EXISTENCE AND ULAM–HYERS STABILITY OF SOLUTIONS FOR SOME BACKWARD IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES

Abdelouahab Mahmoudi     (Dynamical Systems Laboratory, Faculty of Mathematics, University of Sciences and Technology Houari Boumediene (USTHB), Po Box 32, El Alia, Bab Ezzouar 16111, Algiers, Algeria)
Arezki Kessi     (Dynamical Systems Laboratory, Faculty of Mathematics, University of Sciences and Technology Houari Boumediene (USTHB), Po Box 32, El Alia, Bab Ezzouar 16111, Algiers, Algeria)

Abstract


This paper investigates the existence and generalized Ulam–Hyers stability of solutions for a class of nonlinear backward impulsive fractional differential equations in Banach spaces with both local and nonlocal conditions. By reformulating the problem as an equivalent integral equation, we apply classical fixed point theorems, including those of Banach, Schaefer, and Krasnoselskii, to derive sufficient conditions for the existence and uniqueness of solutions. The generalized Ulam–Hyers stability of the solutions is analyzed under suitable assumptions on the nonlinearities. Several illustrative examples are provided to demonstrate the applicability and effectiveness of the theoretical results.

Keywords


Backward impulsive, Caputo derivative, Nonlocal conditions, Ulam stability, Fixed point

Full Text:

PDF

References


  1. Benchohra M., Henderson J., Ntouyas S. Impulsive Differential Equations and Inclusions. Contemp. Math. Appl., vol. 2. New York: Hindawi Publishing Corporation, 2006. 366 p.
  2. Benchohra M., Slimani B.A. Existence and uniqueness of solutions to impulsive fractional differential equations. Electron. J. Differential Equations, 2009. Vol. 2009, No. 10. P. 1–11.
  3. Byszewski L. Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl., 1991. Vol. 162, No. 2. P. 494–505. DOI: 10.1016/0022-247X(91)90164-U
  4. Byszewski L. Existence and uniqueness of a classical solutions to a functional-differential abstract nonlocal Cauchy problem. J. Appl. Math. Stoch. Anal., 1999. Vol. 12, No. 1. P. 91–97. DOI: 10.1155/S1048953399000088
  5. Byszewski L., Lakshmikantham V. Theorem about the existence and uniqueness of solutions of a nonlocal Cauchy problem in a Banach space. Appl. Anal., 1991. Vol. 40, No. 1. P. 11–19. DOI: 10.1080/00036819008839989
  6. Dauer J.P., Mahmudov N.I., Matar M.M. Approximate controllability of backward stochastic evolution equations in Hilbert spaces. J. Math. Anal. Appl., 2006. Vol. 323, No. 1. P. 42–56. DOI: 10.1016/j.jmaa.2005.09.089
  7. Gao H., Li Y., Zhang B. A fixed-point theorem of Krasnoselskii-Schaefer type and its applications in control and periodicity of integral equations. Fixed Point Theory, 2011. Vol. 12, No. 1. P. 91–112.
  8. Guo D., Lakshmikantham V. Nonlinear Problems in Abstract Cones. Notes Rep. Math. Sci. Eng., vol. 5. Boston, San Diego: Academic Press, 1988. 275 p.
  9. Mahmoudi A., Kessi A. Existence and Ulam stability of solution for some backward impulsive differential equations on Banach spaces. Armen. J. Math., 2021. Vol. 13, No. 8. P. 1–21. DOI: 10.52737/18291163-2021.13.8-1-21
  10. Perestyuk N.A., Plotnikov V.A., Samoilenko A.M., Skripnik N.V. Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities. Berlin/Boston: Walter de Gruyter GmbH & Co. KG, 2011. 307 p. DOI: 10.1515/9783110218176
  11. Podlubny I. Fractional Differential Equations. Math. Sci. Eng., vol. 198. San Diego: Academic Press, 1999. 340 p. DOI: 10.1016/S0076-5392(99)X8001-5
  12. Poongodi R., Murugesu R. Existence of solutions for fractional impulsive integro-differential systems. Malaya J. Mat., 2013. Vol. S1. P. 56–65.
  13. Rudin W. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, 1976. 342 p.
  14. Smart D.R. Fixed Point Theorems. Cambridge Tracts in Math., vol. 66. Cambridge: Cambridge University Press, 1980. 100 p.
  15. Wang J.R., Feckan M., Zhou Y. Ulam's type stability of impulsive ordinary differential equations. J. Math. Anal. Appl., 2012. Vol. 395, No. 1. P. 258–264. DOI: 10.1016/j.jmaa.2012.05.040
  16. Wei W., Xiang X., Peng Y. Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization, 2006. Vol. 55, No. 1–2. P. 141–156. DOI: 10.1080/02331930500530401
  17. Zhang S. Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differential Equations, 2006. Vol. 2006, No. 36. P. 1–12.




DOI: http://dx.doi.org/10.15826/umj.2026.1.006

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.