Stability analysis of implicit fractional differential equations with anti-periodic integral boundary value problem

Akbar Zada, Hira Waheed

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


Implicit fractional differential equations; Fixed point theorem; Hyers-Ulam Stability

Mathematics Subject Classification (2010)


26A33; 34A08; 35B40

References


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.

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.

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.

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.

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.

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.

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.

Benchohra, Mouffak, and Jamal E. Lazreg. "On stability for nonlinear implicit fractional differential equations." Matematiche (Catania) 70, no. 2 (2015): 49-61.

Granas, Andrzej, and James Dugundji. Fixed Point Theory. Springer Monographs in Mathematics. New York: Springer-Verlag, 2003.

Hilfer, Rudolf. Applications of Fractional Calculus in Physics. River Edge, New York: World Scientific Publishing Co. Inc., 2000.

Hyers, Donald H. "On the stability of the linear functional equation." Natl. Acad. Sci. USA 27, no. 4 (1941): 222-224.

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.

Kilbas, Anatoly A., Oleg I. Marichev, and Stefan G. Samko. Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland, 1993.

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.

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.

Lakshmikantham, Vangipuram, Sagar Leela, and Jonnalagedda Vasundhara Devi. Theory of Fractional Dynamic Systems, Cambridge: Cambridge Scientific Publishers, 2009.

Lewandowski, Roman and B. Chorazyczewski. "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.

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.

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.

Obłoza, Marta. "Hyers stability of the linear differential equation." Rocznik Nauk.-Dydakt. Prace Mat. 13 (1993): 259-270.

Rus, Joan A. "Ulam stabilities of ordinary differential equations in a Banach space." Carpathian J. Math. 26, no. 1 (2010): 103-107.

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.

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.

Ulam, Stanisław. Problems in Modern Mathematics. New York: John Wiley and sons, 1940.

Vanterler da C. Sousa, Jose, and Edmindo Capelas de Oliveira. "On the psi– fractional integral and applications." Comp. Appl. Math. 38, no. 4 (2019): 22 pp.

Vanterler da C. Sousa, Jose, Kishor D. Kucche and Edmindo Capelas de Oliveira. "Stability of psi-Hilfer impulsive fractional differential equations." Appl. Math. Lett. 88 (2019): 73-80.

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.

Vanterler da C. Sousa, Jose, and Edmindo Capelas de Oliveira, "On the psi–Hilfer fractional derivative." Communication in Nonl. Sci. and Num. Simul. 60 (2018): 72-91.

Wang, JinRong, and Xuezhu Li, "Ulam Hyers stability of fractional Langevin equations." Appl. Math. Comput. 258, no. 1 (2015): 72-83.

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.

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.

Yu, Fajun, "Integrable coupling system of fractional solution equation hierarchy." Physics Letters A 373, no. 41 (2009): 3730-3733.

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.

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.

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.

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.

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.

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.

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.

Zada, Akbar, Omar Shah, and Rahim Shah, "Hyers-Ulam stability of nonautonomous systems in terms of boundedness of Cauchy problems." Appl. Math. Comput. 271 (2015): 512-518.

Zada, Akbar, Shaleena Shaleena, and Tongxing Li. "Stability analysis of higher order nonlinear differential equations in –normed spaces." Math. Methods Appl. Sci. 42, no. 4 (2019): 1151-1166. Cited on 6.


Full Text: PDF

Download statistics: 570

Licencja Creative Commons
This article by Akbar Zada, Hira Waheed is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej