- Research Article
- Open Access

# Variational-Like Inclusions and Resolvent Equations Involving Infinite Family of Set-Valued Mappings

- Rais Ahmad
^{1}Email author and - Mohd Dilshad
^{1}

**2011**:635030

https://doi.org/10.1155/2011/635030

© Rais Ahmad and Mohd Dilshad. 2011

**Received:**18 December 2010**Accepted:**23 December 2010**Published:**29 December 2010

## Abstract

We study variational-like inclusions involving infinite family of set-valued mappings and their equivalence with resolvent equations. It is established that variational-like inclusions in real Banach spaces are equivalent to fixed point problems. This equivalence is used to suggest an iterative algorithm for solving resolvent equations. Some examples are constructed.

## Keywords

- Variational Inequality
- Iterative Algorithm
- Real Banach Space
- Lipschitz Continuity
- Variational Inclusion

## 1. Introduction

The important generalization of variational inequalities, called variational inclusions, have been extensively studied and generalized in different directions to study a wide class of problems arising in mechanics, optimization, nonlinear programming, economics, finance and applied sciences, and so forth; see, for example [1–7] and references theirin. The resolvent operator technique for solving variational inequalities and variational inclusions is interesting and important. The resolvent operator technique is used to establish an equivalence between variational inequalities and resolvent equations. The resolvent equation technique is used to develop powerful and efficient numerical techniques for solving various classes of variational inequalities (inclusions) and related optimization problems.

In this paper, we established a relationship between variational-like inclusions and resolvent equations. We propose an iterative algorithm for computing the approximate solutions which converge to exact solution of considered resolvent equations. Some examples are constructed.

## 2. Formulation and Preliminaries

Definition 2.1.

Let be a real Banach space. Let ; be the single-valued mapping, and let be a set-valued mapping. Then,

and the equality hold if and only if ,

Definition 2.2.

Let , be the single-valued mappings. Then, a set-valued mapping is called -accretive if is -relaxed -accretive and , for every .

Definition 2.4.

Let , , and let be the mappings. Then,

Similarly, we can define Lipschitz continuity in the second argument.

Definition 2.5.

Proposition 2.6 (see [10]).

Example 2.7.

Let , , and for all . Then, is -accretive.

Example 2.8.

Let be -strongly -accretive in the first argument. Then, is -relaxed -accretive for , for .

Let , be an infinite family of set-valued mappings, and let be a nonlinear mapping. Let ; be single-valued mappings, let and be set-valued mappings. Suppose that is -accretive mapping in the first argument. We consider the following problem.

The problem (2.14) is called variational-like inclusions problem.

Special Cases

(i)If , , then problem (2.14) reduces to the problem of finding , , such that

Problem (2.15) is introduced and studied by Wang [11].

(ii)If , , , then problem (2.14) reduces to a problem considered by Chang, et al. [12, 13] that is, find , , such that

It is now clear that for a suitable choice of maps involved in the formulation of problem (2.14), we can drive many known variational inclusions considered and studied in the literature.

In connection with problem (2.14), we consider the following resolvent equation problem.

where is a constant and , where and is the identity mapping. Equation (2.17) is called the resolvent equation problem.

In support of problem (2.17), we have the following example.

Example 2.9.

Let us suppose that , , , , , and .

Then, for , it is easy to check that the resolvent equation problem (2.17) is satisfied.

## 3. An Iterative Algorithm and Convergence Result

We mention the following equivalence between the problem (2.14) and a fixed point problem which can be easily proved by using the definition of resolvent operator.

Lemma 3.1.

Now, we show that the problem (2.14) is equivalent to a resolvent equation problem.

Lemma 3.2.

Let , , , , , , then the following are equivalent:

(i) is a solution of variational inclusion problem (2.14),

(ii) is a solution of the problem (2.17),

Proof.

that is, is a solution of problem (2.17).

that is, is a solution of (2.14).

We now invoke Lemmas 3.1 and 3.2 to suggest the following iterative algorithm for solving resolvent equation problem (2.17).

Algorithm 3.3.

where is the Housdorff metric on .

Continuing the above process inductively, we obtain the following.

Theorem 3.4.

