- Research Article
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Mixed Variational-Like Inequality for Fuzzy Mappings in Reflexive Banach Spaces
Journal of Inequalities and Applications volume 2009, Article number: 209485 (2009)
Abstract
Some existence theorems for the mixed variational-like inequality for fuzzy mappings (FMVLIP) in a reflexive Banach space are established. Further, the auxiliary principle technique is used to suggest a novel and innovative iterative algorithm for computing the approximate solution. Consequently, not only the existence of solutions of the FMVLIP is shown, but also the convergence of iterative sequences generated by the algorithm is also proven. The results proved in this paper represent an improvement of previously known results.
1. Introduction
The concept of fuzzy set theory was introduced by Zadeh [1]. The applications of the fuzzy set theory can be found in many branches of mathematical and engineering sciences including artificial intelligence, control engineering, management sciences, computer science, and operations research [2]. On the other hand, the concept of variational inequality was introduced by Hartman and Stampacchia [3] in early 1960s. These have been extended and generalized to study a wide class of problems arising in mechanics, physics, optimization and control, economics and transportation equilibrium, and so forth. The generalized mixed variational-like inequalities, which are generalized forms of variational inequalities, have potential and significant applications in optimization theory [4, 5], structural analysis [6], and economics [4, 7]. Motivated and inspired by the recent research work going on these two different fields, Chang [8], Chang and Huang [9], Chang and Zhu [10] and Noor [11] introduced and studied the concept of variational inequalities and complementarity problems for fuzzy mappings in different contexts.
It is noted that there are many effective numerical methods for finding approximate solutions of various variational inequalities (e.g., the projection method and its variant forms, linear approximation, descent and Newton's methods), and there are very few methods for general variational-like inequalities. For example, among the most effective numerical technique is the projection method and its variant forms; however, the projection type techniques cannot be extended for constructing iterative algorithms for mixed variational-like inequalities, since it is not possible to find the projection of the solution. Thus, the development of an efficient and implementable technique for solving variational-like inequalities is one of the most interesting and important problems in variational inequality theory. These facts motivated Glowinski et al. [12] to suggest another technique, which does not depend on the projection. The technique is called the auxiliary principle technique.
Recently, the auxiliary principle technique was extended by Huang and Deng [13] to study the existence and iterative approximation of solutions of the set-valued strongly nonlinear mixed variational-like inequality, under the assumptions that the operators are bounded closed values. On the other hand, by using the concept of -strongly mixed monotone of a fuzzy mapping on a bounded closed convex set, the auxiliary principle technique was extended by Chang et al. [14] to study the existence and iterative approximation of solutions of the mixed variational-like inequality problem for fuzzy mappings in a Hilbert space.
In this paper, the mixed variational-like inequality problem for fuzzy mapping (FMVLIP) in a reflexive Banach space is studied, and some existence theorems for the problem are proved. We also prove the existence theorem for auxiliary problem of FMVLIP. Further, by exploiting the theorem, we construct and analyze an iterative algorithm for finding the solution of the FMVLIP. Finally, we discuss the convergence analysis of iterative sequence generated by the iterative algorithm.
2. Preliminaries
Throughout this paper, we assume that is a real Banach space with its topological dual
,
a nonempty convex subset of
,
is the generalized duality pairing between
and
,
is the family of all nonempty bounded and closed subsets of
, and
is the Hausdorff metric on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ1_HTML.gif)
In the sequel we denote the collection of all fuzzy sets on by
. A mapping
from
to
is called a fuzzy mapping. If
is a fuzzy mapping, then the set
, for
, is a fuzzy set in
(in the sequel we denote
by
) and
, for each
is the degree of membership of
in
.
A fuzzy mapping is said to be closed, if, for each
, the function
is upper semicontinuous; that is, for any given net
satisfying
, we have
.
For and
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ2_HTML.gif)
is called a -cut set of
.
A closed fuzzy mapping is said to satisfy condition
, if there exists a function
such that for each
the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ3_HTML.gif)
is a nonempty bounded subset of .
Remark 2.1.
It is worth mentioning that if is a closed fuzzy mapping satisfying condition
, then for each
, the set
. Indeed, let
be a net and
, then
for each
. Since the fuzzy mapping
is closed, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ4_HTML.gif)
This implies that , and so
.
