Multiple positive solutions for system of mixed Hadamard fractional boundary value problems with $ (p_{1}, p_{2}) $-Laplacian operator

Sabbavarapu Nageswara Rao; Abdullah Ali H. Ahmadini; Sabbavarapu Nageswara Rao; Abdullah Ali H. Ahmadini

doi:10.3934/math.2023755

AIMS Mathematics

2023, Volume 8, Issue 6: 14767-14791. doi: 10.3934/math.2023755

Previous Article Next Article

Research article Special Issues

Multiple positive solutions for system of mixed Hadamard fractional boundary value problems with $ (p_{1}, p_{2}) $-Laplacian operator

Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia

Received: 03 March 2023 Revised: 11 April 2023 Accepted: 12 April 2023 Published: 21 April 2023
MSC : 34A08, 34B15, 34B18, 34B27

In this paper, we investigate the existence of positive solutions of a system of Riemann-Liouville Hadamard differential equations with $ p $-Laplacian operators under various combinations of superlinearity and sublinearity. We apply the Guo-Krasnosel'skii fixed point theorem for the proof of the existence results.
- Hadamard fractional system,
- positive solutions,
- boundary value problems,
- $ p $-Laplacian,
- fixed point theorem
Citation: Sabbavarapu Nageswara Rao, Abdullah Ali H. Ahmadini. Multiple positive solutions for system of mixed Hadamard fractional boundary value problems with $ (p_{1}, p_{2}) $-Laplacian operator[J]. AIMS Mathematics, 2023, 8(6): 14767-14791. doi: 10.3934/math.2023755

Related Papers:

Abstract

In this paper, we investigate the existence of positive solutions of a system of Riemann-Liouville Hadamard differential equations with $ p $-Laplacian operators under various combinations of superlinearity and sublinearity. We apply the Guo-Krasnosel'skii fixed point theorem for the proof of the existence results.

