On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions
Abstract
:1. Introduction
- The main motivation for this work is to use the -Caputo fractional derivative to present a new class of N-InI -CFIDE with BCs;
- Moreover, we investigate the existence and uniqueness of the solutions of Equations (1)–(3) using Krasnoselkii’s and Banach’s FPT;
2. Supporting Notes
3. Main Results
4. Example
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Podlubny, I. Fractional Differential Equations; Acadamic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Almedia, R.; Malinowska, A.B.; Odzijewicz, T. Fractional differential equations with dependence on the caputo-katugampola derivatives. J. Comput. Nonlinear Dyn. 2016, 11, 61017. [Google Scholar] [CrossRef] [Green Version]
- Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 460–481. [Google Scholar] [CrossRef] [Green Version]
- Anguraj, A.; Karthikeyan, P.; Rivero, M.; Trujillo, J.J. On new existence results for fractional integro-differential equations with impulsive and integral conditions. Comput. Math. Appl. 2014, 66, 2587–2594. [Google Scholar] [CrossRef]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006; p. 204. [Google Scholar]
- Katugampola, U.N. New approach to a generalized fractional integral. Appl. Math. Comput. 2011, 218, 860–865. [Google Scholar] [CrossRef] [Green Version]
- Karthikeyan, K.; Karthikeyan, P.; Baskonus, H.M.; Ming-Chu, Y.; Venkatachalam, K. Almost sectorial operators on Ψ-Hilfer derivative fractional impulsive integro-differential equations. Math. Methods Appl. Sci. 2022, 45, 8045–8059. [Google Scholar] [CrossRef]
- Karthikeyan, K.; Debbouche, A.; Delfim, F.T. Analysis of Hilfer fractional integro-differential equations with almost sectorial operators. Fractal Fract. 2021, 5, 22. [Google Scholar] [CrossRef]
- Karthikeyan, K.; Karthikeyan, P.; Chalishajar, D.N.; Senthil Raja, D. Analysis on Ψ-Hilfer Fractional Impulsive Differential Equations. Symmetry 2021, 13, 1895. [Google Scholar] [CrossRef]
- Aissani, K.; Benchohra, M.; Benkhettou, N. On fractional integro-differential equations with state-dependent delay and non-instantaneous impulses. CUBO Math. J. 2019, 21, 61–75. [Google Scholar] [CrossRef] [Green Version]
- Wang, Y.; Liang, S.; Wang, Q. Existence results for fractional differential equations with integral and multipoint boundary conditions. Bound. Value Probl. 2018, 4, 2–11. [Google Scholar]
- Karthikeyan, P.; Venkatachalam, K. Some results on multipoint integral boundary value problems for fractional integro-differential equations. Prog. Fract. Differ. Appl. 2021, 2, 1–10. [Google Scholar]
- Kailasavalli, S.; MallikaArjunan, M.; Karthikeyan, P. Existence of solutions for fractional boundary value problems involving integro-differential equations in banach spaces. Nonlinear Stud. 2015, 22, 341–358. [Google Scholar]
- Zada, A.; Ali, S.; Li, Y. Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition. Adv. Differ. Equ. 2017, 2017, 1–27. [Google Scholar] [CrossRef] [Green Version]
- Zada, A.; Ali, S. Stability of integral Caputo type boundary value problem with non instantaneous impulses. Int. J. Appl. Comput. Math. 2019, 5, 1–18. [Google Scholar] [CrossRef]
- Zada, A.; Ali, N.; Riaz, U. Ulam’s stability of multi-point implicit boundary value problems with non-instantaneous impulses. Bollettino dell’Unione Matematica Italiana 2020, 13, 305–328. [Google Scholar] [CrossRef]
- Agarwal, R.; Hristova, S.; O’Regan, D. Non-instantaneous impulses in Caputo fractional differential equations. Fract. Calc. Appl. Anal. 2017, 20, 1–28. [Google Scholar] [CrossRef]
- Abdo, M.S.; Pamchal, S.K.; Saeed, A.M. Fractional boundary value problem with Ψ-Caputo fractional derivative. Proc. Indian Acad. Sci. Math. Sci. 2019, 65, 1–14. [Google Scholar] [CrossRef]
- Karthikeyan, P.; Venkatachalam, K. Existence results for fractional impulsive integro differential equations with integral conditions of Katugampola type. Acta Math. Univ. Comen. 2021, 90, 1–15. [Google Scholar]
- Ben Makhlouf, A.; El-Hady, E.-S. Novel stability results for Caputo fractional differential equations. Math. Probl. Eng. 2021, 2021. [Google Scholar] [CrossRef]
