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Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems
We consider a general variational inequality and fixed point problem, which is to find a point with the property that (GVF): and where is the solution set of some variational inequality is the fixed points set of nonexpansive mapping , and is a nonlinear operator. Assume the solution set of (GVF) is nonempty. For solving (GVF), we suggest the following method , . It is shown that the sequence converges strongly to which is the unique solution of the variational inequality , for all .
Let and be two nonlinear mappings. We concern the following generalized variational inequality of finding such that The solution set of (1.1) is denoted by . It has been shown that a large class of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, physical, mathematical, engineering, and applied sciences can be studied in the unified and general framework of the general variational inequalities (1.1), see [1–16] and the references therein. Noor  has introduced a new type of variational inequality involving two nonlinear operators, which is called the general variational inequality. It is worth mentioning that this general variational inequality is remarkably different from the so-called general variational inequality which was introduced by Noor  in 1988. Noor  proved that the general variational inequalities are equivalent to nonlinear projection equations and the Wiener-Hopf equations by using the projection technique. Using this equivalent formulation, Noor  suggested and analyzed some iterative algorithms for solving the special general variational inequalities and further proved that these algorithms have strong convergence.
For , where is the identity operator, problem (1.1) is equivalent to finding such that which is known as the classical variational inequality introduced and studied by Stampacchia  in 1964. This field has been extensively studied due to a wide range of applications in industry, finance, economics, social, pure and applied sciences. For related works, please see [20–35]. Our main purposes in the present paper is devoted to study this topic.
Motivated and inspired by the works in this field, in this paper, we consider a general variational inequality and fixed point problem, which is to find a point with the property that where is the fixed points set of nonexpansive mapping . Assume the solution set of (GVF) is nonempty. For solving (GVF), we suggest the following method It is shown that the sequence converges strongly to which is the unique solution of the following variational inequality Our results contain some interesting results as special cases.
Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of . Recall that a mapping is said to be nonexpansive if for all . We denote by the set of fixed points of . A mapping is said to be -Lipschitz continuous, if there exists a constant such that for all . A mapping is said to be -inverse strongly -monotone if and only if for some and for all . A mapping is said to be strongly monotone if there exists a constant such that for all .
Let be a mapping of into . The effective domain of is denoted by , that is, . A multivalued mapping is said to be a monotone operator on if and only if for all , and . A monotone operator on is said to be maximal if and only if its graph is not strictly contained in the graph of any other monotone operator on . Let be a maximal monotone operator on and let .
It is well known that, for any , there exists a unique such that We denote by , where is called the metric projection of onto . The metric projection of onto has the following basic properties:(i) for all ;(ii) for every ;(iii) for all , .
It is easy to see that the following is true:
We use the following notation:(i) stands for the weak convergence of to ;(ii) stands for the strong convergence of to .
We need the following lemmas for the next section.
Lemma 2.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonlinear mapping and let the mapping be -inverse strongly -monotone. Then, for any , one has
Proof. Consider the following: If , we have
Lemma 2.2 (see ). Let be a closed convex subset of a Hilbert space . Let be a nonexpansive mapping. Then is a closed convex subset of and the mapping is demiclosed at 0, that is, whenever is such that and , then .
Lemma 2.3 (see ). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all and . Then, .
Lemma 2.4 (see ). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(1);(2) or . Then .
3. Main Results
In this section, we will prove our main results.
Theorem 3.1. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -Lipschitz continuous mapping, be a weakly continuous and -strongly monotone mapping such that . Let be an -inverse strongly -monotone mapping and let be a nonexpansive mapping. Suppose that . Let and . For given , let be a sequence generated by where satisfies and . Then the sequence generated by (3.1) converges strongly to which is the unique solution of the following variational inequality:
Proof. First, we show the solution set of variational inequality (3.2) is singleton. Assume also solves (3.2). Then, we have
It follows that
Since is -strongly monotone, we have
In particular, . By (3.4), we deduce
which implies that because of by the assumption. Therefore, the solution of variational inequality (3.2) is unique.
Pick up any . It is obvious that and . Set . From (2.6), we know for any . Hence, we have From (3.6), (3.8), and Lemma 2.1, we get It follows from (3.1) that This indicates by induction that Hence, is bounded. By (3.6), we have . This implies that is bounded. Consequently, , and are all bounded.
Note that we can rewrite (3.1) as for all . Next, we will use Lemma 2.3 to prove that . In fact, we firstly have It follows that Since and the sequences , and are bounded, we have By Lemma 2.3, we obtain Hence, This together with (3.6) imply that By the convexity of the norm and (3.9), we have From Lemma 2.1, we derive Thus, So, Since and , we obtain Set for all . By using the property of projection, we get It follows that From (3.18) and (3.24), we have Then, we obtain Since , and , we have Next, we prove where is the unique solution of (3.2). We take a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to some point . Without loss of generality, we may assume that . This implies that due to the weak continuity of . Now, we show . First, we note that from (3.15) and (3.27) that . Hence, . By the demiclosedness principle of the nonexpansive mapping (see Lemma 2.2), we deduce . Next, we only need to prove . Set By , we know that is maximal -monotone. Let . Since and , we have From , we get It follows that Then, Since and , we deduce that by taking in (3.33). Thus, by the maximal -monotonicity of . Hence, . Therefore, . From (3.28), we obtain We take in (3.23) to get It follows that Therefore, where and . From condition , we have . By (3.34), we have . We can therefore apply Lemma 2.4 to conclude that and . This completes the proof.
Corollary 3.2. Let be a nonempty closed and convex subset of a real Hilbert space . Let be an -contraction. Let be an -inverse strongly monotone mapping and let be a nonexpansive mapping. Suppose that . Let and . For given , let be a sequence generated by where satisfies and . Then the sequence generated by (3.38) converges strongly to which is the unique solution of the following variational inequality:
The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The first author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Subprojects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University and was supported in part by NSC 100-2221-E-230-012. The second author was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.
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