On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique

Zoubida Bouazza; Sabit Souhila; Sina Etemad; Mohammed Said Souid; Ali Akgül; Shahram Rezapour; Manuel De la Sen; Zoubida Bouazza; Sabit Souhila; Sina Etemad; Mohammed Said Souid; Ali Akgül; Shahram Rezapour; Manuel De la Sen

doi:10.3934/math.2023276

AIMS Mathematics

2023, Volume 8, Issue 3: 5484-5501. doi: 10.3934/math.2023276

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On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique

1.
Department of Informatics, University of Tiaret, Tiaret, Algeria
2.
Laboratoire Matériaux et Structures, Departement of Mathematics, University of Tiaret, Tiaret, Algeria
3.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
4.
Department of Economic Sciences, University of Tiaret, Tiaret, Algeria
5.
Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Türkiye
6.
Mathematics Research Center, Department of Mathematics, Near East University, North Cyprus, Mersin 10, Nicosia 99138, Türkiye
7.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
8.
Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Leioa 48940, Bizkaia, Spain

Received: 16 October 2022 Revised: 03 December 2022 Accepted: 07 December 2022 Published: 19 December 2022
MSC : 34A12, 34A40, 47A10

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.
- fractional variable order,
- Caputo-Hadamard fractional derivative,
- upper-lower solutions,
- existence results
Citation: Zoubida Bouazza, Sabit Souhila, Sina Etemad, Mohammed Said Souid, Ali Akgül, Shahram Rezapour, Manuel De la Sen. On the Caputo-Hadamard fractional IVP with variable order using the upper-lower solutions technique[J]. AIMS Mathematics, 2023, 8(3): 5484-5501. doi: 10.3934/math.2023276

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Abstract

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results.

