Upper and Lower Solution method for Positive solution of generalized Caputo fractional differential equations

Year 2020, Volume: 4 Issue: 4, 279 - 291, 30.12.2020

Jayshree Patil Archana Chaudhari Mohammed Abdo , Basel Hardan

https://doi.org/10.31197/atnaa.709442

Cited By: 5

Abstract

In this research paper, the nonlinear fractional relaxation equation involving the generalized Caputo derivative is reduced to an equivalent integral equation via the generalized Laplace transform. Moreover, the upper and lower solutions method combined with some fixed point theorems, and the properties of the Mittag-Leffler function are applied to investigate the existence and uniqueness of positive solutions for the problem at hand. At the end, to illustrate our results, we give an example.

Keywords

fractional differential equations, existence and upper and lower solutions, Mittag-Leffler function, generalized Laplace transform, fixed point theorems

Supporting Institution

No financial support

Project Number

There is no

Thanks

The authors thank Dr. Babasaheb Ambedkar Marathwada University for its moral support and facilitating some procedures for researchers.

References

• [1] S. Abbas, M. Benchohra and G. M. N. Guerekata, Topics in Fractional Di?erential Equations, Springer, Berlin, 2012.
• [2] M. S. Abdo, A. G. Ibrahim and S. K. Panchal, Nonlinear implicit fractional differential equation involving ψ-Caputo fractional derivative, Proceedings of the Jangjeon Mathematical Society, 2019, 22(3), 387-400.
• [3] M. S. Abdo and S. K. Panchal, Fractional integro-differential equations involving ψ-Hilfer fractional derivative, Advances in Applied Mathematics and Mechanics, 2019, 11(2), 338-359.
• [4] M. S. Abdo and S.K. Panchal, Existence and continuous dependence for fractional neutral functional differential equations, J. Mathematical Model., 2017, 5(2), 153-170.
• [5] M. S. Abdo, K. Shah, S. K. Panchal, H. A. Wahash, Existence and Ulam stability results of a coupled system for terminal value problems involving ψ-Hilfer fractional operator, Adv. Differ. Equ. 2020, 316 (2020). https://doi.org/10.1186/s13662- 020-02775-x.
• [6] M. S. Abdo, H. A. Wahash and S. K. Panchal, Positive solution of a fractional differential equation with integral boundary conditions, Journal of Applied Mathematics and Computational Mechanics,2018, 17(2), 5-15.
• [7] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differ. Equ. 2009, Article ID 981728.
• [8] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul, 2017, 44, 460-481.
• [9] R. Almeida, A. B. Malinowska and T. Odzijewicz, Fractional differential equations with dependence on the Caputo- Katugampola derivative, Journal of Computational and Nonlinear Dynamics, 2016, 11(6).
• [10] A. Ardjouni and A. Djoudi, Existence and uniqueness of positive solutions for first-order nonlinear Liouville-Caputo frac- tional differential equations Sáo Paulo Journal of Mathematical Sciences, 2019, 1-10.
• [11] A. Ardjouni and A. Djoudi, Positive solutions for first-order nonlinear Caputo-Hadamard fractional relaxation differential equations, Kragujevac Journal of Mathematics, 2021, 45(6), 897-908.
• [12] M. Belaid, A. Ardjouni and A. Djoudi, Positive solutions for nonlinear fractional relaxation differential equations, Journal of Fractional Calculus and Applications, 2020, 11(1), 1-10.
• [13] M. Benchohra, S, Hamani and Y. Zhou, Oscillation and nonoscillation for Caputo-Hadamard impulsive fractional differ- ential inclusions Advances in Di?erence Equations, 2019, 2019(1), 1-15.
• [14] M. Benchohra and B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations Electronic J. Diff. Equ. (EJDE), 2009, 10(2009), 1-11.
• [15] A. Chidouh, A. Guezane-Lakoud and R. Bebbouchi, Positive solutions of the fractional relaxation equation using lower and upper solutions Vietnam Journal of Mathematics, 2016, 44(4), 739-748.
• [16] K. Diethelm and A. D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, Forschung und wissenschaftliches Rechnen, 1999, 57-71.
• [17] H. M. Fahad, On ψ-Laplace transform method and its applications to ψ-fractional differential equations, arXiv preprint arXiv:1907.04541, 2019.
• [18] F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete & Continuous Dynamical Systems-S, 709, (2019).
• [19] A. A. Kilbas, H. M. Shrivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
• [20] R. Khaldi and A. Guezane-Lakoud, Upper and Lower Solutions Method for Higher Order Boundary Value Problems, Progress in Fractional Differentiation and Applications, 2017, 3, 53-57.
• [21] K. D. Kucche and A. D. Mali, Initial time di?erence quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative, Computational and Applied Mathematics, 2020, 39(1), 31.
• [22] N. Li and C. Wang, New existence results of positive solution for a class of nonlinear fractional differential equations, Acta Mathematica Scientia, 2013, 33B, 847-854.
• [23] M. A. Malahi, M. S. Abdo and S. K. Panchal, Positive solution of Hilfer fractional differential equations with integral boundary conditions, arXiv: 1910.07887v1[math.GM], 2019.
• [24] D. S. Oliveira and E. C. de Oliveira, Hilfer-Katugampola fractional derivatives, Computational and Applied Mathematics, 2018, 37(3), 3672-3690. [25] S. Peng and J. Wang, Existence and Ulam-Hyers stability of ODEs involving two Caputo fractional derivatives, Electronic J. Qualitat. Theory Diff. Equ., 2015, 2015(52), 1-16.
• [26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [27] J. V. C. Sousa and C. E. de Oliveira, On the ψ-Hilfer fractional derivative, Commun Nonlinear Sci Numer Simul, 2018, 60, 72-91.
• [28] J. V. C. Sousa, D. S. Oliveira and C. E. de Oliveira, On the existence and stability for impulsive fractional integrodifferential equation, Math Methods Appl Sci., 2019, 42(4), 1249-1261.
• [29] D. Vivek, E. Elsayed and K. Kanagarajan, Theory and analysis of ψ-fractional differential equations with boundary condi- tions. Communications in Applied Analysis, 2018, 22, 401-414.
• [30] H. A. Wahash, S. K. Panchal, M. S. Abdo, Positive solutions for generalized Caputo fractional differential equations with integral boundary conditions, Journal of Mathematical Modeling, 8(4), (2020) 393-414.
• [31] Y. Zhou, Basic theory of fractional differential equations, Singapore: World Scientific, 2014.