Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 16 (2021), 275 -- 299

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This work is licensed under a Creative Commons Attribution 4.0 International License.

FURTHER NEW REFINEMENTS IN h-STABILITY CONDITIONS FOR NONLINEAR ABSTRACT DYNAMIC EQUATIONS ON TIME SCALES AND APPLICATIONS

Amira Ayari and Khaled Boukerrioua

Abstract. In this paper, we study the h-stability problems of some classes of dynamic equations as an extension of exponential stability. We derive some sufficient conditions that guarantee h-stability of perturbed dynamic equations using Grönwall-Bihari and Pachpatte type integral inequality approch. Finally, numerical examples are introduced to illustrate the applicability of the main results.

2020 Mathematics Subject Classification: 74H55, 93D20, 26E70, 35A23.
Keywords: Abstract dynamic equation, time scale, integral inequality, semi-group, h-stability.

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References

  1. R.P. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl. 4 (2001) 535--557. MR1859660. Zbl 1021.34005.

  2. B. Aulbach and S. Hilger, A unified approach to continuous and discrete dynamics. Qualitative theory of differential equations (Szeged, 1988), 37-56, Colloq. Math. Soc. Jnos Bolyai, 53, North-Holland, Amsterdam, 1990. Zbl 0713.34050.

  3. B. Ben Nasser, K. Boukerrioua, M. A. Hammami. On the stability of perturbed time scale systems using integral inequalities, Applied Sciences, 16 (2014), 56 - 71. MR3224499. Zbl 1327.74077.

  4. B. Ben Nasser, K. Boukerrioua, M. A. Hammami, On stability and stabilization of perturbed time scale systems with Grönwall inequalities, Zh. Mat. Fiz. Anal. Geom. 11 (2015), 207--235. MR3443272. Zbl 1336.34129.

  5. B. Ben Nasser, K. Boukerrioua, M. Defoort, M. Djemai, M.A. Hammami, State feedback stabilization of a class of uncertain nonlinear systems on nonuniform time domains, Systems Control Lett. 97 (2016), 18--26. MR3571050. Zbl 1350.93067.

  6. B. Ben Nasser, K. Boukerrioua, M. Defoort, M. Djemai, M.A. Hammami and Taous-Meriem Laleg-Kirati, Sufficient conditions for uniform exponential stability and h-stability of some classes of dynamic equations on arbitrary time scales, Nonlinear Analysis: Hybrid Systems 32, 54-64. MR3880199. Zbl 1427.34119.

  7. B. Ben Nasser, K. Boukerrioua, M. Defoort, M. Djemai and M.A. Hammami, Refinements of some Pachpatte and Bihari inequalities on time scales, Nonlinear Dyn. Syst. Theory,2(3) (2017), 807-825. MR3702176. Zbl 1386.26032.

  8. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. Zbl 0978.39001.

  9. M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston,Mass, USA, 2003. Zbl 1025.34001.

  10. M. Bohner, Some oscillation criteria for first order delay dynamic equations, Far East J. Appl. Math. 18(3) (2005), 289-304. Zbl 1080.39005.

  11. M. Bohner and A.A. Martynyuk, Elements of Stability Theory of A.M. Liapunov for Dynamic Equations on Time Scales, Nonlinear Dynamics and Systems Theory, 7 (3) (2007), 225--251. MR2352908. Zbl 1133.93035.

  12. S. K. Choi, D. M. Im, and N. Koo, Stability of Linear Dynamic Systems on Time Scales, Advances in Difference Equations volume 2008, Article ID 670203, 12 pages. MR2393475. Zbl 1151.34322.

  13. S. K. Choi, Y. H. Goo, N. Koo, h-Stability of Dynamic Equations on Time Scales with Nonregressivity, Abstract and Applied Analysis. (2008), Article ID 632474, 13 pages. MR2417228. Zbl 1154.34023.

  14. S. K. Choi, N. J. Koo, D. M. Im, h-Stability for Linear Dynamic Equations on Time Scales, J.Math. Anal. Appl. 324 (2006), 707--720. MR2262503. Zbl 1112.34031.

