The Existence Of Periodic Solutions For Fourth-order Nonlinear Difference Equations

This paper mainly concerns with the existence and multiplicity of periodic solutions for a class of non-linear fourth-order high-dimensional difference equations. The author reduces the problem of finding periodic solutions of difference equations to that of seeking critical points of the corresponding functional. In chapter 1, the author introduces the historical background and the recent development of problems to be studied about fourth-order difference equations in details and some preparations are given here. In chapter 2, firstly, the author establishes the corresponding variational functional and reduces the problem of finding periodic solutions of difference equations to that of seeking critical points of the functional; secondly, some results are obtained when the nonlinear element is superlinear; thirdly, a multiplicity result for periodic solutions is obtained under an assumption weaker than Ambrosetti-Rabinowitz-type condition; at last, when the nonlinear element is odd for the variable, a multiplicity result for periodic solutions is obtained via Clark theorem. In chapter 3, the author studies the existence of periodic solutions when the equation satisfies different sublinear conditions in details. In chapter 4, some new sufficient conditions for the existence of periodic solutions are obtained when the nonlinear element is asymptomatically linear.