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On generalized equilibrium problems and strictly pseudocontractive mappings in Hilbert spaces
Fixed Point Theory and Applications volume 2014, Article number: 145 (2014)
Abstract
In this article, a mean iterative algorithm is investigated for finding a common element in the solution set of generalized equilibrium problems and in the fixed point set of strictly pseudocontractive mappings. Strong convergence of the mean iterative algorithm is obtained in the framework of Hilbert spaces.
1 Introduction-preliminaries
Many important problems have reformulations which require finding solutions of classical variational inequalities, for instance, image recovery, inverse problems, transportation problems, fixed point problems and optimization problems; see [1–11] and the references therein. Equilibrium problems, which include the classical variational inequalities as special cases, have been recently extensively investigated; see [12–26] and the references therein. In this paper, we study an equilibrium problem via fixed point methods. Global convergence of the fixed point algorithm is obtained. Throughout this paper, we always assume that H is a real Hilbert space with the inner product and the norm . Let C be a nonempty closed convex subset of H, and let be a metric projection from H onto C.
Let be a mapping. In this paper, we use to denote the fixed point set of S. Recall that the mapping S is said to be nonexpansive if
S is said to be k-strictly pseudocontractive if there exists a constant such that
The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn [27] in 1967. It is easy to see that the class of strictly pseudocontractive mappings includes the class of nonexpansive mappings as a special case. If , then it is called a pseudocontractive mapping.
Recall that a set-valued mapping is said to be monotone if for all , and imply . A monotone mapping is maximal if the graph of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if, for any , for all implies . Let A be a monotone mapping of C into H, and let be a normal cone to C at , i.e.,
and define a mapping R on C by
Then T is maximal monotone and if and only if for all ; see [28] and the references therein.
Let be a mapping. Recall that A is said to be monotone if
A is said to be inverse-strongly monotone if there exists a constant such that
For such a case, A is also called an α-inverse-strongly monotone mapping. It is easy to see that if A is an inverse-strongly monotone mapping, then the mapping is a strictly pseudocontractive mapping.
Let be an inverse-strongly monotone mapping, and let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers. In this paper, we consider the following generalized equilibrium problem:
In this paper, we use to denote the solution set of problem (1.1).
Next, we give two special cases of problem (1.1).
-
(a)
If , then the generalized equilibrium problem (1.1) is reduced to the following equilibrium problem:
(1.2)
In this paper, we use to denote the solution set of problem (1.2). We remark here that problem (1.2) was first introduced by Fan [29].
(b) If , then problem (1.1) is reduced to the classical variational inequality: Find such that
In this paper, we use to denote the solution set of variational inequality (1.3). It is well know that is a solution to (1.3) if and only if x is a fixed point of the mapping , where is a constant and I is the identity mapping.
To study the generalized equilibrium problem (1.1), we may assume that F satisfies the following conditions:
(A1) for all ;
(A2) F is monotone, i.e., for all ;
(A3) for each ,
(A4) for each , is convex and weakly lower semicontinuous.
Recently, many authors investigated problems (1.1), (1.2) and (1.3) based on iterative methods. In 2003, Takahashi and Toyoda [30] investigated fixed points of nonexpansive mappings and solutions of variational inequality (1.3). They obtained the following results. Let A be an α-inverse-strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that . Let be a sequence generated by
where for some and for some . Then converges weakly to , where .
Recently, Tada and Takahashi [14] investigated fixed points of nonexpansive mappings and solutions of equilibrium problem (1.2). They obtained the following result. Let F be a bifunction from to ℝ satisfying (A1)-(A4), and let S be a nonexpansive mapping of C into H such that . Let and be sequences generated by , and let
where for some and satisfies . Then converges weakly to , where .
In this paper, motivated by the above results, we investigate fixed points of strictly pseudocontractive mappings and solutions of equilibrium problem (1.1). Weak convergence theorems for common solutions are established in Hilbert spaces. Applications of the main results are also provided. In order to prove our main results, we also need the following lemmas.
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a k-strict pseudocontraction with a fixed point. Define by for each . If , then is nonexpansive with .
Lemma 1.2 [1]
Let C be a nonempty closed convex subset of H, and let be a bifunction satisfying (A1)-(A4). Then, for any and , there exists such that
Further, define
for all and . Then the following hold:
-
(a)
is single-valued;
-
(b)
is firmly nonexpansive, i.e., for any ,
-
(c)
;
-
(d)
is closed and convex.
Lemma 1.3 [32]
Let H be a Hilbert space and for all . Suppose that and are sequences in H such that
and
hold for some . Then .
Lemma 1.4 [33]
Let C be a nonempty closed convex subset of a Hilbert space H, and let be a k-strict pseudocontraction. Then
-
(a)
S is -Lipschitz;
-
(b)
is demi-closed, i.e., if is a sequence in C with and , then .
2 Main results
Now, we are in a position to show the main results of the article.
