Title: Lyapunov-type inequalities on fractional q-difference Schrödinger equation with Woods-Saxon potential
Authors: Kuikui Ma; Zhenlai Han
Addresses: School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China ' School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
Abstract: In this paper, the integer order Schrödinger 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 Schrödinger 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.
Keywords: fractional q-difference equations; Lyapunov inequality; boundary value problem.
International Journal of Dynamical Systems and Differential Equations, 2019 Vol.9 No.2, pp.105 - 119
Received: 11 May 2017
Accepted: 04 Jan 2018
Published online: 04 Jul 2019 *