Existence Of Positive Solutions To Nonlinear Equation On Locally Finite Graphs

In this paper we study the existence of solution for nonlinear equation on locally finite graphs G=(V,E).We consider the following two different types of equations.The first case:we consider the existence of solutions for equation-?u+h(x)u=f(x,u),x EV(1)where f:V×R?R,?(x,t)/t is nondecreasing in t and f(x,t)is asymptotically linear in t at infinity.Without assuming(AR)we prove,by using mountain pass theorem,that equation(1)has a positive solution.In the second case:we consider the existence of solutions for nonlinear equations where a(x),b(x):V?R,Fu,Fu,Fv:V×R2?R.When the equation(2)satisfies(AR).Mountain-pass theorem is used to prove that equation has strictly positive solutions.Without assuming(AR)we prove,that equation(2)has strictly positive solutions under suitable conditions.This thesis consist of three chapters.The first chapter is devoted to discuss the intro-duction including research background and prerequisite knowledge.The second chapter deals with the existence of the solutions for the equation under the first case,the main conclusion is theorem 2.1.In the last chapter we research the solutions for nonlinear equations in two cases,the main results are theorem 3.1 and theorem 3.2.