Nonlinear Variational Inequalities Research Articles

Approximating solutions of generalized pseudocontractive variational inequalities by admissible perturbation type iterative methods

In this paper we give the solvability class of generalized strongly nonlinear variational inequalities modified by the use of the new concept of admissible perturbation operator on nonempty closed convex sets in Hilbert spaces.

Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces

In this paper, a new system of generalized nonlinear variational inequalities involving three operators is introduced. A three-step iterative algorithm is considered for the system of generalized nonlinear variational inequalities. Strong convergence theorems of the three-step iterative algorithm are established.MSC:47H05, 47H09, 47J25.

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.

Perturbed projection and iterative algorithms for a system of general regularized nonconvex variational inequalities

The purpose of this paper is to introduce a new system of general nonlinear regularized nonconvex variational inequalities and verify the equivalence between the proposed system and fixed point problems. By using the equivalent formulation, the existence and uniqueness theorems for solutions of the system are established. Applying two nearly uniformly Lipschitzian mappings S1 and S2 and using the equivalent alternative formulation, we suggest and analyze a new perturbed p-step projection iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S1,S2) which is the unique solution of the system of general nonlinear regularized nonconvex variational inequalities. We also discuss the convergence analysis of the proposed iterative algorithm under some suitable conditions. MSC : Primary 47H05; Secondary 47J20, 49J40, 90C33.

Iterative algorithms approach to a general system of nonlinear variational inequalities with perturbed mappings and fixed point problems for nonexpansive semigroups

In this paper, we introduce new iterative algorithms for finding a common element of the set of solutions of a general system of nonlinear variational inequalities with perturbed mappings and the set of common fixed points of a one-parameter nonexpansive semigroup in Banach spaces. Furthermore, we prove the strong convergence theorems of the sequence generated by these iterative algorithms under some suitable conditions. The results obtained in this paper extend the recent ones announced by many others.Mathematics Subject Classification (2010): 47H09, 47J05, 47J25, 49J40, 65J15

Two General Algorithms for Computing Fixed Points of Nonexpansive Mappings in Banach Spaces

Recently, Yao et al. (2011) introduced two algorithms for solving a system of nonlinear variational inequalities. In this paper, we consider two general algorithms and obtain the extension results for computing fixed points of nonexpansive mappings in Banach spaces. Moreover, the fixed points solve the same system of nonlinear variational inequalities.

Algorithms for a System of General Variational Inequalities in Banach Spaces

The purpose of this paper is using Korpelevich′s extragradient method to study the existence problem of solutions and approximation solvability problem for a class of systems of finite family of general nonlinear variational inequality in Banach spaces, which includes many kinds of variational inequality problems as special cases. Under suitable conditions, some existence theorems and approximation solvability theorems are proved. The results presented in the paper improve and extend some recent results.

The Wiener-Hopf Equation Technique for Solving General Nonlinear Regularized Nonconvex Variational Inequalities

AbstractIn this paper, we introduce and study some new classes of extended general nonlinear regularized non-convex variational inequalities and the extended general nonconvex Wiener-Hopf equations, and by the projection operator technique, we establish the equivalence between the extended general nonlinear regularized nonconvex variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear regularized nonconvex variational inequalities. We apply the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S for constructing some new p-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is unique solution of the problem of extended general nonlinear regularized nonconvex variational inequalities. We also consider the convergence analysis of the suggested iterative schemes under some suitable conditions.Mathematics Subject Classification (2010)Primary 47H05; Secondary 47J20, 49J40

Existence and iterative approximations of solutions for strongly nonlinear variational-like inequalities

Abstract. In this paper, we introduce and study a new class of stronglynonlinear variational-like inequalities. Under suitable conditions, we provethe existence of solutions for the class of strongly nonlinear variational-like inequalities. By making use of the auxiliary principle technique, wesuggest an iterative algorithm for the strongly nonlinear variational-likeinequality and give the convergence criteria of the sequences generatedby the iterative algorithm. 1. IntroductionIt is well known that there are lots of iterative type algorithms for ndingthe approximate solutions of various variational inequalities in Hilbert spaces[3] and [8-13]. Among the most e ective numerical technique is the projectionmethod and its variant forms. However, the standard projection technique canno longer be applied to suggest the iterative type algorithm for variational-likeinequalities. This fact motivated Gowinski, Lions and Tremoliers [7] to developthe auxiliary principle technique, which does not depend on the projection.Ding [4,5] and Ding and Tan [6] extended the auxiliary principle technique tosuggest several iterative algorithms to compute approximate solutions for someclasses of general nonlinear mixed variational inequalities and variational-likeinequalities.In this paper, we introduce and study a new class of strongly nonlinearvariational-like inequalities. By applying a result due to Chang [1], we provethe existence of solutions for the class of strongly nonlinear variational-likeinequalities. Using the auxiliary principle technique, we suggest and analyzea new three-step iterative algorithm for solving the class of strongly nonlinearvariational-like inequalities. The convergence criteria of the sequence generatedby the iterative algorithm are given.

