Existence Of Solutions For A2-dimensional Discrete Non-Cooperative System With Resonance

In this paper, existence of solutions for a2-dimensional discrete non-cooperative system of the form is investigated by using variational methods of nonlinear functional analysis and the critical point theory, especially the critical group and the Morse theory. WhereN(≥3) iS a fixed positive integer,[1,N] denotes the discrete interval{1,2,……N},△denotes the forward difference operator defined by The nonlinearities f，g∈C1（R2,R1) satisfy f(0,0)=g(0,0)=0and there exists a function F∈C2(R2,R1) such thatThis paper is composed of three chapters.In Chapter one, we introduce the advanced developments of the solvability of nonlinear discrete systems. At the same time, the significance and the main results of this paper are presented. That is to say, 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, sufficient conditions to guarantee the existence of non-trivial solutions of problem (1.2.1) are obtained by using variational methods. Furthermore, we obtain six theorems for this nonlinear system.In Chapter two, we list some related basic knowledge and conclusions of the critical point theory and some relevant properties of the matrix’s Kronecker product. Meanwhile, we obtain the whole eigenvalues corresponding to the linear eigenvalue problem of peoblem(1.2.1) and give the matrix form of problem (1.2.1). Thus we construct the corresponding energy functional of the problem (1.2.1) and prove several complementary results. They are of fundamental importance in proving our main conclusions of this paper.In Chapter three, by using some corresponding theorems and lemmas collected in Chapter two, we give the proof of the main conclusions via the computation of the critical groups.