Existence of a mild solution to fractional differential equations with $\psi$-Caputo derivative, and its $\psi$-Hölder continuity

Abstract

This paper is devoted to the study existence of locally/globally mild solutions for fractional differential equations with $\psi$-Caputo derivative with a nonlocal initial condition. We firstly establish the local existence by making use usual fixed point arguments, where computations and estimates are essentially based on continuous and bounded properties of the Mittag-Leffler functions. Secondly, we establish the called $\psi$-H\"older continuity of solutions, which shows how $|u(t')-u(t)|$ tends to zero with respect to a small difference $|\psi(t')-\psi(t)|^{\beta}$, $\beta\in(0,1)$. Finally, by using contradiction arguments, we discuss on the existence of a global solution or maximal mild solution with blowup at finite time.

Keywords

• Fractional calculus
• Fractional differential equations
• $\psi$-Caputo derivative
• Fixed point theorem
• Maximal mild solutions
• $\psi$-H\"older continuity.

Kaynakça

1. [1] Stefan G. Samko, Anatoly A. Kilbas and Oleg I. Marichev, Fractional integrals and derivatives, Theory and Applications, Gordon and Breach Science, Naukai Tekhnika, Minsk (1987).
2. [2] I. Podlubny; Fractional differential equations, Academic Press, London, 1999.
3. [3] Rudolf Hilfer, Fractional calculus in Physics, World Scientific, Singapore , 2000.
4. [4] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative J. Comput. Appl. Math. 264 (2014), 65–70
5. [5] Y. Chen, Y. Yan, K. Zhang, On the local fractional derivative Journal of Mathematical Analysis and Applications Volume 362, Issue 1, 2010, Pages 17–33.
6. [6] K. Balachandrana, J.Y. Parkb, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations Nonlinear Analysis 71 (2009) 4471–4475.
7. [7] Yong Zhou, Feng Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Analysis (Real World Applications), 11 (2010), p. 4465-4475.
8. [8] G.M. N’guerekata, A Cauchy problem for some fractional abstract differential equation with nonlocal condition, Nonlinear Analysis 70 (2009) 1873–1876.

9. [9] S. Yang, L. Wang, S. Zhang, Conformable derivative: Application to non-Darcian flow in low-permeability porous media Applied Mathematics Letters, Volume 79, May 2018, Pages 105–110.
10. [10] H. Fazli, J.J. Nieto, F. Bahrami, On the existence and uniqueness results for nonlinear sequential fractional differential equations, Appl. Comput. Math., Vol. 17, No. 1 (2018), pp. 36–47.
11. [11] D. Baleanu, A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel Commun. Nonlinear Sci. Numer. Simul. 59 (2018), 444–462.
12. [12] R. Almeida, A.B. Malinowska, M.T.T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Meth. Appl. Sci. 41 (2018), 336–352.
13. [13] B. Samet, H. Aydi, Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving ψ-Caputo fractional derivative, J. Inequal. Appl. 2018, Paper No. 286, 11 pp.
14. [14] C. Derbazi, Z. Baitiche, Coupled systems of ψ-Caputo differential equations with initial conditions in Banach spaces. Mediterr. J. Math. 17, 169 (2020).
15. [15] M.S. Abdo, S.K. Panchal, A.M. Saeed, Fractional boundary value problem with ψ-Caputo fractional derivative, Proc. Indian Acad. Sci. (Math. Sci), 129:65 (2019).
16. [16] R. Almeida, M. Jleli, B. Samet, A numerical study of fractional relaxation-oscillation equations involving ψ-Caputo fractional derivative. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 113 (2019), 1873–1891.
17. [17] Z. Baitichea, C. Derbazia, M. Benchohra, ψ-Caputo Fractional Differential Equations with Multi-point Boundary Conditions by Topological Degree Theory, Results in Nonlinear Analysis 3 (2020) No. 4, 166–178.
18. [18] C. Thaiprayoon, W. Sudsutad, S.K. Ntouyas, Mixed nonlocal boundary value problem for implicit fractional integro-differential equations via ψ-Hilfer fractional derivative, Advances in Difference Equations, 2021:50.
19. [19] M.S. Abdo, S.K. Panchal, H.S. Hussien, Fractional integro-differential equations with nonlocal conditions and ψ-Hilfer fractional derivative. Mathematical Modelling and Analysis, 2019, 24(4), 564-584.
20. [20] M. A. Almalahi, M.S. Abdo, S.K. Panchal, Existence and Ulam–Hyers–Mittag-Lefer stability results of ψ- Hilfer nonlocal Cauchy problem, Rendiconti del Circolo Matematico di Palermo Series 2, 2021, 70, 57–77.
21. [21] A. Seemab, J. Alzabut, M. ur Rehman, Y. Adjabi, M.S. Abdo, Langevin equation with nonlocal boundary conditions involving a ψ-Caputo fractional operator, https://arxiv.org/abs/2006.00391.
22. [22] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no. 3, 709–722