Extremal Solutions to Periodic Boundary Value Problem of Nabla Integrodifferential Equation of Volterra Type on Time Scales

Yunlong Shi,Junfang Zhao

doi:10.1155/2014/205659

Abstract

We firstly establish some new theorems on time scales, and then, by employing them together with a new comparison result and the monotone iterative technique, we show the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem:u∇(t)=f(t,u,∫0t‍g(t,s)∇s), t∈[0,a]T, u(0)=u(ρ(a)), whereTis a time scale.

Highlights

In this paper, we are concerned with the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem:t u∇ (t) = f (t, u, ∫ g (t, s) ∇s), t ∈ [0, a]T, (1)u (0) = u (ρ (a)), where T is a time scale and f, g satisfy (S1) f ∈ Cld([0, a]T × R × R, R), g ∈ Cld([0, a]T × [0, a]T, R).By proving a new comparison result and developing the monotone iterative technique, we show the extremal solutions of the periodic boundary value problem of nabla integrodifferential equations of Volterra type on time scales.The study of dynamic equations on time scales has been created in order to unify the study of differential and difference equations
We firstly establish some new theorems on time scales, and by employing them together with a new comparison result and the monotone iterative technique, we show the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem: u∇(t) = f(t, u, ∫0t g(t, s)∇s), t ∈ [0, a]T, u(0) = u(ρ(a)), where T is a time scale
We are concerned with the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem: t u∇ (t) = f (t, u, ∫ g (t, s) ∇s), t ∈ [0, a]T, (1)

Summary

Introduction

We are concerned with the existence of extremal solutions to the following nabla integrodifferential periodic boundary value problem:. By proving a new comparison result and developing the monotone iterative technique, we show the extremal solutions of the periodic boundary value problem of nabla integrodifferential equations of Volterra type on time scales. An integrodifferential equation on time scales (including time scale R) finds many applications in various mathematical problems [3] This leads to the extensive study of the existence of extremal solutions to such kind of equations; see Agarwal et al [4], Franco [5], Guo [6], Z. Monotone iterative technique coupled with the method of upper and lower solutions has been widely used in the treatment of existence results of initial and boundary value problems for nonlinear differential equations in recent years.

Preliminary

Some Important Lemmas

Main Results