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On variational inequality, fixed point and generalized mixed equilibrium problems
Journal of Inequalities and Applications volume 2014, Article number: 203 (2014)
Abstract
In this article, variational inequality, fixed point, and generalized mixed equilibrium problems are investigated based on an extragradient iterative algorithm. Weak convergence of the extragradient iterative algorithm is obtained in Hilbert spaces.
1 Introduction
In this paper, we always assume that H is a real Hilbert space with the inner product and the norm , and C is a nonempty, closed, and convex subset of H. ℝ is denoted by the set of real numbers. Let F be a bifunction of into ℝ. Consider the problem: find a p such that
In this paper, the solution set of the problem is denoted by , i.e.,
The above problem is first introduced by Ky Fan [1]. In the sense of Blum and Oettli [2], the Ky Fan problem is also called an equilibrium problem.
Recently, the ‘so-called’ generalized mixed equilibrium problem has been investigated by many authors: The generalized mixed equilibrium problem is to find such that
where is a real valued function and is mapping. We use to denote the solution set of the equilibrium problem. That is,
Next, we give some special cases.
If , then the problem (1.2) is equivalent to find such that
which is called the mixed equilibrium problem.
If , then the problem (1.2) is equivalent to find such that
which is called the mixed variational inequality of Browder type.
If , then the problem (1.2) is equivalent to find such that
which is called the generalized equilibrium problem.
If and , then the problem (1.2) is equivalent to (1.1).
For solving the above equilibrium problems, let us assume that the bifunction satisfies the following conditions:
(A1) , ;
(A2) F is monotone, i.e., , ;
(A3)
(A4) for each , is convex and weakly lower semicontinuous.
Equilibrium problems have intensively been studied. It has been shown that equilibrium problems cover fixed point problems, variational inequality problems, inclusion problems, saddle problems, complementarity problem, minimization problem, and Nash equilibrium problem; see [1–20] and the references therein.
Let be a mapping. In this paper, we use to stand for the set of fixed points. Recall that the mapping S is said to be nonexpansive if
S is said to be κ-strictly pseudocontractive if there exists a constant such that
It is clear that the class of κ-strictly pseudocontractive includes the class of nonexpansive mappings as a special case. The class of κ-strictly pseudocontractive mappings was introduced by Browder and Petryshyn [21]; for existence and approximation of fixed points of the class of mappings, see [22–29] and the references therein.
Let be a mapping. Recall that A is said to be monotone if
A is said to be κ-inverse strongly monotone if there exists a constant such that
It is clear that the κ-inverse being strongly monotone is monotone and Lipschitz continuous.
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 well 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 devoting their efforts to the studies of the existence and convergence of zero points for maximal monotone operators.
Let , . We see that the problem (1.1) is reduced to the following classical variational inequality. Find such that
It is well known that is a solution to (1.6) if and only if x is a fixed point of the mapping , where is a constant, and I is the identity mapping. If C is bounded, closed, and convex, then the solution set of the variational inequality (1.6) is nonempty.
In order to prove our main results, we need the following lemmas.
Lemma 1.1 [21]
Let be a κ-strictly pseudocontractive mapping. Define by for each . Then, as , is nonexpansive such that .
Lemma 1.2 [2]
Let C be a nonempty, closed, and convex subset of H, and 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 [30]
Let A be a monotone mapping of C into H and 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 1.4 [31]
Let be real numbers in such that . Then we have the following:
for any given bounded sequence in H.
Lemma 1.5 [32]
Let for all . Suppose that and are sequences in H such that
and
hold for some . Then .
Lemma 1.6 [21]
Let C be a nonempty, closed, and convex subset of H, and a strictly pseudocontractive mapping. If is a sequence in C such that and , then .
Lemma 1.7 [33]
Let , , and be three nonnegative sequences satisfying the following condition:
where is some nonnegative integer, and . Then the limit exists.
2 Main results
Theorem 2.1 Let C be a nonempty, closed, and convex subset of H, a κ-strictly pseudocontractive mapping with a nonempty fixed point set, and an L-Lipschitz continuous and monotone mapping. Let be a bifunction from to ℝ which satisfies (A1)-(A4), a continuous and monotone mapping, a lower semicontinuous and convex function for each . Assume that is not empty. Let , , and be real number sequences in . Let , be positive real number sequences. Let be a sequence generated in the following manner:
where is such that
Assume that , , , , satisfy the following restrictions:
-
(a)
;
-
(b)
;
-
(c)
, and ;
-
(d)
and , where .
Then the sequence weakly converges to some point .
Proof The proof is split into five steps.
Step 1. Show that the sequence is bounded.
