Positive Periodic Solutions in Shifts δ± for a Class of Higher-Dimensional Functional Dynamic Equations with Impulses on Time Scales

Abstract

Let $\mathbb{T}\subset \mathbb{R}$ be a periodic time scale in shifts ${\delta }_{\pm }$ with period $P\in (t_0,\mathrm{\infty }{)}_{\mathbb{T}}$ and $t_0\in \mathbb{T}$ is nonnegative and fixed. By using a multiple fixed point theorem in cones, some criteria are established for the existence and multiplicity of positive solutions in shifts ${\delta }_{\pm }$ for a class of higher-dimensional functional dynamic equations with impulses on time scales of the following form: $x^{\mathrm{\Delta }}(t)=A(t)x(t)+b(t)f(t,x(g(t))),t\ne t_j,t\in \mathbb{T},x(t_j^{+})=x(t_j^{-})+I_j(x(t_j)),$ where $A(t)=(a_{ij}(t){)}_{n\times n}$ is a nonsingular matrix with continuous real-valued functions as its elements. Finally, numerical examples are presented to illustrate the feasibility and effectiveness of the results.