Convergence Theorems of Common Elements for Pseudocontractive Mappings and Monotone Mappings

Abstract

An algorithm for treating pseudocontractive mappings and monotone mappings is proposed. Convergence analysis of algorithm is investigated in the framework of Hilbert spaces.

1. Introduction

The motivation for common element problem is mainly due to its possible applications to mathematical modeling of concrete complex problems. The common element problems include mini-max problems, complementarily problems, equilibrium problems, common fixed point problems, and variational inequalities as special cases; see [1–7] and the references therein. It is well-known that the convex feasibility problem is a special case of the common zero (fixed) points of nonlinear mappings. And many important problems have reformulations which require finding zero points, for instance, evolution equations, complementarily problems, mini-max problems, and variational inequalities and optimization. For studying zero points of monotone mappings, the most well-known algorithm is the proximal point algorithm; see [8, 9] and the references therein. Regularization methods recently have been investigated for treating zero points of monotone mappings; see [2, 5, 6, 9] and references therein.

In 2010, Takahashi et al. [6] studied zero point problems of the sum of two monotone mappings and fixed point problems of a nonexpansive mapping based on the following iterative algorithm: where is a nonempty closed convex subset of a real Hilbert space , and are real number sequences in , is a positive sequence, is a nonexpansive mapping, is an inverse strongly monotone mapping, is a maximal monotone mapping, and , where is the identity mapping. They proved that the sequence generated in (1) converges strongly to some provided that the control sequences satisfy some restrictions, where is the set fixed points of .

In 2014, Shahzad and Zegeye [5] considered an iterative method for a common point of fixed points of Lipschitzian pseudocontractive mappings and zeros of sum of two monotone mappings based on the projection method in a real Hilbert space. To be more precise, they investigated the following algorithm: where is a nonempty closed convex subset of a real Hilbert space , , , , , and are real number sequences in , is a positive sequence, is a Lipschitzian pseudocontractive mapping, is an inverse strongly monotone mapping, is a maximal monotone mapping, and . They proved that the sequence generated in (2) converges strongly to the minimum-norm point provided that the control sequences satisfy some restrictions.

In this paper, we are concerned with the problem of finding a common element in the intersection , where denotes the fixed point set of the pseudocontractive mapping , , and denotes the zero point set of the sum of an inverse strongly monotone mapping and a maximal monotone mapping . Applications to a common element of the set of common fixed points of Lipschitzian pseudocontractive mappings and solutions of variational inequality for -inverse strongly monotone mappings are included. Our theorems improve and extend those announced by Shahzad and Zegeye [5], Takahashi et al. [6], and other authors with the related interest.

2. Preliminaries

Let be a real Hilbert space with the inner product and the norm . Let be a nonempty closed convex subset of and let be the metric projection from onto . Let be a mapping. In this paper, we use to denote the fixed point set of ; that is, .

Recall that is nonexpansive if is said to be a -strictly pseudocontractive mapping if there exists such that Note that the class of -strictly pseudocontractive mappings includes the class of nonexpansive mappings as a special case. is said to be a pseudocontractive mapping if We note that inequalities (4) and (5) can be equivalently written as for some and respectively. Note that the class of -strictly pseudocontractive mappings is contained in the class of pseudocontractive mappings. We note that the inclusion is proper. We remark that is a -strictly pseudocontractive mapping if and only if is a -inverse strongly monotone mapping and is a pseudocontractive mapping if and only if is a monotone mapping.

Let be a mapping and stands for the zero point set of ; that is, . Recall that is said to be monotone if is said to be -inverse strongly monotone if there exists a constant such that It is not hard to see that -inverse strongly monotone mappings are Lipschitz continuous with constant ; that is, for all .

Recall that the classical variational inequality, denoted by , is to find such that

A multivalued mapping with the domain and the range is said to be monotone if, for , , , and , we have . A monotone mapping is said to be maximal if its graph is not properly contained in the graph of any other monotone mapping. Let be a maximal monotone mapping. Then we can define, for each , a nonexpansive single-valued mapping by . It is called the resolvent of . We know that for all and is firmly nonexpansive.

Lemma 1. Let be a real Hilbert space. Then, for any given , the following inequality holds:

Lemma 2 (see [10]). Let be a convex subset of a real Hilbert space . Let . Then if and only if

Lemma 3 (see [2]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a mapping and let be a maximal monotone mapping. Then .

Lemma 4 (see [11]). Let be a Hilbert space. Let and let be maximal monotone mappings. Suppose that . Then is a maximal monotone mapping.

Lemma 5 (see [4]). Let be a sequence of real numbers. Assume that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all sufficiently large numbers :

Lemma 6 (see [12]). Let be a real Hilbert space. Then, for all and for such that , the following equality holds:

Lemma 7 (see [7]). Let be a nonempty closed convex subset of a real Hilbert Hilbert and let be a continuous pseudocontractive mapping. Then(i) is a closed convex subset of ;(ii) is demiclosed at zero; that is, if is a sequence in such that and as , then .

Lemma 8 (see [13]). Let be a sequence of nonnegative real numbers satisfying the following relation: where and satisfy the following conditions: , , and . Then .

