Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 258543, 6 pages

http://dx.doi.org/10.1155/2013/258543

Research Article

Sensitivity Analysis for Nonlinear Set-Valued Variational Equations in Banach Framework

A. Farajzadeh¹ and Salahuddin²

¹Department of Mathematics, Razi University, Kermanshah 67149, Iran

²Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Received 2 May 2013; Accepted 30 August 2013

Academic Editor: Nelson Merentes

Copyright © 2013 A. Farajzadeh and Salahuddin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is to study the sensitivity analysis for nonlinear set-valued variational equations based on -resolvent operator technique. The obtained results encompass a broad model of results.

1. Introduction

The set-valued inclusions problem, which was initiated by Di Bella [1] and Huang et al. [2, 3], is a useful extension of the mathematical analysis and variational equation and is important context in the set-valued equations. Verma [4], Huang [5], Fang and Huang [6], Lan et al. [7], Khan and Salahuddin [8, 9], Li et al. [10], and Yen and Lee [11] introduced the concepts of -monotone, -monotone, -monotone operator, -accretive mappings, and resolvent operator associated with them, respectively. Dafermos [12] studied the sensitivity property of solutions of particular kinds of variational inequality on parameter which takes values on an open subset of Euclidean space . Tobin [13], Verma [14, 15], Lee and Salahuddin [3], Kyparisis [16], Moudafi [17], Noor [18], Robinson [19], Yen and Lee [11], and Hussain et al. [20] studied the sensitivity analysis of various types of variational inequalities.

In this paper, we present the sensitivity analysis for -accretive variational equations based on the -resolvent operator technique. The obtained results generalize a wide range of results on the sensitivity analysis for nonlinear set-valued variational equations in Banach spaces.

2. Preliminaries

Let be a real Banach space with dual space , and let be the dual pair between and , denote the family of all the nonempty subset of and CB() the family of all nonempty closed bounded subsets of . The generalized duality mapping is defined by

where is constant.

The modulus of smoothness of is the function defined by

A Banach space is called uniformly smooth if

is called -uniformly smooth if there exists a constant such that

Let be a single-valued mapping. The map is called -Lipschitz continuous if there exists a constant such that

Definition 1. Let be a single-valued mapping, and let be a mapping on . A set-valued mapping is said to be

(i)accretive if

(ii)-accretive if

(iii)-strongly accretive if

(iv)-pseudomonotone if

implying

(v)-relaxed accretive if there exists a positive constant such that

Definition 2. A single-valued mapping is said to be

(i)accretive if

(ii)strictly accretive if is accretive and

(iii)-strongly accretive if there exists a constant such that

(iv)-Lipschitz continuous if there exists a constant such that

Definition 3. A single-valued mapping is said to be

(i)-Lipschitz continuous if there exist constants such that

(ii)-relaxed cocoercive with respect to in the second argument if there exists constant and for all , , such that

where are single-valued mappings.

Definition 4. A mapping is said to be maximal -relaxed accretive if

(i) is -accretive; (ii)for and

one has .

Definition 5. Let and be two single-valued mappings; the mapping is said to be -accretive if

(i) is -relaxed accretive; (ii) for .

Note that alternatively, the mapping is said to be -accretive if

(i) is -relaxed accretive; (ii) is -pseudoaccretive for .

The following propositions will be needed in the sequel. For more details one can refer to [7, 10, 15].

Proposition 6. Let be an -strongly accretive single-valued mapping, and let be an -accretive mapping. Let be -Lipschitz continuous single-valued mapping. Then, is maximal -relaxed accretive, and for .

Proposition 7. Let be an -strongly accretive single-valued mapping, and let be an -accretive mapping. In addition, let be -Lipschitz continuous. Then is maximal -accretive for .

Proposition 8. Let be an -strongly accretive mapping, and let be an -accretive mapping. If, in addition, is -Lipschitz continuous, then, the operator is single-valued for .

