Some fractional dynamic inequalities on time scales of Hardy's type

Volume 23, Issue 2, pp 98--109 http://dx.doi.org/10.22436/jmcs.023.02.03

Publication Date: October 15, 2020 Submission Date: July 22, 2020 Revision Date: August 24, 2020 Accteptance Date: September 27, 2020

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Authors

A. G. Sayed - Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt. S. H. Saker - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. A. M. Ahmed - Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt. - Department of Mathematics, College of Science, Jouf University, Sakaka (2014), Kingdom of Saudi Arabia.

Abstract

In this paper, we prove some new fractional dynamic inequalities on time scales of Hardy's type due to Yang and Hwang. The results will be proved by employing the chain rule, Hölder's inequality, and integration by parts on fractional time scales. Several well-known dynamic inequalities on time scales will be obtained as special cases from our results.

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Keywords

• Hardy's inequality
• Yang and Hwang's inequality
• Copson's inequality
• Hölder's inequality
• time scales
• conformable fractional calculus

MSC

•  26A15
•  26D10
•  26D15
•  39A13
•  34A40
•  34N05

References

• [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57--66
  • View Article
  • Google Scholar


• [2] P. Agarwal, A.-A. Hyder, M. Zakarya, Well-posedness of stochastic modified Kawahara equation, Adv. Difference Equ., 2020 (2020), 10 pages
  • View Article
  • MathSciNet


• [3] P. Agarwal, A. Hyder, M. Zakarya, G. AlNemer, C. Cesarano, D. Assante, Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin-Bona-Mahony Equations, Axioms, 8 (2019), 15 pages
  • View Article
  • Google Scholar


• [4] R. P. Agarwal, D. O’Regan, S. H. Saker, Hardy Type Inequalities on Time Scales, Springer, Switzerland (2016)
  • View Article
  • MathSciNet


• [5] A. M. Ahmed, G. AlNemer, M. Zakarya, H. M. Rezk, Some Dynamic Inequalities of Hilbert’s Type, J. Funct. Spaces, 2020 (2020), 13 pages
  • View Article
  • MathSciNet


• [6] A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for conformable fractional integrals, Asian J. Math. Comput. Res., 15 (2017), 205--212
  • View Article
  • Google Scholar


• [7] N. Benkhettou, S. Hassani, D. F. M. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci., 28 (2016), 93--98
  • View Article
  • Google Scholar


• [8] G. Bennett, Some elementary inequalities, Quart. J. Math., 38 (1987), 401--425
  • View Article
  • MathSciNet


• [9] K. Bogdan, B. Dyda, The best constant in a fractional Hardy inequality, Math. Nachr., 284 (2011), 629--638
  • View Article
  • MathSciNet


• [10] M. Bohner, T. S. Hassan, T. Li, Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments, Indag. Math., 29 (2019), 548--560
  • View Article
  • MathSciNet


• [11] M. Bohner, T. Li, Kamenev-type criteria for nonlinear damped dynamic equations, Sci. China Math., 58 (2015), 1445--1452
  • View Article
  • MathSciNet


• [12] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston (2011)
  • View Article
  • Google Scholar


• [13] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston (2003)
  • View Article
  • MathSciNet


• [14] Y.-M. Chu, M. A. Khan, T. Ali, S. S. Dragomir, Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 12 pages
  • View Article
  • Google Scholar


• [15] E. T. Copson, Note on series of positive terms, J. London Math. Soc, 2 (1927), 9--12
  • View Article


• [16] E. T. Copson, Note on series of positive terms, J. London Math. Soc., 3 (1928), 49--51
  • View Article


• [17] E. T. Copson, Some integral inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 75 (1976), 157--164
  • View Article


• [18] H. A. Ghany, A.-A. Hyder, M. Zakarya, Exact solutions of stochastic fractional Korteweg de-Vries equation with conformable derivatives, Chin. Phys. B, 29 (2020), 1--18
  • View Article


• [19] Notes on a theorem of Hilbert, G. H. Hardy, Math. Z., 6 (1920), 314--317
  • View Article


• [20] G. H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math., 54 (1925), 150--156
  • Google Scholar


• [21] G. H. Hardy, Notes on some points in the integral calculus LXIV: Further inequalities between integrals, Messenger Math., 57 (1928), 12--16
  • View Article


