splitting method for solving equilibrium problems

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A Douglas–Rachford splitting method for solving equilibrium problems

2012, Nonlinear Analysis: Theory, Methods & Applications

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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.

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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

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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

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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 $z_{0}\in F(T)\cap\Gamma$ .

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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).

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Uamporn Witthayarat

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Afrah A N Abdou & Yeol Je Cho

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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

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Received : 10 August 2015

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DOI : https://doi.org/10.1186/s13663-015-0448-5

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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

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

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

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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.

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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

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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.

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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).

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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

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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

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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.

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A Douglas–rachford Splitting Method for Solving Equilibrium Problems 1 1 1 This work was supported by CONICYT under grant FONDECYT N o 3120054.

1 Introduction

Assumption 1.1

Problem 1.2

Lemma 1.3 ( cf. ( Opti04 , , Lemma 5.1) )

2 Monotone inclusions and equilibrium problems

2.1 Monotone inclusion associated to equilibrium problems

Proposition 2.1

Theorem 2.2

2.2 Equilibrium problems associated to monotone inclusions

Proposition 2.3

Proposition 2.5

Proposition 2.7

Proposition 2.8

3 Algorithm and convergence

Theorem 3.1

Theorem 3.2

Example 3.4

4 Acknowledgement

Projection methods for solving split equilibrium problems

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Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces

1 Introduction

2 Preliminaries

Assumption 2.3

3 Main results

Theorem 3.1

Corollary 3.2

Corollary 3.3

4 Applications

Theorem 4.1

4.2 Applications to split optimization problems

Theorem 4.2

Acknowledgements

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The modified split generalized equilibrium problem for quasi-nonexpansive mappings and applications

Atid Kangtunyakarn

Introduction

Definition 1.1

Example 1.2

Preliminaries

Main results

Theorem 3.4

Corollary 3.5

Corollary 3.6

Corollary 3.7

Application

Definition 4.1

Theorem 4.4

Example and numerical results

Example 5.1

Example 5.2

Table 1

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Linear approximation method for solving split inverse problems and its applications

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