1. bookVolume 19 (2020): Issue 1 (December 2020)
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11 Dec 2014
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access type Open Access

Stability analysis of implicit fractional differential equation with anti–periodic integral boundary value problem

Published Online: 31 Dec 2020
Page range: 5 - 25
Received: 21 Jan 2019
Accepted: 18 Jul 2019
Journal Details
License
Format
Journal
First Published
11 Dec 2014
Publication timeframe
1 time per year
Languages
English
Abstract

In this manuscript, we study the existence, uniqueness and various kinds of Ulam stability including Ulam–Hyers stability, generalized Ulam– Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers– Rassias stability of the solution to an implicit nonlinear fractional differential equations corresponding to an implicit integral boundary condition. We develop conditions for the existence and uniqueness by using the classical fixed point theorems such as Banach contraction principle and Schaefer’s fixed point theorem. For stability, we utilize classical functional analysis. The main results are well illustrated with an example.

Keywords

[1] Abbas, Saïd et al. Implicit Fractional Differential and Integral Equations: Existence and Stability. Vol. 26 of De Gruyter Series in Nonlinear Analysis and Applications. Walter de Gruyter GmbH & Co KG, 2018. Cited on 6.Search in Google Scholar

[2] Ahmad, Bashir, Ahmed Alsaedi, and Badra S. Alghamdi. “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions.” Nonlinear Anal. Real World Appl. 9, no. 4 (2008): 1727-1740. Cited on 6.Search in Google Scholar

[3] Ahmad, Bashir and Ahmed Alsaedi. “Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions.” Nonlinear Anal. Real World Appl. 10, no. 1 (2009): 358-367. Cited on 6.Search in Google Scholar

[4] Ali, Arshad, Faranak Rabiei, and Kamal Shah. “On Ulams type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions.” J. Nonlinear Sci. Appl. 10, no. 9 (2017): 4760-4775. Cited on 6.Search in Google Scholar

[5] Ali, Zeeshan, Akbar Zada, and Kamal Shah. “Ulam stability results for the solutions of nonlinear implicit fractional order differential equations.” Hacet. J. Math. Stat. 48, no. 4 (2019): 1092-1109. Cited on 7.Search in Google Scholar

[6] Almeida, Ricardo, Nuno R.O. Bastos, and M. Teresa T. Monteiro. “Modeling some real phenomena by fractional differential equations” Math. Methods Appl. Sci., 39 no. 16 (2016): 4846-4855. Cited on 5.Search in Google Scholar

[7] Bagley, Ronald L., and Peter J. Torvik. “On the appereance of fractional derivatives in the behaviour of real materials.” J. Appl. Mech. 51, no. 2 (1984): 294-298. Cited on 5.Search in Google Scholar

[8] Benchohra, Mouffak, and Jamal E. Lazreg. “On stability for nonlinear implicit fractional differential equations.” Matematiche (Catania) 70, no. 2 (2015): 49-61. Cited on 6.Search in Google Scholar

[9] Granas, Andrzej, and James Dugundji. Fixed Point Theory. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003. Cited on 9.Search in Google Scholar

[10] Hilfer, Rudolf. Applications of Fractional Calculus in Physics. River Edge, New York: World Scientific Publishing Co. Inc., 2000. Cited on 5.Search in Google Scholar

[11] Hyers, Donald H. “On the stability of the linear functional equation.” Natl. Acad. Sci. USA 27, no. 4 (1941): 222-224. Cited on 6.Search in Google Scholar

[12] Khan, Rahmat Ali, and Kamal Shah. “Existence and uniqueness of solutions to fractional order multi–point boundary value problems.” Commun. Appl. Anal. 19 (2015): 515-526. Cited on 5.Search in Google Scholar

[13] Kilbas, Anatoly A., Oleg I. Marichev, and Stefan G. Samko. Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland, 1993. Cited on 5.Search in Google Scholar

[14] Kilbas, Anatoly A., Hari M. Srivastava, and Juan J. Trujillo. Theory and Applications of Fractional Diffrential Equations. Vol. 204 of North-Holland Mathematics Studies. Elsevier Science, 2006. Cited on 5 and 7.Search in Google Scholar

[15] Kumam, Poom, et all. “Existence results and Hyers–Ulam stability to a class of nonlinear arbitrary order differential equations”, J. Nonlinear Sci. Appl. 10, no. 6 (2017): 2986-2997. Cited on 6.Search in Google Scholar

[16] Lakshmikantham, Vangipuram, Sagar Leela, and Jonnalagedda Vasundhara Devi. Theory of Fractional Dynamic Systems, Cambridge: Cambridge Scientific Publishers, 2009. Cited on 5.Search in Google Scholar

[17] Lewandowski, Roman and B. Chorążyczewski. “Identification of parameters of the Kelvin–Voight and the Maxwell fractional models, used to modeling of viscoelasti dampers.” Computer and Structures 88, no. 1-2 (2010): 1-17. Cited on 5.Search in Google Scholar

