Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales

Youssef N. Raffoul

Archivum Mathematicum (2016)

  • Volume: 052, Issue: 1, page 21-33
  • ISSN: 0044-8753

Abstract

top
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.

How to cite

top

Raffoul, Youssef N.. "Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales." Archivum Mathematicum 052.1 (2016): 21-33. <http://eudml.org/doc/276749>.

@article{Raffoul2016,
abstract = {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.},
author = {Raffoul, Youssef N.},
journal = {Archivum Mathematicum},
keywords = {necessary; sufficient; time scales; Lyapunov functionals; stability; zero solution},
language = {eng},
number = {1},
pages = {21-33},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales},
url = {http://eudml.org/doc/276749},
volume = {052},
year = {2016},
}

TY - JOUR
AU - Raffoul, Youssef N.
TI - Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales
JO - Archivum Mathematicum
PY - 2016
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 052
IS - 1
SP - 21
EP - 33
AB - 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.
LA - eng
KW - necessary; sufficient; time scales; Lyapunov functionals; stability; zero solution
UR - http://eudml.org/doc/276749
ER -

References

top
  1. Adivar, M., Function bounds for solutions of Volterra integrodynamic equations on time scales, EJQTDE (7) (2010), 1–22. (2010) MR2577160
  2. Adivar, M., Raffoul, Y., 10.1007/s10231-008-0088-z, Ann. Mat. Pura Appl. (4) 188 (4) (2009), 543–559. DOI: http://dx.doi.org/10.1007/s1023-008-0088-z (2009) Zbl1176.45008MR2533954DOI10.1007/s10231-008-0088-z
  3. Adivar, M., Raffoul, Y., 10.1016/j.camwa.2009.03.065, Comput. Math. Appl. 58 (2009), 264–272. (2009) Zbl1189.34143MR2535793DOI10.1016/j.camwa.2009.03.065
  4. Adivar, M., Raffoul, Y., 10.1016/j.camwa.2010.03.025, Comput. Math. Appl. 59 (2010), 3351–3354. (2010) Zbl1198.34150MR2651874DOI10.1016/j.camwa.2010.03.025
  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. (2013) Zbl1313.45008MR3145088
  6. Akın–Bohner, E., Raffoul, Y., Boundeness in functional dynamic equations on time scales, Adv. Difference Equ. (2006), 1–18. (2006) MR2255160
  7. Bohner, M., Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. (2001) Zbl0978.39001MR1843232
  8. Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. (2003) Zbl1025.34001MR1962542
  9. Bohner, M., Raffoul, Y., Volterra Dynamic Equations on Time Scales, preprint. 
  10. Burton, T.A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover, New York, 2005. (2005) Zbl1209.34001MR2761514
  11. Burton, T.A., Stability by Fixed Point Theory for Functional Differential Equations, Dover, New York, 2006. (2006) Zbl1160.34001MR2281958
  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. (2000) MR1844634
  13. Grace, S., Graef, J., Zafer, A., 10.1016/j.aml.2012.10.001, Appl. Math. Lett. 26 (4) (2013), 383–386. (2013) Zbl1261.45005MR3019962DOI10.1016/j.aml.2012.10.001
  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. (2008) MR2548121
  15. Lupulescu, V., Ntouyas, S., Younus, A., Qualitative aspects of a Volterra integro-dynamic system on time scales, EJQTDE (5) (2013), 1–35. (2013) MR3011509
  16. Peterson, A., Raffoul, Y., Exponential stability of dynamic equations on time scales, Adv. Difference Equ. 2 (2005), 133–144. (2005) Zbl1100.39013MR2197128
  17. Peterson, A., Tisdell, C.C., 10.1080/10236190410001652793, J. Differ. Equations Appl. 10 (13–15) (2004), 1295–1306. (2004) Zbl1072.39017MR2100729DOI10.1080/10236190410001652793
  18. Raffoul, Y., Boundedness in nonlinear differential equations, Nonlinear Studies 10 (2003), 343–350. (2003) Zbl1050.34046MR2021322
  19. Raffoul, Y., 10.1216/jiea/1181075297, J. Integral Equations Appl. 16 (4) (2004). (2004) Zbl1090.34056MR2133906DOI10.1216/jiea/1181075297

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.