Existence and approximation of solutions for system of generalized mixed variational inequalities

Fixed Point Theory and Applications, Apr 2013

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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://fixedpointtheoryandalgorithms.springeropen.com/counter/pdf/10.1186/1687-1812-2013-108

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 ρT(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 ρT(y∗) + g(x∗) – g(y∗), x – g(x∗) + ϕ(x) – ϕ(g(x∗)) ≥ , ρT(x∗) + g(y∗) – g(x∗), x – g(y∗) + ϕ(x) – ϕ(g(y∗)) ≥  ρT(y∗, x∗) + g(x∗) – g(y∗), x – g(x∗) ≥ , ρT(x∗, y∗) + g(y∗) – g(x∗), x – g(y∗) ≥  ρT (y∗, x∗) + x∗ – y∗, x – x∗ ≥ , ρT (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 ρT (y∗, x∗) + x∗ – y∗, x – x∗ + ϕ(x) – ϕ(x∗) ≥ , ρT (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)


This is a preview of a remote PDF: https://fixedpointtheoryandalgorithms.springeropen.com/counter/pdf/10.1186/1687-1812-2013-108

Thakur, Balwant Singh, Khan, Mohammad Saeed, Kang, Shin Min. Existence and approximation of solutions for system of generalized mixed variational inequalities, Fixed Point Theory and Applications, 2013, pp. 1-15, Volume 2013, Issue 1, DOI: 10.1186/1687-1812-2013-108