Qualitative Analyses of ψ-Caputo Type Fractional Integrodifferential Equations in Banach Spaces
Abstract - 481


ψ-Fractional operators
Banach’s fixed point theorem
Schaefer’s fixed point theorem
Fractional differential equations

How to Cite

Abdo, M. S. . (2022). Qualitative Analyses of ψ-Caputo Type Fractional Integrodifferential Equations in Banach Spaces. Journal of Advances in Applied & Computational Mathematics, 9, 1–10. https://doi.org/10.15377/2409-5761.2022.09.1


In this research paper, we develop and extend some qualitative analyses of a class of a nonlinear fractional integro-differential equation involving ψ-Caputo fractional derivative (ψ-CFD) and ψ-Riemann-Liouville fractional integral (ψ-RLFI). The existence and uniqueness theorems are obtained in Banach spaces via an equivalent fractional integral equation with the help of Banach’s fixed point theorem (B’sFPT) and Schaefer’s fixed point theorem (S’sFPT). An example explaining the main results is also constructed.



Gaul L, Klein P, Kempfle S. Damping description involving fractional operators. Mech. Systems Signal Processing 1991; 5(2): 81-88. https://doi.org/10.1016/0888-3270(91)90016-X

Glockle WG, Nonnenmacher TF. A fractional calculus approach of self-similar protein dynamics. Biophys J., 1995; 68(1): 46-53. https://doi.org/10.1016/S0006-3495(95)80157-8

Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud, 2006; 204, Elsevier, Amsterdam.

Magin RL. Fractional calculus models of complex dynamics in biological tissues. Computers & Mathematics with Applications, 2010; 59(5): 1586-1593. https://doi.org/10.1016/j.camwa.2009.08.039

Almeida R. A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 2017; 44: 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006

Sousa JV, Oliveira EC. On the Ψ-Hilfer fractional derivative. Communications in Nonlinear Science and Numerical Simulation, 2018; 60: 72-91. https://doi.org/10.1016/j.cnsns.2018.01.005

Alaoui MK, Fayyaz R, Khan A, Shah R, Abdo MS. Analytical Investigation of Noyes-Field Model for Time-Fractional Belousov-Zhabotinsky Reaction. Complexity, 2021: 2021; Article ID 3248376, https://doi.org/10.1155/2021/3248376

Alesemi, M, Iqbal N, Abdo MS. Novel Investigation of Fractional-Order Cauchy-Reaction Diffusion Equation Involving Caputo-Fabrizio Operator. Journal of Function Spaces, 2022: 2022; Article ID 4284060, https://doi.org/10.1155/2022/4284060

Shatanawi W, Abdo MS, Abdulwasaa MA, Shah K, Panchal SK, Kawale SV, Ghadle KP. A fractional dynamics of tuberculosis (TB) model in the frame of generalized Atangana- Baleanu derivative. Results in Physics, 2021; 29: 104739. https://doi.org/10.1016/j.rinp.2021.104739

Jeelani MB, Alnahdi AS, Abdo MS, Abdulwasaa MA, Shah K, Wahash HA. Mathematical Modeling and Forecasting of COVID-19 in Saudi Arabia under Fractal-Fractional Derivative in Caputo Sense with Power-Law. Axioms, 2021; 10(3): 228. https://doi.org/10.3390/axioms10030228

Toufik M, Atangana A. New numerical approximation of fractional derivative with non-local and non-singular kernel, application to chaotic models. Eur. Phys. J. Plus, 2017; 132: 444. https://doi.org/10.1140/epjp/i2017-11717-0

Khan F, Pilz J. Modelling and sensitivity analysis of river flow in the Upper Indus Basin, Pakistan. Int. J. Water, 2018; 12: 1-21. https://doi.org/10.1504/IJW.2018.090184

Ahmad B, Nieto JJ. Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Boundary Value Problems, 2009: 2009; Article ID 708576. https://doi.org/10.1155/2009/708576

Agarwal RP, Lakshmikantham V, Nieto JJ. On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis, 2010; 72: 2859-2862. https://doi.org/10.1016/j.na.2009.11.029

Delbosco D, Rodino L. Existence and uniqueness for a fractional differential equation, Journal of Mathematical Analysis and Applications, 1996; 204: 609-625. https://doi.org/10.1006/jmaa.1996.0456

Lakshmikantham V, Vasundhara DJ. Theory of fractional differential equations in Banach spaces, European Journal of Pure and Applied Mathematics, 2008; 1: 38-45.

Wei Z, Che J. Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, Journal of Mathematical Analysis and Applications, 2010; 367: 260-272. https://doi.org/10.1016/j.jmaa.2010.01.023

Abdo MS, Abdeljawad T, Shah K, Jarad F. Study of impulsive problems under Mittag-Leffler power law. Heliyon, 2020; 6(10): e05109. https://doi.org/10.1016/j.heliyon.2020.e05109

Abdo MS, Panchal SK. Weighted fractional neutral functional differential equations. J. Sib. Fed. Univ. Math. Phys, 2018; 11(5): 535-549. https://doi.org/10.17516/1997-1397-2018-11-5-535-549

Abdo MS, Panchal SK. Existence and continuous dependence for fractional neutral functional differential equations. Journal of Mathematical Modeling, 2017; 5(2): 153-170. https://doi.org/10.21275/v5i2.NOV161682

Abdo MS, Panchal SK, Shah K, Abdeljawad T. Existence theory and numerical analysis of three species prey-predator model under Mittag-Leffler power law. Adv. Differ. Equ., 2020; 2020(249): 1-16. https://doi.org/10.1186/s13662-020-02709-7

Chang YK, Nieto JJ. Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. Numerical Functional Analysis and Optimization, 2009; 30(3): 227-244. https://doi.org/10.1080/01630560902841146

Balachandran K, Chandrasekaran M. The non-local Cauchy problem for semilinear integrodifferential equations with deviating argument. Proceedings of the Edinburgh Mathematical Society, 2001; 44(1): 63-70. https://doi.org/10.1017/S0013091598001060

Balachandran K, Trujillo JJ. The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Theory Methods & Applications, 2010; 72(12): 4587-4593. https://doi.org/10.1016/j.na.2010.02.035

Chang YK, Nieto JJ. Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Numerical Functional Analysis and Optimization, 2009; 30: 227-244. https://doi.org/10.1080/01630560902841146

Balachandran K, Kiruthika S, Trujillo JJ. Existence results for fractional impulsive integrodifferential equations in Banach spaces. Communications in Nonlinear Science and Numerical Simulation, 2011; 16(4): 1970-1977. https://doi.org/10.1016/j.cnsns.2010.08.005

Ahmad B, Luca R. Existence of solutions for sequential fractional integro-differential equations and inclusions with nonlocal boundary conditions. Applied Mathematics and Computation, 2018; 339: 516-534. https://doi.org/10.1016/j.amc.2018.07.025

Abdo MS, Panchal SK. An existence result for fractional integro-differential equations on Banach space. Journal of Mathematical Extension, 2019; 13: 19-33.

Wahash HA, Abdo MS, Panchal SK. A Nonlinear Integro-Differential Equation with Fractional Order and Nonlocal Conditions. Journal of Applied Nonlinear Dynamics, 2020; 9(3): 469-481. https://doi.org/10.5890/JAND.2020.09.009

El-Borai MM. Semigroups and some nonlinear fractional differential equations, Applied Mathematics and Computation, 2004; 149: 823-831. https://doi.org/10.1016/S0096-3003(03)00188-7

Rashid MHM, El-Qaderi Y. Semilinear fractional integrodifferential equations with compact semigroup, Nonlinear Analysis, 2009; 71: 6276-6282. https://doi.org/10.1016/j.na.2009.06.035

El-Sayeed MAA. Fractional order diffusion wave equation, International Journal of Theoretical Physics, 1996; 35: 311-322. https://doi.org/10.1007/BF02083817

Almeida R. Fractional differential equations with mixed boundary conditions. Bulletin of the Malaysian Mathematical Sciences Society, (2017; 1-11.

Almeida R, Malinowska AB, Monteiro MTT. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathematical Methods in the Applied Sciences 2018; 41(1): 336-352. https://doi.org/10.1002/mma.4617

Sousa JV, de Oliveira EC. Stability of the fractional Volterra integro-differential equation by means of Ψ-Hilfer operator. Mathematical Methods in the Applied Sciences, 2019; 42(9): 3033-3043. https://doi.org/10.1002/mma.5563

Sousa JVDC, Oliveira DDS, Capelas de Oliveira E. On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation. Mathematical Methods in the Applied Sciences, 2019; 42(4): 1249-1261. https://doi.org/10.1002/mma.5430

Mali AD, Kucche KD. Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations. Mathematical Methods in the Applied Sciences, 2020; 43(15): 8608-8631. https://doi.org/10.1002/mma.6521

Wahash HA, Abdo MS, Saeed AM, Panchal SK. Singular fractional differential equations with Ψ-Caputo operator and modified Picard's iterative method. Appl. Math. E-Notes, 2020; 20: 215-229.

Abdo MS, Panchal SK. Fractional integro-differential equations involving Ψ-Hilfer fractional derivative. Adv. Appl. Math. Mech, 2019; 11(2): 338-359. https://doi.org/10.4208/aamm.OA-2018-0143

Abdo MS, Ibrahim AG, Panchal SK. Nonlinear implicit fractional derivative. In Proc. Jangjeon Math. Soc., 2019; 22(3): 387-400.

Derbazi C, Baitiche Z, Abdo MS, Abdeljawad T. Qualitative analysis of fractional relaxation equation and coupled system with psi -Caputo fractional derivative in Banach spaces. AIMS Mathematics, 2021; 6: 2486-2509. https://doi.org/10.3934/math.2021151

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