References

[1]	A. Alsaedi, R. Luca, B. Ahmad, Existence of positive solutions for a system of singular fractional boundary value problems with $p$-Laplacian operators, Mathematics., 8 (2020), 1890. https://doi.org/10.3390/math8111890 doi: 10.3390/math8111890
[2]	B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Switzerland: Springer, 2017. https://doi.org/10.1007/978-3-319-52141-1
[3]	B. Ahmad, R. Luca, Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions, Frac. Calc. Appl. Anal., 21 (2018), 423–441. https://doi.org/10.1515/fca-2018-0024 doi: 10.1515/fca-2018-0024
[4]	B. Ahmad, S. K. Ntouyas, A. Alsaedi, A. Albideewi, A study of a coupled system of Hadamard fractional differential equations with nonlocal coupled initial-multipoint conditions, Adv. Differ. Equ., 2021 (2021), 33. https://doi.org/10.1186/s13662-020-03198-4 doi: 10.1186/s13662-020-03198-4
[5]	B. Ahmad, J. Henderson, R. Luca, Boundary value problems for fractional differential equations and systems, World Scientific, 2021.
[6]	B. Ahmad, A. F. Albideewi, S. K. Ntouyas, A. Alsaedi, Existence results for a multi-point boundary value problem of nonlinear sequential Hadamard fractional differential equations, Cubo (Temuco), 23 (2021), 225–237. https://doi.org/10.4067/S0719-06462021000200225 doi: 10.4067/S0719-06462021000200225
[7]	M. Al-Refai, Y. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications, Fract. Calc. Appl. Anal., 17 (2014), 483–498. https://doi.org/10.2478/s13540-014-0181-5 doi: 10.2478/s13540-014-0181-5
[8]	S. Das, Functional fractional calculus for system identification and control, Berlin: Springer, 2008.
[9]	X. Du, Y. Meng, H. Pang, Iterative positive solutions to a coupled Hadamard-type fractional differential system on infinite domain with the multistrip and multipoint mixed boundary conditions, J. Funct. Space., 2020 (2020), 6508075. https://doi.org/10.1155/2020/6508075 doi: 10.1155/2020/6508075
[10]	D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, 1988.
[11]	H. Huang, K. Zhao, X. Liu, On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses, AIMS Math., 7 (2022), 19221–19236. https://doi.org/10.3934/math.20221055 doi: 10.3934/math.20221055
[12]	J. Hadamard, Essai sur létude des fonctions donnees par leur développment de Taylor, J. Math. Pure. Appl., 8 (1892), 101–186.
[13]	J. Hristov, New trends in fractional differential equations with real-world applications in physics, Frontiers Media SA, 2020.
[14]	X. Hao, H. Wang, L. Liu, Y. Cui, Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and $p$-Laplacian operator, Bound. Value Probl., 2017 (2017), 182. https://doi.org/10.1186/s13661-017-0915-5 doi: 10.1186/s13661-017-0915-5
[15]	J. Jiang, D. O'Regan, J. Xu, Z. Fu, Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions, J. Inequal. Appl., 2019 (2019), 204. https://doi.org/10.1186/s13660-019-2156-x doi: 10.1186/s13660-019-2156-x
[16]	J. Jiang, D. O'Regan, J. Xu, Y. Cui, Positive solutions for a Hadamard fractional $p$-Laplacian three-point boundary value problem, Mathematics., 7 (2019), 439. https://doi.org/10.3390/math7050439 doi: 10.3390/math7050439
[17]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[18]	M. Khuddush, K. R. Prasad, P. Veeraiah, Infinitely many positive solutions for an iterative system of fractional BVPs with multistrip Riemann-Stieltjes integral boundary conditions, Afr. Mat., 33 (2022), 91. https://doi.org/10.1007/s13370-022-01026-4 doi: 10.1007/s13370-022-01026-4
[19]	M. Khuddush, K. R. Prasad, D. Leela, Existence theory and stability analysis to the system of infinite point fractional order bvps by multivariate best proximity point theorem, Int. J. Nonlinear Anal. Appl., 13 (2022), 1713–1733. https://doi.org/10.22075/ijnaa.2022.25945.3167 doi: 10.22075/ijnaa.2022.25945.3167
[20]	M. Khuddush, K. R. Prasad, Iterative system of nabla fractional order difference equations with two-point boundary conditions, Appl. Math., 11 (2022), 57–74. https://doi.org/10.13164/ma.2022.06 doi: 10.13164/ma.2022.06
