The Ulam's types stability of non-linear Volterra integro-delay dynamic system with simple non-instantaneous impulses on time scales
Volume 4, Issue 1, pp 13--25
http://dx.doi.org/10.22436/mns.04.01.02
Publication Date: August 07, 2019
Submission Date: September 24, 2018
Revision Date: March 26, 2019
Accteptance Date: May 04, 2019
-
2018
Downloads
-
3996
Views
Authors
Syed Omar Shah
- Department of Mathematics, University of Peshawar, Peshawar 25000.
Akbar Zada
- Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan.
Cemil Tunc
- Department of Mathematics, Faculty of Sciences, Yuzuncu Yil University, 65080 Van, Turkey.
Abstract
This manuscript presents Hyers-Ulam stability and Hyers-Ulam-Rassias stability results of non-linear Volterra integro-delay dynamic system on time scales with non-instantaneous impulses. Picard fixed point theorem is used for obtaining existence and uniqueness of solutions. By means of abstract Gronwall lemma, Gronwall's inequality on time scales and applications of Gronwall's inequality on time scales, we establish Hyers-Ulam stability and Hyers-Ulam-Rassias stability results. There are some primary lemmas, inequalities and relevant assumptions that helps in our stability results.
Share and Cite
ISRP Style
Syed Omar Shah, Akbar Zada, Cemil Tunc, The Ulam's types stability of non-linear Volterra integro-delay dynamic system with simple non-instantaneous impulses on time scales, Mathematics in Natural Science, 4 (2019), no. 1, 13--25
AMA Style
Shah Syed Omar, Zada Akbar, Tunc Cemil, The Ulam's types stability of non-linear Volterra integro-delay dynamic system with simple non-instantaneous impulses on time scales. Math. Nat. Sci. (2019); 4(1):13--25
Chicago/Turabian Style
Shah, Syed Omar, Zada, Akbar, Tunc, Cemil. "The Ulam's types stability of non-linear Volterra integro-delay dynamic system with simple non-instantaneous impulses on time scales." Mathematics in Natural Science, 4, no. 1 (2019): 13--25
Keywords
- Hyers-Ulam stability
- time scale
- impulses
- delay dynamic equation
- Gronwall's inequality
- abstract Gronwall lemma
- Banach fixed point theorem
MSC
References
-
[1]
R. P. Agarwal, A. S. Awan, D. O'Regan, A. Younus, Linear impulsive Volterra integro--dynamic system on time scales, Adv. Difference Equ., 2014 (2014), 17 pages
-
[2]
Z. Ali, A. Zada, K. Shah, Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem, Bound. Value Probl., 2018 (2018), 16 pages
-
[3]
Z. Ali, A. Zada, K. Shah, Ulam stability results for the solutions of nonlinear implicit fractional order differential equations, Hacettepe J. Math. Stat., 2018 (2018), (in press)
-
[4]
C. Alsina, R. Ger, On some inequalities and stability results related to the exponential function, J. Inequal. Appl., 2 (1998), 373--380
-
[5]
S. András, A. R. Mészáros, Ulam--Hyers stability of dynamic equations on time scales via Picard operators, Appl. Math. Comput., 209 (2013), 4853--4864
-
[6]
D. D. Bainov, 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., 42 (1989), 29--32
-
[7]
D. D. Bainov, P. S. Simenov, Systems with impulse effect stability theory and applications, Ellis Horwood Limited, Chichester (1989)
-
[8]
M. Bohner, A. Peterson, Dynamic equations on time scales: an introduction with applications, Birkhäuser, Boston (2001)
-
[9]
M. Bohner, A. Peterson, Advances in dynamics equations on time scales, Birkhäuser, Boston (2003)
-
[10]
J. J. DaChunha, Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math., 176 (2005), 381--410
-
[11]
A. E. Hamza, K. M. Oraby, Stability of abstract dynamic equations on time scales, Adv. Difference Equ., 2012 (2012), 15 pages
-
[12]
S. Hilger, Analysis on measure chains--A unified approach to continuous and discrete calculus, Result math., 18 (1990), 18--56
-
[13]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222--224
-
[14]
S.-M. Jung, Hyers--Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004), 1135--1140
-
[15]
S.-M. Jung, Hyers--Ulam--Rassias stability of functional equations in nonlinear analysis, Springer, New York (2011)
-
[16]
Y. J. Li, Y. Shen, Hyers--Ulam stability of linear differential equations of second order, Appl. Math. Lett., 23 (2010), 306--309
-
[17]
T. X. Li, 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., 2016 (2016), 8 pages
-
[18]
T. X. Li, A. Zada, S. Faisal, Hyers--Ulam stability of nth order linear differential equations, J. Nonlinear Sci. Appl., 9 (2016), 2070--2075
-
[19]
Z. Lin, W. Wei, J. R. Wang, Existence and stability results for impulsive integro-differential equations, Facta Univ. Ser. Math. Inform., 29 (2014), 119--130
-
[20]
V. Lupulescu, A. Zada, Linear impulsive dynamic systems on time scales, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 30 pages
-
[21]
S. I. Nenov, Impulsive controllability and optimization problems in population dynamics, Nonlinear Anal., 36 (1999), 881--890
-
[22]
M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., 13 (1993), 259--270
-
[23]
M. Obłoza, Connections between Hyers and Lyapunov stability of the ordinary differential equations, Rocznik Nauk.-Dydakt. Prace Mat., 14 (1997), 141--146
