Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative

Saima Rashid; Abdulaziz Garba Ahmad; Fahd Jarad; Ateq Alsaadi; Saima Rashid; Abdulaziz Garba Ahmad; Fahd Jarad; Ateq Alsaadi

doi:10.3934/math.2023018

AIMS Mathematics

2023, Volume 8, Issue 1: 382-403. doi: 10.3934/math.2023018

Previous Article Next Article

Research article Special Issues

Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative

1.
Department of Mathematics, Government College University, Faisalabad, Pakistan
2.
Department of Mathematics Programme, National Mathematical Centre, Abuja, Nigeria
3.
Department of Mathematics, Çankaya University, Ankara, Turkey
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
6.
Department of Mathematics and Statistics, College of Science, Taif University, P.O.Box 11099, Taif 21944, Saudi Arabia

Received: 22 July 2022 Revised: 04 September 2022 Accepted: 16 September 2022 Published: 30 September 2022
MSC : 46S40, 47H10, 54H25

This article adopts a class of nonlinear fractional differential equation associating Hilfer generalized proportional fractional ($ GPF $) derivative with having boundary conditions, which amalgamates the Riemann-Liouville $ (RL) $ and Caputo-$ GPF $ derivative. Taking into consideration the weighted space continuous mappings, we first derive a corresponding integral for the specified boundary value problem. Also, we investigate the existence consequences for a certain problem with a new unified formulation considering the minimal suppositions on nonlinear mapping. Detailed developments hold in the analysis and are dependent on diverse tools involving Schauder's, Schaefer's and Kransnoselskii's fixed point theorems. Finally, we deliver two examples to check the efficiency of the proposed scheme.
- existence of solution,
- Hilfer proportional fractional derivative,
- boundary value problem,
- fixed point theory
Citation: Saima Rashid, Abdulaziz Garba Ahmad, Fahd Jarad, Ateq Alsaadi. Nonlinear fractional differential equations and their existence via fixed point theory concerning to Hilfer generalized proportional fractional derivative[J]. AIMS Mathematics, 2023, 8(1): 382-403. doi: 10.3934/math.2023018

Related Papers:

Abstract

This article adopts a class of nonlinear fractional differential equation associating Hilfer generalized proportional fractional ($ GPF $) derivative with having boundary conditions, which amalgamates the Riemann-Liouville $ (RL) $ and Caputo-$ GPF $ derivative. Taking into consideration the weighted space continuous mappings, we first derive a corresponding integral for the specified boundary value problem. Also, we investigate the existence consequences for a certain problem with a new unified formulation considering the minimal suppositions on nonlinear mapping. Detailed developments hold in the analysis and are dependent on diverse tools involving Schauder's, Schaefer's and Kransnoselskii's fixed point theorems. Finally, we deliver two examples to check the efficiency of the proposed scheme.

