Caputo-Katugampola-type implicit fractional differential equation with anti-periodic boundary conditions

Abstract

The given article describes the implicit fractional dierential equation with anti-periodic boundary conditions
in the frame of Caputo-Katugampola fractional derivative. We obtain an analogous integral equation of the
given problem and prove the existence and uniqueness results of such a problem using the Banach and
Krasnoselskii xed point theorems. Further, by applying generalized Gronwall inequality, the Ulam-Hyers
stability results are discussed. To show the eectiveness of the acquired results, convenient examples are
presented.

Keywords

• implicit fractional differential equation
• fractional derivative and fractional integral
• anti-periodic conditions
• fixed point theorem
• Gronwall inequality

References

1. [1] S.Y. Al-Mayyahi, M.S. Abdo, S.S. Redhwan, B.N. Abood, Boundary value problems for a coupled system of Hadamard-type fractional differential equations. IAENG International Journal of Applied Mathematics, 51(1) (2021) 1-10.
2. [2] S. Abbas, M. Benchohra, J.R. Graef, Implicit fractional differential and integral equations. de Gruyter, (2018).
3. [3] R. Almeida, A Gronwall inequality for a general Caputo fractional operator, arXiv preprint arXiv:1705.10079. (2017).
4. [4] E. Alvarez, C. Lizama, R. Ponce, Weighted pseudo anti-periodic solutions for fractional integro-differential equations in Banach spaces, Applied Mathematics and Computation, 259 (2015)164-172.
5. [5] B. Ahmad, J.J. Nieto, Anti-periodic fractional boundary value problems, Computers & Mathematics with Applications, 62 (2011) 1150-1156.
6. [6] M. ALMALAHI, S.K. PANCHAL, Existence and stability results of relaxation fractional differential equations with Hilfer- Katugampola fractional derivative, Advances in the Theory of Nonlinear Analysis and its Application, 4(4) 299-315.
7. [7] M.S. Abdo, S.K. Panchal, Some new uniqueness results of solutions to nonlinear fractional integro-differential equations, Annals of Pure and Applied Mathematics, 16 (1) (2018) 345-352.
8. [8] B.N. Abood, S.S. Redhwan, M.S. Abdo, Analytical and approximate solutions for generalized fractional quadratic integral equation, Nonlinear Functional Analysis and Applications, 26(3) (2021) 497-512.

9. [9] M. Benchohra, S. Bouriah, M.A. Darwish, Nonlinear boundary value problem for implicit differential equations of fractional order in Banach spaces, Fixed Point Theory, 18 (2017) 457-470.
10. [10] M. Benchohra, S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan Journal of Pure and Applied Analysis, 1(1) (2015) 22-37.
11. [11] A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Mathematics. , 5(1) (2019) 259-272.
12. [12] T.A. Burton, C. KirkÙ, A fixed point theorem of Krasnoselskii Schaefer type, Mathematische Nachrichten, 189 (1998), 23-31.
13. [13] M. Benchohra, J.E. Lazreg, Nonlinear fractional implicit differential equations, Communications in Applied Analysis, 17 (2013) 471-482.
14. [14] S. Hamani, W. Benhamida, J. Henderson, Boundary value problems for Caputo-Hadamard fractional differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 2(3) (2015) 138-145.
15. [15] R. Herrmann, Fractional calculus: an introduction for physicists, (2011).
16. [16] G. Jumarie, On the representation of fractional Brownian motion as an integral with respect to (dt) a, Applied Mathematics Letters, 18 (7) (2005) 739-748.
17. [17] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations. Preprint. arXiv:1411.5229. (2014).
18. [18] U.N. Katugampola, A new approach to generalized fractional derivatives, Bulletin of Mathematical Analysis and Applications, 6 (2014) 1-15.
19. [19] U.N. Katugampola, New approach to a generalized fractional integral, Applied Mathematics and Computation, 218 (3) (2011) 860-865.
20. [20] U.N. Katugampola, Mellin transforms of the generalized fractional integrals and derivatives, Applied Mathematics and Computation, 257 (2015) 566-580.
21. [21] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 204 (2006).
22. [22] N.H. Luc, D. Baleanu, N.H. Can, Reconstructing the right-hand side of a fractional subdiffusion equation from the final data, Journal of Inequalities and Applications, 2020(1) (2020) 1-15.
23. [23] A.B. Malinowska, T. Odzijewicz, D.F.M. Torres, Advanced Methods in the Fractional Calculus of Variations, Springer: Berlin. (2015).
24. [24] B. Nghia, Existence of a mild solution to fractional differential equations with ψ-Caputo derivative, and its ψ-Holder continuity, Advances in the Theory of Nonlinear Analysis and its Application, 5(3) 337-350.
25. [25] D.S. Oliveira, E. Capelas, de. Oliveira, Hilfer-Katugampola fractional derivatives, Computational and Applied Mathematics, 37 (2018) 3672-3690.
26. [26] I. Podlubny, Fractional Differential Equations, Academic Press: San Diego, (1999).
27. [27] S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type, AIMS MATHEMATICS, 5(4) (2020) 3714-3730.
28. [28] S.S. Redhwan, S.L. Shaikh, Analysis of implicit Type of a generalized fractional differential equations with nonlinear integral boundary conditions, Journal of Mathematical Analysis and Modeling, 1(1) (2020) 64-76.
29. [29] S.S. Redhwan, S.L. Shaikh, M.S. Abdo, A coupled non-separated system of Hadamard-type fractional dierential equations, Advances in the Theory of Nonlinear Analysis and its Applications, 1(1) (2022) 33-44.
30. [30] S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Some properties of Sadik transform and its applications of fractional-order dynamical systems in control theory, Advances in the Theory of Nonlinear Analysis and its Application, 4(1) (2019) 51-66.
31. [31] S. Redhwan, S.L. Shaikh, Implicit fractional differential equation with nonlocal integral-multipoint boundary conditions in the frame of Hilfer fractional derivative. Journal of Mathematical Analysis and Modeling, 2(1) (2021) 62-71.
32. [32] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian Journal of Mathematics, 26 (2010), 103-107.
33. [33] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives; Yverdon-les-Bains, (1993).
34. [34] J.V.C. Sousa, E.C. Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, arXiv preprint arXiv:1709.03634, (2017).
35. [35] T.N. Thach, T.N. Huy, P.T.M. Tam, M.N. Minh, N.H. Can, Identi?cation of an inverse source problem for time-fractional diffusion equation with random noise, Mathematical Methods in the Applied Sciences, 42(1) (2019) 204-218.