﻿Set-Valued Non-Linear Random Implicit Quasivariational Inclusions

Applied Mathematics
Vol.4 No.3(2013), Article ID:28828,8 pages DOI:10.4236/am.2013.43063

Set-Valued Non-Linear Random Implicit Quasivariational Inclusions

Department of Mathematics, Aligarh Muslim University, Aligarh, India

Received April 18, 2012; revised May 24, 2012; accepted June 2, 2012

Keywords: Set Valued Nonlinear Random Implicit Quasivariational Inclusions; Random Implicit Proximal Operator Equations; Equivalence Relations; Random Iterative Algorithms; Measurable Spaces; Separable Real Hilbert Spaces; Random Strongly Monotone; Random Relaxed Monotone; Random Lipschitz Continuous Mappings

ABSTRACT

In this paper, we propose iterative algorithms for set valued nonlinear random implicit quasivariational inclusions. We define the related random implicit proximal operator equations and establish an equivalence between them. Finally, we prove the existence and convergence of random iterative sequences generated by random iterative algorithms.

1. Introduction

The theory of variational inequality provides a natural and elegant framework for study of many seemingly unrelated free boundary value problems arising in various branches of engineering and mathematical sciences. Variational inequalities have many deep results dealing with nonlinear partial differential equations which play important and fundamental role in general equilibrium theory, economics, management sciences and operations research, see [1-3].

The quasi variational inequalities have been introduced by Bensoussan and Lions [1] and closely related to contact problems with friction in electrostatics and nonlinear random equations frequently arise in biological, physical and system sciences [4,5]. With the emergence of probabilistics functional analysis, the study of random operators became a central topic of this discipline [4,5]. The theory of resolvent operators introduced by Brezis [2] is closely related to the variational inequality problems; for applications we refer to [6-8].

Motivated by recent research work on random variational inequalities [9-14], in this paper we consider a class of set valued nonlinear random implicit quasi variational inclusions and a class of random proximal operator equations and establish an equivalence between them. We use the equivalence to suggest and analyzed some iterative algorithms for finding approximate solution of (1). Further we prove the existence of solution of this class of problem and discuss the convergence of iterative sequences generated by these random iterative algorithms.

2. Preliminaries

Let be a measurable space and a separable real Hilbert space with inner product and norm. We denote, and the class of Borel field in family of all nonempty power subsets of and the family of all nonempty compact subsets of respectively.

Definition 1. A mapping is called measurable if for any,.

Definition 2. A mapping is called a random operator if for any is measurable.

Definition 3. A random operator T is said to be continuous if for any, the mapping is continuous.

Given random set valued mappings

and are the single valued mappings. Let be a random set valued mapping such that for each fix, , is a maximal monotone mapping with

Throughout this paper, we will consider the following set valued nonlinear random implicit quasivariational inclusions for finding,

, , ,

, such that

(1)

where be the bifunction.

In deterministic case, the problem (1) is equivalent to the problem of the Ding and Park [15].

For a suitable choice of the operators A, g, m, f, N, T, V, G, P and E a number of known classes of variational inequalities, quasivariational inclusions can be obtained as special cases of problem, studied previously by many authors including Hassouni and Moudafi [16], Huang [17], Uko [18], Verma [13], Salahuddin [19], and Salahuddin and Ahmad [20].

Definition 4. If A is maximal monotone operator on H, then for a given constant the proximal operator associated with A is defined by

where I is the identity operator. It is also known that the operator A is maximal monotone if and only if the proximal operator JA is defined everywhere on the space. Furthermore, the proximal operator JA is single-valued and nonexpansive, i.e. for all,

Remark 1. Since the operator is a maximal monotone operator with respect to the first argument, we define

the generalized proximal operator associated with

.

Related to the problem (1), we consider the problem of finding, , ,

, , ,

such that

(2)

where be the measurable function and

.

Here stand for an identity operator and

is the random proximal operator. Equation of the type (2) is called random implicit proximal operator equations.

3. Random Iterative Algorithm

In this section, we prove results which will establish eqivalence between the problems (1) and (2). Then we construct a number of iterative algorithms for solving problem (1).

