On the positive solutions for IBVP of conformable differential equations

Mouataz Billah Mesmouli; Taher S. Hassan; Mouataz Billah Mesmouli; Taher S. Hassan

doi:10.3934/math.20231261

AIMS Mathematics

2023, Volume 8, Issue 10: 24740-24750. doi: 10.3934/math.20231261

Previous Article Next Article

Research article Special Issues

On the positive solutions for IBVP of conformable differential equations

Mouataz Billah Mesmouli ^{1
,
,},
Taher S. Hassan ²

1.
Mathematics Department, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 24 March 2023 Revised: 23 May 2023 Accepted: 07 August 2023 Published: 23 August 2023
MSC : 34A08, 34A12, 34B18, 47H10

A problem with integral boundary conditions (IBVP) involving conformable fractional derivatives is considered in this article. The upper and lower solutions technique is used to discuss the existence and uniqueness of positive solutions. The fixed point Theorem of Schauder proves the existence of positive solutions, and the fixed point Theorem of Banach proves the uniqueness of solutions. Our results are illustrated by an example.
- IBVP,
- positive solutions,
- fixed point,
- conformable differential equations
Citation: Mouataz Billah Mesmouli, Taher S. Hassan. On the positive solutions for IBVP of conformable differential equations[J]. AIMS Mathematics, 2023, 8(10): 24740-24750. doi: 10.3934/math.20231261

Related Papers:

Abstract

A problem with integral boundary conditions (IBVP) involving conformable fractional derivatives is considered in this article. The upper and lower solutions technique is used to discuss the existence and uniqueness of positive solutions. The fixed point Theorem of Schauder proves the existence of positive solutions, and the fixed point Theorem of Banach proves the uniqueness of solutions. Our results are illustrated by an example.

References

[1]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[2]	S. Ailing, Z. Shuqin, Upper and lower solutions method and a fractional differential equation boundary value problem, Electron. J. Qual. Theory Differ. Equ., 30 (2009), 1–13. https://doi.org/10.14232/ejqtde.2009.1.30 doi: 10.14232/ejqtde.2009.1.30
[3]	F. M. Al-Askar, W. W. Mohammed, S. K. Samura, M. El-Morshedy, The exact solutions for fractional-stochastic Drinfel'd-Sokolov-Wilson equations using a conformable operator, J. Funct. Space., 2022 (2022), 7133824. https://doi.org/10.1155/2022/7133824 doi: 10.1155/2022/7133824
[4]	D. R. Anderson, D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl., 10 (2015), 109–137.
[5]	S. Asawasamrit, S. K. Ntouyas, P. Thiramanus, J. Tariboon, Periodic boundary value problems for impulsive conformable fractional integrodifferential equations, Bound. Value Probl., 2016 (2016), 122. https://doi.org/10.1186/s13661-016-0629-0 doi: 10.1186/s13661-016-0629-0
[6]	I. Bachar, H. Eltayeb, Positive solutions for a class of conformable fractional boundary value problems, Math. Method. Appl. Sci., 2020 (2020). https://doi.org/10.1002/mma.6637 doi: 10.1002/mma.6637
[7]	H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh, Three-point boundary value problems for conformable fractional differential equations, J. Funct. Space., 2015 (2015), 706383. https://doi.org/10.1155/2015/706383 doi: 10.1155/2015/706383
[8]	A. Batool, I. Talib, M. B. Riaz, C. Tunç, Extension of lower and upper solutions approach for generalized nonlinear fractional boundary value problems, Arab J. Basic Appl. Sci., 29 (2022), 249–256. https://doi.org/10.1080/25765299.2022.2112646 doi: 10.1080/25765299.2022.2112646
[9]	Y. Cenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers' type equations with conformable derivative, Wave. Random Complex, 27 (2017), 103–116. https://doi.org/10.1080/17455030.2016.1205237 doi: 10.1080/17455030.2016.1205237
[10]	K. Diethelm, Multi-Term Caputo fractional differential equations, In: The analysis of fractional differential equations, Berlin: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2_8
[11]	K. Diethelm, H. T. Tuan, Upper and lower estimates for the separation of solutions to fractional differential equations, Fract. Calc. Appl. Anal., 25 (2022), 166–180. https://doi.org/10.1007/s13540-021-00007-x doi: 10.1007/s13540-021-00007-x
[12]	X. Han, H. Gao, Positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation, Math. Probl. Eng., 2011 (2011), 725494. https://doi.org/10.1155/2011/725494 doi: 10.1155/2011/725494
[13]	R. Khaldi, A. Guezane-Lakoud, Upper and lower solutions method for higher order boundary value problems, Progr. Fract. Differ. Appl., 3 (2017), 53–57. http://dx.doi.org/10.18576/pfda/030105 doi: 10.18576/pfda/030105
[14]	R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[15]	N. Khodabakhshi, S. Vaezpour, J. Trujillo, Method of upper and lower solutions for coupled system of nonlinear fractional integro-differential equations with advanced arguments, Math. Slovaca, 67 (2017), 89–98. https://doi.org/10.1515/ms-2016-0250 doi: 10.1515/ms-2016-0250
[16]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[17]	A. Korkmaz, Explicit exact solutions to some one-dimensional conformable time fractional equations, Wave. Random Complex., 29 (2017), 124–137. https://doi.org/10.1080/17455030.2017.1416702 doi: 10.1080/17455030.2017.1416702
[18]	W. W. Mohammed, N. Iqbal, A. M. Albalahi, A. E. Abouelregal, D. Atta, H. Ahmad, et al., Brownian motion effects on analytical solutions of a fractional-space long-short-wave interaction with conformable derivative, Results Phys., 35 (2022), 105371. https://doi.org/10.1016/j.rinp.2022.105371 doi: 10.1016/j.rinp.2022.105371
[19]	K. B. Oldham, J. Spanier, Theory and applications of differentiation and integration to arbitrary order, New York: Academic Press, 1974.
[20]	I. Podlubny, Fractional differential equations, 1999.
[21]	A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional integral and derivatives: Theory and applications, 1993.
[22]	M. Xu, S. Sun, Positivity for integral boundary value problems of fractional differential equations with two nonlinear terms, J. Appl. Math. Comput., 59 (2019), 271–283. https://doi.org/10.1007/s12190-018-1179-7 doi: 10.1007/s12190-018-1179-7
[23]	S. Yang, L. Wang, S. Zhang, Conformable derivative: Application to non-Darcian flow in low-permeability porous media, Appl. Math. Lett., 79 (2018), 105–110. https://doi.org/10.1016/j.aml.2017.12.006 doi: 10.1016/j.aml.2017.12.006
[24]	D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903–917. https://doi.org/10.1007/s10092-017-0213-8 doi: 10.1007/s10092-017-0213-8
[25]	W. Zhong, L. Wang, Positive solutions of conformable fractional differential equations with integral boundary conditions, Bound. Value Probl., 2018 (2018), 137. https://doi.org/10.1186/s13661-018-1056-1 doi: 10.1186/s13661-018-1056-1

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)