[21]	M. Khuddush, S. Kathun, Infinitely many positive solutions and Ulam-Hyers stability of fractional order two-point boundary value problems, J. Anal., 2023 (2023). https://doi.org/10.1007/s41478-023-00549-8 doi: 10.1007/s41478-023-00549-8
[22]	L. S. Leibenson, General problem of the movement of a compressible uid in a porous medium, Izv. Akad. Nauk Kirg. SSSR, 9 (1983), 7–10.
[23]	M. Li, P. Guo, C. Ren, Water resources management models based on two-level linear fractional programming method under uncertainty, J. Water Res. Plan. Man., 141 (2015), 05015001. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000518 doi: 10.1061/(ASCE)WR.1943-5452.0000518
[24]	R. Luca, Positive solutions for a system of fractional differential equations with $p$-Laplacian operator and multi-point boundary conditions, Nonlinear Anal. Model., 23 (2018), 771–801. https://doi.org/10.15388/NA.2018.5.8 doi: 10.15388/NA.2018.5.8
[25]	R. Luca, Positive solutions for a system of Riemann-Liouville fractional differential equations with multi-point fractional boundary conditions, Bound. Value Probl., 2017 (2017), 102. https://doi.org/10.1186/s13661-017-0833-6 doi: 10.1186/s13661-017-0833-6
[26]	R. Luca, On a system of fractional boundary value problems with $p$-Laplacian operator, Dyn. Syst. Appl., 28 (2019), 691–713.
[27]	S. Li, C. Zhai, Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals, J. Inequal Appl., 2019 (2019), 150. https://doi.org/10.1186/s13660-019-2102-y doi: 10.1186/s13660-019-2102-y
[28]	Y. Li, J. Xu, H. Luo, Approximate iterative sequences for positive solutions of a Hadamard type fractional differential system involving Hadamard type fractional derivatives, AIMS Math., 6 (2021), 7229–7250. https://doi.org/10.3934/math.2021424 doi: 10.3934/math.2021424
[29]	A. H. Msmali, Positive solutions for a system of Hadamard fractional $(\varrho_{1}, \varrho_{2}, \varrho_{3})$-Laplacian operator with a parameter in the boundary, AIMS Math., 7 (2022), 10564–10581. https://doi.org/10.3934/math.2022589 doi: 10.3934/math.2022589
[30]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
[31]	I. Podlubny, Fractional differential equations, Academic Press, 1999.
[32]	K. R. Prasad, I. D. Leela, M. Khuddush, Existence and uniqueness of positive solutions for system of $(p, q, r)$-Laplacian fractional order boundary value problems, Adv. Theory Nonlinear Anal. Appl., 5 (2021), 138–157. https://doi.org/10.31197/atnaa.703304 doi: 10.31197/atnaa.703304
[33]	S. Rekhviashvili, A. Pskhu, P. Agarwal, S. Jain, Application of the fractional oscillator model to describe damped vibrations, Turk. J. Phys., 43 (2019), 236–242. https://doi.org/10.3906/fiz-1811-16 doi: 10.3906/fiz-1811-16
[34]	S. N. Rao, A. Ahmadini, Multiple positive solutions for a system of $(p_{1}, p_{2}, p_{3})$-Laplacian Hadamard fractional order BVP with parameters, Adv. Differ. Equ., 2021 (2021), 436. https://doi.org/10.1186/s13662-021-03591-7 doi: 10.1186/s13662-021-03591-7
[35]	S. N. Rao, M. Singh, M. Z. Meetei, Multiplicity of positive solutions for Hadamard fractional differential equations with $p$-Laplacian operator, Bound. Value Probl., 2020 (2020), 43. https://doi.org/10.1186/s13661-020-01341-4 doi: 10.1186/s13661-020-01341-4
[36]	J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in fractional calculus: Theoretical developments and applications in physics and engineering, Dordrecht: Springer, 2007. https://doi.org/10.1007/978-1-4020-6042-7
[37]	A. A. Kilbas, O. I. Marichev, S. G. Samko, Fractional integrals and derivatives: Theory and applications, 1993.
[38]	A. Tudorache, R. Luca, System of Riemann-Liouville fractional differential equations with $p$-Laplacian operators and nonlocal coupled boundary conditions, Fractal Fract., 6 (2022), 610. https://doi.org/10.3390/fractalfract6100610 doi: 10.3390/fractalfract6100610
[39]	A. Tudorache, R. Luca, Positive solutions for a system of Riemann-Liouville fractional boundary value problems with $p$-Laplacian operators, Adv. Differ. Equ., 2020 (2020), 292. https://doi.org/10.1186/s13662-020-02750-6 doi: 10.1186/s13662-020-02750-6
