Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE AND STABILITY RESULTS OF GENERALIZED FRACTIONAL INTEGRODIFFERENTIAL EQUATIONS

  • Kausika, C. (Department of Mathematics Bharathiar Univerity) ;
  • Balachandran, K. (Department of Mathematics Bharathiar Univerity) ;
  • Annapoorani, N. (Department of Mathematics Bharathiar Univerity) ;
  • Kim, J.K. (Department of Mathematics Education Kyungnam University)
  • Received : 2021.02.19
  • Accepted : 2021.04.09
  • Published : 2021.12.15
Download PDF
⟨ Previous Next ⟩

Abstract

This paper gives sufficient conditions to ensure the existence and stability of solutions for generalized nonlinear fractional integrodifferential equations of order α (1 < α < 2). The main theorem asserts the stability results in a weighted Banach space, employing the Krasnoselskii's fixed point technique and the existence of at least one mild solution satisfying the asymptotic stability condition. Two examples are provided to illustrate the theory.

Keywords

Acknowledgement

The work of first author is supported by the Department of Science and Technology, Government of India, Grant No: SR/WOS-A/PM-34/2019(G).

References

  1. N. Abdellouahab, B. Tellab and K. Zennir, Existence and stability results of a nonlinear fractional integro-differential equation with integral boundary conditions, Kragujev. J. Math., 46 (2022), 685-699. https://doi.org/10.46793/KgJMat2205.685A
  2. R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006
  3. R. Almeida, A.B. Malinowska and T. Odzijewicz, Fractional differential equations with dependence on the Caputo-Katugampola derivative, J. Comput. Nonlinear Dyn., 11 (2016), 061017, 11 pages. https://doi.org/10.1115/1.4034432
  4. A. Ardjouni, H. Boulares and Y. Laskri, Stability in higher-order nonlinear fractional differential equations, Acta et Comment. Univ. Tartu. de Math., 22 (2018), 37-47. https://doi.org/10.12697/ACUTM.2018.22.04
  5. K. Balachandran and S. Divya, Controllability of nonlinear neutral fractional integrodifferential systems with infinite delay, J. Appl. Nonlinear Dyn., 6 (2017), 333-344. https://doi.org/10.5890/JAND.2017.09.002
  6. D. Baleanu, G.C. Wu and S.D. Zeng, Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Soliton Fract., 102 (2017), 99-105. https://doi.org/10.1016/j.chaos.2017.02.007
  7. A. Ben Makhlouf, D. Boucenna and M.A. Hammami, Existence and stability results for generalized fractional differential equations, Acta Math. Sci., 40B (2020), 141-154. https://doi.org/10.1007/s10473-020-0110-3
  8. A. Ben Makhlouf and A.M. Nagy, Finite-time stability of linear Caputo-Katugampola fractional-order time delay systems, Asian J. Control, 22 (2020), 297-306. https://doi.org/10.1002/asjc.1880
  9. A. Boulfoul, B. Tellab, N. Abdellouahab and K. Zennir, Existence and uniqueness results for initial value problem of nonlinear fractional integro-differential equation on an unbounded domain in a weighted Banach space, Math. Methods Appl. Sci., 2020 (2020), 1-12.
  10. T.A. Burton and B. Zhang, Fractional differential equations and generalizations of Schaefer's and Krasnoselskii's fixed point theorems, Nonlinear Anal., 75 (2012), 6485-6495. https://doi.org/10.1016/j.na.2012.07.022
  11. C.D. Constantinescu, J.M. Ramirez and W.R. Zhu, An application of fractional differential equations to risk theory, Financ. Stoch., 23 (2019), 1001-1024. https://doi.org/10.1007/s00780-019-00400-8
  12. F. Ge and C. Kou, Stability analysis by Krasnoselskii's fixed point theorem for nonlinear fractional differential equations, Appl. Math. Comput., 257 (2015), 308-316. https://doi.org/10.1016/j.amc.2014.11.109
  13. F. Jarada, T. Abdeljawad and C. Baleanua, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619. https://doi.org/10.22436/jnsa.010.05.27
  14. B. Kamalapriya, K. Balachandran and N. Annapoorani, Existence results for fractional integrodifferential equations, Nonlinear Funct. Anal. Appl., 22(3) (2017), 641-653. https://doi.org/10.22771/NFAA.2017.22.03.12
  15. B. Kamalapriya, K. Balachandran and N. Annapoorani, Existence results of fractional neutral integrodifferential equations with deviating arguments, Discontinuity, Nonlinearity, Complex., 9 (2020), 277-287. https://doi.org/10.5890/DNC.2020.06.008
  16. C. Kausika, Stability of higher order nonlinear implicit fractional differential equations by fixed point technique, J. Vib. Test. Syst. Dyn., 5 (2021), 169-180.
  17. C. Kausika, K. Balachandran and N. Annapoorani, Linearized asymptotic stability of Caputo-Katugampola fractional integro differential equations, Indian J. Ind. Appl. Math., 10 (2019), 217-230. https://doi.org/10.5958/1945-919x.2019.00036.7
  18. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  19. C. Kou, H. Zhou and Y. Yan, Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal., 74 (2011), 5975-5986. https://doi.org/10.1016/j.na.2011.05.074
  20. M.A. Krasnoselskii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl., 10 (1958), 345-409.
  21. J.T. Machado, V. Kiryakova and F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140-1153. https://doi.org/10.1016/j.cnsns.2010.05.027
  22. N. Sene, Stability analysis of the generalized fractional differential equations with and without exogenous inputs, J. Nonlinear Sci. Appl., 12 (2019), 562-572. https://doi.org/10.22436/jnsa.012.09.01
  23. N. Sene and G. Srivastava, Generalized Mittag-Leffler input stability of the fractional differential equations, Symmetry, 11 (2019), 608. https://doi.org/10.3390/sym11050608
  24. V.E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media, Theor. Math. Phys., 158 (2009), 355-359. https://doi.org/10.1007/s11232-009-0029-z
  25. M.D. Tran, V. Ho and H.N. Van, On the stability of fractional differential equations involving generalized Caputo fractional derivative, Math. Probl. Eng., 2020 (2020), Article ID 1680761, 14 pages.