References

[1]	A. O. Akdemir, A. Karaoǧlan, M. A. Ragusa, E. Set, Fractional integral inequalities via Atangana-Baleanu operators for convex and concave functions, J. Funct. Spaces, 2021 (2021), 1055434. https://doi.org/10.1155/2021/1055434 doi: 10.1155/2021/1055434
[2]	M. S. Abdo, Further results on the existence of solutions for generalized fractional quadratic functional integral equations, J. Math. Anal. Model., 1 (2020), 33–46. https://doi.org/10.48185/jmam.v1i1.2 doi: 10.48185/jmam.v1i1.2
[3]	R. Rizwan, A. Zada, X. Wang, Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses, Adv. Differ. Equ., 2019 (2019), 85. https://doi.org/10.1186/s13662-019-1955-1 doi: 10.1186/s13662-019-1955-1
[4]	D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Bound. Value Probl., 2020 (2020), 64. https://doi.org/10.1186/s13661-020-01361-0 doi: 10.1186/s13661-020-01361-0
[5]	A. Zada, J. Alzabut, H. Waheed, I. L. Popa, Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions, Adv. Differ. Equ., 2020 (2020), 64. https://doi.org/10.1186/s13662-020-2534-1 doi: 10.1186/s13662-020-2534-1
[6]	E. Bonyah, C. W. Chukwu, M. L. Juga, Fatmawati, Modeling fractional-order dynamics of Syphilis via Mittag-Leffler law, AIMS Math., 6 (2021), 8367–8389. https://doi.org/10.3934/math.2021485 doi: 10.3934/math.2021485
[7]	M. S. Abdo, T. Abdeljawad, S. M. Ali, K. Shah, F. Jarad, Existence of positive solutions for weighted fractional order differential equations, Chaos Solitons Fract., 141 (2020), 110341. https://doi.org/10.1016/j.chaos.2020.110341 doi: 10.1016/j.chaos.2020.110341
[8]	A. Atangana, S. İ. Araz, Nonlinear equations with global differential and integral operators: existence, uniqueness with application to epidemiology, Results Phys., 20 (2021), 103593. https://doi.org/10.1016/j.rinp.2020.103593 doi: 10.1016/j.rinp.2020.103593
[9]	H. Mohammad, S. Kumar, S. Rezapour, S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons Fract., 144 (2021), 110668. https://doi.org/10.1016/j.chaos.2021.110668 doi: 10.1016/j.chaos.2021.110668
[10]	S. Etemad, I. Iqbal, M. E. Samei, S. Rezapour, J. Alzabut, W. Sudsutad, et al., Some inequalities on multi-functions for applying in the fractional Caputo-Hadamard jerk inclusion system, J. Inequal. Appl., 2022 (2022), 84. https://doi.org/10.1186/s13660-022-02819-8 doi: 10.1186/s13660-022-02819-8
[11]	H. Khan, K. Alam, H. Gulzar, S. Etemad, S. Rezapour, A case study of fractal-fractional tuberculosis model in China: existence and stability theories along with numerical simulations, Math. Comput. Simul., 198 (2022), 455–473. https://doi.org/10.1016/j.matcom.2022.03.009 doi: 10.1016/j.matcom.2022.03.009
[12]	S. Belmor, F. Jarad, T. Abdeljawad, G. Kınıç, A study of boundary value problem for generalized fractional differential inclusion via endpoint theory for weak contractions, Adv. Differ. Equ., 2020 (2020), 348. https://doi.org/10.1186/s13662-020-02811-w doi: 10.1186/s13662-020-02811-w
[13]	S. Rezapour, M. I. Abbas, S. Etemad, N. M. Dien, On a multipoint $p$-Laplacian fractional differential equation with generalized fractional derivatives, Math. Meth. Appl. Sci., 2022. https://doi.org/10.1002/mma.8301 doi: 10.1002/mma.8301
[14]	A. M. Saeed, M. S. Abdo, M. B. Jeelani, Existence and Ulam-Hyers stability of a fractional order coupled system in the frame of generalized Hilfer derivatives, Mathematics, 9 (2021), 2543. https://doi.org/10.3390/math9202543 doi: 10.3390/math9202543
[15]	S. Etemad, I. Avci, P. Kumar, D. Baleanu, S. Rezapour, Some novel mathematical analysis on the fractal-fractional model of the AH1N1/09 virus and its generalized Caputo-type version, Chaos Solitons Fract., 162 (2022), 112511. https://doi.org/10.1016/j.chaos.2022.112511 doi: 10.1016/j.chaos.2022.112511
[16]	J. F. Gómez-Aguilar, Analytical and numerical solutions of nonlinear alcoholism model via variable-order fractional differential equations, Phys. A: Stat. Mech. Appl., 494 (2018), 52–75. https://doi.org/10.1016/j.physa.2017.12.007 doi: 10.1016/j.physa.2017.12.007
