Set-Valued Non-Linear Random Implicit Quasivariational Inclusions ()
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 and
here 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.