New interpretation of topological degree method of Hilfer fractional neutral functional integro-differential equation with nonlocal condition

Kanagaraj Muthuselvan; Baskar Sundaravadivoo; Suliman Alsaeed; Kottakkaran Sooppy Nisar; Kanagaraj Muthuselvan; Baskar Sundaravadivoo; Suliman Alsaeed; Kottakkaran Sooppy Nisar

doi:10.3934/math.2023876

AIMS Mathematics

2023, Volume 8, Issue 7: 17154-17170. doi: 10.3934/math.2023876

Previous Article Next Article

Research article

New interpretation of topological degree method of Hilfer fractional neutral functional integro-differential equation with nonlocal condition

1.
Department of Mathematics, Alagappa University, Karaikudi 630004, Tamil Nadu, India
2.
Applied Sciences Collage, Department of Mathematical Sciences, Umm Al-Qura University, P.O. Box 715, Makkah 21955, Saudi Arabia
3.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia
4.
School of Technology, Woxsen University- Hyderabad-502345, Telangana State, India

Received: 17 January 2023 Revised: 11 March 2023 Accepted: 27 March 2023 Published: 17 May 2023
MSC : 34A08, 34A12, 46B80

This manuscript deals with the concept of Hilfer fractional neutral functional integro-differential equation with a nonlocal condition. The solution representation of a given system is obtained from the strongly continuous operator, linear operator and bounded operator, as well as the Wright type of function. The sufficient and necessary conditions for the existence of a solution are attained using the topological degree method. The uniqueness of the solution is attained by Gronwall's inequality. Finally, we employed some specific numerical computations to examine the effectiveness of the results.
- Hilfer fractional derivative,
- nonlocal condition,
- Kuratowski measure of non-compactness,
- existence and uniqueness,
- Gronwall's inequality
Citation: Kanagaraj Muthuselvan, Baskar Sundaravadivoo, Suliman Alsaeed, Kottakkaran Sooppy Nisar. New interpretation of topological degree method of Hilfer fractional neutral functional integro-differential equation with nonlocal condition[J]. AIMS Mathematics, 2023, 8(7): 17154-17170. doi: 10.3934/math.2023876

Related Papers:

Abstract

This manuscript deals with the concept of Hilfer fractional neutral functional integro-differential equation with a nonlocal condition. The solution representation of a given system is obtained from the strongly continuous operator, linear operator and bounded operator, as well as the Wright type of function. The sufficient and necessary conditions for the existence of a solution are attained using the topological degree method. The uniqueness of the solution is attained by Gronwall's inequality. Finally, we employed some specific numerical computations to examine the effectiveness of the results.