3. Main Results

Theorem 9. Let be a nonempty closed convex subset of a real Hilbert space . Let be Lipschitzian pseudocontractive mappings with Lipschitz constants and , respectively. Let be an -inverse strongly monotone mapping and let be a maximal monotone mapping such that the domain of is subset of . Assume that . Let , where is a positive real number sequence. Given , let be the sequence generated by the following algorithm: Assume that the sequences , , , , , , , and satisfy the following restrictions:(a);(b) and ;(c), and ;(d), , for all ,for some real numbers , where . Then converges strongly to some point , where .

Proof. First, we show that is nonexpansive.
Indeed, we have It follows from restriction (a) that is nonexpansive.
Let . It follows from (5), (16), and Lemmas 3 and 6 that It follows from (5), (16), and Lemma 6 that Similarly, we have that Substituting (19) and (20) into (18), we obtain that In view of restriction (d), we find that for all . It follows from (21) and (22) that Putting , we find that for all .
Indeed, it is clear that . Suppose that for some positive integer . It follows that This finds that is bounded and hence and are bounded.
Let . Then we see that . Put . Using (16), (19), and (20) and Lemmas 1 and 6, we find that which implies from (22) that Now we consider two cases.
Case 1. Suppose that there exists such that is decreasing for all . Then we get that is convergent. It follows from (22) and (26) that as . Also we obtain from (27) that as . In view of the Lipschitz continuity of and (27) and (28), we find that It follows from (27), (29), and (30) that Since is a bounded subset of , we can choose a subsequence of such that and It follows from (31) that . By (27) and Lemma 7, we obtain that and .
Next, we show that .
Notice that It follows from (27) that Hence we get Putting , we find that . Since is monotone, we get that, for any , where . Since , , and as , we have . Thus, letting , we obtain from (27) and (36) that . This means , that is, . Hence we get . This implies from Lemma 2 that On the other hand, we have from (26) that From Lemma 8 and (37), we find that .
Case 2. Suppose that there exists a subsequence of such that for all . By Lemma 5, there exists a nondecreasing sequence such that and for all . From (22) and (26), we have , , and as . Thus, like in Case 1, we obtain and From (26) and (40), we have Applying (41) and , we have as . It implies that as . By (40), we have as .
Therefore, from the above two cases, we can conclude that the sequence converges strongly to . This completes the proof.

From Lemma 4, we have the following result.

Corollary 10. Let be a nonempty closed convex subset of a real Hilbert space such that . Let be Lipschitzian pseudocontractive mappings with Lipschitz constants and , respectively. Let and be maximal monotone mappings such that . Assume that . Let , where is a positive real number sequence. Given , let be the sequence generated by the following algorithm: Assume that the sequences , , , , , , , and satisfy the following restrictions:(a);(b), and ;(c), , and ;(d), for all ,for some real numbers , where . Then converges strongly to some point , where .

Remark 11. If , (the identity mapping), and , then Theorem 9 reduces to Theorem of Shahzad and Zegeye [6]. Thus, Theorem 9 covers Theorem of Shahzad and Zegeye [6] as a special case.

4. Applications

In this section, we will consider equilibrium problems and variational inequalities.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction of into , where denotes the set of real numbers. Recall the following equilibrium problem: find such that We use to denote the solution set of the equilibrium problem. To study the equilibrium problems, we assume that satisfies the following conditions:(A1) for all ;(A2) is monotone, that is, , for all ;(A3)for each , (A4)for each , is convex and lower semicontinuous

Lemma 12 (see [1]). Let be a nonempty closed convex subset of a real Hilbert space 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; that is, for any , (c);(d) is closed and convex.

Lemma 13 (see [13]). Let be a nonempty closed convex subset of a real Hilbert space , let be a bifunction from to which satisfies (A1)–(A4), and let be a multivalued mapping of into itself defined by Then is a maximal monotone mapping with the domain , , and where is defined as in (47).

Now we consider an equilibrium problem. Using Lemmas 12 and 13, the following result holds.

Theorem 14. Let be a nonempty closed convex subset of a real Hilbert space . Let be Lipschitzian pseudocontractive mappings with Lipschitz constants and , respectively. Assume that . Given , let be the sequence generated by the following algorithm: Assume that the sequences , , , , , , , and satisfy the following restrictions:(a) and ;(b), , and ;(c), , and ;(d), , for all ,for some real numbers , where . Then converges strongly to some point , where .
Let be a proper convex lower semicontinuous function. Then the subdifferential of of is defined as follows: From Rockafellar [14], we find that is maximal monotone. It is easy to verify that if and only if . Let be the indicator function of ; that is, Then is a proper convex lower semicontinuous function and is a maximal monotone mapping.

Lemma 15 (see [6]). Let be a nonempty closed convex subset of a real Hilbert space , let be the metric projection from onto , and let be the subdifferential of , where is the indicator function of and let . Then

Now we consider a variational inequality problem.

Theorem 16. Let be a nonempty closed convex subset of a real Hilbert space . Let be Lipschitzian pseudocontractive mappings with Lipschitz constants and , respectively. Let be an -inverse strongly monotone mapping. Assume that . Given , let be the sequence generated by the following algorithm: Assume that the sequences , , , , , , , and satisfy the following restrictions:(a);(b), and ;(c), , and ;(d), , for all ,for some real numbers , where . Then converges strongly to some point , where .

Proof. Put in Theorem 9. Then we get that From Lemma 15, we can conclude the desired conclusion immediately.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.