Strong Convergence Analysis of Iterative Algorithms for Solving Variational Inclusions and Fixed-Point Problems of Pseudocontractive Operators

Abstract

Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.

1. Introduction

Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed and convex subset of . Let and be two nonlinear operators. Recall that the variational inclusion ([1]) is to solve the following problem of finding verifyingHere, use to denote the set of solutions of (1).

Special Case 1. Let be defined bySetting , variational inclusion (1) reduces to find such thatProblem (3) is the well-known variational inequality which has been studied, extended, and developed in a broad category of jobs (see, e.g., [2–14]).

Special Case 2. Let be a proper lower semicontinuous convex function and be the subdifferential of . Setting , variational inclusion (1) reduces to find such thatProblem (4) is called the mixed quasi-variational inequality [15] which is a very significant extension of variational inequality (3) involving the nonlinear function . It is well known that a large number of practical problems arising in various branches of pure and applied sciences can be formulated as the model of mixed quasi-variational inequality (4).
Problem (1) plays a key role in minimization, convex feasibility problems, machine learning, and others. A popular algorithm for solving problem (1) is the forward-backward algorithm [16] generated bywhere is a forward step and is a backward step with . This algorithm is a splitting algorithm which solves the difficulty of calculating of the resolvent of .
Recently, there has been increasing interest for studying common solution problems relevant to (1) (see for example, [17–27]). Especially, Zhao, Sahu, and Wen [28] presented an iterative algorithm for solving a system of variational inclusions involving accretive operators. Ceng and Wen [29] introduced an implicit hybrid steepest-descent algorithm for solving generalized mixed equilibria with variational inclusions and variational inequalities. Li and Zhao [30] considered an iterate for finding a solution of quasi-variational inclusions and fixed points of nonexpansive mappings.
Motivated by the results in this direction, the main purpose of this paper is to research a common solution problem of variational inclusions and fixed point of pseudocontractions. We suggest a Halpern-type algorithm for solving such problem. We show that the proposed algorithm has strong convergence under some mild conditions.

2. Preliminaries

Let be a real Hilbert space. Let be an operator. Write . Recall that is called monotone if , and ,

A monotone operator is maximal monotone if and only if its graph is not strictly contained in the graph of any other monotone operator on .

For a maximal monotone operator on ,(i)Set (ii)Denote its resolvent by which is single-valued from into

It is known that and is firmly nonexpansive, i.e.,for all .

Let be a nonempty closed convex subset of a real Hilbert space . Recall that an operator is said to be(i)-Lipschitz if there exists a positive constant such that If , is nonexpansive.(ii)Pseudocontractive if(iii)Inverse-strongly monotone ifwhere is a constant and is also called -ism.

Recall that the projection is an orthographic projection from onto , which is defined by . It is known that is nonexpansive.

Lemma 1 (see [23, 31]). Let be a nonempty closed convex subset of a real Hilbert space . Let be an -Lipschitz pseudocontractive operator. Then,(i) is demiclosed, i.e., and (ii)For , and , we have

Lemma 2 (see [16, 32]). Let be a real Hilbert space and let be a maximal monotone operator on . Then, we havefor all and .

Lemma 3 (see [33]). Assume that a real number sequence satisfieswhere and satisfy the following conditions:(i)(ii) or Then, .

Lemma 4 (see [8]). Let be a sequence. Assume that there exists at least a subsequence of verifying for all . Let be an integer sequence defined as . Then as and

3. Main Results

Let be a nonempty closed convex subset of a real Hilbert space . Let the operator be an -ism. Let be a maximal monotone operator with . Let be an Lipschitz pseudocontractive operator with . Let and be two sequences. Let and be two constants.

Next, we introduce a Halpern-type algorithm for finding a common solution of variational inclusion (1) and fixed point of pseudocontractive operator .

Algorithm 1. Let be a fixed point. Choose . Set . Step 1. For given , compute by Step 2. Compute by Step 3. Set and return to Step 1.Next, we prove the convergence of Algorithm 1.

Theorem 1. Suppose that . Assume that the following conditions are satisfied:  and   and Then, the sequence generated by Algorithm 1 converges strongly to .

Proof. Let . Set . Since is -ism, we haveBy the nonexpansivity of , we haveUsing Lemma 1, we getThis together with (14) implies thatAccording to (15)-(19), we obtainThen, the sequence is bounded. The sequences and are also bounded.
Again, by (15)-(19), we deduceIt follows thatSince is firmly nonexpansive, using (6), we havewhich leads toCombining (21) with (24), we obtainwhich results in thatNext, we analyze two cases. (i) . (ii) For any , such that .
In case of (i), exists. From (22), we deduceandIt follows from (14) thatOn the basic of (26) and (27), we haveNote that . Thanks to (29) and (30), we derive thatHowever,We haveThis together with (28) implies thatSet . Next, we prove thatSince is bounded, there exists a subsequence of satisfying(1) (hence, by (31))(2)From (34) and Lemma 1, we obtain .
Owing to (29) and (30), we have that andSince , without loss of generality, we assume that . Observe thatApplying Lemma 2, we obtainIt follows thatThanks to (37) and (39), we getNoting that , from (36) and (40), we getBy Lemma 1, we deduce that . Therefore, andFrom (15), we haveApplying Lemma 3 to (43) to deduce .
In case of (ii), let . So, we have . Define an integer sequence , by . It is obvious that and for all . Similarly, we can prove that and Therefore, all weak cluster points . Consequently,Note that . From (43), we deduceIt follows thatCombining (44) and (46), we have and henceFrom (45), we deduce that . This together with (47) implies that . According to Lemma 4, we get . Therefore, and . This completes the proof.