Let be a real reflexive Banach space with the dual space
. In this paper, we devote our study to a class of mixed variational-like inequality problem for fuzzy mappings, which is stated as follows.
Let are two closed fuzzy mappings satisfying the condition
with functions
, respectively.
and
are two single-valued mappings. Let
be a real bifunction. We shall study the following problem :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ5_HTML.gif)
The problem (2.5) is called a fuzzy mixed variational-like inequality problem, and we will denote by the solution set of the problem (2.5).
Now, let us consider some special cases of problem (2.5).
() Let
be two ordinary set-valued mappings, and let
be the mappings as in problem (2.5). Define two fuzzy mappings
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ6_HTML.gif)
where and
are the characteristic functions of the sets
and
, respectively. It is easy to see that
and
both are closed fuzzy mappings satisfying condition
with constant functions
and
, for all
, respectively. Furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ7_HTML.gif)
Thus, problem (2.5) is equivalent to the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ8_HTML.gif)
This kind of problem is called the set-valued strongly nonlinear mixed variational-like inequality, which was studied by Huang and Deng [13], when .
() If
is a Hilbert space, then problem (2.5) collapses to the following problem: Let
are two closed fuzzy mappings satisfying the condition
with functions
, respectively.
are two single-valued mappings. Let
be a real bifunction. We consider the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ9_HTML.gif)
The inequality of type (2.9) was studied by Chang et al. [14] under the additional condition that is a nonempty bounded closed subset of
.
-
(3)
If
is a Hilbert space and f(u,v)=0, then problem (2.5) is equivalent to the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ10_HTML.gif)
This is also a class of special fuzzy variational-like inequalities, which has been studying by many authors.
Evidently, for appropriate and suitable choice of the fuzzy mappings , mappings
, the bifunction
, and the space
, one can obtain a number of the known classes of variational inequalities and variational-like inequalities as special cases from problem (2.5) (see [1, 4, 5, 7–19]).
The following basic concepts will be needed in the sequel.
Definition 2.2.
Let be a nonempty subset of a Banach space
. Let
be two closed fuzzy mappings satisfying the condition
with functions
, respectively. Let
be mappings. Then
(i) is said to be
-cocoercive with respect to the first argument of
, if there exists a constant
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ11_HTML.gif)
for each , and for all
;
(ii) is Lipschitz continuous in the second argument with respect to the fuzzy mapping
, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ12_HTML.gif)
for any and
;
(iii) is
-strongly monotone in the first argument with respect to the fuzzy mapping
if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ13_HTML.gif)
for any and
. Similarly,
-strongly monotone of
in the second argument with respect to the fuzzy mapping
can be defined;
(iv) is said to be
-Lipschitz continuous if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ14_HTML.gif)
for any ;
(v) is Lipschitz continuous, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ15_HTML.gif)
for any .
Definition 2.3.
The bifunction is said to be skew-symmetric, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ16_HTML.gif)
for all .
Remark 2.4.
The skew-symmetric bifunctions have properties which can be considered as an analogs of monotonicity of gradient and nonnegativity of a second derivative for a convex function. As for the investigations of the skew-symmetric bifunction, we refer the reader to [20].
Definition 2.5 (see [15, 21]).
Let be a nonempty convex subset of a Banach space
. Let
be a Fréchet differentiable function and
. Then
is said to be
(i)-convex, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ17_HTML.gif)
for all
(ii)-strongly convex, if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ18_HTML.gif)
for all
Note that if for all
, then
is said to be strongly convex.
Throughout this paper, we shall use the notations "" and "
" for weak convergence and strong convergence, respectively.
Remark 2.6.
() Assume that for each fixed
the mapping
is continuous from the weak topology to the weak topology. Let
and
be fixed, and let
be a functional defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ19_HTML.gif)
Then, it is easy to see that is a weakly continuous functional on
.
() Let
be a Fréchet differentiable function, and let
be a mapping such that
. If
is an
-strongly convex functional with constant
on a convex subset
of
then
is
-strongly monotone with constant
(see [19], Proposition??2.1).
The following lemma due to Zeng et al. [19] will be needed in proving our results.
Lemma 2.7 (see [19, Lemma??2]).
