Geodesic
Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales
Zada, Akbar 1 ; Shah, Syed Omar 1 ; Li, Yongjin 2
1 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2 Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 11, p. 5701-5711.

Voir la notice de l'article provenant de la source International Scientific Research Publications

This paper proves the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales via a fixed point approach. The uniqueness and existence of the solution of nonlinear impulsive Volterra integro-delay dynamic system is proved with the help of Picard operator. The main tools for proving our results are abstract Gronwall lemma and Banach contraction principle. We also make some assumptions along with Lipschitz condition which make our results appropriate for the approach we are using.
DOI : 10.22436/jnsa.010.11.08
Classification : 34N05, 34A37, 45M10, 45J05
Keywords: Hyers-Ulam stability, Hyers-Ulam-Rassias stability, time scale, nonlinear Volterra integro-delay dynamic system

Zada, Akbar  1 ; Shah, Syed Omar  1 ; Li, Yongjin  2

1 Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
2 Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China 
@article{JNSA_2017_10_11_a7,
     author = {Zada, Akbar  and Shah, Syed Omar  and Li, Yongjin },
     title = {Hyers-Ulam stability of nonlinear impulsive {Volterra} integro-delay dynamic system on time scales},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {5701-5711},
     publisher = {mathdoc},
     volume = {10},
     number = {11},
     year = {2017},
     doi = {10.22436/jnsa.010.11.08},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.11.08/}
}
TY  - JOUR
AU  - Zada, Akbar 
AU  - Shah, Syed Omar 
AU  - Li, Yongjin 
TI  - Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales
JO  - Journal of nonlinear sciences and its applications
PY  - 2017
SP  - 5701
EP  - 5711
VL  - 10
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.11.08/
DO  - 10.22436/jnsa.010.11.08
LA  - en
ID  - JNSA_2017_10_11_a7
ER  - 
%0 Journal Article
%A Zada, Akbar 
%A Shah, Syed Omar 
%A Li, Yongjin 
%T Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales
%J Journal of nonlinear sciences and its applications
%D 2017
%P 5701-5711
%V 10
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.11.08/
%R 10.22436/jnsa.010.11.08
%G en
%F JNSA_2017_10_11_a7
Zada, Akbar ; Shah, Syed Omar ; Li, Yongjin . Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 11, p. 5701-5711. doi : 10.22436/jnsa.010.11.08. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.11.08/

[1] Agarwal, R. P.; Awan, A. S.; O’Regan, D.; A. Younus Linear impulsive Volterra integro-dynamic system on time scales, Adv. Difference Equ., Volume 2014 (2014), pp. 1-17 | DOI

[2] Alsina, C.; Ger, R. On some inequalities and stability results related to the exponential function, J. Inequal. Appl., Volume 2 (1998), pp. 373-380 | DOI

[3] András, S.; Mészáros, A. R. Ulam-Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., Volume 209 (2013), pp. 4853-4864 | Zbl | DOI

[4] Baınov, D. D.; A. B. Dishliev Population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population, C. R. Acad. Bulgare Sci., Volume 42 (1989), pp. 29-32

[5] Bohner, M.; Peterson, A. Dynamic equations on time scales: An introduction with applications, Birkhäuser Boston, USA, 2001

[6] Bohner, M.; Peterson, A. Advances in Dynamics Equations on time scales, Birkhäuser Boston, USA, 2003 | DOI

[7] J. J. Dacunha Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math., Volume 176 (2005), pp. 381-410 | DOI

[8] Hamza, A. E.; K. M. Oraby Stability of abstract dynamic equations on time scales, Adv. Difference Equ., Volume 2012 (2012), pp. 1-15 | DOI

[9] Hilger, S. Analysis on measure chains-A unified approach to continuous and discrete calculus, Result math., Volume 18 (1990), pp. 18-56 | DOI | Zbl

[10] Hyers, D. H. On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., Volume 27 (1941), pp. 222-224 | DOI

[11] Jung, S.-M. Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., Volume 17 (2004), pp. 1135-1140 | DOI

[12] Jung, S.-M. Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., Volume 320 (2006), pp. 549-561 | DOI | Zbl

[13] S.-M. Jung Hyers-Ulam stability of linear partial differential equations of first order, Appl. Math. Lett., Volume 22 (2009), pp. 70-74 | DOI | Zbl

[14] Jung, S.-M.; J. Roh The linear differential equations with complex constant coefficients and Schrödinger equations, Appl. Math. Lett., Volume 66 (2017), pp. 23-29 | Zbl | DOI

[15] Li, Y.-J.; Shen, Y. Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett., Volume 23 (2010), pp. 306-309 | DOI

[16] Li, T.-X.; A. Zada Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces, Adv. Difference Equ., Volume 2016 (2016), pp. 1-8 | DOI

[17] Lupulescu, V.; A. Zada Linear impulsive dynamic systems on time scales, Electron. J. Qual. Theory Differ. Equ., Volume 2010 (2010), pp. 1-30

[18] Obłoza, M. Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., Volume 13 (1993), pp. 259-270

[19] M. Obłoza Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.- Dydakt. Prace Mat., Volume 14 (1997), pp. 141-146 | Zbl

[20] Pötzsche, C.; Siegmund, S.; Wirth, F. A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Sys., Volume 9 (2003), pp. 1223-1241 | Zbl | DOI

[21] Rassias, T. M. On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., Volume 72 (1978), pp. 297-300 | DOI

[22] Rus, I. A. Gronwall lemmas: ten open problems, Sci. Math. Jpn., Volume 70 (2009), pp. 221-228 | Zbl

[23] Ulam, S. M. A collection of the mathematical problems, Interscience Publishers, New York, 1960

[24] Ulam, S. M. Problem in modern mathematics, Science Editions John Wiley and Sons, New York, 1964

[25] Zada, A.; Ali, W.; Farina, S. Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Methods Appl. Sci., Volume 40 (2017), pp. 5502-5514 | DOI | Zbl

[26] Zada, A.; Faisal, S.; Y.-J. Li On the Hyers-Ulam stability of first-order impulsive delay differential equations, J. Funct. Spaces, Volume 2016 (2016), pp. 1-6 | Zbl

[27] Zada, A.; Li, T.-X.; Ismail, S.; Shah, O. Exponential dichotomy of linear autonomous systems over time scales, Diff. Equa. Appl., Volume 8 (2016), pp. 123-134 | Zbl

[28] Zada, A.; Shah, S. O.; Ismail, S.; Li, T.-X. Hyers-Ulam stability in terms of dichotomy of first order linear dynamic systems , Punjab Univ. J. Math., Volume 49 (2017), pp. 37-47 | Zbl

[29] Zada, A.; Shah, O.; R. Shah Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput., Volume 271 (2015), pp. 512-518 | DOI

[30] Zada, A.; Khan, F. Ullah; Riaz, U.; T.-X. Li Hyers-Ulam stability of linear summation equations, Punjab Univ. J. Math., Volume 49 (2017), pp. 19-24 | Zbl

[31] Zada, A.; Wang, P.; Lassoued, D.; Li, T.-X. Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems , Adv. Difference Equ., Volume 2017 (2017), pp. 1-7 | DOI

Cité par Sources :