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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
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.
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 ([1–6]). A lot of work and a great deal of algorithms for solving GVI or VI have been introduced and investigated, see, e.g., [7–15]. 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., [16–24]. Recently, iterative algorithms for solving variational inequalities and fixed-point problems have been investigated extensively by many authors [25–33].
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.
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  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 (). Let be a nonempty closed convex subset of a real Hilbert space . Let be an -Lipschitz pseudocontractive operator. Then, and , we havewhere .
Lemma 2 (). 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 (). Let , , and be real number sequences. Suppose the following conditions are satisfied:(i)(ii)(iii) or Then, .
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:
: for all
: and for all
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