Analytic general solutions of nonlinear difference equations

Abstract

There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions of the following nonlinear second order difference equation, $$u(t+2)=f(u(t),u(t+1)),$$ where f(x,y) is an entire function of x, y. The main work of the present paper is obtaining representations of analytic general solutions of the difference equation with new methods of complex analysis.