×

Existence and stability of impulsive coupled system of fractional integrodifferential equations. (English) Zbl 1439.45010

Summary: In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers-Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

MSC:

45J05 Integro-ordinary differential equations
34A08 Fractional ordinary differential equations
45M10 Stability theory for integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dalir M., Bashour M., Applications of fractional calculus, Appl. Math. Sci., 2010, 4, 1021-1032 · Zbl 1195.26011
[2] Khan H., Khan A., Abdeljawad T., Alkhazzan A., Existence results in Banach space for a nonlinear impulsive system, Adv. Difference Equ., 2019, 2019:18 · Zbl 1458.34127
[3] Khan A., Gómez-Aguilar J. F., Khan T. S., Khan H., Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos Solitons Fractals, 2019, 122, 119-128 · Zbl 1448.92307
[4] Khan H., Abdeljawad T., AslamM., Khan R. A., Khan A., Existence of positive solution and Hyers-Ulamstability for a nonlinear singular-delay-fractional differential equation, Adv. Difference Equ., 2019, 2019:104 · Zbl 1459.34024
[5] Khan H., Gómez-Aguilar J. F., Khan A., Khan T. S., Stability analysis for fractional order advection reaction diffusion system, Phys. A, 2019, 521, 737-751 · Zbl 1514.35461
[6] Hilfer R., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000 · Zbl 0998.26002
[7] Meral F., Royston T.,Magin R., Fractional calculus in viscoelasticity: an experimental study, Commun. Nonlinear Sci. Numer. Simul., 2010, 15, 939-945 · Zbl 1221.74012
[8] Benchohra M., Graef J. R., Hamani S., Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal., 2008, 87, 851-863 · Zbl 1198.26008
[9] Abdeljawad T., Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Difference Equ., 2017, 2017:313 · Zbl 1444.26003
[10] Babakhani A., Abdeljawad T., A Caputo fractional order boundary value problem with integral boundary conditions, J. Comput. Anal. Appl., 2013, 15(4), 753-763 · Zbl 1275.34006
[11] Abdeljawad T., Jarad F., Baleanu D., On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China, Mathematics, 2008, 51, 1775-1786 · Zbl 1179.26024
[12] Abdeljawad T., Baleanu D., Jarad F., Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys., 2008, 49(8), 083507-083507-11 · Zbl 1152.81550
[13] Abdeljawad T., Al-Mdallal Q. M., Discrete Mittag-Leffer kernel type fractional difference initial value problems and Gronwalls inequality, J. Comput. Appl. Math., 2018, 339, 218-230 · Zbl 1472.39006
[14] Kilbas A. A., Marichev O. I., Samko S. G., Fractional integrals and derivatives (theory and applications), Switzerland: Gordonand Breach, 1993 · Zbl 0818.26003
[15] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley, 1993 · Zbl 0789.26002
[16] Rehman M., Khan R., A note on boundary value problems for a coupled system of fractional differential equations, Comput. Math Appl., 2011, 61, 2630-2637 · Zbl 1221.34018
[17] 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, 763-774 · Zbl 1461.34014
[18] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, The Netherlands, 2006 · Zbl 1092.45003
[19] Podlubny I., Fractional Differential Equations, Academic Press, New York, 1999 · Zbl 0924.34008
[20] Ahmad B., Nieto J. J., Existence results for nonlinear boundary value problems of fractional integro-differential equations with integral boundary conditions, Boun. Value Prob., 2009, 2009:708576 · Zbl 1167.45003
[21] Chalishajar D. N., Karthikeyan K., Boundary value problems for impulsive fractional evolution integrodifferential equations with Gronwall’s inequality in Banach spaces, J. Dis. Nonl. Compl., 2014, 3, 33-48 · Zbl 1300.34151
