General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of generalized equilibrium problems and the set of common fixed points of a finite family of k-strictly pseudo-contractive nonself mappings. Strong convergence theorems are established under some suitable conditions in a real Hilbert space, which also solves some variation inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

MSC: 47H05, 47H09, 47H10.

Let H be a real Hilbert space with inner product and norm , respectively. Let K be a nonempty closed convex subset of H. Let be a nonlinear mapping and be a bi-function, where denotes the set of real numbers. We consider the following generalized equilibrium problem: Find such that

We use to denote the solution set of the problem (1.1). If , the zero mapping, then the problem (1.1) is reduced to the normal equilibrium problem: Find such that

We use to denote the solution set of the problem (1.2). If , then the problem (1.1) is reduced to the classical variational inequality problem: Find such that

The generalized equilibrium problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games and others (see, e.g., [1-3]).

Recall that a nonself mapping is called a k-strict pseudo-contraction if there exists a constant such that

We use to denote the fixed point set of T, i.e., . As , T is said to be nonexpansive, i.e.,

T is said to be pseudo-contractive if , and is also said to be strongly pseudo-contractive if there exists a positive constant such that is pseudo-contractive. Clearly, the class of k-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions. We remark also that the class of strongly pseudo-contractive mappings is independent of the class of k-strict pseudo-contractions (see, e.g., [4,5]).

Iterative methods for equilibrium problems and fixed point problems of nonexpansive mappings have been extensively investigated. However, iterative schemes for strict pseudo-contractions are far less developed than those for nonexpansive mappings though Browder and Petryshyn [5] initiated their work in 1967; the reason is probably that the second term appearing in the right-hand side of (1.3) impedes the convergence analysis for iterative algorithms used to find a fixed point of the strict pseudo-contraction. On the other hand, strict pseudo-contractions have more powerful applications than nonexpansive mappings do in solving inverse problems; see, e.g., [6-18,20-27] and the references therein. Therefore it is interesting to develop the effective iterative methods for equilibrium problems and fixed point problems of strict pseudo-contractions.

In 2006, Marino and Xu [8] introduced a general iterative method and proved that for a given , the sequence generated by

where T is a self-nonexpansive mapping on H, f is a contraction of H into itself and satisfies certain conditions, B is a strongly positive bounded linear operator on H, converges strongly to , which is the unique solution of the following variational inequality:

and is also the optimality condition for some minimization problem.

Recently, Takahashi and Takahashi [12] considered the equilibrium problem and nonexpansive mapping by viscosity approximation methods. To be more precise, they proved the following theorem.

Theorem of TTLetKbe a nonempty closed convex subset ofH. LetFbe a bi-function fromtosatisfying (A1)-(A4) and letbe a nonexpansive mapping such that. Letbe a contraction and letandbe sequences generated byand

whereandsatisfy

Thenandconverge strongly to, where.

In 2009, Ceng et al. [15] further studied the equilibrium problem and fixed point problems of strict pseudo-contraction mappings T by an iterative scheme for finding an element of . Very recently, by using the general iterative method Liu [16] proposed the implicit and explicit iterative processes for finding an element of and then obtained some strong convergence theorems, respectively. On the other hand, Takahashi and Takahashi [18] considered the generalized equilibrium problem and nonexpansive mapping in a Hilbert space. Moreover, they constructed an iterative scheme for finding an element of and then proved a strong convergence of the iterative sequence under some suitable conditions.

In this paper, inspired and motivated by research going on in this area, we introduce a general iterative method for generalized equilibrium problems and strict pseudo-contractive nonself mappings, which is defined in the following way:

where constant , f is a contraction and A, B are two operators, is a finite family of -strict pseudo-contractions, is a finite sequence of positive numbers, , and are some sequences with certain conditions.

Our purpose is not only to modify the general iterative method to the case of a finite family of -strictly pseudo-contractive nonself mappings, but also to establish strong convergence theorems for a generalized equilibrium problem and -strict pseudo-contractions in a real Hilbert space, which also solves some variation inequality problems. Our theorems presented in this paper improve and extend the corresponding results of [12,15,16,18,20,21,25].

Let K be a nonempty closed convex subset of a real Hilbert H space with inner product and norm , respectively. Recall that a mapping is a contraction, if there exists a constant such that

We use to denote the collection of all contractions on K. The operator is said to be monotone if

is said to be r-strongly monotone if there exists a constant such that

is said to be α-inverse strongly monotone if there exists a constant such that

Recall that an operator B is strongly positive if there exists a constant with the property

To study the generalized equilibrium problem (1.1), we may assume that the bi-function satisfies the following conditions:

(A1) for all ;

(A2) F is monotone, i.e., for all ;

(A3) for each , ;

(A4) for each , is convex and lower semi-continuous.

In order to prove our main results, we need the following lemmas and propositions.

Lemma 2.1[1,3]

Letbe a bi-function satisfying (A1)-(A4). Then, for anyand, there existssuch that

Further, if, then the following hold:

(1) is single-valued;

(2) is firmly nonexpansive, i.e, for all;

(3) ;

(4) is closed and convex.

Lemma 2.2[8]

In the Hilbert spaceH, there hold the following identities:

(i) , ;

(ii) , , .

Lemma 2.3[8]

Assume thatBis a strongly positive linear bounded operator on the Hilbert spaceHwith a coefficientand. Then.

Lemma 2.4[10]

Ifis ak-strict pseudo-contraction, then the fixed point setis closed convex so that the projectionis well defined.

Lemma 2.5[2,10]

Letbe ak-strict pseudo-contraction. For, definebyfor each. ThenSis a nonexpansive mapping such that.

Lemma 2.6[19]

Assumeis a sequence of nonnegative real numbers such that

whereis a sequence inandis a real sequence such that

(i) ;

(ii) or.