  15. J. J. Dacunha, Stability for Time Varying Linear Dynamic Systems on Time Scales, J. Comput. Appl. Math. 176 (2005), 381-410. MR2116401. Zbl 1064.39005.

  16. J.M. Davis, I.A. Gravagne, B.J. Jackson, R.J. Marks II and A.A. Ramos, The Laplace transform on time scales revisited, J. Math. Anal. and Appl. 332 (2007), 1291-1307. MR2324337. Zbl 1120.44001.

  17. A. E. Hamza, M. A. Al-Qubaty, On the Exponential Operator Functions on Time Scales, Advances in Dynamical Systems and Applications, 7 (2012), No. 1, 57-80. MR2911992.

  18. A. E. Hamza, K. M. Oraby, Stability of Abstract Dynamic Equations on Time Scales, Advances in Difference Equations, 2012, 2012:143, 15 pp. MR3016337. Zbl 1347.34136.

  19. A. E. Hamza, K. M. Oraby, Semigroups of Operators and Abstract Dynamic Equations onTime Scales, Applied Mathematics and Computation, 270 (2015), 334--48. MR3406893. Zbl 1410.34271.

  20. A. E.Hamza,K. M. Oraby, Characterizations of stability of abstract dynamic equations on time scales, Commun. Korean Math. Soc, 34(1) (2019), 185-202. MR3912186 . Zbl 1431.34099.

  21. H.R. Henriquez, C. Lizama, J.G. Mesquita, Semigroups on time scales and applications to abstract Cauchy problems, Topological Methods in Nonlinear Analysis 56(1) (2020) 83-115. MR4175072.

  22. A. Martynyuk, Stability theory for dynamic equations on time scales. Systems & Control: Foundations & Applications. Basel: Birkh\ddotauser/Springer,2016. Zbl 1359.34002.

  23. K. M. Oraby, Asymptotic Behavior of Solutions of Dynamic Equations on Time Scales, M.SC thesis, Cairo University, 2012.

  24. B Neggal, K Boukerrioua, B Kilani and I Meziri, h-stability for nonlinear abstract dynamic equations on time scales and applications, Rendiconti del Circolo Matematico di Palermo Series 2 (2019), 1-15.

  25. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlage, New York, 44, 1983. Zbl 0516.47023.

  26. M. Pinto, Perturbations of Asymptotically Stable Differential Systems, Analysis, 4 no. 1-2, (1984), 161--175. MR775553. Zbl 0568.34035.

  27. M. Pinto, Stability of nonlinear differential system. Appl. Anal. 43 (1992) , 1--20. MR1284758. Zbl 0748.34029.

  28. M. Pinto, Asymptotic integration of a system resulting from the perturbation of an h-system. J. Math. Anal. Appl. 131 (1988) , 194--216. MR934441. Zbl 0656.34042.

  29. C.Tunç, A Note on Boundedness of Solutions to a Class of Non-autonomous Differential Equations of Second Order, Appl. Anal. Discrete Math. 4 (2010), 361-372. MR2724643. Zbl 1299.34128.

  30. C. Tunç and O. Tunç, On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, Journal of Advanced Research, 7 (1) (2016), 165--168.

  31. C. Tunç, A note on the stability and boundedness of solutions to non-linear differential systems of second order, Journal of the Association of Arab Universities for Basic and Applied Sciences, 24 (2017), 169--175.



A. Ayari
Lanos Laboratory,
Faculty of Sciences, Badji Mokhtar-Annaba University,
P.O. Box 12, 23000 Annaba, Algeria.
E-mail: ayari.amira1995@gmail.com

K. Boukerrioua
Lanos Laboratory,
Faculty of Sciences, Badji Mokhtar-Annaba University,
P.O. Box 12, 23000 Annaba, Algeria.
E-mail: khaledv2004@yahoo.fr



http://www.utgjiu.ro/math/sma