A coupled non-separated system of Hadamard-type fractional differential equations

Abstract

In this article, we discuss the existence and uniqueness of solutions of a coupled non-separated system for
fractional differential equations involving a Hadamard fractional derivative. The existence and uniqueness
results obtained in the present study are not only new but also cover some results corresponding to special
values of the parameters involved in the Caputo problems. These developed results are obtained by apply-
ing Banach's fixed point theorem and Leray-Schauder's nonlinear alternative. An example is presented to
illustrate our main results.

Keywords

• Fractional differential equations
• fixed point theorems
• boundary conditions Generalized fractional operators

Kaynakça

1. [1] S.Y.A. Al-Mayyahi, M.S. Abdo, S.S. Redhwan, and B.N. Abbood, Boundary value problems for a coupled system of Hadamard-type fractional di?erential equations, IAENG Int. J. Appl. Math. 51 (1) (2020) 1-10.
2. [2] Y. Arioua, N. Benhamidouche, Boundary value problem for Caputo-Hadamard fractional differential equations, Surv. Math. Appl. 12 (2017) 103-115.
3. [3] S. Abbas, M. Benchohra, N. Hamidi and Y. Zhou, Implicit coupled Hilfer-Hadamard fractional differential systems under weak topologies, Adv. Diference Equ. 2018 (1) (2018) 1-17.
4. [4] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Si. Numer. Simul. 44 (2017) 460-481.
5. [5] Y. Adjabi, F. Jarad, D. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo Hadamard fractional derivatives, J. Comput. Anal. Appl. 21 (1) (2016) 661-681.
6. [6] H.H. Alsulami, S.K. Ntouyas, R. P. Agarwal, B. Ahmad, and A. Alsaedi, A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions, Bound. Value Probl. 2017 (1) (2017) 68.
7. [7] B. Ahmad, SK. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Solitons Fractals 83 (2016) 234-241.
8. [8] B. Ahmad, SK. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 266 (2015) 615-622.

9. [9] M.S. Abdo, S.K. Panchal, and H.A. Wahash, Ulam- Hyers-Mittag-Leffler stability for a ψ-Hilfer problem with fractional order and ininite delay, Results in Applied Mathematics 7 (2020) 100115.
10. [10] M.S. Abdo, K. Shah, S.K. Panchal, and H.A. Wahash, Existence and Ulam stability results of a coupled system for terminal value problems involving ψ-Hilfer fractional operator, Adv. Difference Equ. 2020 (1) (2020) 1-21.
11. [11] D.B. Dhaigude, and S.P. Bhairat, Local Existence and Uniqueness of Solution for Hilfer-Hadamard fractional differential problem, Nonlinear Dyn. Syst. Theory 18 (2) (2018) 144-153.
12. [12] V. Daftardar-Gejji, H. Jafari, Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl. 301 (2)(2005) 508-518.
13. [13] A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York (2005).
14. [14] Y. Gambo, F. Jarad, D. Baleanu, F. Jarad, On Caputo modi?cation of the Hadamard fractional derivatives, Adv. Difference Equ. 10 (2014) 1-12.
15. [15] R. Hilfer, Applications of fractional calculus in Physics, World Scientific, Singapore, (2000).
16. [16] F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modi?cation of the Hadamard fractional derivatives, Adv. Difference Equ. 142 (2012) 1-8.
17. [17] H. Jafari, V. Daftardar-Gejji, Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput. 180 (2) (2006) 700-706.
18. [18] H. Jafari, V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math. 196 (2) (2006) 644-651.
19. [19] U. N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 6 (4) (2014) 1-15.
20. [20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier, Amsterdam (2006).
21. [21] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, (1993).
22. [22] L. Palve, M.S. Abdo, S.K. Panchal, Some existence and stability results of Hilfer-Hadamard fractional implicit differential fractional equation in a weighted space, preprint: arXiv, arXiv:1910.08369v1, math.GM. (2019).
23. [23] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999).
24. [24] S.S. Redhwan, M.S. Abdo, K. Shah, T. Abdeljawad, S. Dawood, H.A. Abdo and S.L. Shaikh, Mathematical modeling for the outbreak of the coronavirus (COVID-19) under fractional nonlocal operator, Results in Physics, 19 (2020), 103610.
25. [25] S.S. Redhwan, S.L. Shaikh, M. S. Abdo, Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type, AIMS Math. 5 (4) (2020) 3714-3730.
26. [26] S.S. Redhwan, S.L.Shaikh, Analysis of implicit type of a generalized fractional differential equations with nonlinear integral boundary conditions, J. Math. Anal. Model. 1 (1) (2020) 64-76.
27. [27] S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Some properties of Sadik transform and its applications of fractional-order dynam- ical systems in control theory, Adv. Theory Nonlinear Anal. Appl. 4 (1) (2020) 51-66.
28. [28] J.V.D.C. Sousa, F. Jarad, and T. Abdeljawad, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, Ann. Funct. Anal. 12 (1) (2021) 1-16.
29. [29] J.V.C. Sousa, D.D.S. Oliveira, and E. Capelas de Oliveira, On the existence and stability for noninstantaneous impulsive fractional integrodi?erential equation, Math. Methods Appl. Sci. 42 (4) (2019) 1249-1261.
30. [30] J.V.C. Sousa, and E.C. de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simula. 60 (2018) 72-91.
31. [31] W. Saengthong, E. Thailert, and S.K. Ntouyas, Existence and uniqueness of solutions for system of Hilfer-Hadamard sequential fractional di?erential equations with two point boundary conditions, Adv. Difference Equ. 2019 (1) (2019) 525.
32. [32] B. Senol, C. Yeroglu, Frequency boundary of fractional order systems with nonlinear uncertainties, J. Franklin Inst. 350 (2013) 1908-1925.
33. [33] J.R. Wang, Y. Zhang, Analysis of fractional order differential coupled systems, Math. Methods Appl. Sci. 38 (2015) 3322-3338.
34. [34] J.R. Wang, Y. Zhou, M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl. 64 (2012) 3008-3020.