Unified existence results for nonlinear fractional boundary value problems

Imran Talib; Asmat Batool; Muhammad Bilal Riaz; Md. Nur Alam; Imran Talib; Asmat Batool; Muhammad Bilal Riaz; Md. Nur Alam

doi:10.3934/math.2024202

AIMS Mathematics

2024, Volume 9, Issue 2: 4118-4134. doi: 10.3934/math.2024202

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Unified existence results for nonlinear fractional boundary value problems

1.
Department of Mathematics, Nonlinear Analysis Group (NAG), Virtual University of Pakistan, 54-Lawrence Road, Lahore, Pakistan
2.
Department of Mathematics, University of Management and Technology, 54770 Lahore, Pakistan
3.
IT4Innovations, VSB - Technical University of Ostrava, Ostrava, Czech Republic
4.
Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon
5.
Department of Mathematics, Pabna Unversity of science and Technology, Pabna-6600, Bangladesh

Received: 15 November 2023 Revised: 03 January 2024 Accepted: 09 January 2024 Published: 15 January 2024
MSC : 26A33, 35R11, 45K05, 65M99

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.
Citation: Imran Talib, Asmat Batool, Muhammad Bilal Riaz, Md. Nur Alam. Unified existence results for nonlinear fractional boundary value problems[J]. AIMS Mathematics, 2024, 9(2): 4118-4134. doi: 10.3934/math.2024202

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Abstract

In this work, we focus on investigating the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. By extending the framework of the technique based on well-ordered coupled lower and upper solutions, we guarantee the existence of solutions in a sector defined by these solutions. One notable aspect of our study is that the proposed approach unifies the existence results for the problems that have previously been discussed separately in the literature. To substantiate these findings, we have added three illustrative examples.

References

[1]	A. Gerasimov, A generalization of linear laws of deformation and its application to problems of internal friction, Akad. Nauk SSSR. Prikl. Mat. Meh, 12 (1948), 251–260.
[2]	G. Scott-Blair, The role of psychophysics in rheology, Journal of Colloid Science, 2 (1947), 21–32. http://dx.doi.org/10.1016/0095-8522(47)90007-X doi: 10.1016/0095-8522(47)90007-X
[3]	A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, North-Holland: Elsevier, 2006.
[4]	I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Elsevier, 1999. http://dx.doi.org/10.1016/s0076-5392(99)x8001-5
[5]	M. Benchohra, E. Karapinar, J. Lazreg, A. Salim, Advanced topics in fractional differential equations: a fixed point approach, Cham: Springer, 2023. http://dx.doi.org/10.1007/978-3-031-26928-8
[6]	S. Samko, A. Kilbas, O. Marichev, Fractional integrals and derivatives: theory and applications, Philadelphia: Gordon and Breach Science Publishers, 1993.
[7]	R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), 399–408. http://dx.doi.org/10.1016/S0301-0104(02)00670-5 doi: 10.1016/S0301-0104(02)00670-5
[8]	R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application, J. Comput. Appl. Math., 308 (2016), 39–45. http://dx.doi.org/10.1016/j.cam.2016.05.014 doi: 10.1016/j.cam.2016.05.014
[9]	J. Vanterler da C. Sousa, E. Capelas de Oliveira, On the $\psi$- fractional integral and applications, Comp. Appl. Math., 38 (2019), 4. http://dx.doi.org/10.1007/s40314-019-0774-z doi: 10.1007/s40314-019-0774-z
[10]	J. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850–859. http://dx.doi.org/10.1016/j.amc.2015.05.144 doi: 10.1016/j.amc.2015.05.144
[11]	K. Furati, M. Kassim, N. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616–1626. http://dx.doi.org/10.1016/j.camwa.2012.01.009 doi: 10.1016/j.camwa.2012.01.009
[12]	R. Hilfer, Fractional calculus and regular variation in thermodynamics, Applications of Fractional Calculus in Physics, 9 (2000), 429–463. http://dx.doi.org/10.1142/9789812817747_0009 doi: 10.1142/9789812817747_0009
[13]	I. Ahmed, P. Kumam, F. Jarad, P. Borisut, K. Sitthithakerngkiet, A. Ibrahim, Stability analysis for boundary value problems with generalized nonlocal condition via Hilfer-Katugampola fractional derivative, Adv. Differ. Equ., 2020 (2020), 225. http://dx.doi.org/10.1186/s13662-020-02681-2 doi: 10.1186/s13662-020-02681-2
[14]	S. Asawasamrit, A. Kijjathanakorn, S. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull. Korean Math. Soc., 55 (2018), 1639–1657. http://dx.doi.org/10.4134/BKMS.b170887 doi: 10.4134/BKMS.b170887
[15]	A. Atangana, D. Baleanu, Application of fixed point theorem for stability analysis of a nonlinear Schrodinger with Caputo-Liouville derivative, Filomat, 31 (2017), 2243–2248. http://dx.doi.org/10.2298/FIL1708243A doi: 10.2298/FIL1708243A
[16]	P. Borisut, P. Kumam, I. Ahmed, K. Sitthithakerngkiet, Nonlinear Caputo fractional derivative with nonlocal Riemann-Liouville fractional integral condition via fixed point theorems, Symmetry, 11 (2019), 829. http://dx.doi.org/10.3390/sym11060829 doi: 10.3390/sym11060829
[17]	S. Harikrishnan, K. Shah, D. Baleanu, K. Kanagarajan, Note on the solution of random differential equations via $\psi$-Hilfer fractional derivative, Adv. Differ. Equ., 2018 (2018), 224. http://dx.doi.org/10.1186/s13662-018-1678-8 doi: 10.1186/s13662-018-1678-8
[18]	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. http://dx.doi.org/10.3934/dcdss.2020040 doi: 10.3934/dcdss.2020040
[19]	W. Shammakh, H. Alzumi, Existence results for nonlinear fractional boundary value problem involving generalized proportional derivative, Adv. Differ. Equ., 2019 (2019), 94. http://dx.doi.org/10.1186/s13662-019-2038-z doi: 10.1186/s13662-019-2038-z
[20]	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. http://dx.doi.org/10.1007/s00009-017-1061-0 doi: 10.1007/s00009-017-1061-0
[21]	D. Vivek, K. Shah, K. Kanagarajan, Existence theory and continuation analysis of nonlinear pantograph equations via Hilfer-Hadamard fractional derivative, Dyn. Contin. Discrete Impuls. A-Math. Anal., 25 (2018), 397–417.
[22]	S. Harikrishnan, K. Kanagarajan, E. Elsayed, Existence of solutions of nonlocal initial value problems for differential equations with Hilfer-Katugampola fractional derivative, RACSAM, 113 (2019), 3903. http://dx.doi.org/10.1007/s13398-019-00645-0 doi: 10.1007/s13398-019-00645-0
[23]	A. Wongcharoen, S. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer-type pantograph fractional differential equations and inclusions, Adv. Differ. Equ., 2020 (2020), 279. http://dx.doi.org/10.1186/s13662-020-02747-1 doi: 10.1186/s13662-020-02747-1
[24]	K. Kucche, A. Mali, On the nonlinear (k, $\psi$)-Hilfer fractional differential equations, Chaos Soliton. Fract., 152 (2021), 111335. http://dx.doi.org/10.1016/j.chaos.2021.111335 doi: 10.1016/j.chaos.2021.111335
[25]	I. Ahmed, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Differ. Equ., 2020 (2020), 329. http://dx.doi.org/10.1186/s13662-020-02792-w doi: 10.1186/s13662-020-02792-w

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