Qualitative study of linear and nonlinear relaxation equations with $ \psi $-Riemann-Liouville fractional derivatives

Muath Awadalla; Mohammed S. Abdo; Hanan A. Wahash; Kinda Abuasbeh; Muath Awadalla; Mohammed S. Abdo; Hanan A. Wahash; Kinda Abuasbeh

doi:10.3934/math.20221110

AIMS Mathematics

2022, Volume 7, Issue 11: 20275-20291. doi: 10.3934/math.20221110

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Research article

Qualitative study of linear and nonlinear relaxation equations with $ \psi $-Riemann-Liouville fractional derivatives

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Hafuf, Al Ahsa 31982, Saudi Arabia
2.
Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen
3.
Department of Mathematics, Albaydha University, Albaydha, Yemen

Received: 06 July 2022 Revised: 31 August 2022 Accepted: 13 September 2022 Published: 16 September 2022
MSC : 26A33, 34A08, 34A12, 47H10

In the present paper, we consider the linear and nonlinear relaxation equation involving $ \psi $-Riemann-Liouville fractional derivatives. By the generalized Laplace transform approach, the guarantee of the existence of solutions for the linear version is shown by Ulam-Hyer's stability. Then by establishing the method of lower and upper solutions along with Banach contraction mapping, we investigate the existence and uniqueness of iterative solutions for the nonlinear version with the non-monotone term. A new condition on the nonlinear term is formulated to ensure the equivalence between the solution of the nonlinear problem and the corresponding fixed point. Moreover, we discuss the maximal and minimal solutions to the nonlinear problem at hand. Finally, we provide two examples to illustrate the obtained results.
- relaxation equations,
- fractional derivative,
- lower and upper solution method,
- fixed point theorem
Citation: Muath Awadalla, Mohammed S. Abdo, Hanan A. Wahash, Kinda Abuasbeh. Qualitative study of linear and nonlinear relaxation equations with $ \psi $-Riemann-Liouville fractional derivatives[J]. AIMS Mathematics, 2022, 7(11): 20275-20291. doi: 10.3934/math.20221110

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Abstract

In the present paper, we consider the linear and nonlinear relaxation equation involving $ \psi $-Riemann-Liouville fractional derivatives. By the generalized Laplace transform approach, the guarantee of the existence of solutions for the linear version is shown by Ulam-Hyer's stability. Then by establishing the method of lower and upper solutions along with Banach contraction mapping, we investigate the existence and uniqueness of iterative solutions for the nonlinear version with the non-monotone term. A new condition on the nonlinear term is formulated to ensure the equivalence between the solution of the nonlinear problem and the corresponding fixed point. Moreover, we discuss the maximal and minimal solutions to the nonlinear problem at hand. Finally, we provide two examples to illustrate the obtained results.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[2]	A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional integrals and derivatives: Theory and applications, 1993.
[3]	J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series, SIAM J. Appl. Math., 18 (1970), 658–674. https://doi.org/10.1137/0118059 doi: 10.1137/0118059
[4]	A. Bashir, J. J. Nieto, Existence results for nonlinear boundary value problems of fractional integtodifferential equations with integral boundary conditions, Bound. Value Probl., 2009 (2009), 708576. https://doi.org/10.1155/2009/708576 doi: 10.1155/2009/708576
[5]	S. Zhang, The Existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252 (2000), 804–812. https://doi.org/10.1006/jmaa.2000.7123 doi: 10.1006/jmaa.2000.7123
[6]	V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. Theor., 69 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042 doi: 10.1016/j.na.2007.08.042
[7]	S. Zhang, Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Anal. Theor., 71 (2009), 2087–2093. https://doi.org/10.1016/j.na.2009.01.043 doi: 10.1016/j.na.2009.01.043
[8]	Z. Bai, S. Zhang, S. Sun, C. Yin, Monotone iterative method for fractional differential equations, Electron. J. Differ. Equ., 6 (2016), 8.
[9]	J. J. Nieto, Maximum principles for fractional differential equations derived from Mitta-Leffler functions, Appl. Math. Lett., 23 (2010), 1248–1251. https://doi.org/10.1016/j.aml.2010.06.007 doi: 10.1016/j.aml.2010.06.007
[10]	O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15 (2012), 700-711. https://doi.org/10.2478/s13540-012-0047-7 doi: 10.2478/s13540-012-0047-7
[11]	F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Cont. Dyn. S, 13 (2020), 709.
[12]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Si., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[13]	J. V. C. Sousa, E. C. de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Si., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
[14]	C. Derbazi, Z. Baitiche, M. S. Abdo, T. Abdeljawad, Qualitative analysis of fractional relaxation equation and coupled system with $\psi$-Caputo fractional derivative in Banach spaces, AIMS Math., 6 (2021), 2486–2509. https://doi.org/10.3934/math.2021151 doi: 10.3934/math.2021151
[15]	C. Derbazi, Z. Baitiche, M. S. Abdo, K. Shah, B. Abdalla, T. Abdeljawad, Extremal solutions of generalized Caputo-type fractional-order boundary value problems using monotone iterative method, Fractal Fract., 6 (2022), 146. https://doi.org/10.3390/fractalfract6030146 doi: 10.3390/fractalfract6030146
[16]	M. Awadalla, K. Abuasbeh, M. Subramanian, M. Manigandan, On a system of $\psi$-Caputo hybrid fractional differential equations with Dirichlet boundary conditions, Mathematics, 10 (2022), 1681. https://doi.org/10.3390/math10101681 doi: 10.3390/math10101681
[17]	M. Awadalla, Y. Y. Yameni Noupoue, K. A. Asbeh, Psi-Caputo logistic population growth model, J. Math., 2021 (2021), 8634280. https://doi.org/10.1155/2021/8634280 doi: 10.1155/2021/8634280
[18]	S. M. Ali, M. S. Abdo, Qualitative analysis for multiterm Langevin systems with generalized Caputo fractional operators of different orders, Math. Probl. Eng., 2022 (2022), 1879152. https://doi.org/10.1155/2022/1879152 doi: 10.1155/2022/1879152
[19]	H. A. Wahash, S. K. Panchal, Positive solutions for generalized two-term fractional differential equations with integral boundary conditions, J. Math. Anal. Model., 1 (2020), 47–63. https://doi.org/10.48185/jmam.v1i1.35 doi: 10.48185/jmam.v1i1.35
[20]	M. B. Jeelani, A. M. Saeed, M. S. Abdo, K. Shah, Positive solutions for fractional boundary value problems under a generalized fractional operator, Math. Method. Appl. Sci., 44 (2021), 9524–9540. https://doi.org/10.1002/mma.7377 doi: 10.1002/mma.7377
[21]	J. Patil, A. Chaudhari, M. S. Abdo, B. Hardan, Upper and lower solution method for positive solution of generalized Caputo fractional differential equations, Adv. Theor. Nonlinear Anal. Appl., 4 (2020), 279–291. https://doi.org/10.31197/atnaa.709442 doi: 10.31197/atnaa.709442
[22]	D. Guo, J. Sun, Z. Liu, Functional methods in nonlinear ordinary differential equations, Jinan: Shandong Science and Technology Press, 1995.
[23]	Y. Zhou, Basic theory of fractional differential equations, World Scientific, 2014.

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