
In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159 ...
In this paper, sequential and parallel splitting algorithms are proposed for solving equilibrium problems given by a sum of two functions. The convergence of the sequences generated by the proposed methods is guaranteed by assuming the Hölder continuity of each function. Some preliminary numerical experiences and comparisons are also reported.
Abstract In this paper, we prove a weak convergence theorem for finding a common solution of combination of equilibrium problems, infinite family of nonexpansive mappings, and the modified inclusion problems using inertial forward-backward algorithm. Further, we discuss some applications of our obtained results.
A new splitting algorithm for solving equilibrium problems arising from Nash-Cournot oligopolistic equilibrium problems in electricity markets with non-convex cost functions is discussed and the strong convergence of the proposed algorithm is proved. "In this paper, we discuss a new splitting algorithm for solving equilibrium problems arising from Nash-Cournot oligopolistic equilibrium ...
Such a method, in the context of variational inequalities, is known as a splitting method. This can lead to the development of very efficient methods, since one can treat each part of the original bifunction independently. In the context of variational inequalities splitting methods and related techniques have been studied by many authors.
Convergence analysis for solving equilibrium problems and split feasibility problems in Hilbert spaces. Haiying Li a College of Mathematics and Information Science, Henan Normal University, Xinxiang, People's Republic of China Correspondence [email protected] View further author information,
Projection methods for solving split equilibrium problems Dang Van Hieu , Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam * Corresponding author: [email protected] Received: July 2018 Revised: January 2019 Early access: May 2019 Published: August 2020 Abstract
In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence { x n } $\\{x_{n}\\}$ by the shrinking projection method. Our results improve and extend the previous ...
Projection methods for solving split equilibrium problems DOI: Authors: Dang Van Hieu University of Air Force, Vietnam Abstract The paper considers a split inverse problem involving...
ABSTRACT. In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces.
We introduce a new method of an iterative scheme {x n} for finding a common element of the set of solutions of variational inequality problems and the set of common fixed points of a finite family of quasi-nonexpansive mappings and the set of solutions of the modified split generalized equilibrium problem without assuming a demicloseness ...
We consider a general setting in which the involved sets are represented as level sets of given convex functions, and propose a constructible linear approximation scheme that involves the subgradient of the associated convex functions.
Projection methods for solving split equilibrium problems Dang Van Hieu The paper considers a split inverse problem involving component equilibrium problems in Hilbert spaces. This problem therefore is called the split equilibrium problem (SEP).