Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space

Abstract

Let be a real Banach space with the dual space . Let be a proper functional and let be a bifunction. In this paper, a new concept of -proximal mapping of with respect to is introduced. The existence and Lipschitz continuity of the -proximal mapping of with respect to are proved. By using properties of the -proximal mapping of with respect to , a generalized mixed equilibrium problem with perturbation (for short, GMEPP) is introduced and studied in Banach space . An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space , and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in a Hilbert space.

1. Introduction

Let be a real Banach space with norm and let be its dual space. The value of at will be denoted by . The normalized duality mapping from into the family of nonempty (by Hahn-Banach theorem) weak-star compact subsets of its dual space is defined by

(1.1)

It is known that the norm of is said to be Gateaux differentiable (and is said to be smooth) if

(1.2)

exists for every in the unit sphere of . It is said to be uniformly Gateaux differentiable if for each , this limit is attained uniformly for . The norm of is said to be uniformly Frechet differentiable (and is said to be uniformly smooth) if the limit in (1.2) is attained uniformly for . Every uniformly smooth Banach space is reflexive and has a uniformly Gateaux differentiable norm.

Recall also that if is smooth, then is single-valued and continuous from the norm topology of to the weak star topology of , that is, norm-to-weak continuous. It is also well known that if has a uniformly Gateaux differentiable norm, then is uniformly continuous on bounded subsets of from the strong topology of to the weak star topology of , that is, uniformly norm-to-weak* continuous on any bounded subset of . Moreover, if is uniformly smooth, then is uniformly continuous on bounded subsets of from the strong topology of to the strong topology of , that is, uniformly norm-to-norm continuous on any bounded subset of . See [1] for more details.

It is well known that the variational inequality theory has played an important and powerful role in the studying of a wide class of linear and nonlinear problems arising in many diverse fields of pure and applied sciences, such as mathematical programming, optimization theory, engineering, elasticity theory and equilibrium problems of mathematical economics, and game theory; see, for instance, [1–6] and the references therein.

One of the most interesting and important problems in the theory of variational inequalities is the development of an efficient iterative algorithm to compute approximate solutions. In the setting of Hilbert spaces, one of the most efficient numerical techniques is the projection method and its variant forms; see [4, 6–15]. Since the standard projection method strictly depends on the inner product property of Hilbert spaces, it can no longer be applied for general mixed type variational inequalities in Banach spaces. This fact motivates us to develop alterative methods to study the existence and iterative algorithms of solutions for general mixed variational inequalities in Banach spaces. Recently, [16–18] extended the auxiliary principle technique to study the existence of solutions and to suggest the iterative algorithms for solving various mixed type variational inequalities in Banach spaces. For some related work, we refer to [19–21] and the references therein.

Very recently, inspired by the research work going on in this field, Xia and Huang [14] first introduced a new concept of -proximal mapping for a proper subdifferentiable functional on a Banach space. They proved an existence theorem and Lipschitz continuity of the -proximal mapping. Using the properties of the -proximal mapping, they proved an existence theorem of solutions for a new class of general mixed variational inequalities in a Banach space and suggested an iterative algorithm for computing approximate solutions. Moreover, they gave the strong convergence criteria of the iterative sequence generated by this algorithm. Their results include some known results in [8, 9, 11, 12, 16–18] as special cases.

Let be a real Banach space with the dual space . Let be a proper functional and let be a bifunction. In this paper, motivated by Xia and Huang [14], we first introduce a new concept of -proximal mapping of with respect to . We prove an existence theorem and Lipschitz continuity of the -proximal mapping of with respect to . Utilizing the properties of the -proximal mapping of with respect to , we introduce and consider a generalized mixed equilibrium problem with perturbation (for short, GMEPP) which includes as a special case the general mixed variational inequality studied by Xia and Huang [14]. We show an existence theorem of solutions for this problem under some appropriate conditions. In order to compute approximate solutions of the GMEPP, we propose a new iterative algorithm which includes as a special case the iterative algorithm considered by Xia and Huang [14]. Finally, we establish the strong convergence criteria of the iterative sequence generated by the new algorithm in a uniformly smooth Banach space , and also derive the weak convergence criteria of the iterative sequence generated by this new algorithm in a Hilbert space. Our results are new and represent the improvement, extension, and development of Xia and Huang's results in [14].

2. Preliminaries

Let be a real Banach space with the topological dual space and let be the pairing between and . We write to indicate that the sequence converges weakly to . implies that converges strongly to . Let and denote the family of all subsets of and the family of all nonempty closed bounded subsets of , respectively. Let be a bifunction, let and be single-valued mappings, and let be a proper lower semicontinuous functional. We consider the following generalized mixed equilibrium problem with perturbation (for short, GMEPP): find such that

(2.1)

Some special cases of problem (2.1) are the following.

