Lyapunov-type inequalities on fractional q-difference Schrödinger equation with Woods-Saxon potential
Lyapunov-type inequalities on fractional q-difference Schrödinger equation with Woods-Saxon potential
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DOI：
10.1504/ijdsde.2019.10022222
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发表时间：
2019-06
影响因子：
0.3
通讯作者：
Kuikui Ma;Z. Han
中科院分区：
文献类型：
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作者：
Kuikui Ma;Z. Han
本文将具有Woods-Saxon势的整数阶薛定谔方程推广到分数q-差分场。据我们所知，我们建立了非线性分数阶 q 差分方程的 Lyapunov 型不等式，这是第一个处理非线性分数 q 差分方程的 Lyapunov 型不等式的工作。本文的结果在整数阶情况下甚至是新的。此外，我们进一步研究了具有Woods-Saxon势的非线性分数q-差分薛定谔方程的两点边值问题。通过应用Leray-Schauder度理论，我们得到了解存在的充分条件，与现有文献的结果相比，该条件相对容易验证。利用Leggett-Williams不动点定理，在现有文献研究的此类问题解的存在条件上添加一个不等式，得到该问题解的重数性。作为应用，提出了两个例子来说明我们的主要结果。
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.