[22] Chalishajar D. N., Karthikeyan K., Existence and uniqueness results for boundary value problems of higher order fractional integro-differential equations involving Gronwall’s inequality in Banach spaces, Acta Math. Sci. Ser., 2013, 33, 758-772 · Zbl 1299.34059
[23] Khan A., Shah K., Li Y., Khan T. S., Ulamtype stability for a coupled system of boundary value problems of nonlinear fractional differential, J. Funct. Spaces, 2017, Article ID 3046013 · Zbl 1377.34012
[24] Muslim M., Kumar A., Agarwal R. P., Exact controllability of fractional integro-differential systems of order α ∈ (1, 2] with deviated argument, Analele Universitatii Oradea, XXIV, 2017, 59, 185-194 · Zbl 1389.34239
[25] 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, 615-623 · Zbl 07033240
[26] Ahmad B., Nieto J. J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comp. Math. Appl., 2009, 58, 1838-1843 · Zbl 1205.34003
[27] Laskin N., Fractional market dynamics, Phys. A, 2000, 287, 482-492
[28] Lin W., Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 2007, 332, 709-726 · Zbl 1113.37016
[29] Ravichandran C., Logeswari K., Jarad F., New results on existence in the framework of Atangana-Baleanu derivative for fractional integro-differential equations, Chaos Solitons Fractals, 2019, 125, 194-200 · Zbl 1448.34024
[30] Gambo Y. Y., Ameen R., Jarad F., Abdeljawad T., Existence and uniqueness of solutions to fractional differential equations in the frame of generalized Caputo fractional derivatives, Adv. Difference Equ., 2018, 2018:134 · Zbl 1445.34013
[31] Chalishajar D., Kumar A., Existence, uniqueness and Ulam’s stability of solutions for a coupled system of fractional differential equations with integral boundary conditions, Mathematics, 2018, 6, 96 · Zbl 1405.34006
[32] Alzabut J., Almost periodic solutions for impulsive delay Nicholsons blowflies population model, J. Comput. Appl. Math., 2010, 234, 233-239 · Zbl 1196.34095
[33] Georgieva A., Kostadinov S., Stamov G. T., Alzabut J. O., On L_p(k) equivalence of impulsive differential equations and its applications to partial impulsive differential equations, Adv. Difference Equ., 2012, 2012:144 · Zbl 1347.34098
[34] Lakshmikantham V., Leela S., Vasundhara J., Theory of fractional dynamic systems, Cambridge, UK: Cambridge Academic Publishers, 2009 · Zbl 1188.37002
[35] Lakshmikanthan V., Bainov D. D., Simeonov P.S., Theory of impulsive differential equations, Singapore: World Scientific, 1989 · Zbl 0719.34002
[36] Lupulescu V., Zada A., Linear impulsive dynamic systems on time scales, Electron. J. Qual. Theory Differ. Equ., 2010, 11, 1-30 · Zbl 1201.34144
[37] Tang S., Zada A., Faisal S., El-Sheikh M. M. A., Li T., Stability of higher-order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl., 2016, 9, 4713-4721 · Zbl 1350.34022
[38] 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, 553-560 · Zbl 1401.34091
[39] Zada A., Faisal S., Li Y., On the Hyers-Ulam stability of first order impulsive delay differential equations, J. Funct. Spaces., 2016, Article ID 8164978 · Zbl 1342.34100
[40] Zada A., Riaz U., Khan F. U., Hyers-Ulam stability of impulsive integral equation, Boll. UnioneMat. Ital., 2019, 12(3), 453-467 · Zbl 1436.45001
[41] Zada A.,Mashal A., Stability analysis of n^th order nonlinear impulsive differential equations in quasi-Banach space, Numer. Funct. Anal. Optim., 2019, DOI:10.1080/01630563.2019.1628049 · Zbl 1432.34075
[42] Bainov D., Dimitrova M., Dishliev A., Oscillation of the bounded solutions of impulsive differential difference equations of second order, Appl. Math. Comput., 2000, 114, 61-68 · Zbl 1030.34062
[43] Chernousko F., Akulenko L., Sokolov B., Control of Oscilations, Moskow: Nauka, 1980 · Zbl 0574.49001
[44] Chua L. O., Yang L., Cellular neural networks: applications, IEEE Trans. Circ. syst., 1998, 35, 1273-1290