• [22] S. Hilger, Analysis on measure chains--a unified approach to continuous and discrete calculus, Results. Math., 18 (1990), 18--56
  • View Article
  • MathSciNet


• [23] C. Jiang, A. Zada, M. T. S¸enel, T. Li, Synchronization of bidirectional N-coupled fractional-order chaotic systems with ring connection based on antisymmetric structure, Adv. Difference Equ., 2019 (2019), 16 pages
  • View Article
  • MathSciNet


• [24] M. Jleli, B. Samet, Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions, Math. Inequal. Appl., 18 (2015), 443--451
  • View Article
  • MathSciNet


• [25] V. Kac, P. Cheung, Quantum Calculus, Springer, New York (2001)
  • View Article
  • MathSciNet


• [26] R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65--70
  • View Article
  • MathSciNet


• [27] M. A. Khan, T. Ali, S. S. Dragomir, M. Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals, Rev. R. Acad. Cienc. Exactas Fıs. Nat., Ser. A Mat., 112 (2017), 1033--1048
  • View Article
  • MathSciNet


• [28] L. Leindler, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math., (Szeged), 31 (1970), 279--285
  • MathSciNet
  • Google Scholar


• [29] N. Levinson, Generalization of an inequality of Hardy, Duke Math. J., 31 (1964), 389--394
  • View Article
  • MathSciNet


• [30] E. R. Nwaeze, D. F. M. Torres, Chain rules and inequalities for the BHT fractional calculus on arbitrary timescales, Arab. J. Math., (Springer), 6 (2017), 13--20
  • View Article
  • MathSciNet


• [31] S. H. Saker, Some nonlinear dynamic inequalities on time scales and applications, J. Math. Inequal., 4 (2010), 561--579
  • View Article
  • MathSciNet


• [32] S. H. Saker, Hardy-Leindler type inequalities on Time Scales, Appl. Math. Inf. Sci., 8 (2014), 2975--2981
  • View Article
  • MathSciNet


• [33] S. H. Saker, M. R. Kenawy, G. AlNemer, M. Zakarya, Some Fractional Dynamic Inequalities of Hardy’s Type Via Conformable Calculus, J. Mathematics., 8 (2020), 1--15
  • View Article
  • Google Scholar


• [34] S. H. Saker, R. R. Mahmoud, M. M. Osman, R. P. Agarwal, Some new generalized forms of Hardy’s type inequality on time scales, Math. Inequal. Appl., 20 (2017), 459--481
  • View Article
  • MathSciNet


• [35] S. H. Saker, D. O’Regan, R. Agarwal, Generalized Hardy, Copson, Leindler and Bennett inequalities on time scales, Math. Nachr., 287 (2014), 686--698
  • View Article
  • MathSciNet


• [36] M. Z. Sarikaya, Hardy Type Inequalities for Conformable Fractional Integrals, Konuralp J. Math., 8 (2020), 211--215
  • View Article
  • Google Scholar


• [37] M. Z. Sarikaya, H. Budak, Opial type inequalities for conformable fractional integrals, RGMIA Res. Rep. Collect., 19 (2016), 11 pages
  • Google Scholar


• [38] M. Z. Sarikaya, H. Budak, New inequalities of Opial type for conformable fractional integrals, Turk J. Math., 41 (2017), 1164--1173
  • View Article
  • MathSciNet


• [39] M. Z. Sarikaya, H. Yaldiz, H. Budak, Steffensen’s integral inequality for conformable fractional integrals, Int. J. Anal. Appl., 15 (2017), 23--30
  • View Article
  • Google Scholar


• [40] E. Set, A. Gözpınar, A. Ekinci,, Hermite-Hadamard type inequalities via conformable fractional integrals, Acta Math. Univ. Comen., 86 (2017), 309--320
  • View Article
  • MathSciNet
  • Google Scholar


• [41] G.-S. Yang, D.-Y. Hwang, Generalizations of some reverse integral inequalities, J. Math. Anal. Appl., 233 (1999), 193--204
  • View Article
  • MathSciNet


• [42] C. Yildiz, M. E. Özdemir, H. K. Önelan, Fractional integral inequalities for different functions, New Trends in Math. Sci., 3 (2015), 110--117
  • View Article
  • MathSciNet