Split General Strong Nonlinear Quasi-Variational Inequality Problem

Abstract

We introduce a split general strong nonlinear quasi-variational inequality problem which is a natural extension of a split general quasi-variational inequality problem, split variational inequality problem, and quasi-variational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for the split general strongly nonlinear quasi-variational inequality problem and discuss the convergence criteria of the iterative algorithm. The results presented here generalized, unify, and improve many previously known results for quasi-variational and variational inequality problems.

1. Introduction

Variational inequalities are a very powerful tool of the current mathematical technology and have become a rich source of inspiration for scientists and engineers. These have been extended and generalized to study a wide class of problems arising in mechanics, optimization and control problem, operations research and engineering sciences, and so forth. The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On one hand, it reveals the fundamental facts on the qualitative behavior of solutions to important classes of problems. On the other hand, it enables us to develop highly efficient and powerful numerical methods to solve, for example, obstacle, unilateral, free, and moving boundary value problems. In the last five decades, considerable interest has been shown in developing various classes of variational inequality problems, both for their own sake and for their applications.

An important generalization of the variational inequality problem is the quasi-variational inequality problem introduced and studied by Bensoussan et al. [1] in connection with impulse control problem. Recently, Kazmi [2] introduced and studied the following split general quasi-variational inequality problem (in short, SpGQVIP). For each , let be a Hilbert space with inner product and induced norm , let be a nonempty, closed, and convex set-valued mapping, let and be nonlinear mappings, and let be a bounded linear operator with its adjoint operator Then, the SpGQVIP is to find such that andand such that, , solves SpGQVIP (1a)-(1b) amounts to saying the following: find a solution of general quasi-variational inequality GQVI (1a) whose image under a given bounded linear operator is a solution of GQVIP (1b). If , where is an identity mapping on , for all , and then SpGQVIP (1a)-(1b) is reduced to the following SpVIP. Find such that and such that solvesSpVIP (2a)-(2b) has been introduced and studied by Censor et al. [3]. It is worth mentioning that the SpVIP (2a)-(2b) is quite general and permits split minimization between two spaces so that the imagine of a minimizer of a given function, under a bounded linear operator, is a minimizer of another function and it includes as a special case the split zero problem and the split feasibility problem which have already been studied and used in practice as a model in the intensity-modulated radiation therapy planning; see [4–6] and the references therein.

In this paper, we introduced the following split general strongly nonlinear quasi-variational inequality problem: for each , let be a nonempty, closed, and convex set-valued mapping, let , , and be three nonlinear mappings, and let be a bounded linear operator with its adjoint operator . Then, we consider the problem: find such that and and such that , solvesWe call problem (3a)-(3b) the split general strongly nonlinear quasi-variational inequality problem (in short, SpGSNQVIP).

Remark 1. If , then SpGSNQVIP (3a)-(3b) is reduced to SpGQVIP (1a)-(1b). So SpGSNQVIP (3a)-(3b) is the generalization of SpGQVIP (1a)-(1b).

Remark 2. Note that general strongly nonlinear variational inequality problem , , is an important class of variational inequalities, which is the optimal condition of the following minimization problem:where We denote the solution set of SpGSNQVIP (3a)-(3b) and the solution set of SpGQVIP (1a)-(1b) by and , respectively.

Example 3. Letting , , and ,It is easy to see that satisfies (3a) and satisfies (3b). So and thus .

Example 4. Let with the norm and with the normSetIt is easy to see that the solution sets of (3a) and (3b) are and , respectively. Letting , then is a bounded linear operator. It is obvious that if , then , and so

In this paper, using the projection method, we propose an iterative algorithm for SpGSNQVIP (3a)-(3b) and discuss the convergence of the iterative algorithm. The results presented here generalized, unify, and improve the many previously known results for quasi-variational and variational inequality problems.

2. Iterative Algorithms and Convergence Results

For each , a mapping is said to be the metric projection of on if, for every point , there exists a unique nearest point in denoted by such thatIt is well known that is nonexpansive and satisfiesMoreover, is characterized byFurther, it is easy to see the following fact: satisfied QVIP we find such that Hence, SpGSNQVIP (3a)-(3b) can be reformulated as follows: find with such that andfor .

