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Existence and approximation of solutions for system of generalized mixed variational inequalities
Fixed Point Theory and Applications volume 2013, Article number: 108 (2013)
Abstract
The aim of this work is to study a system of generalized mixed variational inequalities, existence and approximation of its solution using the resolvent operator technique. We further propose an algorithm which converges to its solution and common fixed points of two Lipschitzian mappings. Parallel algorithms are used, which can be used to simultaneous computation in multiprocessor computers. The results presented in this work are more general and include many previously known results as special cases.
MSC:47J20, 65K10, 65K15, 90C33.
1 Introduction and preliminaries
Variational inequality theory was introduced by Stampacchia [1] in the early 1960s. The birth of variational inequality problem coincides with Signorini problem, see [[2], p.282]. The Signorini problem consists of finding the equilibrium of a spherically shaped elastic body resting on the rigid frictionless plane. Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. A variational inequality involving the nonlinear bifurcation, which characterized the Signorini problem with nonlocal friction is: find such that
where is a nonlinear operator and is a continuous bifunction.
Inequality above is called mixed variational inequality problem. It is an useful and important generalization of variational inequalities. This type of variational inequality arise in the study of elasticity with nonlocal friction laws, fluid flow through porus media and structural analysis. Mixed variational inequalities have been generalized and extended in many directions using novel and innovative techniques. One interesting problem is to find common solution of a system of variational inequalities. The existence problem for solutions of a system of variational inequalities has been studied by Husain and Tarafdar [3]. System of variational inequalities arises in double porosity models and diffusion through a composite media, description of parallel membranes, etc.; see [4] for details.
In this paper, we consider the following system of generalized mixed variational inequalities (SGMVI). Find such that
for all and , where are nonlinear mappings and are any mappings.
If are univariate mappings then the problem (SGMVI) reduced to the following. Find such that
for all and .
If and , then the problem (SGMVI) reduces to the following system of mixed variational inequalities considered by [5, 6]. Find such that
for all and .
If K is closed convex set in H and for all , where is the indicator function of K defined by
then the problem (1.1) reduces to the following system of general variational inequality problem: Find such that
for all and . The problem (1.4) with has been studied by [7].
If and , then the problem (1.4) reduces to the following system of general variational inequality problem. Find such that
for all and . The problem (1.5) is studied by Verma [8, 9] and Chang et al. [10].
In the study of variational inequalities, projection methods and its variant form has played an important role. Due to presence of the nonlinear term φ, the projection method and its variant forms cannot be extended to suggest iterative methods for solving mixed variational inequalities. If the nonlinear term φ in the mixed variational inequalities is a proper, convex and lower semicontinuous function, then the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. Hassouni and Moudafi [11] used the resolvent operator technique to study a new class of mixed variational inequalities.
For a multivalued operator , the domain of T, the range of T and the graph of T denote by
and
respectively.
Definition 1.1 T is called monotone if and only if for each , and , , we have
T is maximal monotone if it is monotone and its graph is not properly contained in the graph of any other monotone operator.
is the operator defined by .
Definition 1.2 ([12])
For a maximal monotone operator T, the resolvent operator associated with T, for any , is defined as
It is known that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. Furthermore, the resolvent operator is single-valued and nonexpansive i.e., for all . In particular, it is well known that the subdifferential ∂φ of φ is a maximal monotone operator; see [13].
Lemma 1.3 ([12])
For a given satisfies the inequality
if and only if , where is the resolvent operator and is a constant.
Using Lemma 1.3, we will establish following important relation.
Lemma 1.4 The variational inequality problem (1.1) is equivalent to finding such that
where is the resolvent operator and .
Proof Let be a solution of (1.1). Then for all , we have
which can be written as
using Lemma 1.3 for , we get
i.e.,
This completes the proof. □
Definition 1.5 An operator is said to be
-
(1)
ζ-strongly monotone if for each , there exists a constant such that
for all ;
-
(2)
η-Lipschitz continuous if for each , there exists a constant such that
An operator is said to be
-
(3)
relaxed -cocoercive with respect to the first argument if for each , there exist constants and such that
-
(4)
μ-Lipschitz continuous with respect to the first argument if for each , there exists a constant such that
-
(5)
γ-Lipschitz continuous with respect to the second argument if for each , there exists a constant such that
Lemma 1.6 ([14])
Let and be two nonnegative real sequences satisfying the following conditions:
where is some nonnegative integer, with and , then as .