[40]	A. Tudorache, R. Luca, Positive solutions of a singular fractional boundary value problem with $r$-Laplacian operators, Fractal Fract., 6 (2022), 18. https://doi.org/10.3390/fractalfract6010018 doi: 10.3390/fractalfract6010018
[41]	A. Tudorache, R. Luca, Positive solutions for a system of Riemann-Liouville fractional boundary value problems with $p$-Laplacian operators, Adv. Differ. Equ., 2020 (2020), 292. https://doi.org/10.1186/s13662-020-02750-6 doi: 10.1186/s13662-020-02750-6
[42]	Y. Tian, Z. Bai, S. Sun, Positive solutions for a boundary value problem of fractional differential equation with $p$-Laplacian operator, Adv. Differ. Equ., 2019 (2019), 349. https://doi.org/10.1186/s13662-019-2280-4 doi: 10.1186/s13662-019-2280-4
[43]	G. Wang, T. Wang, On a nonlinear Hadamard type fractional differential equation with $p$-Laplacian operator and strip condition, J. Nonlinear Sci. Appl., 9 (2016), 5073–5081. http://dx.doi.org/10.22436/jnsa.009.07.10 doi: 10.22436/jnsa.009.07.10
[44]	G. T. Wang, K. Pei, R. P. Agarwal, L. H. Zhang, B. Ahmad, Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math., 343 (2018), 230–239. https://doi.org/10.1016/j.cam.2018.04.062 doi: 10.1016/j.cam.2018.04.062
[45]	H. Wang, J. Jiang, Existence and multiplicity of positive solutions for a system of nonlinear fractional multi-point boundary value problems with $p$-Laplacian operator, J. Appl. Anal. Comput., 11 (2021), 351–366. https://doi.org/10.11948/20200021 doi: 10.11948/20200021
[46]	Y. Wang, Multiple positive solutions for mixed fractional differential system with $p$-Laplacian operators, Bound. Value Probl., 2019 (2019), 144. https://doi.org/10.1186/s13661-019-1257-2 doi: 10.1186/s13661-019-1257-2
[47]	Y. Wang, G. Zhao, A comparative study of fractional-order models for lithium-ion batteries using Runge Kutta optimizer and electrochemical impedance spectroscopy, Control Eng. Pract., 133 (2023), 105451. https://doi.org/10.1016/j.conengprac.2023.105451 doi: 10.1016/j.conengprac.2023.105451
[48]	Y. Wang, G. Gao, X. Li, Z. Chen, A fractional-order model-based state estimation approach for lithium-ion battery and ultra-capacitor hybrid power source system considering load trajectory, J. power sources, 449 (2020), 227543. https://doi.org/10.1016/j.jpowsour.2019.227543 doi: 10.1016/j.jpowsour.2019.227543
[49]	J. Xu, J. Jiang, D. O'Regan, Positive solutions for a class of $p$-Laplacian Hadamard fractional three-point boundary value problem, Mathematics., 8 (2020), 308. https://doi.org/10.3390/math8030308 doi: 10.3390/math8030308
[50]	J. Xu, D. O'Regan, Positive solutions for a fractional $p$-Laplacian boundary value problem, Filomat., 31 (2017), 1549–1558.
[51]	J. Xu, L. Liu, S. Bai, Y. Wu, Solvability for a system of Hadamard fractional multi-point boundary value problems, Nonlinear Anal. Model., 26 (2021), 502–521. https://doi.org/10.15388/namc.2021 doi: 10.15388/namc.2021
[52]	F. Yan, M. Zuo, X. Hao, Positive solution for a fractional singular boundary value problem with $p$-Laplacian operator, Bound. Value Probl., 2018 (2018), 51. https://doi.org/10.1186/s13661-018-0972-4 doi: 10.1186/s13661-018-0972-4
[53]	W. Yang, Monotone iterative technique for a coupled system of nonlinear Hadamard fractional differential equations, J. Appl. Math. Comput., 59 (2019), 585–596. https://doi.org/10.1007/s12190-018-1192-x doi: 10.1007/s12190-018-1192-x
[54]	K. Zhao, Existence and UH-stability of integral boundary problem for a class of nonlinear higher-order Hadamard fractional Langevin equation via Mittag-Leffler functions, Filomat, 37 (2023), 1053–1063. https://doi.org/10.2298/FIL2304053Z doi: 10.2298/FIL2304053Z
[55]	W. Zhang, J. Ni, New multiple positive solutions for Hadamard type fractional differential equations with nonlocal conditions on an infinite interval, Appl. Math. Lett., 118 (2021), 107165. https://doi.org/10.1016/j.aml.2021.107165 doi: 10.1016/j.aml.2021.107165

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)