Research Article | Open Access

Li-Jun Zhu, Naseer Shahzad, Asim Asiri, "Iterative Solutions for Solving Variational Inequalities and Fixed-Point Problems", *Journal of Function Spaces*, vol. 2021, Article ID 5511634, 10 pages, 2021. https://doi.org/10.1155/2021/5511634

# Iterative Solutions for Solving Variational Inequalities and Fixed-Point Problems

**Academic Editor:**Huseyin Isik

#### Abstract

In this paper, we are interested in variational inequalities and fixed-point problems in Hilbert spaces. We present an iterative algorithm for finding a solution of the studied variational inequalities and fixed-point problems. We show the strong convergence of the suggested algorithm.

#### 1. Introduction

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex set. Let , , , and be four nonlinear operators. Use to denote the fixed-point set of .

In this paper, we will investigate the following variational inequalities and fixed-point problems of finding a point such thatwhere denotes the solution set of the generalized variational inequality (shortly, GVI) which is to find a point such thatand means the solution set of the variational inequality (shortly, VI) which is to find a point such that

Throughout, we use to denote the solution set of problem (1), that is,

It is well known that variational inequalities play key roles and provide a useful mathematical framework, theory, and method for studying many valuable problems arising in water resources, finance, economics, medical images, and so on ([1â€“6]). A lot of work and a great deal of algorithms for solving GVI or VI have been introduced and investigated, see, e.g., [7â€“15]. Among them, a basic and important algorithm is the projected algorithm which generates a sequence with the formwhere is step-size and is the orthogonal projection.

At the same time, we are also interested in the fixed-point problem of finding a point such that . Iterative solution for solving a fixed-point problem is an active research field, see, e.g., [16â€“24]. Recently, iterative algorithms for solving variational inequalities and fixed-point problems have been investigated extensively by many authors [25â€“33].

Motivated by the work in this direction, in this paper, we devote to research variational inequalities and fixed-point problem (1). We introduce an iterative algorithm for finding a solution of problem (1). We show the strong convergence of the suggested algorithm.

#### 2. Preliminaries

Let be a nonempty closed convex subset of a real Hilbert space . Recall that an operator is said to be(i)strongly monotone if(ii)-inverse strongly -monotone if there exists a constant such that(iii)relaxed -cocoercive [34, 35], if there exist two constants such that

An operator is said to be(i)pseudocontractive [36] if(ii)-Lipschitz ifwhere is a constant

If , then is said to be -contraction. If , then is said to be nonexpansive.

An operator is said to be monotone if for all , , and . A monotone operator on is said to be maximal if its graph is not strictly contained in the graph of any other monotone operator on .

For , there exists a unique point in , denoted by satisfying

Moreover, is firmly nonexpansive, that is,

Further, has the following property:

Lemma 1 ([37]). *Let be a nonempty closed convex subset of a real Hilbert space . Let be an -Lipschitz pseudocontractive operator. Then, and , we havewhere .*

Lemma 2 ([24]). *Let be a nonempty, convex, and closed subset of a Hilbert space . Let be a continuous pseudocontractive operator. Then,*(i)* is closed and convex*(ii)* is demiclosedness, i.e., and imply that *

Lemma 3 ([23]). *Let , , and be real number sequences. Suppose the following conditions are satisfied:*(i)*(ii)**(iii)** or **Then, .*

Lemma 4 ([38, 39]). *Let be a real number sequence. Assume there exists at least a subsequence of such thatfor all . For every , define an integer sequence as*

Then, as and for all ,

#### 3. Main Results

In this section, we present our iterative algorithm and convergence theorem. Let be a nonempty closed convex subset of a real Hilbert space . Assume that(i) is a -contractive operator(ii) is a weakly continuous and -strongly monotone operator such that its rang (iii) is a -inverse strongly -monotone operator(iv) is an -Lipschitz and relaxed -cocoercive operator(v) is an -Lipschitz pseudocontractive operator with

Let , , , and be four real number sequences in and and be two real number sequences in .

Now, we present our algorithm for solving problem (1).

*Algorithm 5. *Let be an initial value. Define the sequence by the following form:

Theorem 6. *Suppose that . Assume that the following conditions are satisfied:**: and **: for all **: and for all **:**: and **Then, the sequence generated by (17) converges strongly to verifying*

*Proof. *Since is -strongly monotone, we can get from (6) thatThus, VI (18) has a unique solution, denoted by . It follows that and . Using inequality (13), we can obtain that for all .

Since is -inverse strongly -monotone, for any , we haveBased on (20), we deduceBy (17), (19), and (21), we deriveAccording to (21) and (23), we obtainApplying Lemma 1 to (17), we haveSince is relaxed -cocoercive and -Lipschitz, for all , we haveSince , . Thus, from (26), we obtainHence,Combining (17), (23), (25), and (28), we obtainBy induction, we haveIt follows thatSo, , , , , and are bounded.

From (17), we haveIt follows thatThanks to (33), we deduceCombining (32) and (34), we obtainBy virtue of (23), we getNow, we consider two cases.*Case 1*. There exists some integer such that is decreasing when . Then, exists. According to (35), (36), and , we haveThis together with implies thatTherefore, by (32), we haveBy (24), we haveIt results in thatHence,Set for all . Using (13) and (21), we haveIt yieldsIn the light of (17) and (44), we haveBased on (40) and (45), we obtainThen,According to , , (39), (42), and (47), we deduceSince , from (38), (39), and (48), we haveFrom (26) and (28), we getIt follows thatwhich together with (49) implies thatTherefore,Since is firmly nonexpansive, from (12) and (28), we have