Definition 9. Let be a single-valued mapping. Let be an -strongly accretive mapping, and let be an -accretive mapping. Then, the generalized resolvent operator is defined by

where is a constant.

Definition 10. The mapping is said to be -strongly accretive with respect to in the first argument if there exists a positive constant such that

3. Sensitivity Analysis

Let , let , and let be three set-valued mappings, and let a nonlinear mapping be an -accretive mapping with respect to first argument, and let be the mapping, where is a nonempty open subset of . Let be the single-valued mappings. Furthermore, let be a nonlinear mapping. Then, the problem of finding an element , , , such that

where is the set-valued parameter is called a generalized set-valued variational inclusions.

The solvability of problem (21) depends on the equivalence between (21) and the problem of finding the fixed point of the associated resolvent operator.

Note that if is -accretive, then the corresponding resolvent operator defined by

where and is an -strongly accretive mapping.

Lemma 11 (see [7, 10, 15]). Let be a real Banach spaces, and let be a -Lipschitz continuous nonlinear mapping. Let be -strongly accretive mapping, and let be -accretive in the first variable with . Then, the resolvent operator associated with for a fixed , defined by

is -Lipschitz continuous, where ; that is,

Lemma 12 (see [7, 10, 15]). Let be a real uniformly smooth Banach spaces, let be the -strongly accretive mapping, and let be an -accretive in the first variable. Let be a -Lipschitz continuous nonlinear mapping; then the following statements are mutually equivalent:

(i)an element , ,  , is a solution to (21);(ii)the mapping defined by

has a fixed point.

Lemma 13 (see [21]). Let be a real uniformly smooth Banach spaces. Then, is -uniformly smooth if and only if there exists a constant such that for all

Theorem 14. Let be a -uniformly smooth Banach spaces, and let be a -Lipschitz continuous nonlinear mapping. Let be the -strongly accretive, and let -Lipschitz continuous mapping and be the -accretive in the first variable with . Let be the set-valued mappings and -Lipschitz continuous with , , and constants, respectively. Let be the -Lipschitz continuous and -Lipschitz continuous mappings. Let be the -Lipschitz continuous with first variable and -Lipschitz continuous in the second variable. Let be the -relaxed cocoercive with respect to in the first argument. Let be the strongly accretive mapping. If the following condition holds:

where is the same as in Lemma 13 and . Consequently, for each , the mapping in the light (28) has a unique fixed point . Hence, in light of Lemma 12, is a unique solution to (21). Thus, one has

Proof. For any elements and , , , , , and , we have

Since , , and are -Lipschitz continuous and is Lipschitz continuous with respect to first and second arguments, , are Lipschitz continuous, we have

Since is -relaxed cocoercive with respect to in the first argument and is -Lipschitz continuous mapping, so we obtain

Since is -strongly accretive and is the -Lipschitz continuous, we have

Combining (31)–(34), we can get

where

It follows from (29) such that ; it concludes the proof.

Theorem 15. Let be a real -uniformly smooth Banach spaces, and let be a -Lipschitz continuous nonlinear mapping. Let be a -strongly accretive and -Lipschitz continuous mapping, and let be -accretive in the first variable with . Let , , and be the set-valued mappings and -Lipschitz continuous with respect to , , and constants, respectively. Let be the -Lipschitz continuous and -Lipschitz continuous mappings, respectively. Let be the -Lipschitz continuous with first variable and -Lipschitz continuous with second variable. Let be the -relaxed cocoercive with respect to in the first variable. Let be the -strongly accretive mapping if the following condition holds:

where is the same as in Lemma 13 and .

If the mapping , , and , , , , and are continuous (or Lipschitz continuous) from to , then the solution of (21) is continuous (or Lipschitz continuous) from to .

Proof. From the hypothesis of theorem, for any , we have

It follows that

Hence, we get

This completes the proof.

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