Skip to main content

Advertisement

We’d like to understand how you use our websites in order to improve them. Register your interest.

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

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 [17] 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 ,   is the topological dual of ,   is the pairing between and ,   is the metric induced by the norm ,   (resp., ) is the family of nonempty (resp., nonempty closed and bounded) subsets of  ,  and is the Housdorff metric on defined by

(21)

where  and. The normalized duality mapping    is defined by

(22)

Definition 2.1.

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

(i)the mapping   is said to be accretive if

(23)

(ii)the mapping    is said to be strictly accretive if

(24)

and the equality hold if and only if ,

(iii)the mapping    is said to be -strongly accretive if for any  ,  there exists   such that

(25)

(iv)the mapping    is said to be  -strongly  -accretive, if there exists a constant   such that

(26)

(v)  the mapping    is said to be  -relaxed  -accretive, if there exists a constant   such that

(27)

Definition 2.2.

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

Proposition 2.3 (see [8, 9]).

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

(28)

Definition 2.4.

Let ,  ,  and let   be the mappings. Then,

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

(29)

(ii)the mapping    is said to be Lipschitz continuous in the first argument with constant    if

(210)

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

(iii)the mapping    is said to be Lipschitz continuous in the th argument with constant   if

(211)

Definition 2.5.

Let   be a strictly -accretive mapping, and let   be an -accretive mapping.  Then, the resolvent operator    is defined by

(212)

Proposition 2.6 (see [10]).

Let    be a real Banach space, and let    be  -Lipschitz continuous; let   be an -strongly -accretive mapping, and let   be an -accretive mapping. Then the resolvent operator   is -Lipschitz continuous, that is,

(213)

where    is a constant.

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.

Find ,,,,,and such that

(214)

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

(215)

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

(216)

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

(217)

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 .

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 ,,,  ,, and , is a solution of (2.14) if and only if it is a solution of the following equation:

(31)

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),

where

(32)

Proof.

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

(33)

using the fact that

(34)

which implies that

(35)

with

(36)

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

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

(37)

from (3.2) and (3.7), we have

(38)

which implies that

(39)

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 ,,,  ,, and . Let

(310)

Take such that

(311)

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

(312)

where    is the Housdorff metric on .

Let

(313)

and take any such that

(314)

Continuing the above process inductively, we obtain the following.

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

(315)
(316)
(317)
(318)
(319)
(320)

where is a constant and .

Theorem 3.4.

Let be a real Banach space.  Let be  -Lipschitz continuous mapping with constants ,, , , respectively. Let   be Lipschitz continuous with constant , let be Lipschitz continuous with constants , ,, respectively, and let    be -strongly  -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 :

(321)

Suppose there exists a such that

(322)

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 Algorithm 3.3, we have

(323)

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

(324)

Since is Lipschitz continuous in all the arguments with constant  ,, respectively,  and using  -Lipschitz continuity of  's with constant  , we have

(325)

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

(326)

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

(327)

By using Proposition 2.3 and  -strong accretiveness of  , we have

(328)

Using (3.28), (3.27) becomes

(329)

where

(330)

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 .

Now, we prove that . In fact, since and

(331)

which implies that  .  As , we have , .

Finally, by the continuity of  , , , ,  and and by Algorithm 3.3, it follows that

(332)

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

(333)

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

References

  1. 1.

    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.2004.02.021

  2. 2.

    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)00028-8

  3. 3.

    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/S1025583402000425

  4. 4.

    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)00048-3

  5. 5.

    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.

  6. 6.

    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-X

  7. 7.

    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)00250-2

  8. 8.

    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.1997.5556

  9. 9.

    Xu HongKun: Inequalities in Banach spaces with applications. Nonlinear Analysis. Theory, Methods & Applications 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-K

  10. 10.

    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.2005.11.036

  11. 11.

    Wang Y-H: The infinite family of generalized set-valued quasi-variation inclusions in Banach spaces. Acta Analysis Functionalis Applicata 2008, 1009–1327.

  12. 12.

    Chang SS: Set-valued variational inclusions in Banach spaces. Journal of Mathematical Analysis and Applications 2000,248(2):438–454. 10.1006/jmaa.2000.6919

  13. 13.

    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.2001.7800

  14. 14.

    Nadler SB: Multivalued contraction mappings. Pacific Journal of Mathematics 1996, 30: 475–488.

Download references

Acknowledgment

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rais Ahmad.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

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