Lyapunov-type inequalities on fractional q-difference Schrödinger equation with Woods-Saxon potential

Abstract

In this paper, the integer order Schrodinger equation with Woods-Saxon potential is extended to the fractional q-difference field. We establish the Lyapunov-type inequalities for nonlinear fractional q-difference equations, to the best of our knowledge, which is the first work dealing with Lyapunov-type inequalities for nonlinear fractional q-difference equations. Results in this paper even are new in integer order case. Moreover, we further investigate the two-point boundary value problem of nonlinear fractional q-difference Schrodinger equation with Woods-Saxon potential. By applying the Leray-Schauder degree theory, we get a sufficient condition of the existence of solutions that is relatively easy to verify compared with the result of existing literature. By utilising the Leggett-Williams fixed point theorem, an inequality is added to the existence condition of solutions of such problem studied in the existing literature, and we get the multiplicity of solutions of this problem. As applications, two examples are presented to illustrate our main results.