Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems

doi:10.1016/j.na.2007.11.015

Nonlinear Analysis: Theory, Methods & Applications

Volume 69, Issue 12, 15 December 2008, Pages 4585-4603

https://doi.org/10.1016/j.na.2007.11.015 Get rights and content

Abstract

We introduced and studied the concept of well-posedness to a generalized mixed variational inequality. Some characterizations are given. Under suitable conditions, we prove that the well-posedness of the generalized mixed variational inequality is equivalent to the well-posedness of the corresponding inclusion problem. We also discuss the relations between the well-posedness of the generalized mixed variational inequality and the well-posedness of the corresponding fixed-point problem. Finally, we derive some conditions under which the generalized mixed variational inequality is well-posed.

Introduction

In 1966, Tykhonov [26] first gave the Tykhonov well-posedness of a minimization problem, which means the existence and uniqueness of minimizers, and the convergence of every minimizing sequence to the unique minimizer. In many practical situations, there are more than one minimizer for a minimization problem. In a natural way, the concept of Tykhonov well-posedness in the generalized sense was introduced, which means the existence of minimizers and the convergence of some subsequence of every minimizing sequence to a minimizer. There is no doubt that the concept of well-posedness is motivated by the numerical methods producing optimizing sequences. Because the concept of well-posedness is an important and powerful tool in the study of optimization problems, various concepts of well-posedness have been introduced and studied widely for minimization problems in past decades. A great deal of effort has gone into those concepts; see, e.g., [1], [6], [10], [18], [24], [26], [29], [30] and the references therein.

In recent years, the concept of well-posedness has been extended to other contexts: variational inequality problems [5], [8], [15], [16], [17], [18], saddle point problems [4], Nash equilibrium problems [17], [19], [20], [21], [22], [23], [25], inclusion problems [13], [14], and fixed-point problems [13], [14], [27], [34]. Concerning the well-posedness of a given variational problem, it is interesting and important to establish its metric characterization, to find conditions under which the problem is well-posed, to investigate its links with the well-posedness of other related problems. Some metric characterizations of various well-posedness were established for minimization problems [6], variational inequalities [5], [8], [15], [16] and Nash equilibrium problems [22]. For the well-posedness conditions of various variational problems, we refer the reader to [5], [6], [8], [15], [16], [23], [25]. The relations between the well-posedness of variational inequalities and the well-posedness of minimization problems were discussed in [5], [16], [18]. Lemaire [13] discussed the relations among the well-posedness of minimization problems, inclusion problems and fixed-point problems. Recently, Lemaire et al. [14] further extended the results in [13] by considering perturbations.

Very recently, Fang, Huang and Yao [31] considered and studied the well-posedness of a mixed variational inequality which includes as a special case the classical variational inequality, and derived some results for the well-posedness of this mixed variational inequality, the corresponding inclusion problem and the corresponding fixed-point problem.

Inspired by Fang, Huang and Yao [31], we extend the concept of well-posedness to a generalized mixed variational inequality which includes as a special case the mixed variational inequality, and give some characterizations of its well-posedness. Under suitable conditions, we prove that the well-posedness of the generalized mixed variational inequality is equivalent to the well-posedness of the corresponding inclusion problem. We also discuss the relations between the well-posedness of the generalized mixed variational inequality and the well-posedness of the corresponding fixed-point problem. Finally, we derive some conditions under which the generalized mixed variational inequality is well-posed.

Section snippets

Preliminaries

Let $H$ be a real Hilbert space with norm $‖ \cdot ‖$ and inner product $〈 \cdot, \cdot 〉$ . For convenience, we denote strong (resp. weak) convergence by $\to$ (resp. $⇀$ ). Let $F : H \to 2^{H}$ be a nonempty-valued multifunction, $A : H \to H$ be a single-valued mapping and $φ : H \to R \cup {+ \infty}$ be a proper, convex and lower semicontinuous functional. Denote by dom $φ$ the domain of $φ$ , i.e., $dom φ = {x \in H : φ (x) < + \infty} .$ Consider the following generalized mixed variational inequality associated with $(F, A, φ)$ : $GMVI (F, A, φ) : find x \in H such that for some u \in F (x), 〈 A u, x - y 〉 + φ (x)$

Well-posedness and metric characterization

In this section we introduce some concepts of well-posedness of the generalized mixed variational inequality and establish their metric characterizations. Let $α \geq 0$ be a given number and let $H, A, F$ and $φ$ be defined as in the previous section.