Lemma 1. If, , ,

, , ,

is a random solution set of problem (1) if and only if, , ,

, , ,

such that

(3)

where and are two measurable functions andhere I stand for identity functions.

Proof. Let, , ,

, , ,

be the random solution set of (1). Then for a given measurable function,

where be the measurable mapping. This completes the proof.

Theorem 1. The random problem (1) has a random solution set, , ,

, , ,

if and only if random problem (2) has a random solution set, ,

, , ,

, where

(4)

and

(5)

Proof. Let, , ,

, , ,

be a random solution set of (1). Then for a measurable function,

Take,

Then,

Thus,

That is,

where, completing the proof of Theorem 1.

Theorem 1 implies that random problems (1) and (2) are equivalent, which allows us to suggest a number of iterative algorithms for solving problem (1). For a suitable rearrangement of the terms of the random Equation (2), we suggest the following algorithms:

1) The random problem (2) can be written as

which implies that

This random fixed point formulation allow us to suggest the following random iterative algorithm.

Algorithm 1. For given, , , ,

, ,

, compute, ,

, , , and by random iterative schemes,

(6)

(7)

2) The random problem (5) can be written as

This random fixed point formulation is used to suggest the following algorithm:

Algorithm 2. For given, , ,

, ,

, compute, ,

, , , and by random iterative schemes,

(8)

(9)

4. Main Result

First we recall the following well known concepts.

Definition 5. A random mapping is said to be 1) random strongly monotone, if there exists a measurable function such that

2) random Lipschitz continuous, if there exists a measurable function such that

Definition 6. A random mapping is said to be random relaxed monotone with respect to the first argument of a random mapping

, if there exists a measurable function such that

for all .

Definition 7. A random mapping is said to be random relaxed Lipschitz continuous with respect to the second argument of random mapping if there exists a measurable mapping such that

for all .

Definition 8. A random set-valued mapping is said to be -Lipschitz continuous if there exists a measurable function such that

where is a Hausdorff metric on.

Theorem 2. Let the random bifunction be random Lipschitz continuous with respect to first and second arguments with measurable functions respectively and random set valued mappings be the random -Lipschitz continuous with random coefficients respectively. Let the random mappings be random Lipschitz continuous with random coefficients respectively and the random mapping g is the random strongly monotone with respect to the measurable map. A random bifunction is randomly relaxed monotone with respect to the first argument of with random coefficient and relaxed Lipschitz continuous with respect to the second argument of with random coefficient. Let be such that for each fixed, , be a random maximal monotone satisfying

for all

. Suppose that for any fix, ,

(10)

(11)

where be the measurable map, then there exists such that

(12)

where, ,

,

Then there exist, ,

, , ,

, satisfying (2) and (5) and random sequences, , , , ,

and generated by Algorithm 1, converge strongly to, , , , , and in for each.

Proof. From Algorithm 1, we have

(13)

Since N is randomly Lipschitz continuous with respect to first and second argument and V and are random - Lipschitz continuous, we have

(14)

Again are randomly Lipschitz continuous and are random -Lipschitz continuous, we have

(15)

(16)

(17)

(18)

Since is random strongly monotone and from (18), we have

(19)

Since is random relaxed monotone with respect to first argument and random relaxed Lipschitz continuous with second argument, we have

(20)

From (13)-(15), (19) and (20), we have

(21)

where and

Also from (6), (10), (11), (15) and (19), we obtain

which implies that

(22)

Adding (21) and (22), we obtain

(23)

where,

From (12), it follows that for each. Consequently from (23), we see that the random sequence is a Cauchy sequence in for each, that is there exists for fix with as. From (22), we know that the random sequence is a Cauchy sequence in

that is there exists with for each fix. Also from the random Lipschitz continuity, we have

which implies that the random sequences, , , and are Cauchy sequences in Assume that

, ,

, and

as, for each fix.

Now by using the random continuity of the random operators, and Algorithm 1, we have

Now we show that In fact

where,

Since the random sequences and are Cauchy sequences, it follows from the above inequality that. This implies that

. In a similar way, we can show that

, ,

and.

By Theorem 1, it follows that, ,

, , ,

and, which satisfies the inequality (1) and for fix, ,

, , ,

, and

strongly in the required result.