Based on the above-mentioned arguments, we propose the following iterative algorithm for approximating a solution to SpGSNQVIP (3a)-(3b).

Let be a sequence such that , and let , , and be the parameters with positive values.

Algorithm 5. Given , compute the iterative sequence by the iterative schemes:for all ,

If , then Algorithm 5 is reduced to the following iterative algorithm for SpGQVIP (1a)-(1b).

Algorithm 6. Given , compute the iterative sequence by the iterative schemes:for all ,

If , , where is a nonempty closed convex subset of , and then Algorithm 5 is reduced to the following iterative algorithm for SpVIP (2a)-(2b).

Algorithm 7. Given , compute the iterative sequence by the iterative schemes:for all ,

Remark 8. Algorithms 6 and 7 are proposed by Kazmi in [2] and [7], respectively. Note that Algorithm 5 concludes them as special cases.

In order to obtain our main results, we need the following assumption, definition, and lemmas.

Assumption 9. For all , the operator satisfies the following condition:for some constant .

Definition 10. A nonlinear mapping is said to be (i)-strongly monotone with respect to if there exists a constant such that(ii)-Lipschitz continuous if there exists a constant such that

Remark 11. If , where is an identity mapping on , then Definition 10(i) is reduced to the definition of -strong monotonicity of .

Lemma 12. Let be a real Hilbert space. Then, the following inequalities hold:(1),(2).

Lemma 13 (see [8]). Assume that is a sequence of nonnegative numbers such that , where is a sequence in and is a sequence such that(i);(ii)or ; then .

Now we study the convergence of Algorithm 5 for SpGSNQVIP (3a)-(3b).

Theorem 14. For each , let be a nonempty, closed, and convex set-valued mapping, and let be -Lipschitz continuous such that is -strongly monotone, where is the identity mapping on . Let be -strongly monotone with respect to and -Lipschitz continuous. Let be -Lipschitz continuous and let be a bounded linear operator and be its adjoint mapping. Suppose is a solution to SpGSNQVIP (3a)-(3b) and Assumption 9 holds. Then, the sequence generated by Algorithm 5 convergences strongly to provided that the constants and satisfy the following conditions:

Proof. Since is a solution of SpGSNQVIP (3a)-(3b), such that and for . It follows from Algorithm 5(13a), Assumption 9, and (20) thatNoting that is -strongly monotone with respect to and -Lipschitz continuous and is -Lipschitz continuous, we have Combining (22) and (23), we getSince is -strongly monotone, by virtue of Lemma 12(1), we havewhich implies thatIt follows from (24) and (26) that we havewhere
Similarly, we obtainwhere
Furthermore, in view of Algorithm 5(13c), we haveNote that is a bounded linear operator with and the given condition on , we getAnd using (29), we haveFrom (30)–(32),where It follows from the conditions on , , and that Thus, and for So it follows from Lemma 13 that converges strongly to as . Since is continuous, it follows from (24), (27), (28), and (29) thatThis completes the proof.

If , then Theorem 14 reduced to the following result of the convergence of Algorithm 6 for SpGQVIP (1a)-(1b).

Corollary 15. For each , let be a nonempty, closed, and convex set-valued mapping, and let be -Lipschitz continuous such that is -strongly monotone, where is the identity mapping on . Let be -strongly monotone with respect to and -Lipschitz continuous. Let be a bounded linear operator and be its adjoint mapping. Suppose is a solution to SpGQVIP (1a)-(1b) and Assumption 9 holds. Then, the sequence generated by Algorithm 6 convergences strongly to provided that the constants and satisfy the following conditions:If , where is a nonempty closed and convex subset of , , and , then Theorem 14 reduces to the following convergence result of Algorithm 7 for SpVIP (2a)-(2b).

Corollary 16. For each , let be a nonempty, closed, and convex subset of Let be -strongly monotone and -Lipschitz continuous. Let be a bounded linear operator and be its adjoint operators. Suppose is a solution to SpVIP (2a)-(2b). Then, the sequence generated by Algorithm 7 converges strongly to provided that the constants and satisfy the following conditions:

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Project no. 11371070.