-
[24]
C. Pötzsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time--invariant systems on time scales, Discrete Contin. Dyn. Syst., 9 (2003), 1223--1241
-
[25]
T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297--300
-
[26]
I. A. Rus, Gronwall lemmas: ten open problems, Sci. Math. Jpn., 70 (2009), 221--228
-
[27]
S. O. Shah, A. Zada, Uniform exponential stability for time varying linear dynamic systems over time scales, J. Anal. Number Theory, 5 (2017), 115--118
-
[28]
R. Shah, A. Zada, A fixed point approach to the stability of a nonlinear volterra integrodifferential equations with delay, Hacet. J. Math. Stat., 47 (2018), 615--623
-
[29]
S. O. Shah, A. Zada, Connections between Ulam--Hyers stability and uniform exponential stability of time varying linear dynamic systems over time scales, Sohag J. Math., 6 (2019), 1--4
-
[30]
S. O. Shah, A. Zada, Hyers--Ulam stability of non--linear Volterra integro--delay dynamic system with fractional integrable impulses on time scales, IJMSI., (), (in press)
-
[31]
S. O. Shah, A. Zada, A. E. Hamza, Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales, Qual. Theory Dyn. Syst., 2019 (2019), 16 pages
-
[32]
S. H. Tang, A. Zada, S. Faisal, M. M. A. El-Sheikh, T. X. Li, Stability of higher--order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl., 9 (2016), 4713--4721
-
[33]
S. M. Ulam, A Collection of Mathematical Problems, Interscience Publisheres, New York-London (1960)
-
[34]
S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, New York (1964)
-
[35]
X. Wang, M. Arif, A. Zada, $\beta$--Hyers--Ulam--Rassias stability of semilinear nonautonomous impulsive system, Symmetry, 11 (2019), 18 pages
-
[36]
J. R. Wang, M. Fečkan, Y. Zhou, Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395 (2012), 258--264
-
[37]
J. R. Wang, M. Fečkan, Y. Zhou, On the stability of first order impulsive evolution equations, Opuscula Math., 34 (2014), 639--657
-
[38]
J. R. Wang, A. Zada, W. Ali, Ulam's--type stability of first--order impulsive differential equations with variable delay in quasi--Banach spaces, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 553--560
-
[39]
J. R. Wang, Y. R. Zhang, A class of nonlinear differential equations with fractional integrable impulses, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3001--3010
-
[40]
A. Younus, D. O'Regan, N. Yasmin, S. Mirza, Stability criteria for nonlinear volterra integro--dynamic systems, Appl. Math. Inf. Sci., 11 (2017), 1509--1517
-
[41]
B. Zada, Uniform exponential stability in the sense of Hyers and Ulam for periodic time varying linear systems, Differ. Equ. Appl., 10 (2018), 227--234
-
[42]
A. Zada, S. Ali, Stability analysis of multi--point boundary value problem for sequential fractional differential equations with non--instantaneous impulses, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 763--774
-
[43]
A. Zada, W. Ali, S. Farina, Hyers--Ulam stability of nonlinear differential equations with fractional integrable impulses, Math. Methods Appl. Sci., 40 (2017), 5502--5514
-
[44]
A. Zada, S. Ali, Y. J. Li, Ulam--type stability for a class of implicit fractional differential equations with non--instantaneous integral impulses and boundary condition, Adv. Difference Equ., 2017 (2017), 26 pages
-
[45]
A. Zada, W. Ali, C. Park, Ulam's type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari's type, Appl. Math. Comput., 350 (2019), 60--65
-
[46]
A. Zada, U. Riaz, F. Ullah Khan, Hyers--Ulam stability of impulsive integral equations, Boll. Unione Mat. Ital., 2018 (2018), 1--15
-
[47]
A. Zada, S. O. Shah, Hyers--Ulam stability of first--order non--linear delay differential equations with fractional integrable impulses, Hacettepe J. Math. Stat., 47 (2018), 1196--1205
-
[48]
A. Zada, S. O. Shah, S. Ismail, T. X. Li, Hyers--Ulam stability in terms of dichotomy of first order linear dynamic systems, Punjab Univ. J. Math. (Lahore), 49 (2017), 37--47
-
[49]
A. Zada, S. O. Shah, Y. J. Li, Hyers--Ulam stability of nonlinear impulsive Volterra integro--delay dynamic system on time scales, J. Nonlinear Sci. Appl., 10 (2017), 5701--5711
-
[50]
A. Zada, O. Shah, R. Shah, Hyers--Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems, Appl. Math. Comput., 271 (2015), 512--518
-
[51]
A. Zada, S. Shaleena, T. X. Li, Stability analysis of higher order nonlinear differential equations in $\beta$--normed spaces, Math. Methods Appl. Sci., 42 (2019), 1151--1166
-
[52]
A. Zada, F. Ullah Khan, U. Riaz, T. X. Li, Hyers--Ulam stability of linear summation equations, Punjab Univ. J. Math. (Lahore), 49 (2017), 19--24
-
[53]
A. Zada, P. U. Wang, D. Lassoued, T. X. Li, Connections between Hyers--Ulam stability and uniform exponential stability of 2--periodic linear nonautonomous systems, Adv. Difference Equ., 2017 (2017), 7 pages
-
[54]
A. Zada, M. Yar, T. X. Li, Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions, Ann. Univ. Paedagog. Crac. Stud. Math., 17 (2018), 103--125