STABILITY ANALYSIS OF CAUSAL INTEGRAL EVOLUTION IMPULSIVE SYSTEMS ON TIME SCALES

doi:10.1007/s10473-021-0310-2

Abstract

Abstract: In this article, we present the existence, uniqueness, Ulam-Hyers stability and Ulam-Hyers-Rassias stability of semilinear nonautonomous integral causal evolution impulsive integro-delay dynamic systems on time scales, with the help of a fixed point approach. We use Grönwall's inequality on time scales, an abstract Gröwall's lemma and a Picard operator as basic tools to develop our main results. To overcome some difficulties, we make a variety of assumptions. At the end an example is given to demonstrate the validity of our main theoretical results.

Key words: Time scale, Ulam-Hyers stability, impulses, semilinear nonautonomous system, Gr?nwall's inequality, dynamic system

CLC Number:

34N05

Jiafa XU, Bakhtawar PERVAIZ, Akbar ZADA, Syed Omar SHAH. STABILITY ANALYSIS OF CAUSAL INTEGRAL EVOLUTION IMPULSIVE SYSTEMS ON TIME SCALES[J].Acta mathematica scientia,Series B, 2021, 41(3): 781-800.

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References

[1] Agarwal R P, Arshad S, Lupulescu V. Evolution equations with causal operators. Differ Equ Appl, 2015, 7(1):15-26
[2] Ryan E P, Sangwin C J. Systems of controlled functional differential equations and adaptive tracking. Syst Control Lett, 2002, 47:365-374
[3] Agarwal R P, Awan A S, ÓRegan D, Younus A. Linear impulsive Volterra integro-dynamic system on time scales. Adv Differ Equ, 2014, 2014(6):1-17
[4] Ahmad M, Zada A, Alzabut J. Stability analysis of a nonlinear coupled implicit switched singular fractional differential system with p-Laplacian. Adv Differ Equ, 2019, 2019(436):1-22
[5] Ahmad M, Zada A, Alzabut J. Hyers-Ulam stability of a coupled system of fractional differential equations of Hilfer-Hadamard type. Demonstr Math, 2019, 52(1):283-295
[6] Ali Z, Kumam P, Shah K, Zada A. Investigation of Ulam stability results of a coupled system of nonlinear implicit fractional differential equations. Mathematics, 2019, 7(4):1-26
[7] Alsina C, Ger R. On some inequalities and stability results related to the exponential function. J Inequal Appl, 1998, 2:373-380
[8] Corduneanu C. Functional equations with causal operators. Stability and Control Taylor and Francis London:Theory Methods and applications, 2002
[9] Corduneanu C. A modified LQ-optimal control problem for causal functional differential equations. Nonlinear Dyn Syst Theory, 2004, 4:139-144
[10] András S, Mészáros A R. Ulam-Hyers stability of dynamic equations on time scales via Picard operators. Appl Math Comput, 2013, 209:4853-4864
[11] Bainov D D, Dishliev A. Population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population. Comp Rend Bulg Scie, 1989, 42:29-32
[12] Bainov B B, Simenov P S. Systems with impulse effect stability theory and applications. Ellis Horwood Limited:Chichester UK, 1989
[13] Bedivan D M, O'Regan D. The set of solutions for abstract Volterra equations in Lp([0, a], Rn). Appl Math Lett, 1999, 12:7-11
[14] O'Regan D. A note on the topological structure of the solutions set of abstract Volterra equations//Proceedings of the Royal Irish Academy-Section. Mathematical and Physical Sciences, 1999:99
[15] O'Regan D, Precup R. Existence criteria for integral equations in Banach spaces. J Ineq Appl, 2001, 6:77-97
[16] Ryan E P, Sangwin C J. Controlled functional differential equations and adaptive tracking. System Control Letters, 2002, 47:365-374
[17] Zhukovskii E S, Alves M J. Abstract Volterra Operators. Russian Mathematics, 2008, 52:1-4
