Abstract
In this paper, a hybrid algorithm is investigated for solving common solutions of a generalized equilibrium problem, a variational inequality, and fixed point problems of an asymptotically strict pseudocontraction. Weak convergence theorems are established in the framework of real Hilbert spaces.
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1 Introduction
Monotone variational inequalities recently have been investigated as an effective and powerful tool for studying a wide class of real world problems which arise in economics, finance, image reconstruction, ecology, transportation, and network; see [1–9] and the references therein. Monotone variational inequalities, which include many important problems in nonlinear analysis and optimization, such as the Nash equilibrium problem, complementarity problems, fixed point problems, saddle point problems, and game theory recently have been extensively studied based on projection methods. Many well-known problems can be studied by using methods which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. Krasnoselskii-Mann iteration, which is also known as a one-step iteration, is a classic algorithm to study fixed points of nonlinear operators. However, Krasnoselskii-Mann iteration only enjoys weak convergence for nonexpansive mappings; see [10] and the references therein.
The purposes of this paper is to study common solutions of a generalized equilibrium problem, a variational inequality, and fixed point problems of an asymptotically strict pseudocontraction based on a hybrid algorithm. Weak convergence theorems are established in the framework of real Hilbert spaces. The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a hybrid algorithm is introduced and the convergence analysis is given. Weak convergence theorems are established in a real Hilbert space.
2 Preliminaries
From now on, we always assume that H is a real Hilbert space with the inner product and the norm , C is a nonempty closed convex subset of H and denotes the metric projection from H onto C.
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, we also call it an α-inverse-strongly monotone mapping.
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 . 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 ; see [6] and the references therein.
Recall that the classical variational inequality problem is to find such that
It is known that is a solution to (2.1) if and only if x is a fixed point of the mapping , where is a constant and I is the identity mapping. Projection methods recently have been studied for variational inequality (2.1); see [11–22] and the references therein.
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 if
S is said to be asymptotically nonexpansive if there exists a sequence with such that
S is said to be κ-strictly pseudocontractive if there exists a constant such that
The class of strict pseudocontractions was introduced by Browder and Petryshyn [23]. It is clear that every nonexpansive mapping is a 0-strict pseudocontraction.
T is said to be an asymptotically κ-strict pseudocontraction if there exists a sequence with and a constant such that
The class of asymptotically strict pseudocontractions was introduced by Qihou [24]. It is clear that every asymptotically nonexpansive mapping is an asymptotically 0-strict pseudocontraction.
Let F be a bifunction of into ℝ, where ℝ denotes the set of real numbers and is an inverse-strongly monotone mapping. In this paper, we consider the following generalized equilibrium problem:
In this paper, the set of such is denoted by , i.e.,
To study the generalized equilibrium problem (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.
If , then the generalized equilibrium problem (2.2) is reduced to the following equilibrium problem:
In this paper, the set of such is denoted by , i.e.,
If , then the generalized equilibrium problem (2.2) is reduced to the classical variational inequality (2.1).
Recently, equilibrium problems (2.2) and (2.3) have been investigated by many authors; see [25–31] and the references therein. Motivated by the research going on in this direction, we study a hybrid algorithm for solving common solutions of variational inequality (2.1), generalized equilibrium problem (2.2), and fixed points of an asymptotically strict pseudocontraction. Possible computation errors are taken into account. Weak convergence theorems are established in the framework of real Hilbert spaces.
In order to prove our main results, we also need the following lemmas.
Lemma 2.1 [32]
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 [24]
Let C be a nonempty closed convex subset of a Hilbert space H and be an asymptotically strict pseudocontraction. Then is demi-closed, that is, if is a sequence in C with and , then .
Lemma 2.3 [33]
Let H be a Hilbert space and for all . Suppose that and are sequences in H such that
and
hold for some . Then .
Lemma 2.4 [34]
Let , , and be three nonnegative sequences satisfying the following condition:
where is some nonnegative integer, and . Then the limit exists.
3 Main results
Theorem 3.1 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 an α-inverse-strongly monotone mapping, and let be a β-inverse-strongly monotone mapping. Let be an asymptotically κ-strict pseudocontraction with the sequence such that . Assume that is not empty. Let , , , and be real number sequences in . Let and be two positive real number sequences. Let be a sequence generated in the following process:
where is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
-
(a)
;
-
(b)
and ;
-
(c)
;
-
(d)
and ,
where p, q, b, s, , r, are real constants. Then converges weakly to some point in Ω.
