Set-Valued Non-Linear Random Implicit Quasivariational Inclusions

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.


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][2][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][7][8].
Motivated by recent research work on random variational inequalities [9][10][11][12][13][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.

Preliminaries
Let   ,   be a measurable space and H a separable real Hilbert space with inner product .,. and norm . .We denote   Throughout this paper, we will consider the following set valued nonlinear random implicit quasivariational inclusions for finding t   ,   , 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 0   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 J A is defined everywhere on the space.Furthermore, the proximal operator J A is single-valued and nonexpansive, i.e. for all , x y H  ,
is a maximal monotone operator with respect to the first argument, we define the generalized proximal operator associated with where be the measurable function and Here I stand for an identity operator and is the random proximal operator.Equation of the type ( 2) is called random implicit proximal operator equations.

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).
here I stand for identity functions.
 be the random solution set of (1).Then for a given measurable function

A v t t A v t t u t t J g t x t m t u t t f t w t N a t b t t g t x t t m t u t t J g t x t m t u t t f t w t N a t b t t t g t x t
where be the measurable mapping.This completes the proof. and  be a random solution set of (1).Then for a measurable function , .
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 This random fixed point formulation allow us to suggest the following random iterative algorithm.
2) The random problem (5) can be written as .
This random fixed point formulation is used to suggest the following algorithm: .

Main Result
First we recall the following well known concepts.Definition 5. A random mapping : g H H    is said to be 1) random strongly monotone, if there exists a measurable function such that , , for all , , ,     be the measurable map, then there exists ,  where, satisf ying ( 2) and ( 5) and random se- Since N is randomly Lipschitz continuous with respect to first and second argument and V and are random -Lipschitz continuous, we have .