Let be a nonempty convex subset of a topological vector space
and let
be such that
(i)for each is lower semicontinuous on each nonempty compact subset of
;
(ii)for each finite set and for each
;
(iii)there exists a nonempty compact convex subset of
such that for some
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ20_HTML.gif)
Then there exists , such that
??for all
.
We also need the following lemma.
Lemma 2.8 (see [22]).
Let be a complete metric space and let
and
be any real number. Then, for every
there exists
such that
.
In the sequel, we assume that and
satisfy the following assumption.
Assumption 2.9.
Let be two mappings satisfying the following conditions:
(a) for each
;
(b)for each fixed is a concave function;
(c)for each fixed , the functional
is weakly lower semicontinuous function from
to
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ21_HTML.gif)
Remark 2.10.
It follows from Assumption 2.9(a) that and
.
3. The Existence Theorems
Theorem 3.1.
Let be a real reflexive Banach space with the dual space
, and
be a nonempty convex subset of
. Let
be two closed fuzzy mappings satisfying the condition
with functions
, respectively. Let
, and
. Let
be skew-symmetric and weakly continuous such that
and
is a proper convex, for each
. Suppose that
(i) is
-cocoercive with respect to the first argument of
with constant
;
(ii) is Lipschitz continuous with constant
;
(iii) is Lipschitz continuous and
-strongly monotone in the second argument with respect to
with constant
and
, respectively.
If Assumption 2.9 is satisfied, then .
Proof.
For any , we define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ22_HTML.gif)
where .
Observe that, by is weakly continuous functional and since each fixed
the functional
is weakly lower semicontinuous, we have the functional
is weakly lower semicontinuous for each
. This shows that condition (i) in Lemma 2.7 holds. Next, we claim that
satisfies condition (ii) in Lemma 2.7. If it is not true, then there exist a finite set
and
, such that
for all
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ23_HTML.gif)
This gives
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ24_HTML.gif)
Note that for each is a convex functional, that is
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ25_HTML.gif)
From Assumption 2.9, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ26_HTML.gif)
which is a contradiction. Thus condition (ii) in Lemma 2.7 holds. Since for each is a proper convex weakly lower semicontinuous functional and int
, the element
int
can be found. Moreover, by Proposition I.2.6 of Pascali and Sburlan [23, page 27],
is subdifferentiable at
. This means
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ27_HTML.gif)
Since is skew-symmetric, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ28_HTML.gif)
Letting and
be fixed, by using conditions (ii) and (iv) and equality
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ29_HTML.gif)
Define and
. Then
is a weakly compact convex subset of
. Furthermore, it is easy to see that
for all
. Thus, condition (iii) of Lemma 2.7 is satisfied. By Lemma 2.7, there exists
such that
for all
, this means that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ30_HTML.gif)
where . Hence,
is a solution of the fuzzy variational like inequality (2.5), that is,
. This completes the proof.
Remark 3.2.
If all assumptions to Theorem 3.1 hold and is
-strongly monotone in the first argument with respect to
with constant
, then the solution of problem (2.5) is unique up to the element
. Indeed, supposing that
and
are elements in
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ32_HTML.gif)
Taking in (3.10) and
in (3.11) and adding two inequalities, since
is skew-symmetric, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ33_HTML.gif)
Using this one, in view of Remark 2.10, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ34_HTML.gif)
Since is
-strongly monotone in the first argument with respect to
with the constant
and
-strongly monotone in the second argument with respect to
with constant
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ35_HTML.gif)
Since , we must have
.
4. Convergence Analysis
4.1. Auxiliary Problem and Algorithm
In this section, we extend the auxiliary principle technique to study the fuzzy mixed variational-like inequality problem (2.5) in a reflexive Banach space . First, we give the existence theorem for the auxiliary problem for the problem (2.5). Consequently, we construct the iterative algorithm for solving the problem of type (2.5).
Let be a mapping, let
be a given Fréchet differentiable
-convex functional, and let
be a given positive real number. Given
, we consider the following problem
: find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ36_HTML.gif)
The problem is called the auxiliary problem for fuzzy mixed variational-like inequality problem (2.5).
Theorem 4.1.
If the conditions of Theorem 3.1 hold and for each fixed is continuous from the weak topology to the weak topology. If the function
is
-strongly convex with constant
and the functional
is weakly upper semicontinuous on
for each
, then the auxiliary problem
has a unique solution.
