| Title:
|
Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales (English) |
| Author:
|
Raffoul, Youssef N. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
52 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
21-33 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this research we establish necessary and sufficient conditions for the stability of the zero solution of scalar Volterra integro-dynamic equation on general time scales. Our approach is based on the construction of suitable Lyapunov functionals. We will compare our findings with known results and provides application to quantum calculus. (English) |
| Keyword:
|
necessary |
| Keyword:
|
sufficient |
| Keyword:
|
time scales |
| Keyword:
|
Lyapunov functionals |
| Keyword:
|
stability |
| Keyword:
|
zero solution |
| MSC:
|
34D05 |
| MSC:
|
34N05 |
| MSC:
|
39A12 |
| MSC:
|
45D05 |
| idZBL:
|
Zbl 06562206 |
| idMR:
|
MR3475110 |
| DOI:
|
10.5817/AM2016-1-21 |
| . |
| Date available:
|
2016-02-29T18:30:35Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144837 |
| . |
| Reference:
|
[1] Adivar, M.: Function bounds for solutions of Volterra integrodynamic equations on time scales.EJQTDE (7) (2010), 1–22. MR 2577160 |
| Reference:
|
[2] Adivar, M., Raffoul, Y.: Existence results for periodic solutions of integro-dynamic equations on time scales.Ann. Mat. Pura Appl. (4) 188 (4) (2009), 543–559. DOI: http://dx.doi.org/10.1007/s1023-008-0088-z Zbl 1176.45008, MR 2533954, 10.1007/s10231-008-0088-z |
| Reference:
|
[3] Adivar, M., Raffoul, Y.: Stability and periodicity in dynamic delay equations.Comput. Math. Appl. 58 (2009), 264–272. Zbl 1189.34143, MR 2535793, 10.1016/j.camwa.2009.03.065 |
| Reference:
|
[4] Adivar, M., Raffoul, Y.: A note on “Stability and periodicity in dynamic delay equations”.Comput. Math. Appl. 59 (2010), 3351–3354. Zbl 1198.34150, MR 2651874, 10.1016/j.camwa.2010.03.025 |
| Reference:
|
[5] Adivar, M., Raffoul, Y.: Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation.An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat 21 (3) (2013), 17–32. Zbl 1313.45008, MR 3145088 |
| Reference:
|
[6] Akın–Bohner, E., Raffoul, Y.: Boundeness in functional dynamic equations on time scales.Adv. Difference Equ. (2006), 1–18. MR 2255160 |
| Reference:
|
[7] Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications.Birkhäuser, Boston, 2001. Zbl 0978.39001, MR 1843232 |
| Reference:
|
[8] Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales.Birkhäuser, Boston, 2003. Zbl 1025.34001, MR 1962542 |
| Reference:
|
[9] Bohner, M., Raffoul, Y.: Volterra Dynamic Equations on Time Scales.preprint. |
| Reference:
|
[10] Burton, T.A.: Stability and Periodic Solutions of Ordinary and Functional Differential Equations.Dover, New York, 2005. Zbl 1209.34001, MR 2761514 |
| Reference:
|
[11] Burton, T.A.: Stability by Fixed Point Theory for Functional Differential Equations.Dover, New York, 2006. Zbl 1160.34001, MR 2281958 |
| Reference:
|
[12] Eloe, P., Islam, M., Zhang, B.: Uniform asymptotic stability in linear Volterra integrodifferential equations with applications to delay systems.Dynam. Systems Appl. 9 (2000), 331–344. MR 1844634 |
| Reference:
|
[13] Grace, S., Graef, J., Zafer, A.: Oscillation of integro-dynamic equations on time scales.Appl. Math. Lett. 26 (4) (2013), 383–386. Zbl 1261.45005, MR 3019962, 10.1016/j.aml.2012.10.001 |
| Reference:
|
[14] Kulik, T., Tisdell, C.: Volterra integral equations on time scales: basic qualitative and quantitative results with applications to initial value problems on unbounded domains.Int. J. Difference Equ. 3 (1) (2008), 103–133. MR 2548121 |
| Reference:
|
[15] Lupulescu, V., Ntouyas, S., Younus, A.: Qualitative aspects of a Volterra integro-dynamic system on time scales.EJQTDE (5) (2013), 1–35. MR 3011509 |
| Reference:
|
[16] Peterson, A., Raffoul, Y.: Exponential stability of dynamic equations on time scales.Adv. Difference Equ. 2 (2005), 133–144. Zbl 1100.39013, MR 2197128 |
| Reference:
|
[17] Peterson, A., Tisdell, C.C.: Boundedness and uniqueness of solutions to dynamic equations on time scales.J. Differ. Equations Appl. 10 (13–15) (2004), 1295–1306. Zbl 1072.39017, MR 2100729, 10.1080/10236190410001652793 |
| Reference:
|
[18] Raffoul, Y.: Boundedness in nonlinear differential equations.Nonlinear Studies 10 (2003), 343–350. Zbl 1050.34046, MR 2021322 |
| Reference:
|
[19] Raffoul, Y.: Boundedness in nonlinear functional differential equations with applications to Volterra integrodifferential.J. Integral Equations Appl. 16 (4) (2004). Zbl 1090.34056, MR 2133906, 10.1216/jiea/1181075297 |
| . |
