Positive periodic solutions in shifts δ ± for a nonlinear first-order functional dynamic equation on time scales

Erbil Çetin

doi:10.1186/1687-1847-2014-76

Abstract

Let T⊂R be a periodic time scale in shifts δ ± with period P∈ [ t 0 , ∞ ) T . We consider the existence of positive periodic solutions in shifts δ ± for the nonlinear functional dynamic equation of the form x Δ (t)=−a(t) x σ (t)+λb(t)f ( t , x ( h ( t ) ) ) ,t∈T using the cone theory techniques. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales.MSC:34N05, 39A12, 35B10.

Highlights

Functional differential equations include many mathematical ecological and population models, such as the Lasota-Wazewska model [ – ], Nicholson’s blowflies model [, – ], the model for blood cell production [, ] etc
Chow [ ], Freedman and Wy [ ], Hadeler and Tomiuk [ ], Kuang [ ], Wang [ ], Weng and Sun [ ] and many others studied the existence of at least one and at least two positive periodic solutions of nonlinear first-order differential equations using the fixed point theorem of cone expansion and the cone compression method, the upper and lower solution method and iterative technique [ ]
It has been observed that very few papers exist in the literature on the existence of at least three and the nonexistence of a nonnegative periodic solution for first-order differential equations

Summary

Introduction

Functional differential equations include many mathematical ecological and population models, such as the Lasota-Wazewska model [ – ], Nicholson’s blowflies model [ , , – ], the model for blood cell production [ , , , ] etc. In Section , we state some facts about exponential function on time scales, the new periodicity concept for time scales and some important theorems which will be needed to show the existence and nonexistence of periodic solutions in shifts δ±. The following definitions, lemmas, corollaries and examples are about the shift operators and the new periodicity concept for time scales which can be found in [ ]. [ ] Let T∗ be a nonempty subset of the time scale T including a fixed number t ∈ T∗ such that there exist operators δ± : [t , ∞)T × T∗ → T∗ satisfying the following properties:. [ ] The following time scales are periodic in the sense of shift operators given in Definition .

Notice that the time scale in Example

We define

Aλ xn