(1)If a real Hilbert space, an identity mapping on , and is a lower semicontinuous and convex functional, then GMEPP (2.1) reduces to the generalized mixed equilibrium problem with perturbation considered by Ceng et al. [22].

(2)If a real Hilbert space, an identity mapping on , , and is a lower semicontinuous and convex functional, then GMEPP (2.1) reduces to the generalized mixed equilibrium problem considered by Peng and Yao [23].

(3)If , then GMEPP (2.1) reduces to the general mixed variational inequality problem (for short, GMVIP) considered by Xia and Huang [14].

(4)If a real Hilbert space, , and is a proper convex lower semicontinuous functional, then GMEPP (2.1) was studied by many authors (see, e.g., [1, 17–19]).

We first recall the following definitions and some known results.

Definition 2.1.

Let be a set-valued mapping, and let and be two single-valued mappings. We say that

(i) is -strongly monotone with constant if, for any ,

(2.2)

(ii) is -strongly monotone if, for any , and ,

(2.3)

(iii) is -Lipschitz continuous with constant if, for all ,

(2.4)

where is the Hausdorff metric on ;

(iv) is -strongly accretive (where ) if, for any , there exists such that

(2.5)

where is the normalized duality mapping defined by

(2.6)

Definition 2.2.

Let be a Banach space with the dual space , let be a bifunction, and let be a proper functional. If, for , there exists a such that

(2.7)

then is said to be -subdifferentiable at . We denote by the set of such elements , that is,

(2.8)

The set is said to be the -subdifferential of at . If there exists the -subdifferential at each , then is said to be -subdifferentiable. The mapping is said to be the -subdifferential of .

Definition 2.3.

Let be a Banach space with the dual space , and let be a proper subdifferentiable functional. If for any given and any constant , there is a unique satisfying

(2.9)

then the mapping , denoted by , is said to be an -proximal mapping of with respect to .

Remark 2.4.

From Definitions 2.2 and 2.3 it follows that is -subdifferentiable at each the range of . If is additionally -subdifferentiable at each , then there exists the -subdifferential at each ; that is, is -subdifferentiable. Observe that and so

(2.10)

In particular, whenever , then the concept of -proximal mapping of with respect to reduces to the one of -proximal mapping of by Xia and Huang [14, Definition ]. In this case, is rewritten as . By the definition of the subdifferential, we know that and so

(2.11)

Lemma 2.5 (see [24]).

Let be a nonempty convex subset of a topological vector space and let be such that

(i)for each is lower semicontinuous on each nonempty compact subset of ;

(ii)for each nonempty finite set and for each with and ,

(2.12)

(iii)there exist a nonempty compact convex subset of and a nonempty compact subset of such that for each , there is an with .

Then there exists such that for all .

Recall now that satisfies Opial's property [25] provided that, for each sequence in , the condition implies

(2.13)

It is known [25] that each enjoys this property, while does not unless . It is known [26] that any separable Banach space can be equivalently renormed so that it satisfies Opial's property.

Furthermore, recall that in a Hilbert space, there holds the following equality:

(2.14)

for all and .

In order to investigate the generalized mixed equilibrium problem (2.1) with perturbation, we will need the following conditions on mapping and bifunction in the sequel:

(H1);

(H2) is monotone, that is, ;

(H3) for each is convex and lower semicontinuous;

(H4) the following equilibrium problem (for short, EP) or generalized equilibrium problem (for short, GEP) has a solution in :

(EP) Find such that for all , or

(GEP) Find such that for all .

Now we give some sufficient conditions which guarantee the existence and Lipschitz continuity of an -proximal mapping of with respect to in a reflexive Banach space .

Theorem 2.6.

Let be a reflexive Banach space with the dual space , let be a bifunction satisfying conditions (H1)–(H4), let be an -strongly monotone and continuous mapping, and let be a lower semicontinuous subdifferentiable proper functional. Then for any given and any , there exists a unique such that

(2.15)

that is, and the -proximal mapping of with respect to is well defined. If is additionally -subdifferentiable at each , then is -subdifferentiable and .

Proof.

For any given and any , define a functional as follows:

(2.16)

By the continuity of and the lower semicontinuity of and , we know that the function is lower semicontinuous on for each fixed .

Now, let us show that satisfies condition (ii) of Lemma 2.5. If it is false, then there exist a finite set and with and such that

(2.17)

Since is subdifferentiable at , there exists a point such that

(2.18)

From (H2) it follows that

(2.19)

Utilizing the convexity of , we get from (H1)

(2.20)

which leads to a contradiction. Therefore, satisfies condition (ii) of Lemma 2.5.