References

[1]	M. Nazeer, F. Hussain, M. Ijaz Khan, Asad-ur-Rehman, E. R. El-Zahar, Y. M. Chu, et al., Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel, Appl. Math. Comput., 420 (2022), 126868. https://doi.org/10.1016/j.amc.2021.126868 doi: 10.1016/j.amc.2021.126868
[2]	Y. M. Chu, B. M. Shankaralingappa, B. J. Gireesha, F. Alzahrani, M. Ijaz Khan, S. U. Khan, Combined impact of Cattaneo-Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano-material surface, Appl. Math. Comput., 419 (2022), 126883. https://doi.org/10.1016/j.amc.2021.126883 doi: 10.1016/j.amc.2021.126883
[3]	Y. M. Chu, U. Nazir, M. Sohail, M. M. Selim, J. R. Lee, Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach, Fractal Fract., 5 (2021), 119. https://doi.org/10.3390/fractalfract5030119 doi: 10.3390/fractalfract5030119
[4]	T. H. Zhao, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, et al., A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math., 20 (2021), 160–176.
[5]	Z. Denton, A. S. Vatsala, Fractional integral inequalities and applications, Comput. Math. Appl., 59 (2010), 1087–1094. https://doi.org/10.1016/j.camwa.2009.05.012 doi: 10.1016/j.camwa.2009.05.012
[6]	R. Khalil, M. Al 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
[7]	R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci., 44 (2017), 460–481. https://doi.org/10.1016/j.cnsns.2016.09.006 doi: 10.1016/j.cnsns.2016.09.006
[8]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[9]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000.
[10]	A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948–956. https://doi.org/10.1016/j.amc.2015.10.021 doi: 10.1016/j.amc.2015.10.021
[11]	S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, Inequalities by means of generalized proportional fractional integral operators with respect to another function, Mathematics, 7 (2020), 1225. https://doi.org/10.3390/math7121225 doi: 10.3390/math7121225
[12]	R. A. Yan, S. R. Sun, Z. L. Han, Existence of solutions of boundary value problems for caputo fractional differential equations on time scales, Bull. Iranian Math. soc., 42 (2016), 247–262.
[13]	A. Atangana, D. Baleanu, Application of fixed point theorem for stability analysis of a nonlinear Schördinger with Caputo-Liouville derivative, Filomat, 31 (2017), 2243–2248. https://doi.org/10.2298/FIL1708243A doi: 10.2298/FIL1708243A
[14]	F. Jarad, S. Harikrishnan, K. Shah, K. Kanagarajan, Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative, Discrete Cont. Dyn. S, 13 (2020), 723–739. https://doi.org/10.3934/dcdss.2020040 doi: 10.3934/dcdss.2020040
[15]	O. A. Arqub, Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method, Int. J. Numer. Method. H., 30 (2020), 4711–4733. https://doi.org/10.1108/HFF-10-2017-0394 doi: 10.1108/HFF-10-2017-0394
[16]	S. Djennadi, N. Shawagfeh, M. Inc, M. S. Osman, J. F. Gómez-Aguilar, O. A. Arqub, The Tikhonov regularization method for the inverse source problem of time fractional heat equation in the view of ABC-fractional technique, Phys. Scr., 96 (2021), 094006. https://doi.org/10.1088/1402-4896/ac0867 doi: 10.1088/1402-4896/ac0867
[17]	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
[18]	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
[19]	D. R. Anderson, D. J. Ulness, Newly defined conformable derivatives, Adv. Dyn. Syst. Appl. 10 (2015), 109–137. https://doi.org/10.13140/RG.2.1.1744.9444 doi: 10.13140/RG.2.1.1744.9444
[20]	T. H. Zhao, W. M. Qian, Y. M. Chu, Sharp power mean bounds for the tangent and hyperbolic sine means, J. Math. Inequal., 15 (2021), 1459–1472. https://doi.org/10.7153/jmi-2021-15-100 doi: 10.7153/jmi-2021-15-100
[21]	S. Rashid, E. I. Abouelmagd, S. Sultana, Y. M. Chu, New developments in weighted $n$-fold type inequalities via discrete generalized ${\rm{\hat h}}$-proportional fractional operators, Fractals, 30 (2022), 2240056. https://doi.org/10.1142/S0218348X22400564 doi: 10.1142/S0218348X22400564
[22]	S. Rashid, E. I. Abouelmagd, A. Khalid, F. B. Farooq, Y. M. Chu, Some recent developments on dynamical $\hbar$-discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels, Fractals, 30 (2022), 2240110. https://doi.org/10.1142/S0218348X22401107 doi: 10.1142/S0218348X22401107
[23]	M. Al Qurashi, S. Rashid, S. Sultana, H. Ahmad, K. A. Gepreel, New formulation for discrete dynamical type inequalities via $h$-discrete fractional operator pertaining to nonsingular kernel, Math. Biosci. Eng., 18 (2021), 1794–1812. https://doi.org/10.3934/mbe.2021093 doi: 10.3934/mbe.2021093
[24]	S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, AIMS Math., 6 (2021), 4507–4525. https://doi.org/10.3934/math.2021267 doi: 10.3934/math.2021267
[25]	F. Jarad, T. Abdeljawad, J. Alzabut, Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top., 226 (2017), 3457–3471. https://doi.org/10.1140/epjst/e2018-00021-7 doi: 10.1140/epjst/e2018-00021-7
[26]	S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y. M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), 2240026. https://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
[27]	K. Karthikeyan, P. Karthikeyan, H. M. Baskonus, K. Venkatachalam, Y. M. Chu, Almost sectorial operators on $\Psi$-Hilfer derivative fractional impulsive integro-differential equations, Math. Method. Appl. Sci., 45 (2022), 8045–8059. https://doi.org/10.1002/mma.7954 doi: 10.1002/mma.7954
[28]	S. Rashid, S. Sultana, Y. Karaca, A. Khalid, Y. M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 30 (2022), 2240026. https://doi.org/10.1142/S0218348X22400266 doi: 10.1142/S0218348X22400266
[29]	S. Rashid, F. Jarad, M. A. Noor, Grüss-type integrals inequalities via generalized proportional fractional operators, RACSAM, 114 (2020), 93. https://doi.org/10.1007/s13398-020-00823-5 doi: 10.1007/s13398-020-00823-5
[30]	I. Ahmad, P. Kumam, F. Jarad, P. Borisut, Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Differ. Equ., 2020 (2020), 329. https://doi.org/10.1186/s13662-020-02792-w doi: 10.1186/s13662-020-02792-w
[31]	K. Shah, D. Vivek, K. Kanagarajan, Dynamics and stability of $\alpha$-fractional pantograph equations with boundary conditions, Bol. Soc. Paran. Mat., 39 (2021), 43–55. https://doi.org/10.5269/bspm.41154 doi: 10.5269/bspm.41154
[32]	D. Vivek, K. Kanagarajan, E. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions, Mediterr. J. Math., 15 (2018), 15. https://doi.org/10.1007/s00009-017-1061-0 doi: 10.1007/s00009-017-1061-0
[33]	O. A. Arqub, Reproducing Kernel algorithm for the analytical-numerical solutions of nonlinear systems of singular periodic boundary value problems, Math. Probl. Eng., 2015 (2015), 518406. https://doi.org/10.1155/2015/518406 doi: 10.1155/2015/518406
[34]	S. Djennadi, N. Shawagfeh, O. A. Arqub, A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations, Chaos Soliton. Fract., 150 (2021), 111127. https://doi.org/10.1016/j.chaos.2021.111127 doi: 10.1016/j.chaos.2021.111127
[35]	W. Shammakh, H. Z. Alzum, Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative, Adv. Differ. Equ., 2019 (2019), 94. https://doi.org/10.1186/s13662-019-2038-z doi: 10.1186/s13662-019-2038-z
[36]	A. Granas, J. Dugundi, Fixed point theory, New York: Springer, 2003.
[37]	M. Krasnoselskii, Two remarks about the method of successive approximations, Uspekhi Mat. Nauk, 10 (1955), 123–127.

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)