[17]	H. G. Sun, W. Chen, H. Wei, Y. Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185–192. https://doi.org/10.1140/epjst/e2011-01390-6 doi: 10.1140/epjst/e2011-01390-6
[18]	D. Tavares, R. Almeida, D. F. M. Torres, Caputo derivatives of fractional variable order Numerical approximations, Commun. Nonlinear Sci. Numer. Simul., 35 (2016), 69–87. https://doi.org/10.1016/j.cnsns.2015.10.027 doi: 10.1016/j.cnsns.2015.10.027
[19]	J. V. da C. Sousa, E. C. de Oliverira, Two new fractional derivatives of variable order with non-singular kernal and fractional differential equation, Comp. Appl. Math., 37 (2018), 5375–5394. https://doi.org/10.1007/s40314-018-0639-x doi: 10.1007/s40314-018-0639-x
[20]	J. Yang, H. Yao, B. Wu, An efficient numberical method for variable order fractional functional differential equation, Appl. Math. Lett., 76 (2018), 221–226. https://doi.org/10.1016/j.aml.2017.08.020 doi: 10.1016/j.aml.2017.08.020
[21]	J. H. An, P. Y. Chen, P. Chen, Uniqueness of solutions to initial value problem of fractional differential equations of variable-order, Dyn. Syst. Appl., 28 (2019), 607–623.
[22]	Z. Bouazza, S. Etemad, M. S. Souid, S. Rezapour, F. Martínez, M. K. A. Kaabar, A study on the solutions of a multiterm FBVP of variable order, J. Funct. Spaces, 2021 (2021), 9939147. https://doi.org/10.1155/2021/9939147 doi: 10.1155/2021/9939147
[23]	A. Benkerrouche, M. S. Souid, K. Sitthithakerngkiet, A. Hakem, Implicit nonlinear fractional differential equations of variable order, Bound. Value Probl., 2021 (2021), 64. https://doi.org/10.1186/s13661-021-01540-7 doi: 10.1186/s13661-021-01540-7
[24]	A. Refice, M. S. Souid, I. Stamova, On the boundary value problems of Hadamard fractional differential equations of variable order via Kuratowski MNC technique, Mathematics, 9 (2021), 1134. https://doi.org/10.3390/math9101134 doi: 10.3390/math9101134
[25]	S. Hristova, A. Benkerrouche, M. S. Souid, A. Hakem, Boundary value problems of Hadamard fractional differential equations of variable order, Symmetry, 13 (2021), 896. https://doi.org/10.3390/sym13050896 doi: 10.3390/sym13050896
[26]	S. G. Samko, B. Ross, Integration and differentiation to a variable fractional order, Integr. Trans. Spec. F., 1 (1993), 277–300. https://doi.org/10.1080/10652469308819027 doi: 10.1080/10652469308819027
[27]	S. Zhang, S. Li, L. Hu, The existeness and uniqueness result of solutions to initial value problems of nonlinear diffusion equations involving with the conformable variable derivative, RACSAM, 113 (2019), 1601–1623. https://doi.org/10.1007/s13398-018-0572-2 doi: 10.1007/s13398-018-0572-2
[28]	S. Rezapour, M. S. Souid, Z. Bouazza, A. Hussain, S. Etemad, On the fractional variable order thermostat model: existence theory on cones via piece-wise constant functions, J. Funct. Spaces, 2022 (2022), 8053620. https://doi.org/10.1155/2022/8053620 doi: 10.1155/2022/8053620
[29]	S. Rezapour, Z. Bouazza, M. S. Souid, S. Etemad, M. K. A. Kaabar, Darbo fixed point criterion on solutions of a Hadamard nonlinear variable order problem and Ulam-Hyers-Rassias stability, J. Funct. Spaces, 2022 (2022), 1769359. https://doi.org/10.1155/2022/1769359 doi: 10.1155/2022/1769359
[30]	A. Ben Makhlouf, A novel finite time stability analysis of nonlinear fractional-order time delay systems: a fixed point approach, Asian J. Control, 24 (2022), 3580–3587. https://doi.org/10.1002/asjc.2756 doi: 10.1002/asjc.2756
[31]	A. Ben Makhlouf, Partial practical stability for fractional‐order nonlinear systems, Math. Meth. Appl. Sci., 45 (2022), 5135–5148. https://doi.org/10.1002/mma.8097 doi: 10.1002/mma.8097
[32]	A. Ben Makhlouf, D. Baleanu, Finite time stability of fractional order systems of neutral type, Fractal Fract., 6 (2022), 289. https://doi.org/10.3390/fractalfract6060289 doi: 10.3390/fractalfract6060289
[33]	H. Arfaoui, A. Ben Makhlouf, Stability of a time fractional advection-diffusion system, Chaos, Solitons Fract., 157 (2022), 111949. https://doi.org/10.1016/j.chaos.2022.111949 doi: 10.1016/j.chaos.2022.111949