[18] Li, Tongxing, and Akbar Zada. “Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces.” Adv. Difference Equ. paper no. 156 (2016): 8pp. Cited on 6.Search in Google Scholar

[19] Li, Yan, YangQuan Chen, and Igor Podlubny. “Stability of fractional–order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability.” Comput. Math. Appl. 59, no. 5 (2010): 1810-1821. Cited on 6.Search in Google Scholar

[20] Obłoza, Marta. “Hyers stability of the linear differential equation.” Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993): 259-270. Cited on 6.Search in Google Scholar

[21] Rus, Joan A. “Ulam stabilities of ordinary differential equations in a Banach space.” Carpathian J. Math. 26, no. 1 (2010): 103-107. Cited on 8.Search in Google Scholar

[22] Shah, Rahim, and Akbar Zada. “A fixed point approach to the stability of a nonlinear volterra integrodiferential equation with delay.” Hacettepe J. Math. Stat. 47, no. 3 (2018): 615-623. Cited on 6 and 9.Search in Google Scholar

[23] Shah, Syed Omar, Akbar Zada, and Alaa E. Hamza. “Stability analysis of the first order non–linear impulsive time varying delay dynamic system on time scales.” Qual. Theory Dyn. Syst. 18, no. 3 (2019): 825-840. Cited on 6.Search in Google Scholar

[24] Ulam, Stanisław. Problems in Modern Mathematics. New York: John Wiley and sons, 1940. Cited on 6.Search in Google Scholar

[25] Vanterler da C. Sousa, Jose, and Edmindo Capelas de Oliveira. “On the ψ– fractional integral and applications.” Comp. Appl. Math. 38, no. 4 (2019): 22 pp. Cited on 6.Search in Google Scholar

[26] Vanterler da C. Sousa, Jose, Kishor D. Kucche and Edmindo Capelas de Oliveira. “Stability of ψ–Hilfer impulsive fractional differential equations.” Appl. Math. Lett. 88 (2019): 73-80. Cited on 6.Search in Google Scholar

[27] Vanterler da C. Sousa, Jose, and Edmindo Capelas de Oliveira, “Ulam–Hyers stability of a nonlinear fractional Volterra integro–differential equation.” Appl. Math. Lett. 81 (2018): 50-56. Cited on 6.Search in Google Scholar

[28] Vanterler da C. Sousa, Jose, and Edmindo Capelas de Oliveira, “On the ψ–Hilfer fractional derivative.” Communication in Nonl. Sci. and Num. Simul. 60 (2018): 72-91. Cited on 6.Search in Google Scholar

[29] Wang, JinRong, and Xuezhu Li, “Ulam Hyers stability of fractional Langevin equations.” Appl. Math. Comput. 258, no. 1 (2015): 72-83. Cited on 6.Search in Google Scholar

[30] Wang, JinRong, Linli Lv, and Yong Zho, “Ulam stability and data dependec for fractional differential equations with Caputo derivative.” Elec. J. Qual. Theory. Diff. Equns. 63, no. 1 (2011): 1-10. Cited on 6.Search in Google Scholar

[31] Wang, JinRong, Akbar Zada, and Wajid 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, no. 5 (2018): 553-560. Cited on 6.Search in Google Scholar

[32] Yu, Fajun, “Integrable coupling system of fractional solution equation hierarchy.” Physics Letters A 373, no. 41 (2009): 3730-3733. Cited on 5.Search in Google Scholar

[33] Zada, Akbar, and Sartaj 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, no. 7 (2018): 763-774. Cited on 6.Search in Google Scholar

[34] Zada, Akbar, Sartaj Ali, and Yongjin 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): Paper No. 317 26pp. Cited on 6.Search in Google Scholar

[35] Zada, Akbar, Wajid Ali and Syed Farina, “Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses.” Math. Meth. App. Sci. 40, no. 15 (2017): 5502-5514. Cited on 6.Search in Google Scholar

[36] Zada, Akbar, Wajid Ali, and Choonkil 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. Cited on 6.Search in Google Scholar

[37] Zada, Akbar, and Syed Omar Shah, “Hyers-Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses.” Hacet. J. Math. Stat. 47, no. 5 (2018): 1196-1205. Cited on 6.Search in Google Scholar

[38] Zada, Akbar, Omar Shah, and Rahim Shah, “Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems.” Appl. Math. Comput. 271 (2015): 512-518. Cited on 5.Search in Google Scholar

[39] Zada, Akbar, Shaleena Shaleena, and Tongxing Li. “Stability analysis of higher order nonlinear differential equations in β –normed spaces.” Math. Methods Appl. Sci. 42, no. 4 (2019): 1151-1166. Cited on 6.Search in Google Scholar

[40] Zada, Akbar, Mohammad Yar, and Tongxing 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. Cited on 6.Search in Google Scholar

[41] Zada, Akbar, Peiguang Wang, Dhaou Lassoued and Tongxing Li, “Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems.” Adv. Difference Equ. 2017 (2017): Paper No. 192. Cited on 6.Search in Google Scholar

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