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

*Fixed Point Theory and Applications*
**volumeÂ 2011**, ArticleÂ number:Â 635030 (2011)

## 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.

## 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 [1â€“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

Throughout the paper, unless otherwise specified, we assume that is a real Banach space with its norm ,â€‰â€‰ is the topological dual of ,â€‰â€‰ is the pairing between and ,â€‰â€‰ is the metric induced by the norm ,â€‰â€‰ (resp., ) is the family of nonempty (resp., nonempty closed and bounded) subsets ofâ€‰â€‰,â€‰â€‰and is the Housdorff metric on defined by

whereâ€‰â€‰and. The normalized duality mappingâ€‰â€‰â€‰â€‰is defined by

Definition 2.1.

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

(i)the mappingâ€‰â€‰ is said to be accretive if

(ii)the mappingâ€‰â€‰â€‰â€‰is said to be strictly accretive if

and the equality hold if and only if ,

(iii)the mappingâ€‰â€‰â€‰â€‰is said to be -strongly accretive if for anyâ€‰â€‰, â€‰there existsâ€‰â€‰ such that

(iv)the mappingâ€‰â€‰â€‰â€‰is said to beâ€‰â€‰-stronglyâ€‰â€‰-accretive, if there exists a constant â€‰â€‰such that

(v)â€‰â€‰the mappingâ€‰â€‰â€‰â€‰is said to beâ€‰â€‰-relaxedâ€‰â€‰-accretive, if there exists a constant â€‰â€‰such that

Definition 2.2.

Letâ€‰,â€‰â€‰â€‰â€‰be the single-valued mappings. Then, a set-valued mapping is called -accretive ifâ€‰â€‰â€‰â€‰isâ€‰â€‰-relaxedâ€‰â€‰-accretiveâ€‰â€‰andâ€‰â€‰,â€‰â€‰for everyâ€‰â€‰.

Let be a real Banach space, and let be the normalized duality mapping. Then, for any

Definition 2.4.

Let ,â€‰â€‰,â€‰â€‰and letâ€‰â€‰ be the mappings. Then,

(i)the mapping is said to be Lipschitz continuous with constant if

(ii)the mappingâ€‰â€‰â€‰â€‰is said to be Lipschitz continuous in the first argument with constantâ€‰â€‰â€‰â€‰if

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

(iii)the mappingâ€‰â€‰â€‰â€‰is said to be Lipschitz continuous in the th argument with constantâ€‰â€‰ if

Definition 2.5.

Letâ€‰â€‰ be a strictly -accretive mapping, and let â€‰â€‰be an -accretive mapping. â€‰Then, the resolvent operatorâ€‰â€‰â€‰â€‰is defined by

Proposition 2.6 (see [10]).

Letâ€‰â€‰â€‰â€‰be a real Banach space, and letâ€‰â€‰â€‰â€‰beâ€‰â€‰-Lipschitz continuous; let â€‰â€‰be an -strongly -accretive mapping, and let â€‰â€‰be an -accretive mapping. Then the resolvent operatorâ€‰â€‰ is -Lipschitz continuous, that is,

whereâ€‰â€‰ â€‰is a constant.

Example 2.7.

Let , ,â€‰â€‰andâ€‰â€‰for all. Then, is -accretive.

Example 2.8.

Let â€‰â€‰be -stronglyâ€‰â€‰-accretive in the first argument. Then,â€‰â€‰â€‰â€‰is -relaxedâ€‰â€‰-accretive forâ€‰â€‰, for .

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

Find ,,,,,and such that

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

Special Cases

(i)Ifâ€‰â€‰,, â€‰â€‰then problem (2.14) reduces to the problem of finding ,, â€‰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.

Find , , ; , , such that

where is a constant and , where andâ€‰â€‰â€‰â€‰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 , ,â€‰â€‰,â€‰â€‰,â€‰â€‰, and .

We define for ,,, and .

(i),

(ii),

(iii),

(iv), for all ,

(v), for all ,

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.

Let where ,,,â€‰â€‰,, and , is a solution of (2.14) if and only if it is a solution of the following equation:

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

Lemma 3.2.

Let ,,,â€‰â€‰,,,then the following are equivalent:

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

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

where

Proof.

Let be a solution of the problem (2.14), then by Lemma 3.1, it is a solution of the problem

using the fact that

which implies that

with

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

Conversly, let be a solution of problem (2.17), then

from (3.2) and (3.7), we have

which implies that

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.

For a given ,,,â€‰â€‰,, and . Let

Take such that

Since for each , â€‰,, and by Nadler's theorem [14] there exist ,,, and such that

whereâ€‰â€‰â€‰â€‰is the Housdorff metric on .

Let

and take any such that

Continuing the above process inductively, we obtain the following.

For any ,,,â€‰â€‰,,and Compute the sequences, ,,â€‰â€‰,,, by the following iterative schemes:

where is a constant and .

Theorem 3.4.

Let be a real Banach space. â€‰Let beâ€‰â€‰-Lipschitz continuous mapping with constants ,, , , respectively. Let â€‰â€‰be Lipschitz continuous with constant , let be Lipschitz continuous with constants , ,, respectively, and letâ€‰â€‰â€‰â€‰be -stronglyâ€‰â€‰-accretive mapping. Suppose that are mappings such that is Lipschitz continuous with constant and is Lipschitz continuous in both the argument with constant and , respectively. Let be -accretive mapping in the first argument such that the following holds for :

Suppose there exists a such that

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

Proof.

From Algorithm 3.3, we have

By using the Lipschitz continuty of , , and with constants ,, and , respectively, and by Algorithm 3.3, we have

Since is Lipschitz continuous in all the arguments with constantâ€‰â€‰,, respectively,â€‰â€‰and usingâ€‰â€‰-Lipschitz continuity ofâ€‰â€‰'s with constantâ€‰â€‰, we have

Since is a Lipschitz continuous in both the arguments with constant , respectively, and and are -Lipschitz continuous with constant and , respectively, we have

Combining (3.24), (3.25), and (3.26) with (3.23), we have

By using Proposition 2.3 andâ€‰â€‰-strong accretiveness ofâ€‰â€‰, we have

Using (3.28), (3.27) becomes

where

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

Now, we prove that . In fact, since and

which implies thatâ€‰â€‰. â€‰As , we have , .

Finally, by the continuity ofâ€‰â€‰, , , ,â€‰â€‰and and by Algorithm 3.3, it follows that

From (3.32), and Lemma 3.2, it follows that

that is, is a solution of resolvent equation poblem (2.17).

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## Acknowledgment

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

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Ahmad, R., Dilshad, M. Variational-Like Inclusions and Resolvent Equations Involving Infinite Family of Set-Valued Mappings.
*Fixed Point Theory Appl* **2011**, 635030 (2011). https://doi.org/10.1155/2011/635030

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