From (H4) we know that the following equilibrium problem (for short, EP) or generalized equilibrium problem (for short, GEP) has a solution :

(EP), for all or

(GEP) for all

Since is subdifferentiable at , there exists a point such that

(2.21)

It follows that

(2.22)

Next, we discuss two cases.

Case 1.

If EP has solution , then from (2.22) we have

(2.23)

Let

(2.24)

Then and are both weakly compact convex subset of . For each , there exists a point such that and so all conditions of Lemma 2.5 are satisfied. By Lemma 2.5, there exists a point such that for all , that is,

(2.25)

Case 2.

If GEP has solution , then from (2.22) we have

(2.26)

Let

(2.27)

Then and are both weakly compact convex subset of . For each , there exists a point such that and so all conditions of Lemma 2.5 are satisfied. By Lemma 2.5, there exists a point such that for all , that is, (2.25) holds.

Now let us show that is a unique solution of auxiliary equilibrium problem (2.15). Suppose that are arbitrary two solutions of auxiliary equilibrium problem (2.15). Then,

(2.28)

(2.29)

Taking in (2.28) and in (2.29) and adding these inequalities we have from the monotonicity of

(2.30)

This together with the -strong monotonicity of implies that

(2.31)

and so . Therefore, and the -proximal mapping of with respect to is well defined. In the meantime, it is known that is -subdifferentiable at each . If is additionally -subdifferentiable at each , then by Remark 2.4 we obtain that is -subdifferentiable and . This completes the proof.

Corollary 2.7 (see [14, Theorem ]).

Let be a reflexive Banach space with the dual space , let be a lower semicontinuous subdifferentiable proper functional, and let be an -strongly monotone and continuous mapping. Then for any given and any , there exists a unique such that

(2.32)

that is, and the -proximal mapping of is well defined.

Proof.

Putting in Theorem 2.6, we obtain the desired result.

Remark 2.8.

Theorem 2.6 shows that if is a bifunction satisfying conditions (H1)–(H4) and is a lower semicontinuous subdifferentiable proper functional, then for any strongly monotone and continuous mapping , the -proximal mapping of with respect to is well defined. Furthermore, if is additionally -subdifferentiable at each , then for each

(2.33)

is the unique solution of auxiliary equilibrium problem (2.15).

Theorem 2.9.

Let be a reflexive Banach space with the dual space , let be an -strongly monotone and continuous mapping, let be a bifunction satisfying conditions (H1)–(H4), and let be a lower semicontinuous subdifferentiable proper functional. If is -subdifferentiable at each , then is -subdifferentiable and the -proximal mapping of with respect to is -Lipschitz continuous. Furthermore, if additionally the -subdifferential for is -strongly monotone, then the -proximal mapping of with respect to is -Lipschitz continuous.

Proof.

First, utilizing Theorem 2.6 we know that whenever is -subdifferentiable at each , is -subdifferentiable. For any , let and . Then and . By the definition of the -subdifferential of , we have

(2.34)

(2.35)

Taking in (2.34) and in (2.35) and adding these inequalities, we obtain

(2.36)

Utilizing condition (H2) we get and hence

(2.37)

Since is -strongly monotone,

(2.38)

which implies that is -Lipschitz continuous.

Now we suppose that the -subdifferential for is -strongly monotone. Then

(2.39)

Since is -strongly monotone,

(2.40)

It follows that

(2.41)

Thus, is -Lipschitz continuous.

Corollary 2.10 (see [14, Theorem ]).

Let be a reflexive Banach space with the dual space , let be an -strongly monotone and continuous mapping, and let be a lower semicontinuous subdifferentiable proper functional. Then the -proximal mapping is -Lipschitz continuous. Furthermore, if the subdifferential for is -strongly monotone, then the -proximal mapping is -Lipschitz continuous.

Proof.

Putting in Theorem 2.9, we derive the desired result.

3. Existence and Algorithm

We first transfer GMEPP (2.1) into a fixed point problem.

Theorem 3.1.

Let be a reflexive Banach space with the dual space , let be an -strongly monotone and continuous mapping, let be a bifunction satisfying conditions (H1)–(H4), and let be a lower semicontinuous subdifferentiable proper functional. If is -subdifferentiable at each , then is a solution of the GMEPP (2.1) if and only if satisfies the following relation:

(3.1)

where is the -proximal mapping of with respect to and is a constant.

Proof.

Since is -subdifferentiable at each , in terms of Theorem 2.6, is -subdifferentiable.

