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On asymptotically strict pseudocontractions and equilibrium problems
Journal of Inequalities and Applications volume 2013, Article number: 251 (2013)
Abstract
In this paper, an equilibrium problem and a fixed point problem of an asymptotically strict pseudocontraction are investigated based on hybrid iterative algorithms. Strong convergence theorems of common solutions to the equilibrium problem and the fixed point problem are obtained in the framework of real Hilbert spaces.
MSC:47H09, 47H10.
1 Introduction
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; see [1–12] and the references therein. Equilibrium problems include fixed point problems, variational inequality problems, variational inclusion problems, saddle point problems, the Nash equilibrium problem, complementarity problems and so on. For the solutions of equilibrium problems, there are several algorithms to solve the problem. The classical algorithm is the Krasnoselskii-Mann iterative algorithm. However, the Krasnoselskii-Mann iterative algorithm is weak convergence for the solutions of equilibrium problems. Haugazeau’s projection method [13] recently has been considered for the approximation of solutions of equilibrium problems and fixed point problems. The advantage of the projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions. The aim of this paper is to study an equilibrium problem and a fixed point problem of an asymptotically strict pseudocontraction based on hybrid iterative algorithms and establish a strong convergence theorem of common solutions in the framework of Hilbert spaces.
2 Preliminaries
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, let be a monotone mapping and F be a bifunction of into ℝ, where ℝ denotes the set of real numbers.
In this paper, we consider the following equilibrium problem.
The set of such an is denoted by , i.e.,
If , then the problem (2.1) is reduced to the following:
The set of such an is denoted by , i.e.,
If , the problem (2.1) is reduced to the classical variational inequality problem.
To study the problems (2.1) and (2.2), 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.
Recall that a mapping A is said to be monotone iff
A is said to be strongly monotone iff there exists a constant such that
For such a case, A is said to be α-strongly-monotone. A is said to be inverse-strongly monotone iff there exists a constant such that
For such a case, A is said to be α-inverse-strongly monotone.
Recall that a set-valued mapping is said to be monotone iff, for all , and imply . 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 .
Let be a nonlinear mapping. In this paper, we use to denote the fixed point set of S.
Recall that S is said to be nonexpansive iff
Recall that S is said to be asymptotically nonexpansive iff there exists a sequence with as such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [14] in 1972. Since 1972, a number of authors have studied the convergence problems of the iterative processes for such a class of mappings.
Recall that S is said to be strictly pseudocontractive iff there exists a constant such that
For such a case, S is also said to be a κ-strict pseudocontraction. The class of strict pseudocontractions is introduced by Browder and Petryshyn [15] in 1967. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction. We also remark that if , then S is said to be pseudocontractive.
Recall that S is said to be an asymptotically strict pseudocontraction iff there exist a sequence with as and a constant such that
For such a case, S is also said to be an asymptotically κ-strict pseudocontraction. The class of asymptotically strict pseudocontractions is introduced by Qihou [16] in 1996. It is clear that every asymptotically nonexpansive mapping is an asymptotical 0-strict pseudocontraction. Every nonexpansive mapping is an asymptotically nonexpansive mapping with the sequence . We also remark that if , then S is said to be an asymptotically pseudocontractive mapping which was introduced by Schu [17] in 1991.
Recently, many authors considered the equilibrium problems (2.1), (2.2) and fixed point problems based on hybrid iterative methods; see, for instance, [18–30]. In this paper, motivated by these recent results, we consider the shrinking projection algorithm to solve the solutions of the equilibrium problem (2.1) and the fixed point problem of an asymptotically strict pseudocontraction. It is proved that the sequence generated in the purposed iterative process converges strongly to some common element in the solution set of the equilibrium problem (2.1) and in the fixed point set of an asymptotically strict pseudocontraction.
In order to prove our main results, we also need the following lemmas.
Lemma 2.1 [31]
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 2.2 [32]
In a real Hilbert space, the following inequality holds:
Lemma 2.3 [33]
Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let be an asymptotically κ-strict pseudocontraction with the sequence . Then
-
(a)
is closed and convex;
-
(b)
S is L-Lipschitz continuous.
Lemma 2.4 [33]
Let H be a real Hilbert space, let C be a nonempty closed convex subset of H and let be an asymptotically strict pseudocontraction. Then the mapping is demiclosed at zero, that is, if is a sequence in C such that and , then .
3 Main results
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction from to ℝ which satisfies (A1)-(A4) and let be a -inverse-strongly monotone mapping for each , where is some positive integer. Let be an asymptotically κ-strict pseudocontraction. Assume that is nonempty and bounded. Let and be sequences in and let be a positive sequence. Let be a sequence in for each such that . Let be a sequence generated in the following manner:
where and . Assume that the control sequences , , , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
and for each .
Then the sequence converges strongly to some point , where .
