Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

Abstract

We introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

1. Introduction

Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . We denote weak convergence and strong convergence by notations and , respectively. Recall that a mapping is an -contraction on if there exists a constant such that for all . Let be a mapping. In the sequel, we will use to denote the set of fixed points of ; that is, . In addition, let a mapping be callednonexpansive, if , for all . It is well known that if is nonempty, bounded, closed, and convex and is a nonexpansive self-mapping on , then is nonempty; see, for example, [1]. Recall that a mapping is called strictly pseudo-contraction if there exists a constant such that

(1.1)

where denotes the identity operator on . Note that if , then is a nonexpansive mapping. The class of strict pseudo-contractions is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of fixed points for strict pseudo-contractions. In 1967, Browder and Petryshyn [2] introduced a convex combination method to study strict pseudo-contractions in Hilbert spaces. On the other hand, Marino and Xu [3] and Zhou [4] developed some iterative scheme for finding a fixed point of a strict pseudo-contraction mapping. More precisely, take and define a mapping by

(1.2)

where is a strict pseudo-contraction. Under appropriate restrictions on , it is proved that the mapping is nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudo-contractions.

Let be a proper extended real-valued function and let be a bifunction of into such that , where is the set of real numbers and . Ceng and Yao [5] considered the following mixed equilibrium problems for finding such that

(1.3)

The set of solutions of (1.3) is denoted by , that is,

(1.4)

We see that is a solution of a problem (1.3) that implies that .

Special Examples

• (1) If , then the mixed equilibrium problem (1.3) becomes to be the equilibrium problem which is to find such that

  

  (1.5)

The set of solutions of (1.5) is denoted by .

• (2) If and for all , where is a nonlinear mapping, then problem (1.5) becomes to be the variational inequality problems which is to find such that

  

  (1.6)

The set of solutions of (1.6) is denoted by . The variational inequality has been extensively studied in the literature. See, for example, [6–8] and the references therein.

The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.3). Some authors have proposed some useful methods for solving the and ; see, for instance [5, 9–27]. In 1997, Combettes and Hirstoaga [10] introduced an iterative scheme of finding the best approximation to initial data when is nonempty and proved a strong convergence theorem. Next, we recall some definitions.

Definition 1.1.

Let be nonlinear mappings. Then is called

(1)monotone if

(1.7)

(2)-strongly monotone if there exists a constant such that

(1.8)

(3)-Lipschitz continuous if there exists a constant such that

(1.9)

(4)-inverse strongly monotone if there exists a constant such that

(1.10)

Remark 1.2.

It is obvious that any -inverse strongly monotone mappings is monotone and -Lipschitz continuous.

• (5) A set-valued mapping is called amonotone if, for all , and imply .

• (6) A monotone mapping is a maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies .

Let be a monotone map of into , -Lipschitz continuous mapping and let be the normal cone to when , that is,

(1.11)

and define a mapping on by

(1.12)

Then is the maximal monotone and if and only if ; see [28].

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities for -inverse strongly monotone, Takahashi and Toyoda [29] first introduced the following iterative scheme:

(1.13)

where is an -inverse strongly monotone, is a sequence in (0, 1), and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.13) converges weakly to some .

Further, Y. Yao and J.-C. Yao [30] introduced the following iterative scheme:

(1.14)

where is an -inverse strongly monotone, , , are three sequences in [0, 1], and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.14) converges strongly to some .

A map is said to be strongly positive if there exists a constant such that

(1.15)

A typical problem is to minimize a quadratic function over the set of the fixed points of some nonexpansive mapping on a real Hilbert space :

(1.16)

where is some linear, is the fixed point set of a nonexpansive mapping on and is a point in . Let be a strongly positive linear bounded map on with coefficient . In 2006, Marino and Xu [31] studied the following general iterative method:

(1.17)

They proved that if the sequence of parameters appropriate conditions, then the sequence generated by (1.17) converges strongly to . Recently, Plubtieng and Punpaeng [32] proposed the following iterative algorithm:

(1.18)

They proved that if the sequences and of parameters satisfy appropriate condition, then both sequences and converge to the unique solution of the variational inequality

(1.19)

which is the optimality condition for the minimization problem

(1.20)

where is a potential function for (i.e., for ).