[18] Karakostas G. Uniform asymptotic stability of causal operator equations. J Integral Equ, 1983, 5:59-71
[19] Tonelli L. Sulle equazioni funzionali di Vollterra. Bulletin of Callcutta Mathematical Sociaty, 1930, 20:31-48
[20] Bohner M, Peterson A. Dynamic equations on time scales:an introduction with applications. USA:Birkhäuser Boston Mass, 2001
[21] Bohner M, Peterson A. Advances in dynamics equations on time scales. USA:Birkhäuser Boston Mass, 2003
[22] Dachunha J J. Stability for time varying linear dynamic systems on time scales. J Comput Appl Math, 2005, 176(2):381-410
[23] Hamza A, Oraby K M. Stability of abstract dynamic equations on time scales. Adv Differ Equ, 2012, 2012(143):1-15
[24] Hilger S. Analysis on measure chains-A unified approach to continuous and discrete calculus. Result math, 1990, 18(1/2):18-56
[25] Hyers D H. On the stability of the linear functional equation. Proc Nat Acad Sci USA, 1941, 27(4):222-224
[26] Jung S -M. Hyers-Ulam stability of linear differential equations of first order. Appl Math Lett, 2004, 17(10):1135-1140
[27] Jung S -M. Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis. New York:Springer Optim Appl Springer, 2011:48
[28] Li Y, Shen Y. Hyers-Ulam stability of linear differential equations of second order. Appl Math Lett, 2010,23(3):306-309
[29] Li T, Zada A. Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv Differ Equ, 2016, 2016(153):1-8
[30] Lupulescu V, Zada A. Linear impulsive dynamic systems on time scales. Electron J Qual Theory Differ Equ, 2010, 2010(11):1-30
[31] Nenov S I. Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal Theory Methods Appl, 1999, 36(7):881-890
[32] Ob loza M. Hyers stability of the linear differential equation. Rocznik Nauk:Dydakt Prace Mat, 1993:259-270
[33] Ob loza M. Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk:Dydakt Prace Mat, 1997, 14:141-146
[34] Pötzsche C, Siegmund S, Wirth F. A spectral characterization of exponential stability for linear timeinvariant systems on time scales. Discrete Contin Dyn Sys, 2003, 9(5):1223-1241
[35] Rassias T M. On the stability of linear mappings in Banach spaces. Proc Amer Math Soc, 1978, 72(2):297-300
[36] Riaz U, Zada A, Ali Z, Ahmad M, Xu J, Fu Z. Analysis of nonlinear coupled systems of impulsive fractional differential equations with hadamard derivatives. Math Probl Eng, 2019, 2019:1-20
[37] Riaz U, Zada A, Ali Z, Cui Y, Xu J. Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives. Adv Differ Equ, 2019, 2019(226):1-27
[38] Ahangar R. Nonanticipating dynamical model and optimal control. Appl Math Lett, 1989, 2:15-18
[39] Rizwan R, Zada A. Nonlinear impulsive Langevin equation with mixed derivatives. Math Meth App Sci, 2020, 43(1):427-442
[40] Rizwan R, Zada A, Wang X. Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses. Adv Differ Equ, 2019, 2019(85):1-31
[41] Shah R, Zada A. A fixed point approach to the stability of a nonlinear Volterra integrodifferential equations with delay. Hacettepe J Math Stat, 2018, 47(3):615-623
[42] Shah S O, Zada A, Hamza A E. Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales. Qual Theory Dyn Syst, 2019, 18(3):825-840
[43] Shah S O, Zada A. On the stability analysis of non-linear Hammerstein impulsive integro-dynamic system on time scales with delay. Punjab Univ J Math, 2019, 51(7):89-98
[44] Shah S O, Zada A. Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales. Appl Math Comput, 2019, 359:202-213
[45] Stamov G, Alzabut J. Almost periodic solutions of impulsiveintegro-differential neural networks. Math Model Anal, 2010, 15(4):505-516
