Existence Of Solutions For P-laplacian Elliptic Inequalities

Variational inequality is an important part of nonlinear analysis.It has wide applications in control problem,differential equation,mechanics,optimization theory,economic mathe-matics and game theory.To study existence and uniqueness of nonzero solutions conditions for different types variational inequalities is fundamental contents of variational inequalities theory.We will study existence of nonzero solutions for variational inequalities in a reflexive smooth Bana-ch space.Firstly,we establish fixed point index theory and degree theory for variational inequalities and respectively proved to study existence of nonzero solutions.Sec-ondly,variational inequality fixed point index theory and degree theory applied to study existence of nonzero positive weak solutions for p-Laplacian elliptic inequalities.This paper is organized as follows:In chapter 1,we make a brief introduce background and present situation of research for variational inequalities.Moreover,we also introduce some basic conceptions which are used is this dissertation.In chapter 2,we convert existence of nonzero solutions for variational inequalities prob-lem into a existence of fixed point problem.By establish completely continuous mapping fixed point index theory in a reflexive smooth Banach space and results on existence of nonzero solutions for variational inequalities are proved by using theory of fixed point index.In chapter 3,we convert existence of nonzero solutions for variational inequalities prob?lem into operator equations,that is,0?J?(x)+ NK(x)-A(x)problem.According to compact perturbations with(S)+ mappings research result,we define a degree function,that is,d(NK +J?-A,DK,0),and results on existence of nonzero solutions for variational inequalities are proved by using theory of degree.In chapter 4,we applied of variational inequality fixed point index theory to study exis-tence of nonzero positive weak solutions for p-Laplacian elliptic inequalities,and respectively proved existence of nonzero positive weak solutions for p-Laplacian elliptic inequalities.The main feature of this paper is firstly to use topological degree method to study existence of nonzero positive weak solutions for p-Laplacian elliptic inequalities and extended conclusion of cite 14.