On the other hand, for finding a common element of the set of fixed points of a -strict pseudo-contraction mapping and the set of solutions of an equilibrium problem in a real Hilbert space, Liu [33] introduced the following iterative scheme:

(1.21)

where is a -strict pseudo-contraction mapping and , are sequences in [0, 1]. They proved that under certain appropriate conditions over , , and , the sequences and converge strongly to some , which solves some variational inequality problems.

In 2008, Ceng and Yao [5] introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem (1.3) in Hilbert spaces and obtained the strong convergence theorem which used the following condition:

• (H) is -strongly convex with constant and its derivative is sequentially continuous from weak topology to strong topology. We note that the condition (H) for the function is a very strong condition. We also note that the condition (H) does not cover the case and for each . Very recently, R. Wangkeeree and R. Wangkeeree [34] introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a -strict pseudo-contraction mapping, and the set of solutions of the variational inequality for an inverse strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the condition (H) for the sequences generated by these processes.

In 2009, Qin et al. [35] introduced a general iterative scheme for finding a common element of the set of common solution of generalized equilibrium problems, the set of a common fixed point of a family of infinite nonexpansive mappings in Hilbert spaces. Let be the sequence generated iterative by the following algorithm:

(1.22)

They proved that under certain appropriate conditions imposed on , , and , the sequence generated by (1.22) converges strongly to , where .

In the present paper, motivated and inspired by Qin et al. [35], Plubtieng and Punpaeng [32], Peng and Yao [17], R. Wangkeeree and R. Wangkeeree [34], and Y. Yao and J.-C. Yao [30], we introduce a new approximation iterative scheme for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem, and the set of common solutions of the variational inequalities with inverse strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. Moreover, we apply our results for solving convex feasibility problems in Hilbert spaces. The results in this paper extend and improve some well-known results in [17, 30, 32, 34, 35].

2. Preliminaries

Let be a real Hilbert space and be a closed convex subset of . In a real Hilbert space , it is well known that

(2.1)

for all and .

For any , there exists a unique nearest point in , denoted by , such that

(2.2)

The mapping is called the metric projection of onto .

It is well known that is a firmly nonexpansive mapping of onto , that is,

(2.3)

Further, for any and , if and only if , for all .

Moreover, is characterized by the following properties: and

(2.4)

(2.5)

for all .

It is easy to see that the following is true:

(2.6)

The following lemmas will be useful for proving the convergence result of this paper.

Lemma 2.1 (see [36]).

Let be an inner product space. Then, for all and with , one has

(2.7)

Lemma 2.2 (see [31]).

Assume that is a strongly positive linear bounded operator on with coefficient and . Then .

Lemma 2.3 (see [4]).

Let be a nonempty closed convex subset of a real Hilbert space and let be a -strict pseudo-contraction with a fixed point. Then is closed and convex. Define by for each . Then is nonexpansive such that .

Lemma 2.4 (see [37]).

Let be a uniformly convex Banach spaces, be a nonempty closed convex subset of and be a nonexpansive mapping. Then is demi-closed at zero.

Lemma 2.5 (see [38]).

Let be a nonempty closed convex subset of strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose is nonempty. Let be a sequence of positive numbers with . Then a mapping on can be defined by

(2.8)

for is well defined, nonexpansive and holds.

In order to solve the mixed equilibrium problem, the following assumptions are given for the bifunction , and the set :

(A1) for all ;

(A2) is monotone, that is, for all ;

(A3) for each , ;

(A4) for each is convex and lower semicontinuous;

(A5) for each is weakly upper semicontinuous;

(B1) for each and , there exist abounded subset and such that for any ,

(2.9)

(B2) is a bounded set.

Lemma 2.6 (see [39]).

Let be a nonempty closed convex subset of . Let be a bifunction satisfies (A1)–(A5) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:

(2.10)

for all . Then, the following holds:

(i)for each , ;

(ii) is single-valued;

(iii) is firmly nonexpansive, that is, for any ,

(2.11)

(iv);

(v) is closed and convex.

Remark 2.7.

If , then is rewritten as .

Remark 2.8.

We remark that Lemma 2.6 is not a consequence of Lemma 3.1 in [5], because the condition of the sequential continuity from the weak topology to the strong topology for the derivative of the function does not cover the case .

