Existence of positive periodic solutions in shifts δ± for a nonlinear first order functional dynamic equation on time scales

Abstract

Let T ? R be a periodic time scale in shifts ?? with period P ? [t0,?)T. In this paper we consider the nonlinear functional dynamic equation of the form x?(t) = a(t)x(t)- ?b(t) f (x(h(t))), t ? T. By using the Krasnoselski?, 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 ?? 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.