Proof First, we show that the sequences , , and are bounded. Let be fixed arbitrarily. For any , we see that
Using the restriction (d), we see that . This implies that is nonexpansive. In the same way, we find that is also nonexpansive. Using the restriction (c), we obtain that
It follows that
This implies from Lemma 2.4 that exists. This shows that is bounded, so are and . From (3.2), we have
It follows that
With the aid of the restrictions (b) and (d), we find that
Since is firmly nonexpansive, we have
which implies that
Hence, we find from (3.2) that
Therefore, we obtain that
From the restrictions (b) and (d), we find from (3.3) that
It follows from (3.1) that
Hence, we have
This implies that
Using the restrictions (b) and (d), we obtain that
Since is firmly nonexpansive, we find that
which implies that
It follows that
which yields that
Using the restrictions (b) and (d), we find from (3.5) that
It follows from (3.4) and (3.6) that
Since is bounded, we see that there exists a subsequence of which converges weakly to ξ. Let T be a maximal monotone mapping defined by
For any given , we have . Since , by the definition of , we have . Since , we see that and hence
It follows that
Since converges weakly to ξ and B is -Lipschitz continuous, we see that . Notice that T is maximal monotone and hence . This shows that . From (3.6), we see that converges weakly to ξ. It follows that
From condition (A2), we see that
Replacing n by , we arrive at
For t with and , let . Since and , we have . In view of (3.8), we find that
Using (3.6), we have . Since A is monotone, we see that . It follows from condition (A4) that
Using conditions (A1) and (A4), we see from (3.9) that
which yields that
Letting , we find
which implies that .
Now, we are in a position to show . Since exists, we may assume that . Put . It follows from (3.2) that and . On the other hand, we have
Using Lemma 2.3, we obtain that . Note that
Hence, we have . Note that . Since S is Lipschitz continuous, we have . Further, we find that . Using Lemma 2.2, we see that . This proves that .
Finally, we show that the sequence converges weakly to ξ. Assume that there exists another subsequence of such that converges weakly to η. In the same way, we find . If , we see from the Opial condition [35] that
This derives a contradiction. Hence, we have . This implies that . This completes the proof. □
Remark 3.2 The key of the weak convergence of the algorithm is due to the fact that A is inverse-strongly monotone, which yields that is nonexpansive. The nonexpansivity of the mapping plays an important role in this theorem. Therefore, it is of interest to relax the monotonicity of A such that the algorithm is still weakly convergent.
Next, we give some subresults of Theorem 3.1. If S is asymptotically nonexpansive, we find the following result.
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). Let be an α-inverse-strongly monotone mapping, and let be a β-inverse-strongly monotone mapping. Let be an asymptotically nonexpansive mapping with the sequence such that . Assume that is not empty. Let , , and be real number sequences in . Let and be two positive real number sequences. Let be a sequence generated in the following process:
where is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
-
(a)
;
-
(b)
and ;
-
(c)
and ,
where p, q, s, , r, are real constants. Then converges weakly to some point in Ω.
Further, if S is an identity mapping, we have the following result.
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). Let be an α-inverse-strongly monotone mapping, and let be a β-inverse-strongly monotone mapping. Assume that is not empty. Let , , and be real number sequences in . Let and be two positive real number sequences. Let be a sequence generated in the following process:
where is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
-
(a)
;
-
(b)
and ;
-
(c)
and ,
where p, q, s, , r, are real constants. Then converges weakly to some point in Ω.
Next, we give a result on variational inequality (2.1).
Corollary 3.5 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an α-inverse-strongly monotone mapping, and let be a β-inverse-strongly monotone mapping. Assume that is not empty. Let , , and be real number sequences in . Let and be two positive real number sequences. Let be a sequence generated in the following process:
where is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
-
(a)
;
-
(b)
and ;
-
(c)
and ,
where p, q, s, , r, are real constants. Then converges weakly to some point in Ω.
Proof Putting , we see that
is equivalent to
This implies that . Let and S be the identity. Then we can obtain from Theorem 3.1 the desired results immediately. □
Finally, we consider solving common fixed points of a pair of strict pseudocontractions.
Corollary 3.6 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be an α-strict pseudocontraction, and let be a β-strict pseudocontraction. Assume that is not empty. Let , , and be real number sequences in . Let and be two positive real number sequences. Let be a sequence generated in the following process:
where is a bounded sequence in C. Assume that the control sequences satisfy the following restrictions:
-
(a)
;
-
(b)
and ;
-
(c)
and ,
where p, q, s, , r, are real constants. Then converges weakly to some point in Ω.
Proof Put , and . It follows that A is -inverse-strongly monotone and B is -inverse-strongly monotone. We also have and . In view of Theorem 3.1, we find the desired result immediately. □
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Hao, Y. Weak convergence theorems of a hybrid algorithm in Hilbert spaces. J Inequal Appl 2014, 378 (2014). https://doi.org/10.1186/1029-242X-2014-378
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DOI: https://doi.org/10.1186/1029-242X-2014-378