Then.

Proposition 2.1 (See, e.g., Acedo and Xu [20])

LetKbe a nonempty closed convex subset of the Hilbert spaceH. Given an integer, assume thatis a finite family of-strict pseudo-contractions. Suppose thatis a positive sequence such that. Thenis ak-strict pseudo-contraction with.

Proposition 2.2 (See, e.g., Acedo and Xu [20])

Letandbe given as in Proposition 2.1 above. Then.

Theorem 3.1LetKbe a nonempty closed convex subset of the Hilbert spaceHandbe a bi-function satisfying (A1)-(A4). LetAbe anα-inverse strongly monotone mapping andBbe a strongly positive bounded linear operator onHwith. Assume thatbe a finite family of-strict pseudo-contractions such that. Supposewith a coefficientandare finite sequences of positive numbers such thatfor all, for a given point, , and, the following control conditions are satisfied:

(i) , and;

(ii) , and;

(iii) ;

(iv) and.

Then the sequencegenerated by (1.4) converges strongly to, which solves the variational inequality

Proof Putting , we have is a k-strict pseudo-contraction and by Proposition 2.1 and 2.2, where .

First, we show that the mapping is nonexpansive. Indeed, for each , we have

It follows from the condition that the mapping is nonexpansive. From Lemma 2.1, we see that . Note that can be rewritten as and for each as .

From (1.4), condition (ii) and Lemma 2.2, we have

By , we obtain

This together with (3.1), we see that

Furthermore, by Lemma 2.3, we have

It follows from induction that

which gives that sequence is bounded, and so are and .

Define a mapping for each . Then is nonexpansive. Indeed, by using (1.3), Lemma 2.2 and condition (ii), we have for all that

which shows that is nonexpansive.

Next, we show that . From (1.4) and Lemma 2.3, we have

where . Moreover, we note that and

where . On the other hand, we note that

Putting and in (3.6) respectively, we have

It follows from (A2) that

and hence

Since , we assume that there exists a real number μ such that for all . Consequently, we have

and hence

where . Combining (3.4), (3.5) and (3.8), we have

It follows from and Lemma 2.6 that

Moreover, we observe that

It follows from and (3.9) that

For , we note that and

which implies that

From (1.4), (3.2) and (3.11), we have

Using and (3.9) again, we obtain

By the nonexpansion of , we have

This together with (3.9) and (3.12), we obtain

Furthermore, we note that

It follows from condition (ii) that

On the other hand, by condition (iii), we may assume that as for every . It is easily seen that each and . Define , then is a k-strict pseudo-contraction such that by Proposition 2.1 and 2.2. Consequently,

which implies that

Combining (3.14) and (3.15), we obtain

Define by . By condition (ii) again, we have . Then, S is nonexpansive with by Lemma 2.5. Notice that

It follows from (3.13), (3.15) and (3.16) that

Now we claim that , where with being the fixed point of the contraction on H defined by

where . Indeed, by Lemma 2.1 and 2.3, we have

for all . Since , it follows that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point such that

By Lemma 2.2 and (3.10), we have

where as . Observe B is strongly positive, we obtain

Combining (3.18) and (3.19), we have

It follows that

Let in (3.20) and note that as yields

where is an appropriate positive constant such that for all and . Taking from (3.21), we have

On the other hand, we have

It follows that

Therefore, from (3.22) and , we have

Finally, we prove that as . From (1.4) and (3.2) again, we have

It follows that

From , condition (i) and (3.23), we can arrive at the desired conclusion by Lemma 2.6. This completes the proof. □

Theorem 3.2LetKbe a nonempty closed convex subset of the Hilbert spaceHandbe a bi-function satisfying (A1)-(A4). LetAbe anα-inverse strongly monotone mapping, with a coefficientandBbe a strongly positive bounded linear operator onHwithand. Letbe ak-strict pseudo-contraction such that. Letbe a sequence generated byin the following manner:

whereandare two sequences in, constant. If the following control conditions are satisfied:

(i) , and;

(ii) , and.

Then the sequenceconverges strongly to, which solves the variational inequality

Proof Putting and , i.e., , the desired conclusion follows immediately from Theorem 3.1. This completes the proof. □

Theorem 3.3LetKbe a nonempty closed convex subset of the Hilbert spaceHandbe a bi-function satisfying (A1)-(A4). Letwith a coefficientandBbe a strongly positive bounded linear operator onHwithand. Letbe ak-strict pseudo-contraction such that. Letbe a sequence generated byin the following manner:

whereandare two sequences in, sequence. If the following control conditions are satisfied:

(i) , and;

(ii) , and;

(iii) and.

Then the sequenceconverges strongly to, which solves the variational inequality

Proof Putting and , i.e., the generalized equilibrium problem (1.1) reduces to the normal equilibrium problem (1.2), the desired conclusion follows immediately from Theorem 3.1. This completes the proof. □

Remark 3.1 Theorem 3.1 and 3.2 improve and extend the main results of Takahashi and Takahashi [18] and Qin et al. [21] in different directions.

Remark 3.2 Theorem 3.3 is mainly due to Liu [16], which improves and extends the main results of Takahashi and Takahashi [12].

Remark 3.3 If and , then the algorithm (1.4) reduces to approximate the fixed point of k-strict pseudo-contractions, which includes the general iterative method of Marino and Xu [8] and the parallel algorithm of Acedo and Xu [20] as special cases.

The authors declare that they have no competing interests.

Wen, DJ carried out the primary studies for the generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions, participated in the design of iterative methods and drafted the manuscript. Chen YA participated in the convergence analysis and coordination. All authors read and approved the final manuscript.

Supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Changging (CSTC 2012jjA00039) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 110701).

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