Nonlinear second-order \(q\)-difference equations with three-point boundary conditions (Q2514058)

This paper studies the existence of solutions for the nonlinear second-order \(q\)-difference equation with three-point boundary conditions \[ D_q^2x(t)=f(t, x(t)), \;\;t\in J=[0, T], \] \[ x(0)=\beta D_p x(\eta), \;\;x(T)=\alpha \int_0^\eta x(s)d_rs, \] where \(0<p,q,r<1,\;f\in C([0, T]\times \mathbb{R},\;\mathbb{R}),\;0<\eta <T\) and \(\alpha,\;\beta\) are given constants. The authors provide a brief literature review in the first section. Then in the second section they give a few necessary definitions and some preliminary results. In the lemma of this section, they obtain an equivalent integral equation to the second-order \(q\)-difference equation. The main results of the paper are in the third section. First, the authors obtain the sufficient conditions under which the second-order \(q\)-difference equation has a unique solution by using Banach's fixed point theorem. Then, using Krasnoselskii's fixed point theorem, they get the sufficient conditions for the existence of solutions. Furthermore, based on Leray-Schauder degree theory, the authors achieve another set of sufficient conditions to guarantee the existence of solutions. Two examples are given in the last section to illustrate the main results.