Abstract
In this paper, we study monotone variational inequalities and generalized equilibrium problems. Weak convergence theorems are established based on a fixed point method in the framework of Hilbert spaces.
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1 Introduction and preliminaries
It is well known that many nonlinear problems can be reduced to the search for solutions of monotone variational inequalities. Fixed point methods are often used for finding and approximating such solutions. In 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 and convex subset of H. Let be a mapping. In this paper, we use to stand for the set of fixed points. Recall that S is said to be nonexpansive iff , . S is said to be κ-strictly pseudocontractive iff there exists a constant such that
The class of strict pseudocontractions was introduced by Browder and Petryshyn [1]. It is clear that the class of κ-strict pseudocontractions includes the class of nonexpansive mappings as a special case.
Let be a mapping. Recall that A is said to be monotone iff , . A is said to be α-inverse strongly monotone iff there exists a constant such that
It is clear that α-inverse strongly monotone is monotone and Lipschitz continuous.
Recall that the classical variational inequality, denoted by , is to find such that
One can see that variational inequality (1.1) is equivalent to a fixed point problem. The element is a solution of variational inequality (1.1) iff satisfies the fixed point equation , where is a constant. This alternative equivalent formulation plays a significant role in the studies of variational inequalities and related optimization problems. If A is α-inverse-strongly monotone and , then the mapping is nonexpansive; see [2] and the references therein.
A set-valued mapping is said to be monotone if, for all , and imply . A monotone mapping is maximal if the graph of T 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 . The class of monotone operators is one of the most important classes of operators. Within the past several decades, many authors have been devoted to the studies on the existence and convergence of different schemes for zero points for maximal monotone operators; see [3–15] and the references therein.
Let ℝ denote the set of real numbers, and let F be a bifunction of into ℝ. Recall that the following generalized equilibrium problem is to find an x such that
In this paper, the solution set of the equilibrium problem is denoted by , i.e.,
To study equilibrium problem (1.2), 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 lower semi-continuous.
If , the generalized equilibrium problem is reduced to variational inequality (1.1). If , the generalized equilibrium problem is reduced to the following equilibrium problem: find an x such that
In this paper, the solution set of the equilibrium problem is denoted by , i.e.,
Equilibrium problems, which were introduced by Blum and Oettli [16], have intensively been studied. It has been shown that equilibrium problems cover fixed point problems, variational inequality problems, inclusion problems, saddle point problems, complementarity problem, minimization problem, and Nash equilibrium problem. Equilibrium problem has emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Recently, many authors studied the numerical solution of equilibriums (1.2) and (1.3) based on iterative methods. Convergence theorems are established in a different framework of spaces; see [17–24] and the references therein.
In this paper, an iterative algorithm is investigated for monotone variational inequalities and generalized equilibrium problems. Weak convergence of the algorithm is obtained in the framework of Hilbert spaces.
Recall that a space is said to satisfy Opial’s condition [25] if, for any sequence with , where ⇀ denotes the weak convergence, the inequality
holds for every with . Indeed, the above inequality is equivalent to the following:
Recall that a function is lower semi-continuous at some point if . It is known that the norm is lower semi-continuous. In order to prove our main results, we also need the following lemmas.
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 1.2 [4]
Let A be a monotone mapping of C into H and be the normal cone to C at , i.e.,
and define a mapping T on C by
Then T is maximal monotone and if and only if for all .
Lemma 1.3 [26]
Let , , and be three nonnegative sequences satisfying the following condition:
where is some nonnegative integer, and . Then the limit exists.
Lemma 1.4 [27]
Let be real numbers in such that . Then we have the following:
for any given bounded sequence in H.
2 Main results
Theorem 2.1 Let C be a nonempty closed and convex subset of H, and let be an L-Lipschitz continuous and monotone mapping. Let be a bifunction from to ℝ which satisfies (A1)-(A4), and let be a -inverse strongly monotone mapping for each . Assume that is not empty. Let , be real number sequences in . Let and be positive real number sequences. Let be a sequence generated in the following manner:
Assume that , , , and satisfy the following restrictions:
-
(a)
;
-
(b)
, and ;
-
(c)
, and , where .
Then the sequence weakly converges to some point .
