Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale. (English) Zbl 1226.34062

Summary: Let \(\mathbb T\) be a periodic time scale. The purpose of this paper is to use a modification of Krasnosel’skii’s fixed point theorem due to T. A. Burton to show that the totally nonlinear dynamic equation with functional delay
\[ x^\Delta(t)=-a(t)x^3(\sigma(t))+G(t,x^3(t), x^3(t-r(t))),\quad t\in\mathbb T, \]
has a periodic solution. We invert this equation to construct a sum of a compact map and a large contraction, which is suitable for applying the Burton-Krasnosel’skii theorem. Finally, an example is given to illustrate our result.


34K13 Periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations
34N05 Dynamic equations on time scales or measure chains