Existence and approximation of solutions for system of generalized mixed variational inequalities
Fixed Point Theory and Applications
Existence and approximation of solutions for system of generalized mixed variational inequalities
Balwant Singh Thakur 0
Mohammad Saeed Khan 1
Shin Min Kang 2
0 School of Studies in Mathematics, Pt. Ravishankar Shukla University , Raipur, 492010 , India
1 Department of Mathematics and Statistics, Sultan Qaboos University , PCode 123 Al-Khod, P.O. Box 36, Muscat, Sultanate of Oman
2 Department of Mathematics and RINS, Gyeongsang National University , Jinju, 660-701 , Korea
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
system of generalized mixed variational inequality; fixed point problem; resolvent operator technique; relaxed cocoercive mapping; maximal monotone operator; parallel iterative algorithm
1 Introduction and preliminaries
Variational inequality theory was introduced by Stampacchia [] in the early s. The
birth of variational inequality problem coincides with Signorini problem, see [, p.].
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 x ∈ H such that
Tx, y – x + ϕ(y, x) – ϕ(x, x) ≥ ,
∀y ∈ H,
where T : H → H is a nonlinear operator and ϕ(·, ·) : H × H → R ∪ {+∞} 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 [].
System of variational inequalities arises in double porosity models and diffusion through
a composite media, description of parallel membranes, etc.; see [] for details.
In this paper, we consider the following system of generalized mixed variational
inequalities (SGMVI). Find x∗, y∗ ∈ H such that
ρT(x∗, y∗) + g(y∗) – g(x∗), x – g(y∗) + ϕ(x) – ϕ(g(y∗)) ≥
(.)
for all x ∈ H and ρ, ρ > , where T, T : H × H → H are nonlinear mappings and g, g :
H → H are any mappings.
If T, T : H → H are univariate mappings then the problem (SGMVI) reduced to the
following. Find x∗, y∗ ∈ H such that
ρT(y∗) + g(x∗) – g(y∗), x – g(x∗) + ϕ(x) – ϕ(g(x∗)) ≥ ,
ρT(x∗) + g(y∗) – g(x∗), x – g(y∗) + ϕ(x) – ϕ(g(y∗)) ≥
ρT(y∗, x∗) + g(x∗) – g(y∗), x – g(x∗) ≥ ,
ρT(x∗, y∗) + g(y∗) – g(x∗), x – g(y∗) ≥
ρT (y∗, x∗) + x∗ – y∗, x – x∗ ≥ ,
ρT (x∗, y∗) + y∗ – x∗, x – y∗ ≥
then the problem (.) reduces to the following system of general variational inequality
problem: Find x∗, y∗ ∈ K such that
for all x ∈ K and ρ, ρ > . The problem (.) with g = g has been studied by [].
If T = T = T and g = g = I, then the problem (.) reduces to the following system of
general variational inequality problem. Find x∗, y∗ ∈ K such that
ρT (y∗, x∗) + x∗ – y∗, x – x∗ + ϕ(x) – ϕ(x∗) ≥ ,
ρT (x∗, y∗) + y∗ – x∗, x – y∗ + ϕ(x) – ϕ(y∗) ≥
δK (x) = ⎧⎨ , if x ∈ K ;
⎩ +∞, otherwise,
for all x ∈ H and ρ, ρ > .
If T = T = T and g = g = I, then the problem (SGMVI) reduces to the following system
of mixed variational inequalities considered by [, ]. Find x∗, y∗ ∈ H such that
for all x ∈ H and ρ, ρ > .
If K is closed convex set in H and ϕ(x) = δK (x) for all x ∈ K , where δK is the indicator
function of K defined by
⎧
⎨
⎩
⎧
⎨
⎩
for all x ∈ K and ρ, ρ > . The problem (.) is studied by Verma [, ] and Chang et al.
[].
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 pro (...truncated)