Proof.
Let and
be fixed. Define a functional
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ37_HTML.gif)
Note that, for each fixed , the functional
is weakly upper semicontinuous on
,
is continuous from the weak topology to the weak topology, and
is weakly continuous. Thus, it is easy to see that for each fixed
the function
is weakly lower semicontinuous continuous on each weakly compact subset of
and so condition (i) in Lemma 2.7 is satisfied. We claim that condition (ii) in Lemma 2.7 holds. If this is false, then there exist a finite set
and a
with
and
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ38_HTML.gif)
By Assumption 2.9, in light of Remark 2.10, together with the convexity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ39_HTML.gif)
which is a contradiction. Thus, condition (ii) in Lemma 2.7 is satisfied. Note that the -strong convexity of
implies that
is
-strongly monotone with constant
; see Remark 2.6(ii). By using the similar argument as in the proof of Theorem 3.1, we can readily prove that condition (iii) of Lemma 2.7 is also satisfied. By Lemma 2.7 there exists a point
, such that
for all
. This implies that
is a solution to the problem
.
Now we prove that the solution of problem is unique. Let
and
be two solutions of problem (4.1). Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ40_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ41_HTML.gif)
Taking in (4.5) and
in (4.6), and adding these two inequalities, since
and
is skew-symmetric, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ42_HTML.gif)
Thus, by is
-strongly monotone, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ43_HTML.gif)
This implies that and the proof is completed.
By virtue of Theorem 4.1, we now construct an iterative algorithm for solving the fuzzy mixed variational-like inequalities problem (2.5) in a reflexive Banach space .
Let be fixed. For given
, from Theorem 4.1, there is
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ44_HTML.gif)
Since , by Lemma 2.8, there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ45_HTML.gif)
Again by Theorem 4.1, there is such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ46_HTML.gif)
Since , by Lemma 2.8, there exist
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ47_HTML.gif)
Continuing in this way, we can obtain the iterative algorithm for solving problem (2.5) as follows.
Algorithm 4.2.
Let be fixed. For given
there exist the sequences
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ48_HTML.gif)
4.2. Convergence Theorems
Now, we shall prove that the sequences and
generated by Algorithm 4.2 converge strongly to a solution of problem (2.5).
Theorem 4.3.
Suppose that conditions of Theorem 4.1 hold, and the mapping are Lipschitzian continuous fuzzy mappings with Lipschitzian constant
and
, respectively. If
then the iterative sequences
obtained from Algorithm 4.2 converge strongly to a solution of problem (2.5).
Proof.
Let . Define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ49_HTML.gif)
By the -strong convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ50_HTML.gif)
Note that for all
and
is skew-symmetric. Since
and
, from the
-strong convexity of
, and Algorithm 4.2 with
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ51_HTML.gif)
where .
Consider
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ52_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ53_HTML.gif)
Since , the inequality (4.18) implies that the sequence
is strictly decreasing (unless
) and is nonnegative by (4.15). Hence it converges to some number. Thus, the difference of two consecutive terms of the sequence
goes to zero, and so the sequence
converges strongly to
. Further, from Algorithm 4.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ54_HTML.gif)
These imply that and
are Cauchy sequence in
, since
is a convergence sequence. Thus, we can assume that
and
(as
). Noting
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ55_HTML.gif)
Hence we must have . Similarly, we can obtain
. Finally, we will show that
. In regarded of Assumption 2.9(c), for each fixed
we have that the functional
is an upper semicontinuous functional; this together with the weak continuity of the function
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F209485/MediaObjects/13660_2009_Article_1920_Equ56_HTML.gif)
This implies that , and the proof is completed.
Remark 4.4.
() Theorems 3.1 and 4.3 are the extension of the results by Chang et al. [14], from Hilbert setting to a general reflexive Banach space, but it is worth noting that the bounded condition of the convex set
is not imposed here.
() Since every set-valued mapping is the fuzzy mapping, hence, all results obtained in this paper are still hold for any set-valued mappings
.
Thus, our results can be view as a refinement and improvement of the previously known results for variational inequalities.
References
Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.
Zimmermann H-J: Fuzzy Set Theory and Its Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1988.