Assume that satisfies relation (3.1). Noting , we know that relation (3.1) holds if and only if

(3.2)

By the definition of the -subdifferential of with respect to , the above relation holds if and only if

(3.3)

that is,

(3.4)

Thus, is a solution of GMEPP (2.1) if and only if satisfies (3.1). This completes the proof.

Remark 3.2.

Relation (3.1) can be written as

(3.5)

where .

Remark 3.3.

By Theorem 2.6, we can choose a strongly monotone and Lipschitz continuous mapping such that it is easy to compute the values of the -proximal mapping of with respect to . Theorem 3.1 shows that, by using the mapping , GMEPP (2.1) can be transferred into a fixed point problem (3.5). Based on these observations, we can suggest the following new and general iterative algorithms for computing the approximate solutions of GMEPP (2.1) in reflexive Banach spaces.

Lemma 3.4 (see [27]).

Let be a real Banach space and let be the normalized duality mapping. Then for any , the following inequality holds:

(3.6)

We now use Theorem 3.1 to construct the following algorithms for solving the GMEPP (2.1) in Banach spaces.

Algorithm 3.5.

Let be two single-valued mappings, let be a single-valued mapping with let be an -strongly monotone and -Lipschitz continuous mapping, let be a bifunction satisfying conditions (H1)–(H4), and let be a lower semicontinuous subdifferentiable proper functional. For any given , an iterative sequence is defined by

(3.7)

where and for all . Algorithm 3.5 is called the Mann-type iterative algorithm.

Algorithm 3.6.

Let and be the same as in Algorithm 3.5. For any given , the iterative sequences and are defined by

(3.8)

where for all . Algorithm 3.6 is called the Ishikawa-type iterative algorithm.

Remark 3.7.

If for all , then Algorithm 3.6 reduces to Algorithm 3.5. Whenever , Algorithms 3.5 and 3.6 reduce to Algorithms 3.1 and 3.2 by Xia and Huang [14], respectively.

Now we prove an existence theorem of solutions for GMEPP (2.1).

Theorem 3.8.

Let be a reflexive Banach space with the dual space let be two continuous mappings, and let be a continuous mapping. Let be -strongly monotone and continuous, let be a bifunction satisfying conditions (H1)–(H4), and let be a lower semicontinuous subdifferentiable proper functional. If is -subdifferentiable at each , and the ranges and are totally bounded, then there exists which is a solution of GMEPP (2.1).

Proof.

Define by

(3.9)

By Theorem 2.9, the mapping is Lipschitz continuous. Since the ranges and are totally bounded, we know that the range is also totally bounded in ; that is, is totally bounded in . Thus, is a compact subset of . Since and are continuous, so does . By Schauder fixed point theorem, has a fixed point . It follows from Theorem 3.1 that is a solution of GMEPP (2.1). This completes the proof.

Remark 3.9.

From the proof of Theorem 3.8, we know that in [14, Theorem ], the assumption of the boundedness of the ranges and cannot guarantee that all the conditions of Schauder fixed point theorem are satisfied. Thus, in [14, Theorem ], the assumption the ranges and are bounded should be replaced by the stronger condition the ranges and are totally bounded. Here the well-known Schauder fixed point theorem is stated as follows.

Let be a Banach space and let be a nonempty closed convex subset of . Assume that is a continuous mapping such that the closure is a compact subset in . Then has a fixed point in , that is, .

In order to give some sufficient conditions, which guarantee the convergence of the iterative sequences generated by Algorithm 3.6, we will need the following lemma in the sequel.

Lemma 3.10 (see [28]).

Let be a sequence of nonnegative real numbers satisfying the condition

(3.10)

where and are sequences of real numbers such that

(i) and , or equivalently,

(3.11)

(ii), or

is convergent.

Then, .

Theorem 3.11.

Let be a uniformly smooth Banach space with the dual space let be -Lipschitz continuous, let be -Lipschitz continuous, and let be -strongly accretive and -Lipschitz continuous. Suppose that is -strongly monotone and -Lipschitz continuous, is a bifunction satisfying conditions (H1)–(H4), and is a lower semicontinuous subdifferentiable proper functional which is -subdifferentiable at each . Let and be two sequences in with , and . Assume that the -subdifferential of is -strongly monotone, the ranges , and are totally bounded, and where

(3.12)

Then for any given , the sequence defined by Algorithm 3.6 converges strongly to a solution of GMEPP (2.1).

Proof.