[46] Stamov G, Stamova I, Alzabut J. Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations. Nonlinear Anal Hybri, 2012, 6(2):818-823
[47] Ulam S M. A collection of the mathematical problems. London:Interscience Publisheres New York, 1960
[48] Ulam S M. Problem in modern mathematics. Science Editions J Wiley and Sons Inc Nework, 1964
[49] Wang X, Arif M, Zada A. β-Hyers-Ulam-Rassias stability of semilinear nonautonomous impulsive system. Symmetry, 2019, 231(11):1-18
[50] Wang J, Fečkan M, Tian Y. Stability analysis for a general class of non-instantaneous impulsive differential equations. Mediterr J Math, 2017, 14(46):1-21
[51] Wang J, Fečkan M, Zhou Y. Ulam's type stability of impulsive ordinary differential equations. J Math Anal Appl, 2012, 395:258-264
[52] Wang J, Fečkan M, Zhou Y. On the stability of first order impulsive evolution equations. Opuscula Math, 2014, 34(3):639-657
[53] Wang J, Li X. A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr J Math, 2016, 13:625-635
[54] Wang J, Zada A, Ali W. Ulam's-type stability of first-order impulsive differential equations with variable delay in quasi-Banach spaces. Int J Nonlin Sci Num, 2018, 19(5):553-560
[55] Wang J, Zada A, Waheed H. Stability analysis of a coupled system of nonlinear implicit fractional antiperiodic boundary value problem. Math Meth App Sci, 2019, 42(18):6706-6732
[56] Wang J, Zhang Y. A class of nonlinear differential equations with fractional integrable impulses. Com Nonl Sci Num Sim, 2014, 19:3001-3010
[57] Younus A, O'Regan D, Yasmin N, Mirza S. Stability criteria for nonlinear Volterra integro-dynamic systems. Appl Math Inf Sci, 2017, 11(5):1509-1517
[58] Zada A, Alam L, Kumam P, Kumam W, Ali G, Alzabut J. Controllability of impulsive non-linear delay dynamic systems on time scale. IEEE Access, 2020. Doi:ACCESS-2020-14723
[59] Zada A, Ali S. Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses. Int J Nonlinear Sci Numer Simul, 2018, 19(7):763-774
[60] Zada A, Ali W, Farina S. Hyers-Ulam stability of nonlinear differential equations with fractional integrable impulses. Math Methods Appl Sci, 2017, 40(15):5502-5514
[61] Zada A, Ali W, Park C. Ulam's type stability of higher order nonlinear delay differential equations via integral inequality of Grönwall-Bellman-Bihari's type. Appl Math Comput, 2019, 350:60-65
[62] Zada A, Alzabut J, Waheed H, Popa I L. Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions. Adv Differ Equ, 2020, 2020(64):1-50
[63] Zada A, Mashal A. Stability analysis of nth order nonlinear impulsive differential equations in quasi-Banach space. Numer Func Anal Opt, 2020, 41(3):294-321
[64] Zada A, Shah O, Shah R. Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl Math Comput, 2015, 271:512-518
[65] Zada A, Shah S O. Hyers-Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses. Hacettepe J Math Stat, 2018, 47(5):1196-1205
[66] Zada A, Shah S O, Li Y. Hyers-Ulam stability of nonlinear impulsive Volterra integro-delay dynamic system on time scales. J Nonlinear Sci Appl, 2017, 10(11):5701-5711
[67] Zada A, Shaleena S, Li T. Stability analysis of higher order nonlinear differential equations in β-normed spaces. Math Meth App Sci, 2019, 42(4):1151-1166
[68] Zada A, Waheed H, Alzabut J, Wang X. Existence and stability of impulsive coupled system of fractional integrodifferential equations. Demonstr Math, 2019, 52(1):296-335
[69] Makhlouf A B, Boucenna D, Hammami M A. Existence and stability results for generalized fractional differential equations. Acta Mathematica Scientia, 2020, 40:141-154