Applications Of Variational Methods To Boundary Value Problem Of Difference Equation

In this paper, using variational methods, we consider the following boundary value problems andBy defining an appropriate space, and establishing the variational functionals corresponding to the problems (P) and (Pλ,μ), we obtain the solutions to the problems of (P) and (Pλ,μ) as the critical points of their corresponding functionals in the certain space.Firstly, we present the background of problem and state the main work of this thesis. We also list some preliminary knowledge which is needed later.Secondly, we consider the existence of periodic solutions to the problem (P). First, using the direct method, we obtain one solution of (P). Next, we discuss the existence of multiple solutions for (P), with the aid of linking theorem, symmetric mountain pass lemma, etc. Finally, we give some examples to illustrate our results.Last, we study the existence of multiple solutions for (Pλ,μ). Via critical point theory, when parameters λ,μ belong to some suitable intervals,(Pλ,μ) possesses at least three solutions. Further, in certain range, if f and g are all greater than0, considering the auxiliary problem of (Pλ,μ), we verify that (Pλ,μ) has at least three positive solutions, which are uniformly bounded.