Proof First, we show that is closed and convex for each . It is easy to see that is closed for each . We only show that is convex for each . Note that is convex. Suppose that is convex for some positive integer i. Next, we show that is convex for the same i. Note that
is equivalent to
Take and in and put . It follows that , ,
and
Combining (3.2) with (3.3), we can obtain that , that is, . In view of the convexity of , we see that . This shows that . This concludes that is closed and convex for each . Notice that is nonexpansive. Indeed, for any , we find from the restriction (b) that
It follows that
Put for each . Next, we show that for all . It is easy to see that . Suppose that for some integer . We intend to claim that for the same h. For any , we have from Lemma 2.2 and the restriction (a) that
This shows that . This proves that for all . Since and , we have that
It follows that
On the other hand, for any , we see that . In particular, we have
This shows that the sequence is bounded. In view of (3.6), we see that exists. It follows from (3.5) that
which implies that
In view of , we find that
This combines with (3.7) yielding that
Notice that . Combining (3.7) with (3.8), we find that
Note that
In view of the restriction (a), we obtain from (3.9) that
Notice that
and hence
It follows from Lemma 2.2 and the restriction (a) that
It follows that
This implies that
In view of the restrictions (a) and (b), we find from (3.9) that
On the other hand, we find from (3.11) and (3.12) that
This implies that
In view of the restrictions (a) and (b), we obtain from (3.9) and (3.13) that
Since is bounded, there exists a subsequence of such that . Next, we show that . Note that
It follows from (3.14) that
On the other hand, we have
From (3.10) and (3.15), we find that
Note that
which yields that
In view of the restriction (a), we find from (3.16) that
Note that
It follows from (3.7) and (3.17) that
With the aid of Lemma 2.4, we find that . Next, we prove . In view of (3.14), we find that for each . From (3.14) and the restriction (b), we see that
Notice that
From (A2), we see that
Replacing n by , we arrive at
For any t with and , let . Since and , we have . It follows from (3.19) that
Since is Lipschitz continuous, we find from (3.14) that as . From the monotonicity of , we get that
In view of (A4), we find from (3.20) that
With the aid of (A1), (A4), we obtain from (3.21) that
which implies that
Letting in the above inequality, we arrive at
This shows that , . This completes the proof that . Put , we obtain that
which yields that
It follows that converges strongly to . Therefore, we can conclude that the sequence converges strongly to . This completes the proof. □
Based on Theorem 3.1, we have the following results.
Corollary 3.2 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) and let be a ξ-inverse-strongly monotone mapping. Let be an asymptotically κ-strict pseudocontraction. Assume that is nonempty and bounded. Let and be sequences in and be a positive sequence. Let be a sequence generated in the following manner:
where and . Assume that the control sequences , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
.
Then the sequence converges strongly to some point , where .
Proof Putting , and in Theorem 3.1, we see that . With the aid of Theorem 3.1, we can easily conclude the desired conclusion. □
If S is asymptotically nonexpansive, then Corollary 3.2 is reduced to the following.
Corollary 3.3 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) and let be a ξ-inverse-strongly monotone mapping. Let be an asymptotically nonexpansive mapping. Assume that is nonempty and bounded. Let and be sequences in and let be a positive sequence. Let be a sequence generated in the following manner:
where and . Assume that the control sequences , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
.
Then the sequence converges strongly to some point , where .
Putting in Corollary 3.3, we have the following.
Corollary 3.4 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) and let be a ξ-inverse-strongly monotone mapping. Let be an asymptotically nonexpansive mapping. Assume that is nonempty and bounded. Let be a sequence in and be a positive sequence. Let be a sequence generated in the following manner:
where and . Assume that the control sequences and satisfy the following restrictions:
-
(a)
;
-
(b)
.
Then the sequence converges strongly to some point , where .
Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a -inverse-strongly monotone mapping, for each , where is some positive integer. Let be an asymptotically κ-strict pseudocontraction. Assume that is nonempty and bounded. Let and be sequences in and be a positive sequence. Let be a sequence in for each such that . Let be a sequence generated in the following manner:
where and . Assume that the control sequences , , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
and for each .
Then the sequence converges strongly to some point , where .
Proof In Theorem 3.1, put for all . From
we have
This implies that
In view of Theorem 3.1, we can immediately obtain the desired conclusion. This completes the proof. □
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction from to ℝ which satisfies (A1)-(A4) for each , where is some positive integer. Let be an asymptotically κ-strict pseudocontraction. Assume that is nonempty and bounded. Let and be sequences in and be a positive sequence. Let be a sequence in for each such that . Let be a sequence generated in the following manner:
where and . Assume that the control sequences , , , and satisfy the following restrictions:
-
(a)
, ;
-
(b)
and for each .
Then the sequence converges strongly to some point , where .
Proof In Theorem 3.1, put . Then, for any , we see that
Let be a sequence satisfying the restriction , where . Then we can obtain the desired conclusion easily from Theorem 3.1. This completes the proof. □
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Cheng, P., Wu, H. On asymptotically strict pseudocontractions and equilibrium problems. J Inequal Appl 2013, 251 (2013). https://doi.org/10.1186/1029-242X-2013-251
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DOI: https://doi.org/10.1186/1029-242X-2013-251