Proof We first show that the sequence is bounded. Set , where . Letting , we see that
Since A is L-Lipschitz continuous and , we find that
It follows that
On the other hand, we obtain from the restriction (c) that
Substituting (2.2) into (2.1), we obtain that . This in turn implies from the restriction (c) that
It follows from Lemma 1.3 that exists. This in turn shows that is bounded. From (2.3), we also have that
This implies from the restrictions (a) and (c) that
Since , we find from (2.4) . Hence, we have
Notice that
This implies that
It follows from Lemma 1.4 that
By use of (2.3), we find that
This implies that
Since is firmly nonexpansive, we find from the convexity of that . It follows that
Since is bounded, we may assume that a subsequence of converges weakly to ξ. It follows that converges weakly to ξ for each . Next, we show that . Since , we have
From assumption (A2), we see that
Replacing n by , we arrive at
For with and , let . Since and , we have . It follows that
It follows from (A4) and (2.6) that
From (A1) and (A4), we see
which yields that
Letting in the above inequality, we arrive at
This completes the proof that . Next, we show that . In fact, let T be the maximal monotone mapping defined by
For any given , we have . So, we have for all . On the other hand, we have . We obtain that and hence . In view of the monotonicity of A, we see that
On the other hand, we see that . It follows that
Notice that
Combining (2.5) with (2.8), one finds
This in turn implies that . It follows that . Notice that T is maximal monotone and hence . This shows from Lemma 1.2 that . This completes the proof that .
Finally, we show that the whole sequence weakly converges to ξ. Let be another subsequence of converging weakly to , where . In the same way, we can show that . Since the space H enjoys Opial’s condition [25], we, therefore, obtain that
This is a contradiction. Hence . This completes the proof. □
If , we have the following result on equilibrium problem (1.3).
Corollary 2.2 Let C be a nonempty closed and convex subset of H, and let be an L-Lipschitz continuous and monotone mapping. Let be a bifunction from to ℝ which satisfies (A1)-(A4) for each . Assume that is not empty. Let , be real number sequences in . Let and be positive real number sequences. Let be a sequence generated in the following manner:
Assume that , , , and satisfy the following restrictions:
-
(a)
;
-
(b)
, and ;
-
(c)
, and , where .
Then the sequence weakly converges to some point .
Corollary 2.3 Let C be a nonempty closed and convex subset of H, and let be an L-Lipschitz continuous and monotone mapping. Assume that is not empty. Let be a real number sequence in , and let be a positive real number sequence. Let be a sequence generated in the following manner:
Assume that and satisfy the following restrictions:
-
(a)
;
-
(b)
, where .
Then the sequence weakly converges to some point .
Proof Put for all , and . Notice that , and , we easily find from Corollary 2.2 the desired conclusion. □
Letting , , and in Γ, we have the extragradient-type algorithm (see Table 1).
Corollary 2.4 Let C be a nonempty closed and convex subset of H, and let be an L-Lipschitz continuous and monotone mapping. Assume that is not empty. Let be a real number sequence in . Let and be positive real number sequences. Let be a sequence generated in the following manner:
Assume that and satisfy the following restrictions:
-
(a)
;
-
(b)
, where .
Then the sequence weakly converges to some point .
Letting , and in Γ, we have the Mann-type algorithm (see Table 2).
Corollary 2.5 Let C be a nonempty closed and convex subset of H. Let F be a bifunction from to ℝ which satisfies (A1)-(A4) such that is not empty. Let be a real number sequence in . Let be a positive real number sequence. Let be a sequence generated in the following manner:
Assume that and satisfy the following restrictions:
-
(a)
;
-
(b)
.
Then the sequence weakly converges to some point .
3 Conclusion
Generalized equilibrium problems have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, and network. The problems include variational inequality problems, saddle point problems, complementarity problems, Nash equilibrium problem in noncooperative games, and others as special cases. In this paper, we study monotone variational inequalities and generalized equilibrium problems based on a fixed point method. The extragradient algorithm is valid for a family of infinite bifunctions. We analyze the convergence of the algorithm and establish a weak convergence theorem for solutions under mild restrictions imposed on the control sequences.
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The author thanks the Fundamental Research Funds for the Central Universities (2014ZD44). The author is very grateful to the editor and anonymous reviewers for their suggestions which improved the contents of the article.
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Wang, S. Monotone variational inequalities, generalized equilibrium problems and fixed point methods. Fixed Point Theory Appl 2014, 236 (2014). https://doi.org/10.1186/1687-1812-2014-236
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DOI: https://doi.org/10.1186/1687-1812-2014-236