On the solutions of the second-order (p,q)-difference equation with an application to the fixed-point theory

Abstract

In this paper, we examined the existence and uniqueness of solutions to the second-order $ (p, q) $-difference equation with non-local boundary conditions by using the Banach fixed-point theorem. Moreover, we introduced a special case of this equation called the Euler-Cauchy-like $ (p, q) $-difference equation and provide its solution. We also studied the oscillation of solutions for this equation in $ (p, q) $-calculus and proved the $ (p, q) $-Sturm-type separation theorem and $ (p, q) $-Kneser theorem about the oscillation of solutions.

Keywords

• (p， q) -derivative
• second-order difference equation
• time scale
• oscillations
• fixed-points