Abstract
In this paper, we consider the boundary value problem for a 2n th-order nonlinear difference equation containing both advance and retardation. By using the critical point theory, some sufficient conditions of the existence of solutions of the boundary value problem are obtained. The proof is based on the linking theorem. An example is given to illustrate our results.
Highlights
By using various techniques such as critical point theory, fixed point theory, topological degree theory and coincidence degree theory, a great deal of works have been done on the existence of solutions to boundary value problems of difference equations
The critical point theory seems to be a powerful tool to solving this problem
Compared to the boundary value problems of lower order difference equations [, – ], the study of boundary value problems of higher order difference equations is relatively less, especially the works done by using the critical point theory
Summary
1 Introduction Consider the following nth-order nonlinear difference equation: n rt–n nxt–n + (– )n+ f (t, xt+ , xt, xt– ) = , t = , , . By using various techniques such as critical point theory, fixed point theory, topological degree theory and coincidence degree theory, a great deal of works have been done on the existence of solutions to boundary value problems of difference equations (see [ – ] and references therein). There is still spacious room to explore the boundary value problems of higher-order difference equations.
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