Several iterative algorithms have been devised to study existence and approximation of different classes of variational inequalities. Most of them are sequential iterative methods, when we implement such algorithms on computers, then only one processor is used at a time. Availability of multiprocessor computers enabled researchers to develop iterative algorithms having the parallel characteristics. Lions [15] studied a parallel algorithm for a solution of parabolic variational inequalities. Bertsekas and Tsitsiklis [16, 17] developed parallel algorithm using the metric projection. Recently, Yang et al. [7] studied parallel projection algorithm for a system of nonlinear variational inequalities.
2 Existence and convergence
Lemma 1.4 established the equivalence between the fixed-point problem and the variational inequality problem (1.1). Using this equivalence in this section, we construct a parallel iterative algorithm to approximate the solution of the problem (1.1) and study the convergence of the sequence generated by the algorithm.
Algorithm 2.1 For arbitrary chosen points , compute the sequences and such that
where is the resolvent operator and , is positive real numbers.
Theorem 2.2 Let H be a real Hilbert space. Let and be mappings such that is relaxed -cocoercive, -Lipschitz continuous with respect to the first argument, -Lipschitz continuous with respect to the second argument and is -Lipschitz continuous, -strongly monotone mapping for . Assume that the following assumptions hold:
where .
Then there exist , which solves the problem (1.1). Moreover, the iterative sequences and generated by the Algorithm 2.1 converges to and , respectively.
Proof Using (2.1), we have
Since is relaxed -cocoercive and -Lipschitz continuous in the first argument, we have
Since is -Lipschitz continuous and -strongly monotone,
Similarly,
By -Lipschitz continuity of with respect to second argument,
It follows from (2.2)-(2.6) that
where and .
Similarly, we get
where and .
Now (2.7) and (2.8) imply
where by assumption. Hence and are both Cauchy sequences in H, and converges to and converges to . Since , , , and are all continuous, we have
The result follows from Lemma 1.4. This completes the proof. □
If are univariate mappings, then the Algorithm 2.1 reduces to the following.
Algorithm 2.3 For arbitrary chosen points , compute the sequences and such that
where is the resolvent operator and , is positive real numbers.
Theorem 2.4 Let H be a real Hilbert space. Let be mappings such that is relaxed -cocoercive, -Lipschitz continuous and is -Lipschitz continuous, -strongly monotone mapping for . Assume that the following assumptions hold:
where .
Then there exist , which solves the problem (1.2). Moreover the iterative sequences and generated by the Algorithm 2.3 converges to and , respectively.
3 Relaxed algorithm and approximation solvability
Lemma 1.4 implies that the system of general mixed variational inequality problem (1.1) is equivalent to the fixed-point problem. This alternative equivalent formulation is very useful for a numerical point of view. In this section, we construct a relaxed iterative algorithm for solving the problem (1.1) and study the convergence of the iterative sequence generated by the algorithm.
Algorithm 3.1 For arbitrary chosen points , compute the sequences and such that
where is the resolvent operator, , are sequences in and , is positive real numbers.
We first prove a result, which will be helpful to prove main result of this section.
Lemma 3.2 Let H be a real Hilbert space. Let and be sequences in H such that
for some , where and are sequences in such that and . Then and converges to and , respectively.
Proof Now, define the norm on by
Then is a Banach space. Hence, (3.2) implies that
Using Lemma 1.6, we get
Therefore, sequences and converges to and , respectively. This completes the proof. □
We now present the approximation solvability of the problem (1.1).
Theorem 3.3 Let H be a real Hilbert space H. Let and be mappings such that is relaxed -cocoercive, -Lipschitz continuous with respect to the first argument, -Lipschitz continuous with respect to the second argument and is -Lipschitz continuous, -strongly monotone mapping for . Suppose that be a solution of the problem (1.1) and , are sequences in . Assume that the following assumptions hold:
-
(i)
,
-
(ii)
,
-
(iii)
and ,
where
Then the sequences and generated by the Algorithm 3.1 converges to and , respectively.
Proof From Theorem 2.2 the problem (1.1) has a solution in H. By Lemma 1.4, we have
To prove the result, we first evaluate for all . Using (3.1), we obtain
Since is relaxed -cocoercive and -Lipschitz mapping with respect to the first argument, we have
Since is -Lipschitz continuous and -strongly monotone,
Similarly, we have
By -Lipschitz continuity of with respect to second argument,
By (3.4)-(3.8), we have
where and .