Lemma 2.9 (see [40]).

Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .

Lemma 2.10 (see [41]).

Assume that is a sequence of nonnegative real numbers such that

(2.12)

where is a sequence in and is a sequence in such that

(1),

(2) or .

Then .

Lemma 2.11.

Let be a real Hilbert space. Then for all ,

(2.13)

3. Main Results

In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem and the set of a common solutions of the variational inequalities with inverse strongly monotone mappings in a real Hilbert space.

Theorem 3.1.

Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunctions from to satisfying (A1)–(A5) and let be a proper lower semicontinuous and convex function. Let be an -inverse strongly monotone mapping and be an -inverse strongly monotone mapping. Let be a contraction mapping with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let be a -strict pseudo-contraction with a fixed point. Define a mapping by , for all . Assume that

(3.1)

Assume that either (B1) or (B2). Let be a sequence generated by the following iterative algorithm:

(3.2)

where , , , , and are sequences in and , are positive sequences. Assume that the control sequences satisfy the following restrictions:

(C1),

(C2) and ,

(C3),

(C4),

(C5), , where are two positive constants,

(C6), and , for some .

Then, converges strongly to a point which is the unique solution of the variational inequality

(3.3)

or equivalent , where is a metric projection mapping form onto .

Proof.

Since , as , we may assume, without loss of generality, that for all . By Lemma 2.2, we know that if , then . We will assume that . Since is a strongly positive bounded linear operator on , we have

(3.4)

Observe that

(3.5)

so this shows that is positive. It follows that

(3.6)

We divide the proof into seven steps.

Step 1.

We claim that the mapping where has a unique fixed point.

Since be a contraction of into itself with . Then, we have

(3.7)

Since , it follows that is a contraction of into itself. Therefore the Banach Contraction Mapping Principle implies that there exists a unique element such that .

Step 2.

We claim that is nonexpansive.

Indeed, from the -inverse strongly monotone mapping definition on and condition (C5), we have

(3.8)

where , for all implies that the mapping is nonexpansive and so is, .

Step 3.

We claim that is bounded.

Indeed, let and Lemma 2.6, we obtain

(3.9)

Note that and , we have

(3.10)

Since and are nonexpansive and from (2.6), we have

(3.11)

From Lemma 2.3, we have that is nonexpansive with . It follows that

(3.12)

It follows that

(3.13)

By simple induction, we have

(3.14)

Hence, is bounded, so are , , , , , , , and .

Step 4.

We claim that .

Observing that and , by the nonexpansiveness of , we get

(3.15)

Similarly, let and , we have

(3.16)

From and , we compute

(3.17)

Similarly, we have

(3.18)

Observing that

(3.19)

we obtain

(3.20)

Substituting (3.17) and (3.18) into (3.20), we have

(3.21)

where is an appropriate constant such that ,,, ,.

Putting , for all , we have

(3.22)

Then, we compute

(3.23)

It follows from (3.21) and (3.23), that

(3.24)

This together with (C2), (C3), (C4), and (C6) imply that

(3.25)

Hence, by Lemma 2.9, we obtain as . It follows that

(3.26)

So, we also get

(3.27)

Observe that

(3.28)

By condition (C2) and (3.26), we have

(3.29)

Step 5.

We claim that the following statements hold:

(s1);

(s2);

(s3);

(s4).

Indeed, pick any , to obtain

(3.30)

Therefore,

(3.31)

Similarly, we have

(3.32)

Note that

(3.33)

Substituting (3.31) and (3.32) into (3.33), we obtain

(3.34)

From Lemma 2.1, (3.2) and (3.34), we obtain

(3.35)

It follows that

(3.36)

From (C2), (C6), and (3.26), we also have

(3.37)

Similarly, using (3.35) again, we have

(3.38)

From (C2), (C6), and (3.26), we also have

(3.39)

From (3.37) and (3.39), we have

(3.40)

For , we compute

(3.41)

Similarly, we have

(3.42)

Substituting (3.41) and (3.42) into (3.33), we also have

(3.43)

On the other hand, we note that

(3.44)

It follows that

(3.45)

From (C2), (C5), (C6), and (3.26), we have

(3.46)

Thanks to (3.44), we also have

(3.47)

From (C2), (C5), (C6), and (3.26), we obtain

(3.48)

Observe that

(3.49)

and hence

(3.50)

Similarly, we can obtain that

(3.51)

Substituting (3.50) and (3.51) into (3.33), we also have

(3.52)

On the other hand, we have

(3.53)

and hence

(3.54)

From (C2), (C6), (3.26), (3.46), and (3.48), we also have

(3.55)

Similarly, using (3.53) again, we can prove

(3.56)

From (3.39) and (3.55), we also have

(3.57)

From (3.37) and (3.56), we have

(3.58)

Step 6.

