Two New Integral Inequalities In Time Spaces

Gronwall-Bellman plays an important role in the research of boundedness,global existence,stability of solutions of differential and integral equations as well as difference equations.On the other hand,1980,Hilger created the theory of time scales as a theory capable to contain both difference and differential calculus in a consistent way.Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales.In these investigations, integral inequalities on time scales have been paid much attention by many authors.So,in this paper,two new integral inequalities on time scales are established: a new Gronwall-Bellman-type integral inequalities in two independent variables, a class of nonlinear delayed integral inequalities.Two integral inequalities have been studied in this article and I made some new conclusions.This article is divided into the following three chapters according to the contents:Chapter 1 Preference,this chapter gives some notes,de?nitions which are commonly used on time scales,and some theorems used in this paper.Chapter 2 In this chapter,I establish some new Gronwall-Bellman-type integral inequalities in two independent variables containing integration on in?nite intervals on time scales.I improve the studied integral inequality:∫∞∫∞where supx∈Tk(x) = ∞, u, a, f, g ∈ Crd(T × T, R+).to:∫∫where p, q are constants with p ≥ q > 0.On the other hand,I establish a more generalized inequality:∫∞∫∞∫∞implies on the other hand, u(x, y)also satis?es:In the last part,I give some example applications.It is the results of this paper come to practical example that is the meaning of studying inequality lies in.Chapter 3 In this chapter,I establish a new class of nonlinear delayed integral inequalities on time scales:∫∫satis?es:where u, a, b, fi(i = 1, 2) ∈ Crd(T0, R+),with a, bare nondecreasing,b(t) ≥ 1. f1(t), f2(t)is nonnegative function on t. τ ∈(T0, T), τ(t) ≤ t.-∞ < α = inf τ(t), t ∈ T0≤ t0,? ∈ Crd([α, t0] ∩ T, R+).p, q are constants with p > q > 0.Since, one generalized integral inequality is established:∫∫satis?es where u, a, b, f1, f2∈ Crd(T0, R+),w ∈ C(R+, R+),with a, b, ware nondecreasing function.η ∈C(R+, R+)is increasing function.τi∈(T0, T), τi≤ t, i = 1, 2.-∞ < α = inf{min{τi(t), i =1, 2}, t ∈ T0} ≤ t0.p, q are constants with p > q > 0.Another generalized inequality is considered in the last part of this part:∫Similary,some example applications are given in this chapter.This chapter generalizes the main results of [28].