Periodic Solutions in Shifts δ ± for a Nonlinear Dynamic Equation on Time Scales

Abstract

Let $𝕋\subset ℝ$ be a periodic time scale in shifts ${\delta }_{\pm }$. We use a fixed point theorem due to Krasnosel'skiĭ to show that nonlinear delay in dynamic equations of the form $x^{\mathrm{\Delta }}(t)=-a(t)x^{\sigma }(t)+b(t)x^{\mathrm{\Delta }}({\delta }_{-}(k,t)){\delta }_{-}^{\mathrm{\Delta }}(k,t)+q(t,x(t),x({\delta }_{-}(k,t))),t\in 𝕋$, has a periodic solution in shifts ${\delta }_{\pm }$. We extend and unify periodic differential, difference, $h$-difference, and $q$-difference equations and more by a new periodicity concept on time scales.