We claim that , where is the unique solution of the variational inequality , for all .

To show this inequality, we choose a subsequence of such that

(3.59)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that . We claim that .

(a1) First, we prove that .

Assume also that and .

Define a mapping by

(3.60)

where , , and , for some . From Lemma 2.5, we have that is nonexpansive with

(3.61)

Notice that

(3.62)

where is an appropriate constant such that

(3.63)

From (C6), (3.37), (3.39), and (3.29), we obtain

(3.64)

By Lemma 2.4, we have , that is, .

(a2) Now, we prove that .

Define a mapping by

(3.65)

where , , and , for some . From Lemma 2.5, we have that is nonexpansive with

(3.66)

On the other hand, we have

(3.67)

where is an appropriate constant such that

(3.68)

From (C6) and (3.29), we obtain

(3.69)

Since is a contraction with the coefficient , there exists a unique fixed point. We use to denote the unique fixed point to the mapping , that is, . Since is bounded, There exists a subsequence of which converges weakly to . Without loss of generality, we may assume that . It follows from (3.69), that

(3.70)

It follows from Lemma 2.4, we obtain that . Hence , where . From (3.59) and (2.4), we arrive at

(3.71)

On the other hand, we have

(3.72)

From (3.26) and (3.71), we obtain that

(3.73)

Step 7.

We claim that .

Indeed, by (3.2) and using Lemmas 2.2 and 2.11, we observe that

(3.74)

which implies that

(3.75)

Taking

(3.76)

Then we can rewrite (3.75) as

(3.77)

We have . Applying Lemma 2.10 to (3.77), we conclude that converges strongly to in norm. This completes the proof.

If the mapping is nonexpansive, then . We can obtain the following result from Theorem 3.1 immediately.

Corollary 3.2.

Let be a nonempty closed convex subset of a real Hilbert space . Let and be two bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let be an -inverse strongly monotone mapping and be an -inverse strongly monotone mapping. Let be a contraction mapping with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let a nonexpansive mapping with a fixed point. Assume that

(3.78)

Assume that either or . Let be a sequence generated by the following iterative algorithm:

(3.79)

where , , , , and are sequences in and , are positive sequences. Assume that the control sequences satisfy the following restrictions:

(C1),

(C2) and ,

(C3),

(C4),

(C5), , where are two positive constants,

(C6), and , for some .

Then, converges strongly to a point which is the unique solution of the variational inequality

(3.80)

or equivalent , where is a metric projection mapping form onto .

Finally, we consider the following convex feasibility problem :

(3.81)

where is an integer and each is assumed to be the of solutions of equilibrium problem with the bifunction and the solution set of the variational inequality problem. There is a considerable investigation on in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [42, 43], computer tomography [44], and radiation therapy treatment planning [45].

The following result can be concluded from Theorem 3.1 easily.

Theorem 3.3.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction from to satisfying (A1)–(A4) and let be a proper lower semicontinuous and convex function. Let be an -inverse strongly monotone mapping for each . Let be a contraction mapping with coefficient and let be a strongly positive linear bounded operator on with coefficient and . Let be a -strict pseudo-contraction with a fixed point. Define a mapping by , for all . Assume that

(3.82)

Assume that either or . Let be a sequence generated by the following iterative algorithm:

(3.83)

where such that , are positive sequences and , are sequences in . Assume that the control sequences satisfy the following restrictions:

(C1) and ,

(C2),

(C3), for each ,

(C4), where is some positive constant for each ,

(C5), for each .

Then, converges strongly to a point which is the unique solution of the variational inequality

(3.84)

or equivalent , where is a metric projection mapping form onto .