Nonlinear Dynamical Integral Inequalities in Two Independent Variables and Their Applications

Yuangong Sun

doi:10.1155/2011/320794

Abstract

In this paper, we investigate some nonlinear dynamical integral inequalities involving the forward jump operator in two independent variables. These inequalities provide explicit bounds on unknown functions, which can be used as handy tools to study the qualitative properties of solutions of certain partial dynamical systems on time scales pairs.

Highlights

Theory of dynamical equations on time scales, which goes back to Hilger’s landmark paper 1, has received considerable attention in recent years
Since dynamical integral inequalities usually can be used as handy tools to study the qualitative theory of dynamical equations on time scales, many researchers devoted to the study of different types of integral inequalities on time scales
The main purpose of this paper is to investigate several nonlinear integral inequalities in two independent variables on time scale pairs, which can be used to estimate explicit bounds of solutions of certain partial dynamical equations on time scales

Summary

Introduction

Theory of dynamical equations on time scales, which goes back to Hilger’s landmark paper 1 , has received considerable attention in recent years. To the best of our knowledge, the theory of partial dynamic equations on time scales has received less attention 20–24. The main purpose of this paper is to investigate several nonlinear integral inequalities in two independent variables on time scale pairs, which can be used to estimate explicit bounds of solutions of certain partial dynamical equations on time scales. Unlike some existing results in the literature e.g., 12 , the integral inequalities considered in this paper involve the forward jump operator σ t and σ s on a pair of time scales T and T, which results in difficulties in the estimation on the explicit bounds of unknown functions u t, s for t ∈ T and s ∈ T. For an excellent introduction to the calculus on time scales, we refer the reader to monographs 2, 3

Problem Statements

Main Results

Applications