By Theorem 3.8 and the assumptions in Theorem 3.11, we know that the solution set of GMEPP (2.1) is nonempty. Let be a solution of GMEPP (2.1). Since and , we can choose a constant such that

(3.13)

By Algorithm 3.6, we have

(3.14)

Let

(3.15)

Then

(3.16)

It follows from Theorem 2.9 that is Lipschitz continuous. Since the ranges , and are totally bounded, we know that the set

(3.17)

is bounded. Let

(3.18)

This implies that

(3.19)

Since ,

(3.20)

It follows that

(3.21)

By induction we can prove that

(3.22)

On the other hand, by Lemma 3.4,

(3.23)

Now we consider the third term on the right-hand side of (3.23). From (3.19) and (3.22) it follows that

(3.24)

Since is a uniformly smooth Banach space, the normalized duality mapping is uniformly norm-to-norm continuous on any bounded subset of . Hence it is easy to see that

(3.25)

Let

(3.26)

Since is bounded,

(3.27)

Next we consider the second term on the right-hand side of (3.23). Since is a solution of GMEPP (2.1), by Theorem 3.1, we have

(3.28)

It follows that

(3.29)

where

(3.30)

Substituting (3.27) and (3.29) into (3.23), we have

(3.31)

Next we make an estimation for . Indeed,

(3.32)

Substituting (3.32) into (3.31) and simplifying it, we have

(3.33)

From condition (3.13), we get . Since , and , we deduce that and

(3.34)

Thus, utilizing Lemma 3.10 we conclude that as . This completes the proof.

Remark 3.12.

By the careful analysis of the proof of Theorem 3.11, we can see readily that the following conditions are used to derive the following conclusion:

(i)the normalized duality mapping is uniformly norm-to-norm continuous on any bounded subset of ;

(ii)the constant in Algorithm 3.6 satisfies the inequality .

In addition, we apply Lemma 3.4 to derive the strong convergence of the iterative sequence generated by Algorithm 3.6. Moreover, we simplify the original proof by Xia and Huang [14, Theorem 3.11] to a great extent. Therefore, Theorem 3.11 is a generalization and modification of Xia and Huang's Theorem [14].

Remark 3.13.

We would like to point out that, in Theorem 3.11, the functional may not be convex, the mappings and may not have any monotonicity, and their domains and ranges are reflexive Banach space and the dual space of , respectively. Hence Theorem 3.11 improves and generalizes some known results in [8, 9, 11, 16, 17]. Furthermore, the argument methods presented in this paper are quite different from those in [8–11, 13, 14, 16, 21].

Finally, we give a weak convergence theorem for the iterative sequence generated by Algorithm 3.6 in a real Hilbert space. However, we first recall the following lemmas.

Lemma 3.14 (see [29, page 303]).

Let and be sequences of nonnegative real numbers satisfying the inequality

(3.35)

If converges, then exists.

Lemma 3.15 (see [30]).

Demiclosedness Principle. Assume that is a nonexpansive self-mapping of a nonempty closed convex subset of a Hilbert space . If has a fixed point, then is demiclosed. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that . Here is the identity operator of .

Theorem 3.16.

Let be a real Hilbert space, let be -Lipschitz continuous, let be -Lipschitz continuous, and let be -strongly monotone and -Lipschitz continuous. Suppose that is -strongly monotone and -Lipschitz continuous, is a bifunction satisfying conditions (H1)–(H4), and is a lower semicontinuous subdifferentiable proper functional which is -subdifferentiable at each . Let and be two sequences in such that . Assume that the -subdifferential of is -strongly monotone, the ranges , and are totally bounded, and where

(3.36)

Then for any given , the sequence defined by Algorithm 3.6 converges weakly to a solution of GMEPP (2.1).

Proof.

By Theorem 3.8 and the assumptions in Theorem 3.16, we know that the solution set of GMEPP (2.1) is nonempty. Let be a solution of GMEPP (2.1). It is easy to see that

(3.37)

Define a mapping as follows:

(3.38)

where . Since the -subdifferential of is -strongly monotone, by Theorem 2.9 we conclude that is -Lipschitz continuous.

Observe that for all

(3.39)

where and . This implies that is nonexpansive on . It is easy to see that the set of GMEPP (2.1) coincides with .

Note that

(3.40)

This together with Lemma 3.14 implies that exists. And also, from (3.40) we have

(3.41)

Since , we get

(3.42)

Now, let us show that . Indeed, let . Then there exists a subsequence of such that . Since , by Lemma 3.15 we know that .

Further, let us show that is a singleton. Indeed, let be another subsequence of such that . Then is also a fixed point of . If , by Opial's property of , we reach the following contradiction:

(3.43)

This implies that is a singleton. Consequently, converges weakly to a fixed point of , that is, a solution of GMEPP (2.1). This completes the proof.