REFERENCES

1. C. Baiocchi and A. Capelo, “Variational and Quasi-Variational Inequalities,” John Wiley and Sons, New York, 1984.
2. H. Brezis, “Operateurs Maximmaux Monotones,” North Holland, Amsterdam, 1973.
3. F. Giannessi and A. Maugeri, “Variational Inequalities and Network Equilibrium Problems,” Plenum Press, New York, 1995.
4. C. P. Toskos and V. J. Padgett, “Random Integral Equations with Applications to Stochastic Systems,” SpringerVerlag, Berlin and New York, 1971.
5. C. P. Toskos and V. J. Padgett, “Random Integral Equations with Applications in Life Sciences and Engineering,” Academic Press, New York, 1974.
6. M. A. Noor, “Some Recent Advances in Variational Inequalities II,” New Zealand Journal of Mathematics, Vol. 26, No. 2, 1997, pp. 229-255.
7. M. A. Noor, “Generalized Set-Valued Variational Inclusions and Resolvent Equations,” Journal of Mathematical Analysis and Application, Vol. 228, No. 1, 1998, pp. 206- 220. doi:10.1006/jmaa.1998.6127
8. S. H. Shim, S. M. Kang, N. J. Huang and Y. J. Cho, “Generalized Set-Valued Strongly Nonlinear Quasi-Variational Inclusions,” Indian Journal of Pure Applied Mathematics, Vol. 31, No. 9, 2000, pp. 1113-1122.
9. S. S. Chang and N. J. Huang, “Generalized Random Multivalued Quasi Complementarity Problems,” Indian Journal of Mathematics, Vol. 35, No. 3, 1993, pp. 305-320.
10. N. J. Huang, X. Long and Y. J. Cho, “Random Generalized Nonlinear Variational Inclusions,” Bulletin of Korean Mathematical Society, Vol. 34, No. 4, 1997, pp. 603-615.
11. T. Hussain, E. Tarafdar and X. Z. Yuan, “Some Results on Random Generalized Games and Random Quasi-Variational Inequalities,” Far East Journal Mathematical Society, Vol. 2, No. 1, 1994, pp. 35-55.
12. N. X. Tan, “Random Quasi-Variational Inequalities,” Mathematische Nachrichten, Vol. 125, 1986, pp. 319-328.
13. R. U. Verma, “On Generalized Variational Inequalities Involving Relaxed Lipschitz and Relaxed Monotone Operators,” Journal of Mathematical Analysis and Application, Vol. 23, 1997, pp. 389-392.
14. X. Z. Yuan, “Non Compact Random Generalized Games and Random Quasi-Variational Inequalities,” Journal of Applied Stochastic Analysis, Vol. 7, No. 4, 1994, pp. 467- 486. doi:10.1155/S1048953394000377
15. X. P. Ding and Y. J. Park, “A New Class of Generalized Nonlinear Implicit Quasi-variational Inclusions with Fuzzy Mappings,” Journal of Computer and Applied Mathematics, Vol. 138, No. 2, 2002, pp. 243-257. doi:10.1016/S0377-0427(01)00379-X
16. A. Hassouni and A. Moudafi, “A Perturbed Algorithm for Variational Inclusions,” Journal of Mathematical Analysis and Application, Vol. 185, No. 3, 1994, pp. 706-712. doi:10.1006/jmaa.1994.1277
17. N. J. Huang, “Generalized Nonlinear Variational Inclusions with Noncompact Valued Mappings,” Applied Mathematics Letter, Vol. 9, No. 3, 1996, pp. 25-29. doi:10.1016/0893-9659(96)00026-2
18. L. U. Uko, “Strongly Nonlinear Generalized Equations,” Journal of Mathematical Analysis and Application, Vol. 220, No. 2, 1998, pp. 65-76. doi:10.1006/jmaa.1997.5796
19. Salahuddin, “Some Aspects of Variational Inequalities,” Ph.D. Thesis, Aligarh Muslim University, Aligarh, 2000.
20. Salahuddin and M. K. Ahmad, “On Generalized Multivalued Random Variational Like Inclusions,” Applied Mathematics, Vol. 2, No. 8, 2011, pp. 1011-1018. doi:10.4236/am.2011.28140