Generalized System for Relaxed Cocoercive and Involving Projective Nonexpansive Mapping Variational Inequalities

Firstly, the concept of projective nonexpansive mappings is presented in this paper. The approximate solvability of a generalized system for relaxed cocoercive and involving projective nonexpansive mapping nonlinear variational inequalities in Hilbert spaces is studied, based on the convergence of projection methods. The results presented in this paper extend and improve the main results of many authors.

Regularity for nonlinear variational inequalities of hyperbolic type

In this paper, we deal with the regularity for nonlinear variational inequalities of second order in Hilbert spaces with more general conditions on the nonlinear terms and without condition of the compactness of the principal operators. We also obtain the norm estimate of a solution of the given nonlinear equation on C ( [ 0 , T ] ; V ) ∩ C 1 ( ( 0 , T ] ; H ) ∩ C 2 ( ( 0 , T ] ; V ∗ ) by using the results of its corresponding hyperbolic semilinear part.

Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces

AbstractIn this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied. To overcome the difficulties, due to the presence of a proper convex lower semi-continuous function, φ and a mapping g, which appeared in the considered problem, we have used some applications of the resolvent operator technique. We would like to point out that although many authors have proved results for finding solutions of the systems of nonlinear set-valued (mixed) variational inequalities problems, it is clear that it cannot be directly applied to the problems that we have considered in this paper because of φ and g.2000 AMS Subject Classification: 47H05; 47H09; 47J25; 65J15.

Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces

In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality.

Homogenization of a class of nonlinear variational inequalities with applications in fluid film flow

The authors consider the homogenization of a class of nonlinear variational inequalities, which include rapid oscillations with respect to a parameter. The homogenization of the corresponding class of differential equations is also studied. The results are applied to some models for the pressure in a thin fluid film fluid between two surfaces which are in relative motion. This is an important problem in the lubrication theory. In particular, the analysis includes the effects of surface roughness on both faces and the phenomenon of cavitation. Moreover, the fluid can be modeled as Newtonian or non-Newtonian by using a Rabinowitsch fluid model.

Controllability for Nonlinear Variational Inequalities of Parabolic Type

This paper deal with the approximate controllability for the nonlinear functional differential control problem governed by the variational inequality. Sufficient conditions for the approximate controllability of the system are discussed under the bounded condition on the controller operator independent of the the time interval. We also prove the regularity and norm estimations for solutions of the given problems.

Global solvability of a model for grain boundary motion with constraint

We consider a model for grain boundary motion with constraint. In composite material science it is very important to investigate the grain boundary formation and its dynamics. In this paper we study a phase-filed model of grain boundaries, which is a modified version of the one proposed by R. Kobayashi, J.A. Warren and W.C. Carter [18]. The model is described as a system of a nonlinear parabolic partial differential equation and a nonlinear parabolic variational inequality. The main objective of this paper is to show the global existence of a solution for our model, employing some subdifferential techniques in the convex analysis.

A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces

The purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem. Under suitable conditions, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces. The results presented in the paper improve and extend some recent results.

Про одну задачу оптимального керування в коефіцієнтах для нелінійних еліптичних варіаційних нерівностей

In this paper we study an optimal control problem for a nonlinear elliptic variational inequality with generalized solenoidal coefficients which we adopt as controls in L °°(fi). We prove the existence of optimal solution of the stated problem.

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.

Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems

In this paper, we introduce an iterative method for finding a common element of the set of solutions of the generalized equilibrium problems, the set of solutions for the systems of nonlinear variational inequalities problems and the set of fixed points of nonexpansive mappings in Hilbert spaces. Furthermore, we apply our main result to the set of fixed points of an infinite family of strict pseudo-contraction mappings. The results obtained in this paper are viewed as a refinement and improvement of the previously known results.