References

[1]	R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210. https://doi.org/10.1122/1.549724 doi: 10.1122/1.549724
[2]	C. M. Ionescu, A. Lopes, D. Copot, J. A. Tenreiro Machado, J. H. T. Bates, The role of fractional calculus in modelling biological phenomena: A review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141–159. https://doi.org/10.1016/j.cnsns.2017.04.001 doi: 10.1016/j.cnsns.2017.04.001
[3]	A. A. Kilbas, H. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, 2006.
[4]	R. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), 1–104.
[5]	I. Podlubny, Fractional Differential Equations, Academic Press, 1998.
[6]	H. Gou, Monotone iterative technique for Hilfer fractional evolution equations with nonlocal conditions, Bull. des Sci. Math., 167 (2021), 102946. https://doi.org/10.1016/j.bulsci.2021.102946 doi: 10.1016/j.bulsci.2021.102946
[7]	K. Kavitha, V. Vijayakumar, K. S. Nisar, A. Shukla, W. Albalawi, A. H. Abdel-Aty, Existence and controllability of Hilfer fractional neutral differential equations with time delay via sequence method, AIMS Mathematics, 7 (2022), 12760–12780. https://doi.org/10.3934/math.2022706 doi: 10.3934/math.2022706
[8]	T. A. Faree, S. K. Panchal, Existence of solution for impulsive fractional differential equations via topological degree method, J. Korean Soc. Ind. Appl. Math., 25 (2021), 16–25. https://doi.org/10.12941/jksiam.2021.25.016 doi: 10.12941/jksiam.2021.25.016
[9]	H. A. Hammad, H. Aydi, M. Zayed, Involvement of the topological degree theory for solving a tripled system of multi-point boundary value problems, AIMS Mathematics, 8 (2023), 2257–2271. https://doi.org/10.3934/math.2023117 doi: 10.3934/math.2023117
[10]	E. S. Abada, H. A. Lakhal, M. E. Maouni, Topological degree method for fractional Laplacian system, Bull. Math. Anal. Appl., 13 (2021), 10–19.
[11]	A. Amara, S. Etemad, S. Rezapour, Topological degree theory and Caputo-Hadamard fractional boundary value problems, Adv. Differ. Equ., 2020 (2020), 369. https://doi.org/10.1186/s13662-020-02833-4 doi: 10.1186/s13662-020-02833-4
[12]	Z. Baitiche, C. Derbazi, M. Benchohra, $\psi$-Caputo fractional differential equations with multi-point boundary conditions by topological degree theory, Results. Nonlinear Anal., 4 (2020), 167–178.
[13]	L. Wang, X. B. Shu, Y. Cheng, R. Cui, Existence of periodic solutions of second-order nonlinear random impulsive differential equations via topological degree theory, Results Appl. Math., 12 (2021), 100215. https://doi.org/10.1016/j.rinam.2021.100215 doi: 10.1016/j.rinam.2021.100215
[14]	J. W. He, L. Zhang, Y. Zhou, B. Ahmad, Existence of solutions for fractional difference equations via topological degree methods, Adv. Differ. Equ., 2018 (2018), 153. https://doi.org/10.1186/s13662-018-1610-2 doi: 10.1186/s13662-018-1610-2
[15]	K. Muthuselvan, B. Sundaravadivoo, Analyze existence, uniqueness and controllability of impulsive fractional functional differential equations, Tbil. Math. J., 10 (2021).
[16]	U. Riaz, A. Zada, Analysis of $(\alpha, \beta)$-order coupled implicit Caputo fractional differential equations using topological degree method, Int. J. Nonlinear Sci. Numer. Simul., 22 (2020), 897–915. https://doi.org/10.1515/ijnsns-2020-0082 doi: 10.1515/ijnsns-2020-0082
[17]	A. Ullah, K. Shah, T. Abdeljawad, R. Ali Khan, I. Mahariq, Study of impulsive fractional differential equation under Robin boundary conditions by topological degree method, Bound. value. probl., 2020 (2020), 98. https://doi.org/10.1186/s13661-020-01396-3 doi: 10.1186/s13661-020-01396-3
[18]	A. El Mfadel, S. Melliani, E. M'hamed, Existence results for nonlocal Cauchy problem of nonlinear $\psi$-Caputo type fractional differential equations via topological degree methods, Adv. Theory Nonlinear Anal. Appl., 6 (2022), 270–279. https://doi.org/10.31197/atnaa.1059793 doi: 10.31197/atnaa.1059793
[19]	K. S. Nisar, K. Jothimani, C. Ravichandran, D. Baleanu, D. Kumar, New approach on controllability of Hilfer fractional derivatives with nondense domain, AIMS Mathematics, 7 (2022), 10079–10095. https://doi.org/10.3934/math.2022561 doi: 10.3934/math.2022561
[20]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000. https://doi.org/10.1142/3779
[21]	Y. Zhou, J. Wang, L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific, 2016. https://doi.org/10.1142/10238
[22]	K. Balachandran, J. P. Dauer, Elements of Control Theory, Narosa Publishing House, 1999.
[23]	H. M. Ahmed, M. M. El-Borai, H. M. El-Owaidy, A. S. Ghanem, Impulsive Hilfer fractional differential equations, Adv. Differ. Equ., 2018 (2018), 226. https://doi.org/10.1186/s13662-018-1679-7 doi: 10.1186/s13662-018-1679-7
[24]	C. Allalou, K. Hilal, Weak solution to $p(x)$-Kirchoff type problems under no-flux boundary condition by topological degree, Bol. da Soc. Parana. de Mat., 41 (2023), 1–12. https://doi.org/10.5269/bspm.63341 doi: 10.5269/bspm.63341
[25]	S. W. Ahmad, M. Sarwar, K. Shah, T. Abdeljawad, Study of a coupled system with sub-strip and Multi-Valued boundary conditions via topological degree theory on an infinite domain, Symmetry, 14 (2022), 841. https://doi.org/10.3390/sym14050841 doi: 10.3390/sym14050841
[26]	D. Luo, Q. Zhu, Z. Luo, A novel result on averaging principle of stochastic Hilfer-type fractional system involving non-Lipschitz coefficients, Appl. Math. Lett., 122 (2021), 107549. https://doi.org/10.1016/j.aml.2021.107549 doi: 10.1016/j.aml.2021.107549
[27]	Y. K. Ma, K. Kavitha, W. Albalawi, A. Shukla, K. S. Nisar, V. Vijayakumar, An analysis on the approximate controllability of Hilfer fractional neutral differential systems in Hilbert spaces, Alex. Eng. J., 61 (2022), 7291–7302. https://doi.org/10.1016/j.aej.2021.12.067 doi: 10.1016/j.aej.2021.12.067
[28]	M. El Ouaarabi, C. Allalou, M. Melliani, Existence of a weak solutions to a class of nonlinear parabolic problems via topological degree method, Gulf. J. math., 14 (2023), 148–159. https://doi.org/10.56947/gjom.v14i1.1091 doi: 10.56947/gjom.v14i1.1091
[29]	K. Shah, M. Sher, A. Ali, T. Abdeljawad, On degree theory for non-monotone type fractional order delay differential equations, AIMS Mathematics, 7 (2022), 9479–9492. https://doi.org/10.3934/math.2022526 doi: 10.3934/math.2022526

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)