Discrete Dynamics in Nature and Society
Volume 2019 (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

Haidong Liu

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

Correspondence should be addressed to Haidong Liu; tomlhd983@163.com

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 [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 scales

where are rd-continuous functions and are positive constants such that .

Tian et al. [37] investigated the nonlinear integral inequality on time scales

where 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 that

where is continuous at , with , and are rd-continuous on . Then

implies

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

implies

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 satisfies

then

where

Proof. From Lemma 1 and (10), we get that

where is defined as in (16). Denote

From the assumptions on , (18) and (19), we obtain that is nondecreasing and

Combining (19) and (20), we have that

Applying Lemma 2 on the right side of (21), we get that

where , and are defined as in (12), (14), and (15). From (13), we get

and by (22), we obtain

Hence

equivalently

where is defined as in (17). Note that is rd-continuous and ; from Lemma 4 and (26), we have that

This 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 satisfies

then

where

and is defined as in (14).

Proof. From Lemma 1 and (34), we get that

where is defined as in (16). Denote

From the assumptions on , (42) and (43), we obtain that is nondecreasing and

Combining (43) and (44), we have that

Applying 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 satisfies

then

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 conditions

where , and are rd-continuous functions and , and are constants satisfying (i) or (ii) .
If is a solution of (50) and for , then

where

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).

References

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.

M. Adivar and Y. N. Raffoul, “Existence results for periodic solutions of integro-dynamic equations on time scales,” Annali di Matematica Pura ed Applicata, vol. 188, no. 4, pp. 543–559, 2009.

F. M. Atici, D. C. Biles, and A. Lebedinsky, “An application of time scales to economics,” Mathematical and Computer Modelling, vol. 43, no. 7-8, pp. 718–726, 2006.

H. D. Liu and F. W. Meng, “Some new generalized Volterra-Fredholm type discrete fractional sum inequalities and their applications,” Journal of Inequalities and Applications, vol. 2016, article 213, p. 16, 2016.

A. Dogan, “On the existence of positive solutions of the p-Laplacian dynamic equations on time scales,” Mathematical Methods in the Applied Sciences, vol. 40, no. 12, pp. 4385–4399, 2017.

A. Dogan, “Positive solutions of the p-Laplacian dynamic equations on time scales with sign changing nonlinearity,” Electronic Journal of Differential Equations, vol. 39, pp. 1–17, 2018.

H. D. Liu and F. W. Meng, “Existence of positive periodic solutions for a predator-prey system of Holling type IV function response with mutual interference and impulsive effects,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 138984, 12 pages, 2015.

D. L. Zhao and H. D. Liu, “Coexistence in a two species chemostat model with Markov switchings,” Applied Mathematics Letters, vol. 94, pp. 266–271, 2019.

L. Erbe, B. G. Jia, and A. Peterson, “On the asymptotic behavior of solutions of Emden-Fowler equations on time scales,” Annali di Matematica Pura ed Applicata, vol. 191, no. 2, pp. 205–217, 2012.

H. D. Liu and F. W. Meng, “Interval oscillation criteria for second-order nonlinear forced differential equations involving variable exponent,” Advances in Difference Equations, no. 291, p. 14, 2016.

M. Federson, J. G. Mesquita, and A. Slavik, “Measure functional differential equations and functional dynamic equations on time scales,” Journal of Differential Equations, vol. 252, no. 6, pp. 3816–3847, 2012.

E. Tunç and H. D. Liu, “Oscillatory behavior for second-order damped differential equation with nonlinearities including Riemann-Stieltjes integrals,” Electronic Journal of Differential Equations, vol. 2018, article 54, p. 12, 2018.

H. D. Liu, “Some new integral inequalities with mixed nonlinearities for discontinuous functions,” Advances in Difference Equations, no. 22, p. 16, 2018.

L. Erbe, B. Jia, and A. Peterson, “Belohorec-type oscillation theorem for second order sublinear dynamic equations on time scales,” Mathematische Nachrichten, vol. 284, no. 13, pp. 1658–1668, 2011.

D. Hu and Z. Zhang, “Four positive periodic solutions to a Lotka-Volterra cooperative system with harvesting terms,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 1115–1121, 2010.