Then, there exist , , and , , and that satisfy resolvent equation problem (2.17). The iterative sequences , , , , and , , generated by Algorithm 3.3 converge strongly to , , , , , , respectively.

Proof.

From (3.22), we have , and consequently is a Cauchy sequence in . Since is a Banach space, there exists such that . From (3.28), we know that is also a Cauchy sequence in . Therefore, there exists such that . Since the mappings 's, , and are -Lipschitz continuous, it follows from (3.16)–(3.19) of Algorithm 3.3 that , , , and are also Cauchy sequences. We can assume that , , , and .

## Declarations

### Acknowledgment

This work is supported by Department of Science and Technology, Government of India, under Grant no. SR/S4/MS: 577/09.

## Authors’ Affiliations

## References

- Ahmad R, Siddiqi AH, Khan Z:
**Proximal point algorithm for generalized multivalued nonlinear quasi-variational-like inclusions in Banach spaces.***Applied Mathematics and Computation*2005,**163**(1):295–308. 10.1016/j.amc.2004.02.021MATHMathSciNetView ArticleGoogle Scholar - Ahmad R, Ansari QH:
**An iterative algorithm for generalized nonlinear variational inclusions.***Applied Mathematics Letters*2000,**13**(5):23–26. 10.1016/S0893-9659(00)00028-8MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Huang NJ, Cho YJ:
**Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings.***Journal of Inequalities and Applications*2002,**7**(6):807–828. 10.1155/S1025583402000425MATHMathSciNetGoogle Scholar - Agarwal RP, Cho YJ, Huang NJ:
**Sensitivity analysis for strongly nonlinear quasi-variational inclusions.***Applied Mathematics Letters*2000,**13**(6):19–24. 10.1016/S0893-9659(00)00048-3MATHMathSciNetView ArticleGoogle Scholar - Chang S-S, Cho YJ, Zhou H:
*Iterative Methods for Nonlinear Operator Equations in Banach Spaces*. Nova Science Publishers, Huntington, NY, USA; 2002:xiv+459.MATHGoogle Scholar - Chen J-Y, Wong N-C, Yao J-C:
**Algorithm for generalized co-complementarity problems in Banach spaces.***Computers & Mathematics with Applications*2002,**43**(1–2):49–54. 10.1016/S0898-1221(01)00270-XMATHMathSciNetView ArticleGoogle Scholar - Ding XP, Luo CL:
**Perturbed proximal point algorithms for general quasi-variational-like inclusions.***Journal of Computational and Applied Mathematics*2000,**113**(1–2):153–165. 10.1016/S0377-0427(99)00250-2MATHMathSciNetView ArticleGoogle Scholar - Verma RamU:
**On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators.***Journal of Mathematical Analysis and Applications*1997,**213**(1):387–392. 10.1006/jmaa.1997.5556MATHMathSciNetView ArticleGoogle Scholar - Xu HongKun:
**Inequalities in Banach spaces with applications.***Nonlinear Analysis. Theory, Methods & Applications*1991,**16**(12):1127–1138. 10.1016/0362-546X(91)90200-KMATHMathSciNetView ArticleGoogle Scholar - Lan H-Y, Cho YJ, Verma RU:
**Nonlinear relaxed cocoercive variational inclusions involving****-accretive mappings in Banach spaces.***Computers & Mathematics with Applications*2006,**51**(9–10):1529–1538. 10.1016/j.camwa.2005.11.036MATHMathSciNetView ArticleGoogle Scholar - Wang Y-H:
**The infinite family of generalized set-valued quasi-variation inclusions in Banach spaces.***Acta Analysis Functionalis Applicata*2008, 1009–1327.Google Scholar - Chang SS:
**Set-valued variational inclusions in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**248**(2):438–454. 10.1006/jmaa.2000.6919MATHMathSciNetView ArticleGoogle Scholar - Chang SS, Kim JK, Kim KH:
**On the existence and iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces.***Journal of Mathematical Analysis and Applications*2002,**268**(1):89–108. 10.1006/jmaa.2001.7800MATHMathSciNetView ArticleGoogle Scholar - Nadler SB:
**Multivalued contraction mappings.***Pacific Journal of Mathematics*1996,**30:**475–488.MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.