Existence Of Nontrivial Solutions For Fourth-order Discrete Problems With Resonance

In this paper,the existence of nontrivial solutions for fourth-order discrete resonant boundary value problem of the form is discussed by means of variational methods in nonlinear functional analysis, combining with the critical point theory,especially the critical groups and Morse theory,where△denotes the forward difference operator,i.e. Au(k-1)=u(k)-u(k-1),△nu(k)=△(△n-1u(k)),Z[a+1,b+1]={a+1,a+2,…,b+1},a,b are integers and a<b,η,ξ are real parameters,f(k,·)∈C1(R1,R1),F(k,0)=0and satisfies asymptotic linear conditions and This paper is composed of four chapters.In chapter one, we introduce the background and the research methods for fourth-order discrete boundary value problems. At the same time, the significance of study and main results of this paper are presented. That is, by way of variational methods, under the following cases of resonance both at zero and at infinity, non-resonance both at zero and at infinity, resonance at zero but non-resonance at infinity and non-resonance at zero but resonance at infinity, some sufficient conditions to guarantee the existence of non-trivial solutions are obtained.There are four existence theorems of nontrivial solutions to give in the whole paper.In chapter two, we list some related definitions and conclusions of the critical point theory.In chapter three, we give the variational frameworke of problem(1.2.1) and obtain the all eigenvalues corresponding to the linear eigenvalue problem of (1.2.1).In chapter four, by using of some corresponding theorems and lemmas collected in Chapter two and three, we give the proof of the main conclusions via the computations of the critical groups.