Hartman P, Stampacchia G: On some non-linear elliptic differential-functional equations. Acta Mathematica 1966, 115: 271–310. 10.1007/BF02392210
Tian GQ: Generalized quasi-variational-like inequality problem. Mathematics of Operations Research 1993,18(3):752–764. 10.1287/moor.18.3.752
Yao JC: The generalized quasi-variational inequality problem with applications. Journal of Mathematical Analysis and Applications 1991,158(1):139–160. 10.1016/0022-247X(91)90273-3
Panagiotopoulos PD, Stavroulakis GE: New types of variational principles based on the notion of quasidifferentiability. Acta Mechanica 1992,94(3–4):171–194. 10.1007/BF01176649
Cubiotti P: Existence of solutions for lower semicontinuous quasi-equilibrium problems. Computers & Mathematics with Applications 1995,30(12):11–22. 10.1016/0898-1221(95)00171-T
Chang SS: Variational Inequalities and Complementarity Problems Theory and Applications. Shanghai Scientific and Technological Literature, Shanghai, China; 1991.
Chang S, Huang NJ: Generalized complementarity problems for fuzzy mappings. Fuzzy Sets and Systems 1993,55(2):227–234. 10.1016/0165-0114(93)90135-5
Chang S, Zhu YG: On variational inequalities for fuzzy mappings. Fuzzy Sets and Systems 1989,32(3):359–367. 10.1016/0165-0114(89)90268-6
Noor MA: Variational inequalities for fuzzy mappings—I. Fuzzy Sets and Systems 1993,55(3):309–312. 10.1016/0165-0114(93)90257-I
Glowinski R, Lions J-L, Trémolières R: Numerical Analysis of Variational Inequalities, Studies in Mathematics and Its Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 1981:xxix+776.
Huang N, Deng C: Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Mathematical Analysis and Applications 2001,256(2):345–359. 10.1006/jmaa.2000.6988
Chang SS, O'Regan D, Tan KK, Zeng LC: Auxiliary principle and fuzzy variational-like inequalities. Journal of Inequalities and Applications 2005,2005(5):479–494. 10.1155/JIA.2005.479
Ansari QH, Yao JC: Iterative schemes for solving mixed variational-like inequalities. Journal of Optimization Theory and Applications 2001,108(3):527–541. 10.1023/A:1017531323904
Fang YP, Huang NJ: Variational-like inequalities with generalized monotone mappings in Banach spaces. Journal of Optimization Theory and Applications 2003,118(2):327–338. 10.1023/A:1025499305742
Liu Z, Chen Z, Kang SM, Ume JS: Existence and iterative approximations of solutions for mixed quasi-variational-like inequalities in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008,69(10):3259–3272. 10.1016/j.na.2007.09.015
Zeng LC: Iterative approximation of solutions to generalized set-valued strongly nonlinear mixed variational-like inequalities. Acta Mathematica Sinica 2005,48(5):879–888.
Zeng L-C, Ansari QH, Yao J-C: General iterative algorithms for solving mixed quasi-variational-like inclusions. Computers & Mathematics with Applications 2008,56(10):2455–2467. 10.1016/j.camwa.2008.05.016
Antipin AS: Iterative gradient prediction-type methods for computing fixed points of extremal mapping. In Parametric Optimization and Related Topics, IV, Approximate Optimization. Volume 9. Edited by: Guddat J, Jonden HTh, Nizicka F, Still G, Twitt F. Peter Lang, Frankfurt am Main, Germany; 1997:11–24.
Hanson MA: On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications 1981,80(2):545–550. 10.1016/0022-247X(81)90123-2
Rus IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca, Romania; 2001:198.
Pascali D, Sburlan S: Nonlinear Mappings of Monotone Type. Martinus Nijhoff, The Hague, The Netherlands; Sijthoff & Noordhoff International, Alphen aan den Rijn, The Netherlands; 1978:x+341.
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The authors wish to express their gratitude to the referees for a careful reading of the manuscript and helpful suggestions. This research is supported by the Centre of Excellence in Mathematics, the commission on Higher Education, Thailand.
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Kumam, P., Petrot, N. Mixed Variational-Like Inequality for Fuzzy Mappings in Reflexive Banach Spaces. J Inequal Appl 2009, 209485 (2009). https://doi.org/10.1155/2009/209485
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DOI: https://doi.org/10.1155/2009/209485