Define , . Next, we prove that the bifunction satisfies the conditions (A1)-(A4). Therefore, generalized mixed equilibrium problem is equivalent to the following equilibrium problem: find such that , . It is clear that satisfies (A1). Next, we prove is monotone. Since is a continuous and monotone operator, we find from the definition of G that
Next, we show satisfies (A3), that is,
Since is continuous and is lower semicontinuous, we have
Next, we show that, for each , is a convex and lower semicontinuous. For each , for all and for all , since satisfies (A4) and is convex, we have
Thus, is convex. Similarly, we find that is also lower semicontinuous. Put and . Letting , we see that
Notice that A is L-Lipschitz continuous and . It follows that
It follows that
On the other hand, we have
where . Substituting (2.2) into (2.1), we obtain
Putting , we find from Lemma 1.1 that is nonexpansive and . It follows that
It follows from Lemma 1.7 that the exists. This shows that is bounded. Since is bounded, we may assume that a subsequence of converges weakly to ξ.
Step 2. Show that
From (2.3), we find that . In view of the restrictions (b) and (d), we see that . Since , we have that . It follows that
Notice that
This implies that . Since , where , we find that
It follows that
This implies that . In view of the restrictions (a) and (c), we find that
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
In view of the monotonicity of A, we see that
in view of . It follows from (2.5) that . Notice that . It follows that
This in turn implies that . It follows that . Notice that T is maximal monotone and hence . This shows from Lemma 1.3 that .
Step 3. Show that .
It follows from (2.5) that converges weakly to ξ for each . Since , we have
From the assumption (A2), we see that
In view of the assumption (A4), we find from (2.5) that , . For with and , let , for each . Since and , we have . It follows that . Notice that
which yields , . Letting , one sees that , . This implies that for each . This proves that .
Step 4. Show that .
Since exists, we put . It follows that
Notice that . From Lemma 1.5, we see that
Since
we find from (2.6) and (2.7) that
In view of , we find from (2.8) that . This implies from Lemma 1.6 that . This completes the proof that .
Step 5. 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, we, therefore, obtain
This is a contradiction. Hence . This completes the proof. □
3 Applications
In this section, we consider solutions of the mixed equilibrium problem (1.3), which includes the Ky Fan inequality as a special case.
The so-called mixed equilibrium problem is to find such that
The mixed equilibrium problem includes the Ky Fan inequality, fixed point problems, saddle problems, and complementary problems as special cases.
Theorem 3.1 Let C be a nonempty, closed, and convex subset of H, a κ-strictly pseudocontractive mapping with a nonempty fixed point set, and a L-Lipschitz continuous and monotone mapping. Let be a bifunction from to ℝ which satisfies (A1)-(A4), and a lower semicontinuous and convex function for each . Assume that is not empty. Let , and be real number sequences in . Let , be positive real number sequences. Let be a sequence generated in the following manner:
where is such that
Assume that , , , , satisfy the following restrictions:
-
(a)
;
-
(b)
;
-
(c)
, and ;
-
(d)
and , where .
Then the sequence weakly converges to some point .
Proof If , we draw the desired conclusion immediately from Theorem 2.1. □
Further, if S is nonexpansive, we find from Theorem 3.1 the following result.
Corollary 3.2 Let C be a nonempty, closed, and convex subset of H, a nonexpansive mapping with a nonempty fixed point set, and an L-Lipschitz continuous and monotone mapping. Let be a bifunction from to ℝ which satisfies (A1)-(A4), and a lower semicontinuous and convex function for each . Assume that is not empty. Let , , and be real number sequences in . Let , be positive real number sequences. Let be a sequence generated in the following manner:
where is such that
Assume that , , , , satisfy the following restrictions:
-
(a)
;
-
(b)
, and ;
-
(c)
and , where .
Then the sequence weakly converges to some point .
If , we find from Theorem 2.1 the following result.
Theorem 3.3 Let C be a nonempty, closed, and convex subset of H, a κ-strictly pseudocontractive mapping with a nonempty fixed point set. Let be a bifunction from to ℝ which satisfies (A1)-(A4), a continuous and monotone mapping, a lower semicontinuous and convex function for each . Assume that is not empty. Let , , and be real number sequences in . Let be a positive real number sequence. Let be a sequence generated in the following manner:
where is such that
Assume that , , , and satisfy the following restrictions:
-
(a)
;
-
(b)
;
-
(c)
, and ;
-
(d)
.
Then the sequence weakly converges to some point .
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Li, D.F., Zhao, J. On variational inequality, fixed point and generalized mixed equilibrium problems. J Inequal Appl 2014, 203 (2014). https://doi.org/10.1186/1029-242X-2014-203
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DOI: https://doi.org/10.1186/1029-242X-2014-203