H. D. Liu and F. W. Meng, “Some new nonlinear integral inequalities with weakly singular kernel and their applications to FDEs,” Journal of Inequalities and Applications, no. 209, p. 17, 2015.

A. Slavik, “Averaging dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 996–1012, 2012.

H. D. Liu, “Lyapunov-type inequalities for certain higher-order difference equations with mixed non-linearities,” Advances in Difference Equations, no. 229, p. 14, 2018.

Z. Q. Zhang and J. D. Cao, “Periodic solutions for complex-valued neural networks of neutral type by combining graph theory with coincidence degree theory,” Advances in Difference Equations, vol. 2018, no. 261, p. 23, 2018.

B. Karpuz, “Volterra theory on time scales,” Results in Mathematics, vol. 65, no. 3-4, pp. 263–292, 2014.

S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten [Ph.D. thesis], Universität Würzburg, 1988.

R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities & Applications, vol. 4, no. 4, pp. 535–557, 2001.

D. R. Anderson, “Nonlinear dynamic integral inequalities in two independent variables on time scale pairs,” Advances in Dynamical Systems and Applications (ADSA), vol. 3, no. 1, pp. 1–13, 2008.

H. D. Liu, “An improvement of the Lyapunov inequality for certain higher order differential equations,” Journal of Inequalities and Applications, no. 215, p. 9, 2018.

L. Du and R. Xu, “Some new Pachpatte type inequalities on time scales and their applications,” Journal of Mathematical Inequalities, vol. 6, no. 2, pp. 229–240, 2012.

F. Meng and J. Shao, “Some new Volterra-Fredholm type dynamic integral inequalities on time scales,” Applied Mathematics and Computation, vol. 223, pp. 444–451, 2013.

W. N. Li and W. Sheng, “Some nonlinear integral inequalities on time scales,” Journal of Inequalities and Applications, Article ID 70465, p. 15, 2007.

W. N. Li and M. Han, “Bounds for certain nonlinear dynamic inequalities on time scales,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 897087, 14 pages, 2009.

E. A. Bohner, M. Bohner, and F. Akin, “Pachpatte inequalities on time scale,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 1, article no. 6, 2005.

S. H. Saker, “Some nonlinear dynamic inequalities on time scales,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 633–645, 2011.

Y. Sun and T. Hassan, “Some nonlinear dynamic integral inequalities on time scales,” Applied Mathematics and Computation, vol. 220, pp. 221–225, 2013.

Q. Feng, F. Meng, and B. Zheng, “Gronwall-Bellman type nonlinear delay integral inequalities on time scales,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 772–784, 2011.

H. D. Liu, “On some nonlinear retarded Volterra-Fredholm type integral inequalities on time scales and their applications,” Journal of Inequalities and Applications, no. 211, p. 19, 2018.

D. L. Zhao, S. L. Yuan, and H. D. Liu, “Random periodic solution for a stochastic SIS epidemic model with constant population size,” Advances in Difference Equations, no. 64, p. 9, 2018.

H. D. Liu, “A class of retarded Volterra-Fredholm type integral inequalities on time scales and their applications,” Journal of Inequalities and Applications, no. 293, p. 15, 2017.

Y. Z. Tian, Y. L. Cai, L. Z. Li, and T. X. Li, “Some dynamic integral inequalities with mixed nonlinearities on time scales,” Journal of Inequalities and Applications, vol. 2015, article 12, pp. 1–10, 2015.

H. D. Liu and F. W. Meng, “Nonlinear retarded integral inequalities on time scales and their applications,” Journal of Mathematical Inequalities, vol. 12, no. 1, pp. 219–234, 2018.

J. Wang, F. Meng, and J. Gu, “Estimates on some power nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales,” Advances in Difference Equations, vol. 2017, article 257, p. 16, 2017.

J. Gu and F. W. Meng, “Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales,” Applied Mathematics and Computation, vol. 245, pp. 235–242, 2014.

W. N. Li and W. Sheng, “Some Gronwall type inequalities on time scales,” Journal of Mathematical Inequalities, vol. 4, no. 1, pp. 67–76, 2010.

F. C. Jiang and F. W. Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 479–486, 2007.