Existence of Positive Periodic Solutions in Shifts δ± for a Nonlinear First Order Functional Dynamic Equation on Time Scales

Abstract

Let $\mathbb{T}\subset\mathbb{R}$ be a periodic time scale in
shifts $\delta_{\pm}$ with period $P\in[t_0,\infty)_\mathbb{T}$. In
this paper we consider the
nonlinear functional dynamic equation of the form
\begin{eqnarray*} x^\nabla(t)=a(t)x(t)-\lambda b(t)f(x(h(t))),\ \ t\in \mathbb{T}. \end{eqnarray*}
By using the Krasnosel’ski\u{\i}, Avery-Henderson and Leggett-Williams fixed point theorems, we present different sufficient conditions for the nonexistence and
existence of at least one, two or three positive periodic solutions in shifts $\delta_\pm$ of the above problem on
time scales. We extend and unify periodic differential, difference, $h$-difference and $q$-difference equations and more by a new periodicity concept on time scales.