Similarly, we have
where and .
Now (3.9) and (3.10) imply
where
By the assumptions and Lemma 3.2, we get that the sequences and converges to and , respectively. This completes the proof. □
Remark 3.4 Theorem 3.3 extend and generalize the main result in [5], which itself is a extension and improvement of the main result in Chang et al. [10].
If are univariate mappings, then the Algorithm 3.1 reduces to the following.
Algorithm 3.5 For arbitrary chosen points , compute the sequences and such that
where is the resolvent operator, , are sequences in and , is positive real numbers.
As a consequence of Theorem 3.3, we have following result.
Corollary 3.6 Let H be a real Hilbert space H. Let be mappings such that is relaxed -cocoercive, -Lipschitz continuous and is -Lipschitz continuous, -strongly monotone mapping for . Suppose that be a solution of the problem (1.2) and , are sequences in . Assume that the following assumptions hold:
-
(i)
,
-
(ii)
,
-
(iii)
and ,
where
Then the sequences and generated by the Algorithm 3.5 converges to and , respectively.
4 Algorithms for common element
Now, we consider, the approximation solvability of the system (1.1) which is also a common fixed point of two Lipschitzian mappings. We propose a relaxed two-step algorithm, which can be applied to the approximation of solution of the problem (1.1) and common fixed point of two Lipschitzian mappings.
Algorithm 4.1 For arbitrary chosen points , compute the sequences and such that
where is the resolvent operator, , are sequences in and , be positive real numbers.
Let denote the set of fixed points of the mapping , i.e., , and the set of solutions of the problem (1.1).
Theorem 4.2 Let H be a real Hilbert space H. Let and be mappings such that is relaxed -cocoercive, -Lipschitz continuous with respect to the first argument, -Lipschitz continuous with respect to the second argument and is -Lipschitz continuous, -strongly monotone mapping for . Let be -Lipschitzian mapping for with , , are sequences in . Assume that the following assumptions hold:
-
(i)
,
-
(ii)
,
-
(iii)
and ,
where and
If , then the sequences and generated by the Algorithm 4.1 converges to and , respectively, such that and .
Proof Let us have and . By Lemma 1.4, we have
Also since , we have
To prove the result, we first evaluate for all . Using (4.1), we obtain
Using the arguments as in the proof of Theorem 3.3, from (4.2) we get that
where and .
Similarly, we get
where and .
Adding (4.3) and (4.4), taking we get
where
By the assumptions and Lemma 3.2, we get that the sequences and converges to and , respectively. This completes the proof. □
A mapping is said to be asymptotically λ-strictly pseudocontractive [18] if there exist a sequence with such that
for some , for all and .
Kim and Xu [19] proved that, if is an asymptotically λ-strictly pseudocontractive mapping, then is a Lipschitzian mapping with Lipschitz constant
for each integer .
Also if , then for all integer .
Assume that is asymptotically -strictly pseudocontractive mappings for with . Now generate sequence and by Algorithm 4.1 with and for some integer . Theorem 4.2 can be applied to study approximate solvability of the problem (1.1) and common fixed points of two asymptotically strictly pseudocontractive mappings.
A mapping is said to be asymptotically nonexpansive [20] if there exists a sequence with such that for all and . Clearly every asymptotically nonexpansive mapping is an asymptotically 0-strictly pseudocontractive mapping. Theorem 4.2 can be applied to study approximate solvability of the problem (1.1) and common fixed points of two asymptotically nonexpansive mappings.
Remark 4.3 An important feature of the algorithms used in the paper is its suitability for implementing on multiprocessor computer. Assume that and are given, in order to get the new iterative point; we can set one processor of computer to compute and other processor to compute , i.e., and are computed parallel, which will take less time then computing and in a sequence using a single processor; we refer [16, 17, 21–23] and references therein for more examples and ideas of the parallel iterative methods.
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Thakur, B.S., Khan, M.S. & Kang, S.M. Existence and approximation of solutions for system of generalized mixed variational inequalities. Fixed Point Theory Appl 2013, 108 (2013). https://doi.org/10.1186/1687-1812-2013-108
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DOI: https://doi.org/10.1186/1687-1812-2013-108