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

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 [1–21]). 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 [23–41] 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 where , and are defined as in (36), (38), and (40). By a similar argument with Theorem 5 in the remaining proof of theorem, one can prove (35). This completes the proof.

If we let , , , and in Theorem 9, then we obtain the following corollary.

Corollary 10. Assume that , and are defined the same as in Theorem 9 and for . Suppose that satisfiesthen where and , , and are defined the same as in (30), (31), and (32).

Remark 11. Corollary 10 extends Theorem  2.2 in [37] to the cases and .

Remark 12. If , and , then Theorem 9 reduces to Theorem  3.8 in [28]. If and , then Theorem 9 reduces to Theorem  2.11 in [29].

4. Applications

In this section, we apply our results to study the boundedness of the solutions of a dynamic equation on time scales.

Example 13. Consider the following dynamic equation on time scales: where and are constants, is continuous function satisfying and for , and and are continuous functions.

Theorem 14. Suppose that the functions and in (50) satisfy the conditionswhere , and are rd-continuous functions and , and are constants satisfying (i) or (ii) .
If is a solution of (50) and for , thenwhere

Proof. The equivalent integral equation of (50) is denoted by Using the assumptions (52) and (53), we have Then, a suitable application of Theorem 5 to (61) yields (54).

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The author is indebted to the anonymous referees for their valuable suggestions and helpful comments which helped improve the paper significantly. This research was supported by the Natural Science Foundation of Shandong Province (China) (No. ZR2018MA018) and the National Natural Science Foundation of China (Nos. 11671227 and 61873144).