[45] Stamov G. Tr., Alzabut J. O., Almost periodic solutions of impulsive integrodifferential neural networks, Math. Model. Analysis, 2010, 15, 505-516 · Zbl 1219.45003
[46] Popov E., The Dynamics of Automatic Control Systems, Moskow: Goste-hizdat, 1964
[47] Andronov A., Witt A., Haykin S., Oscilation Theory, Moskow: Nauka, 1981
[48] Zavalishchin S., Sesekin A., Impulsive Processes: Models and Applications, Moskow: Nauka, 1991 · Zbl 0745.47042
[49] Babitskii V., Krupenin V., Vibration in Strongly Nonlinear Systems, Moskow: Nauka, 1985 · Zbl 0593.70024
[50] Stamov G. Tr., Alzabut J. O., Atanasov P., Stamov A. G., Almost periodic solutions for an impulsive delay model of price fluctuations in commodity markets, Nonlinear Anal. RWA, 2011, 12, 3170-3176 · Zbl 1231.34124
[51] Wang J., Zhang Y., Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 2014, 63(8), 1181-1190 · Zbl 1296.34034
[52] 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. Difference Equ., 2016, 2016:153 · Zbl 1419.39038
[53] 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., 2017, 2017:192 · Zbl 1422.34172
[54] Ali S., Abdeljawad T., Shah K., Jarad F., Arif M., Computation of iterative solutions along with stability analysis to a coupled system of fractional order differential equations, Adv. Difference Equ., 2019, 2019:215 · Zbl 1459.34006
[55] Ali A., Shah K., Jarad F., Gupta V., Abdeljawad T., Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations, Adv. Difference Equ., 2019, 2019:101 · Zbl 1459.34180
[56] Jarad F., Abdeljawad T., Hammouch Z., On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 2018, 117, 16-20 · Zbl 1442.34016
[57] Asma, Ali A., Shah K., Jarad F., Ulam-Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions, Adv. Difference Equ., 2019, 2019:7 · Zbl 1458.34126
[58] Ameena R., Jaradb F., Abdeljawad T., Ulam stability for delay fractional differential equations with a generalized Caputo derivative, Filomat, 2018, 32, 5265-5274 · Zbl 1513.34296
[59] Hyers D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 1941, 27, 222-224 · JFM 67.0424.01
[60] Ulam S. M., Problems in Modern Mathematics, Courier Corporation, 2004.
[61] Zada A., Shah S. O., Shah R., Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problem, Appl. Math. Comput., 2015, 271, 512-518 · Zbl 1410.39049
[62] 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, 1196-1205 · Zbl 1488.34396
[63] Wang J., Zhou Y., Wei W., Study in fractional differential equations by means of topological degree methods, Num. Func. Anal. Opti., 2012, 33, 216-238 · Zbl 1242.34012
[64] Tian Y., Ba Z., Impulsive boundary value problem for differential equationswith fractional order, Differ. Equ. Dyn. Syst., 2013, 21, 253-260 · Zbl 1273.34015
[65] Zhang X., Zhu C., Wu Z., Solvability for a coupled system of fractional differential equations with impulsis at resonance, Boun. Value Prob., 2013, 2013:80 · Zbl 1296.34044
[66] Shah K., Khalil H., Khan R. A., Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations, Elsevier Science B. V., Amsterdam, The Netherlands, 2015, 77, 240-246 · Zbl 1353.34028
[67] Benchohra M., Lazreg J. E., On the stability of nonlinear implicit fractional differential equations, LeMatematiche, 2015, 70, 49-61 · Zbl 1339.34007
[68] Zada A., Ali S., Li Y., Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Difference Equ., 2017, 2017:317 · Zbl 1444.34083
[69] Guo D., Lakshmikantham V., Nonlinear Problems in Abstract Cone, Academic Press, Orlando, 1988 · Zbl 0661.47045
[70] Yurko V. A., Boundary value problems with discontinuity conditions in an interior point of the interval, J. Diff. Equa., 2000, 36, 1266-1269 · Zbl 0991.34028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.