Sensitivity analysis for a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces

doi:10.1016/j.amc.2009.05.043

Applied Mathematics and Computation

Volume 215, Issue 2, 15 September 2009, Pages 716-726

https://doi.org/10.1016/j.amc.2009.05.043 Get rights and content

Abstract

In this paper, using the concept of $P$ - $η$ -proximal-point mapping introduced by Kazmi and Bhat [11], we study the existence and sensitivity analysis of the solution set of a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameters. The approach used in this paper may be treated as an extension and unification of approaches for studying sensitivity analysis for various important classes of variational inequalities given in [1,2,4,12,14–16,21–24].

Introduction

In recent years, much attention has been given to develop general for the sensitivity analysis of solution set of various classes of variational inequality (inclusion) problems. From the mathematical and engineering point of view, sensitivity properties of various classes of variational inequality problems can provide new insight concerning the problems being studied and can stimulate ideas for solving problems. The sensitivity analysis of solution set for variational inequality (inclusion) problems has been studied extensively by many authors using quite different methods. By using the projection technique, Defermos [4], Mukherjee and Verma [19], Noor [21] and Yen [26] studied the sensitivity analysis of solution of some classes of variational inequality problems with single-valued mappings. By using the implicit function approach that makes use of normal mappings, Robinson [25] studied the sensitivity analysis of solutions for variational inequality problems in finite-dimensional spaces. By using proximal-point (resolvent) mapping technique, Adly [1] and Agarwal et al. [2] studied the sensitivity analysis of solution of some classes of quasi-variational inclusions with single-valued mappings. Also, by using projection and proximal-point techniques, Ding and Luo [7], Liu et al. [18], Park and Jeong [23] and Ding [6], studied the behaviour and sensitivity analysis of solution set for some classes of generalized variational inequality (inclusion) problems with set-valued mappings. It is worth mentioning that most of the results in this direction have been obtained in the setting of Hilbert space.

Recently, Kazmi and Khan [12], [14], [15] and Kazmi et al. [16] have studied behavior and sensitivity analysis of solution set of system of some classes of variational inequality (inclusion) problems by using $P$ - $η$ -proximal-point mappings in the setting of Banach spaces involving set-valued mappings.

Motivated by recent research work in this direction, we consider a system of parametric general quasi-variational-like inequality problems (in short, SPGQVLIP) in uniformly smooth Banach spaces. Further by using, $P$ - $η$ -proximal point mappings introduced by Kazmi and Bhat [11], we study the existence and sensitivity analysis of the solution set of SPGQVLIP. Furthermore, the Lipschitz continuity of the solution set of SPSQVLIP is proved under some suitable conditions. The theorems presented in this paper improve and generalize the corresponding results of [1], [2], [4], [12], [14], [15], [16], [21], [24].

Section snippets

Preliminaries

Let $E$ be a real Banach space equipped with the norm $‖ \cdot ‖$ . Let $〈 \cdot, \cdot 〉$ denote the dual pair between $E$ and its dual space $E^{*}$ ; let $C (E)$ denote the family of all nonempty compact subsets of $E$ ; let $2^{E}$ denote the power set of $E$ ; let $H (\cdot, \cdot)$ be the Hausdorff metric on $C (E)$ defined by $H (A, B) = \max \{\sup_{x \in A} \inf_{y \in B} d (x, y), \sup_{y \in B} \inf_{x \in A} d (x, y)\}, A, B \in C (E),$ and let $J : E \to 2^{E^{*}}$ be the normalized duality mapping defined by $J (u) = \{f \in E^{*}, 〈 f, u 〉 = ‖ u ‖^{2}, ‖ u ‖ = ‖ f ‖_{E^{*}}\}, \forall u \in E .$ It is well known that if $E$ is smooth, then $J$ is single-valued and if $E \equiv H$ ,

System of parametric generalized quasi-variational inequality problems

Throughout rest of this paper, unless otherwise stated, for each $i = 1, 2$ , we assume that $E_{i}$ is uniformly smooth Banach space with norm $‖ \cdot ‖_{i}$ , and denote the duality pairing between $E_{i}$ and its dual $E_{i}^{*}$ by $〈 \cdot, \cdot 〉_{i}$ .

Let $Ω_{1}$ and $Ω_{2}$ be nonempty open subsets of $E_{1}$ and $E_{2}$ , respectively, in which parameters $λ$ and $μ$ takes the values; let $g_{i} : E_{i} \times Ω_{i} \to E_{i}$ ; $η_{i} : E_{i} \times E_{i} \to E_{i}$ be single-valued mappings such that $g_{i} \neq 0$ and $g_{1} (x, λ) \in dom η_{1} (m_{1}, \cdot)$ and $g_{2} (y, μ) \in dom η_{2} (m_{2}, \cdot)$ for all $m_{1} \in E_{1}, m_{2} \in E_{2}$ ; let $A, C : E_{1} \times Ω_{1} \to E_{1}^{★}$ and $B, D : E_{2} \times Ω_{2} \to E_{2}^{★}$ be

Sensitivity analysis of solution set $S (λ, μ)$

Now, we shall study the behavior and sensitivity analysis of the solution set of SPGQVLIP (3.1) and further, under suitable conditions, we shall discuss Lipschitz continuity of the solution set with respect to the parameters.

Theorem 4.1

For each $i = 1, 2$ , let $E_{i}$ be uniformly smooth Banach space with $ρ_{E_{i}} (t) ⩽ c_{i} t^{2}$ for some $c_{i} > 0$ ; let the mappings $P_{i} : E_{i} \to E_{i}^{★}$ and $g_{i} : E_{i} \times Ω_{i} \to E_{i}$ such that $g_{i}$ is $s_{i}$ -strongly accretive and $(L_{g_{i}}, l_{g_{i}})$ -mixed Lipschitz continuous and $P_{i} \circ g_{i}$ be $(L_{P_{i} \circ g_{i}}, l_{P_{i} \circ g_{i}})$ -mixed Lipschitz continuous; let the

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¹: Supported by NBHM, Department of Atomic Energy, Government of India under Grants-in-aid for Post-doctoral fellowship (Reference no. 40/2/2007-R&D II/3910).

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Sensitivity analysis for a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces

Abstract

Introduction

Section snippets

Preliminaries

System of parametric generalized quasi-variational inequality problems

Sensitivity analysis of solution set S(λ,μ)

J. Math. Anal. Appl.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

Appl. Math. Comput.

Nonlinear Anal.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Comput. Math. Appl.

J. Math. Anal. Appl.

Appl. Math. Lett.

Comput. Math. Appl.

Sensitivity analysis of solution set $S (λ, μ)$