Definition 3.1

A sequence ${x_{n}} \subset H$ is said to be an $α$ -approximating sequence for $GMVI (F, A, φ)$ if there exist a sequence ${u_{n}} \subset H$ with $u_{n} \in F (x_{n}) (\forall n \geq 1)$ and a sequence ${ϵ_{n}}$ of nonnegative numbers with $ϵ_{n} \to 0$ such that $x_{n} \in dom φ, 〈 A u_{n}, x_{n} - y 〉 + φ (x_{n}) - φ (y) \leq \frac{α}{2} {‖ x_{n} - y ‖}^{2} + ϵ_{n}, \forall y \in H, n \geq 1 .$ If $α_{1} > α_{2} \geq 0$ ,

Links with well-posedness of inclusion problems

In this section we shall investigate the relations between the well-posedness of generalized mixed variational inequalities and the well-posedness of corresponding inclusion problems. In what follows we always denote by $\to$ and $⇀$ the strong convergence and weak convergence, respectively. Let $M : H \to 2^{H}$ be a set-valued mapping. The inclusion problem associated with $M$ is defined by $IP (M) : find x \in H such that 0 \in M (x) .$

Definition 4.1

See [13], [14]

A sequence ${x_{n}} \subset H$ is called an approximating sequence for $IP (M)$ if $d (0, M (x_{n})) \to 0$ , or

Links with well-posedness of fixed-point problems

In this section, we shall investigate the relations between the well-posedness of generalized mixed variational inequalities and the well-posedness of the corresponding fixed-point problems. Let $T : H \to 2^{H}$ be a set-valued mapping. The fixed-point problem associated with $T$ is defined by $FP (T) : find x \in H such that x \in T (x) .$

We first recall some concepts.

Definition 5.1

See [13], [14]

A sequence ${x_{n}} \subset H$ is called an approximating sequence for $FP (T)$ if there exists a sequence ${y_{n}} \subset H$ with $y_{n} \in T (x_{n}) (n \geq 1)$ such that $‖ x_{n} - y_{n} ‖ \to 0$ as $n \to \infty$ .

Definition 5.2

See [13], [14]

We say that $FP$

Conditions for well-posedness

In this section, we shall prove that under suitable conditions the well-posedness of the generalized mixed variational inequality is equivalent to the existence and uniqueness of its solutions, and the well-posedness in the generalized sense is equivalent to the existence of its solutions.

Theorem 6.1

Let $A : H \to H$ be weakly continuous, let $F : H \to 2^{H}$ be a nonempty weakly compact-valued multifunction which is $H$ -hemicontinuous and monotone with respect to $A$ , and let $φ : H \to R \cup {+ \infty}$ be proper, convex and lower

Conclusions

In this paper we introduce some concepts of well-posedness for generalized mixed variational inequalities. In Section 3, we establish some metric characterizations of strong $α$ -well-posedness. In Section 4, we discuss the connections between the strong (weak) well-posedness of generalized mixed variational inequalities and strong (weak) well-posedness of inclusion problems. In Section 5, we further investigate the relationships between the strong (weak) well-posedness of generalized mixed

Acknowledgements

The first author’s research was partially supported by the National Natural Science Foundation of China (No. 10771141) and Leading Academic Discipline Project in Shanghai (No. T0401). The research of the second author was partially supported by a grant from the National Science Council.

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Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems

Abstract

Introduction

Section snippets

Preliminaries

Well-posedness and metric characterization

Links with well-posedness of inclusion problems

See [13], [14]

Links with well-posedness of fixed-point problems

See [13], [14]

See [13], [14]

Conditions for well-posedness

Conclusions

Acknowledgements

J. Math. Anal. Appl.

Nonlinear Anal. TMA

Comput. Math. Appl.

Metrically well-set minimization problems

Appl. Math. Optim.

Operateurs Maximaux Monotone et Semigroups de Contractions dans les Espaces de Hilbert

On the subdifferentiability of convex functions

Proc. Amer. Math. Soc.

Well-posed saddle point problems

New concepts of well-posedness for optimization problems with variational inequality constraints

JIPAM. J. Inequal. Pure Appl. Math.

Stability of new implicit iteration procedures for a class of nonlinear set-valued mixed variational inequalities

Z. Angew. Math. Mech.

Parametric well-posedness for variational inequalities defined by bifunctions

Comput. Math. Appl.

Existence of solutions of generalized vector variational inequalities in reflexive Banach spaces

J. Global Optim.

Extended and strong extended well-posedness of set-valued optimization problems

Math. Methods Oper. Res.

Iterative schemes for solving mixed variational-like inequalities

J. Optim. Theory Appl.

Well-posedness, conditioning, and regularization of minimization, inclusion, and fixed-point problems

Plisk. Stud. Math. Bulg.

Well-posedness by perturbations of variational problems

J. Optim. Theory Appl.