On Retarded Integral Inequalities for Dynamic Systems on Time Scales

Abstract

The object of this paper is to establish some nonlinear retarded inequalities on time scales which can be used as handy tools in the theory of integral equations with time delays.

1. Introduction

Integral inequalities play an important role in the qualitative analysis of differential and integral equations. The well-known Gronwall inequality provides explicit bounds for solutions of many differential and integral equations. On the basis of various initiatives, this inequality has been extended and applied to various contexts (see, e.g., [1–4]), including many retarded ones (see, e.g., [5–9]).

Recently, Ye and Gao [7] obtained the following.

Theorem A. Let , , , , and with where . Then, the following assertions hold.
Suppose that . Then, where , , , , and If, in addition, and are nondecreasing -functions, then
Suppose that . Then, where , , , , , and If, in addition, and are nondecreasing -functions, then

In this paper, we will further investigate functions satisfying the following more general inequalities: where is any time scale, , , , , and are real-valued nonnegative rd-continuous functions defined on , and are positive constants, , , , and .

First, we make a preliminary definition.

Definition 1. We say that a function is regressive provided that holds, where is graininess function; that is, . The set of all regressive and rd-continuous functions will be denoted by .

2. Main Results

For convenience, we first cite the following lemma.

Lemma 2 (see [10]). Let , , ; then for any .

Lemma 3. Let , , , , for all , is rd-continuous on , and and are real constants. If is rd-continuous and then for and for .
Furthermore, if and are nondecreasing with , then where .

Proof. Let . Then, , and is positive, nondecreasing for . By Lemma 2, we get for . Multiplying (16) by , we get Integrating both sides from to , we obtain For , , so Using (18) and (19), we get for .
Noting that , inequalities (13) and (14) follow.
Finally, if and are nondecreasing, then for , by (14), we have If , by (13), The proof is complete.

Theorem 4. Assume that satisfies condition (8), , , , ; then where , , and + .
If, in addition, and are nondecreasing, and , then where

Proof. The second inequality in (23) is obvious. Next, we will prove the first inequality in (23). For , using Hölder’s inequality with indices and , we obtain from (8) By Jensen’s inequality , we get For the first integral in (26), we have the estimate Hence, and so Let ; then we have For , we have ; that is, . By Lemma 3, we get Hence, the first inequality in (23) follows.
Finally, if and are nondecreasing, and , by Lemma 3, we have The proof is complete.

Lemma 5. Let , , , , , and and let be rd-continuous on , where and are real constants. If is rd-continuous and then for and for .
Furthermore, if and are nondecreasing with , then