[34]	R. Almeida, Caputo-Hadamard fractional derivatives of variable order, Numer. Funct. Anal. Opt., 38 (2017), 1–19. https://doi.org/10.1080/01630563.2016.1217880 doi: 10.1080/01630563.2016.1217880
[35]	A. Ben Makhlouf, L. Mchiri, Some results on the study of Caputo-Hadamard fractional stochastic differential equations, Chaos Solitons Fract., 155 (2022), 111757. https://doi.org/10.1016/j.chaos.2021.111757 doi: 10.1016/j.chaos.2021.111757
[36]	K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Nonlocal fuzzy fractional stochastic evolution equations with fractional Brownian motion of order $(1, 2)$, AIMS Math., 7 (2022), 19344–19358. https://doi.org/10.3934/math.20221062 doi: 10.3934/math.20221062
[37]	K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Local and global existence and uniqueness of solution for class of fuzzy fractional functional evolution equation, J. Funct. Spaces, 2022 (2022), 7512754. https://doi.org/10.1155/2022/7512754 doi: 10.1155/2022/7512754
[38]	A. Khan, R. Shafqat, A. U. K. Niazi, Existence results of fuzzy delay impulsive fractional differential equation by fixed point theory approach, J. Funct. Spaces, 2022 (2022), 4123949. https://doi.org/10.1155/2022/4123949 doi: 10.1155/2022/4123949
[39]	K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Local and global existence and uniqueness of solution for time-fractional fuzzy Navier-Stokes equations, Fractal Fract., 6 (2022), 330. https://doi.org/10.3390/fractalfract6060330 doi: 10.3390/fractalfract6060330
[40]	R. Shafqat, A. U. K. Niazi, M. Yavuz, M. B. Jeelani, K. Saleem, Mild solution for the time-fractional Navier-Stokes equation incorporating MHD effects, Fractal Fract., 6 (2022), 580. https://doi.org/10.3390/fractalfract6100580 doi: 10.3390/fractalfract6100580
[41]	R. Shafqat, A. U. K. Niazi, M. B. Jeelani, N. H. Alharthi, Existence and uniqueness of mild solution where $\alpha \in (1, 2)$ for fuzzy fractional evolution equations with uncertainty, Fractal Fract., 6 (2022), 65. https://doi.org/10.3390/fractalfract6020065 doi: 10.3390/fractalfract6020065
[42]	S. N. Rao, A. H. Msmali, M. Singh, A. Ali, H. Ahmadini, Existence and uniqueness for a system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions, J. Funct. Spaces, 2020 (2020), 8821471. https://doi.org/10.1155/2020/8821471 doi: 10.1155/2020/8821471
[43]	C. Derbazi, H. Hammouche, Caputo-Hadamard fractional differential equations with nonlocal fractional integro-differential boundary conditions via topological degree theory, AIMS Math., 5 (2020), 2694–2709. https://doi.org/10.3934/math.2020174 doi: 10.3934/math.2020174
[44]	M. Gohar, C. Li, Z. Li, Finite difference methods for Caputo-Hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194. https://doi.org/10.1007/s00009-020-01605-4 doi: 10.1007/s00009-020-01605-4
[45]	Y. Bai, H. Kong, Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions, J. Nonlinear Sci. Appl., 10 (2017), 5744–5752. http://dx.doi.org/10.22436/jnsa.010.11.12 doi: 10.22436/jnsa.010.11.12
[46]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differenatial equations, North-Holland Mathematics Studies, Vol. 204, Amsterdam: Elsevier Science B.V., 2006.
[47]	I. Podlubny, Fractional differential equations, New York: Academic Press, 1998.
[48]	R. Almeida, D. F. M. Torres, Computing Hadamard type operators of variable fractional order, Appl. Math. Comput., 257 (2015), 74–88. https://doi.org/10.1016/j.amc.2014.12.071 doi: 10.1016/j.amc.2014.12.071
[49]	O. Kahouli, D. Boucenna, A. Ben Makhlouf, Y. Alruwaily, Some new weakly singular integral inequalities with applications to differential equations in frame of tempered $\chi$-fractional derivatives, Mathematics, 10 (2022), 3792. https://doi.org/10.3390/math10203792 doi: 10.3390/math10203792
[50]	A. Ben Makhlouf, D. Boucenna, A. M. Nagy, L. Mchiri, Some weakly singular integral inequalities and their applications to tempered fractional differential equations, J. Math., 2022 (2022), 1682942. https://doi.org/10.1155/2022/1682942 doi: 10.1155/2022/1682942

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