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Discrete Dynamics in Nature and Society
Volume 2019, Article ID 9860302, 9 pages
https://doi.org/10.1155/2019/9860302
Research Article

Some New Half-Linear Integral Inequalities on Time Scales and Applications

School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

Correspondence should be addressed to Haidong Liu; moc.361@389dhlmot

Received 13 April 2019; Accepted 23 May 2019; Published 16 June 2019

Academic Editor: Zhengqiu Zhang

Copyright © 2019 Haidong Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main aim of this paper is to establish some new half-linear integral inequalities on time scales. Our results not only extend and complement some known integral inequalities, but also provide a handy tool for the study of qualitative properties of solutions of some dynamic equations.

1. Introduction

The study of dynamic equations on time scales is now an important object of research and has been extensively studied in recent years (see [121]). This is due to the fact that the theory of time scales which was introduced by Stefan Hilger [22] can unify and extend the difference and differential calculus in a consistent way.

Because dynamic inequalities play an important role in the study of qualitative properties of solutions of dynamic equations on time scales, many authors have expounded on various classes of dynamic inequalities in recent years; see [2341] and the references cited therein. For instance, in 2013, Sun and Hassan [32] investigated the nonlinear integral inequality on time scaleswhere are rd-continuous functions and are positive constants such that .

Tian et al. [37] investigated the nonlinear integral inequality on time scaleswhere and are real constants and are rd-continuous functions.

We note that the inequalities (1) and (2) have been proved in the cases when and , respectively. So it would be interesting to find the explicit bound for of (2) in the following two cases when or .

In the present paper, we study some new half-linear integral inequalities on time scales. Our results not only complement the results established in [37] in the sense that the results can be applied in cases when or , but also furnish a handy tool for the study of qualitative properties of solutions of some complicated dynamic equations.

2. Preliminaries

In what follows, we always assume that , is an arbitrary time scale. The following lemmas are useful in the proof of the main results of this paper.

Lemma 1. Let , and be given; then for each ,holds for the cases when or .

Proof. If , then it is easy to see that the inequality (3) holds. So we only prove that the inequality (3) holds in the case of . For the case , set , where and . Letting , we get . Since , ; , , attains its maximum at and Thus, (3) holds. For the case , by a similar argument with the case , we can get that (3) holds. The proof is complete.

Lemma 2 (see [42]). Assuming that , and , then for any ,

Lemma 3 (see [1, Theorem 1.117]). Suppose that for each there exists a neighborhood of , independent of , such thatwhere is continuous at , with , and are rd-continuous on . Thenimplies

Lemma 4 (see [1, Theorem 6.1]). Suppose that and are rd-continuous functions and . Thenimplies

3. Main Results

In this section, we deal with some half-line inequalities on time scales. For convenience, we always assume that

Theorem 5. Assume that , are rd-continuous functions, , for , is continuous function satisfying and for , , and are constants satisfying (i) or (ii) .
Suppose that satisfiesthenwhere

Proof. From Lemma 1 and (10), we get thatwhere is defined as in (16). DenoteFrom the assumptions on , (18) and (19), we obtain that is nondecreasing andCombining (19) and (20), we have thatApplying Lemma 2 on the right side of (21), we get thatwhere , and are defined as in (12), (14), and (15). From (13), we get and by (22), we obtain Hence equivalentlywhere is defined as in (17). Note that is rd-continuous and ; from Lemma 4 and (26), we have thatThis and (20) imply the desired inequality (11). This completes the proof.

If we let , , and in Theorem 5, then we obtain the following corollary.

Corollary 6. Assume that and are defined the same as in Theorem 5 and for . Suppose that satisfies then where

Remark 7. Corollary 6 extends Theorem  2.1 in [37] to the cases and .

Remark 8. If , and , then Theorem 5 reduces to Theorem  3.2 in [28]. If , and , then Theorem 5 reduces to Theorem  2.4 in [29].

Theorem 9. Assume that , , and are defined the same as in Theorem 5, and for . Let and be defined as in Lemma 3 such that and for and (5) holds. Suppose that satisfiesthenwhereand is defined as in (14).

Proof. From Lemma 1 and (34), we get thatwhere is defined as in (16). DenoteFrom the assumptions on , (42) and (43), we obtain that is nondecreasing andCombining (43) and (44), we have thatApplying Lemma 2 on the right side of (45), we have that