- El-hady, E.S.; Ben Makhlouf, A. A novel stability analysis for the Darboux problem of partial differential equations via fixed point theory. AIMS Math. 2021, 6, 12894–12901. [Google Scholar] [CrossRef]
- Brociek, R.; Chmielowska, A.; Słota, D. Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation. Fractal Fract. 2020, 4, 39. [Google Scholar] [CrossRef]
- Brociek, R.; Wajda, A.; Lo Sciuto, G.; Słota, D.; Capizzi, G. Computational Methods for Parameter Identification in 2D Fractional System with Riemann-Liouville Derivative. Sensors 2022, 22, 3153. [Google Scholar] [CrossRef] [PubMed]
- Brociek, R.; Slota, D.; Król, M.; Matula, G.; Kwaśny, W. Modeling of heat distribution in porous aluminum using fractional differential equation. Fractal Fract. 2017, 1, 17. [Google Scholar] [CrossRef] [Green Version]
- Gupta, V.; Dabas, J. Nonlinear fractional boundary value problem with not-instantaneous impulse. Aims Math. 2020, 2, 365–376. [Google Scholar] [CrossRef]
- Long, C.; Xie, J.; Chen, G.; Luo, D. Integral boundary value problem for fractional order Differential equations with non-instantaneous impulses. Int. J. Math. Anal. 2020, 14, 251–266. [Google Scholar] [CrossRef]
- Salim, A.; Benchohra, M.; Graef, J.R.; Lazreg, J.E. Boundary value problem for fractional order generalized Hilfer-type fractional derivative with non-instantaneous impulses. Fractal Fract. 2021, 5, 1. [Google Scholar] [CrossRef]
- Yu, X. Existence and β-Ulam-Hyers stability for a class of fractional differential equations with non-instantaneous impulses. Adv. Differ. Equ. 2015, 2015, 1–13. [Google Scholar] [CrossRef] [Green Version]
- Zhang, X.; Agarwal, P.; Liu, Z.; Zhang, X.; Ding, W.; Ciancio, A. On the fractional differential equations with not instantaneous impulses. Open Phys. 2016, 14, 676–684. [Google Scholar]
- Zhu, B.; Han, B.; Liu, L.; Yu, W. On the fractional partial integro-differential equations of mixed type with non-instantaneous impulses. Bound. Value Probl. 2020, 1, 1–12. [Google Scholar] [CrossRef]
- Dhaigude, D.B.; Gore, V.S.; Kundgar, P.D. Existence and uniqueness of solution of nonlinear boundary value problems for Ψ-Caputo fractional differential equations. Malaya J. Mat. 2021, 1, 112–117. [Google Scholar]
- Benchohra, M.; Hamani, S.; Ntouyas, S.K. Boundary value problems for differential equations with fractional order. Surv. Math. Its Appl. 2008, 3, 1–12. [Google Scholar]
- Mahmudov, N.; Emin, S. Fractional-order boundary value problems with katugampola fractional integral conditions. Adv. Differ. Equ. 2018, 2018, 1–17. [Google Scholar] [CrossRef] [Green Version]
- Sousa, J.V.D.C.; de Oliveira, E.C. A Gronwall inequality and the Cauchy-type problem by means of Ψ-Hilfer operator. Differ. Equ. Appl. 2017, 1, 87–106. [Google Scholar]
- El-hady, E.S.; Ben Makhlouf, A.; Boulaaras, S.; Mchiri, L. Ulam-Hyers-Rassias Stability of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative. J. Funct. Spaces 2022, 2022. [Google Scholar] [CrossRef]
- Diaz, J.B.; Margolis, B. A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74, 305–309. [Google Scholar] [CrossRef] [Green Version]
- Burton, T.A. A fixed-point theorem of Krasnoselskii. Appl. Math. Lett. 1998, 11, 85–88. [Google Scholar] [CrossRef]
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Arul, R.; Karthikeyan, P.; Karthikeyan, K.; Geetha, P.; Alruwaily, Y.; Almaghamsi, L.; El-hady, E.-s. On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions. Symmetry 2023, 15, 5. https://doi.org/10.3390/sym15010005
Arul R, Karthikeyan P, Karthikeyan K, Geetha P, Alruwaily Y, Almaghamsi L, El-hady E-s. On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions. Symmetry. 2023; 15(1):5. https://doi.org/10.3390/sym15010005
Chicago/Turabian StyleArul, Ramasamy, Panjayan Karthikeyan, Kulandhaivel Karthikeyan, Palanisamy Geetha, Ymnah Alruwaily, Lamya Almaghamsi, and El-sayed El-hady. 2023. "On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions" Symmetry 15, no. 1: 5. https://doi.org/10.3390/sym15010005
APA StyleArul, R., Karthikeyan, P., Karthikeyan, K., Geetha, P., Alruwaily, Y., Almaghamsi, L., & El-hady, E.-s. (2023). On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions. Symmetry, 15(1), 5. https://doi.org/10.3390/sym15010005