Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time

Abstract

Due to the essential difficulty of establishing an appropriate variational framework on a suitable working space, how to apply the critical point theory for showing the existence and multiplicity of periodic solutions of continuous-time difference equations remains a completely open problem. New ideas including gluing arguments are introduced in this work to overcome such a difficulty. This enables us to employ the critical point theory to construct uncountably many periodic solutions for a class of superlinear continuous-time difference equations without assuming symmetry properties on the nonlinear terms. The obtained solutions are piecewise differentiable in some cases, distinguishing continuous-time difference equations from ordinary differential equations qualitatively. To the best of our knowledge, this is the first time in the literature that the critical point theory has been used for such types of problems. Our work may open an avenue for studying discrete nonlinear systems with continuous time via the critical point theory.

Keywords

• Difference equation
• continuous time
• periodic solution
• multiplicity
• superlinear nonlinearity
• critical point theory