Theorem 2.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a λ-inverse-strongly monotone mapping, and let F be a bifunction from to ℝ which satisfies (A1)-(A4). Let be a k-strict pseudocontraction. Assume that is not empty. Let , , and be sequences in . Let be a sequence in , and let be a bounded sequence in C. Let be a sequence generated in the following manner:
Assume that the sequences , , , and satisfy the following restrictions: , , and . Then the sequence converges weakly to some point , where .
Proof First, we show that the sequences and are bounded. Putting , we see from Lemma 1.1 that is nonexpansive and . Note that
This proves that the mapping is also nonexpansive. Fixing , we find from Lemma 1.2 that . Since
we find that
This implies that exists. This shows that is bounded, so is . Since is convex, we find that
It follows that
This yields that
Using Lemma 1.2, we see that
This implies that
Since is convex, we find that
It follows that
Using the restrictions imposed on the sequences, we obtain from (2.2) that
Since is bounded, we see that there exits a subsequence of which converges weakly to . Using (2.3), we also find that converges weakly to . Note that
From (A2), we see that
Replacing n by , we arrive at
For t with and , let . Since and , we have . It follows from (2.4) that
Using (2.3), we have as . Using the monotonicity of T, we see that . It follows from (A4) that
Using (A1) and (A4), we see from (2.6) that
It follows that . Letting in the above inequality, we arrive at . Hence, .
Next, we are in a position to show that . Note that exists. We may assume that . Note that
Since
we find that . Since , we find that . Using Lemma 1.3, we obtain that . In view of
it follows that . Note that . Using Lemma 1.4, we find that . It follows from Lemma 1.4 that . This proves that . Assume that there exits another subsequence of such that converges weakly to . We can find that . If , we get from the Opial condition [34] that
This derives a contradiction. Hence, we have . This implies that . The proof is completed. □
From Theorem 2.1, the following result is not hard to derive.
Corollary 2.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from to ℝ which satisfies (A1)-(A4). Let be a k-strict pseudocontraction. Assume that is not empty. Let , , and be sequences in . Let be a positive number sequence, and let be a bounded sequence in C. Let be a sequence generated in the following manner:
Assume that the sequences , , , and satisfy the following restrictions: , , and . Then the sequence converges weakly to some point , where .
Corollary 2.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a λ-inverse-strongly monotone mapping, and let F be a bifunction from to ℝ which satisfies (A1)-(A4). Let be a nonexpansive mapping. Assume that is not empty. Let , and be sequences in . Let be a positive real number sequence, and let be a bounded sequence in C. Let be a sequence generated in the following manner:
Assume that the sequences , , and satisfy the following restrictions: , and . Then the sequence converges weakly to some point , where .
3 Applications
The computation of common fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively; for more details, see [35, 36] and the references therein.
First, we consider the following common fixed point problem.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a -strict pseudocontraction for each , where N is some positive integer. Assume that is not empty. Let , and be sequences in . Let be a bounded sequence in C. Let be a sequence generated in the following manner:
Assume that the sequences , , , and satisfy the following restrictions: , , and . Then the sequence converges weakly to some point , where .
Proof Using the definition of strict pseudocontractions, we see that a mapping T is said to be a k-strict pseudocontraction iff
Define a mapping by . Next, we prove that and S is a k-strict pseudocontraction, where . Note that
This proves that S is a k-strict pseudocontraction, where . Next, we show that . It is clear to see that . It suffices to prove that . Let and write . Let . For any and , we have
This shows that
Since , we find that . This proves that . Since x is a fixed point of S, we obtain . This proves that . Putting , and , we find from Theorem 2.1 the desired conclusion immediately. □
Next, we study an optimization problem: Find a minimizer of a convex and lower semicontinuous functional defined on a closed convex subset C of a Hilbert space H.
We denote by Δ the set of solutions of the optimization problem. Let R be a bifunction from to R defined by . We consider the following equilibrium problem:
It is obvious that . In addition, we also find that satisfies the conditions (A1)-(A4).
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a convex and lower semicontinuous functional defined on C with a nonempty minimizer set. Let , and be sequences in . Let be a positive real number sequence, and let be a bounded sequence in C. Let be a sequence generated in the following manner:
Assume that the sequences , , and satisfy the following restrictions: , , and . Then the sequence converges weakly to some point , where .
Remark 3.3 A special form of the optimization problem is to take , which is known as the minimum norm point problem. We also remark here that if we take and , then we easily obtain convergence theorems of solutions of variational inequality (1.3).
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The main idea of this paper was proposed by CH. All authors participated in the research and performed steps of the proof in this research. All authors read and approved the final manuscript.
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Huang, C., Ma, X. On generalized equilibrium problems and strictly pseudocontractive mappings in Hilbert spaces. Fixed Point Theory Appl 2014, 145 (2014). https://doi.org/10.1186/1687-1812-2014-145
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DOI: https://doi.org/10.1186/1687-1812-2014-145