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., [214]).

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, [1727]). 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 thatIf , 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 byStep 2. Compute byStep 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.

Remark 1. Since the pseudocontractive operator is nonexpansive, Theorem 1 still holds if is nonexpansive.

Remark 2. Assumption imposed on parameter is essential and we do not add any other assumptions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


Zhangsong Yao was partially supported by the Grant 19KJD100003. Ching-Feng Wen was partially supported by the Grant of MOST 109-2115-M-037-001.