Existence Results for a Class of $\psi$-Hilfer Fractional Hybrid Differential Equations

Abstract

This study investigates the existence and uniform local attractiveness of solutions for a class of fractional $\psi$-Hilfer hybrid differential equations within Banach algebras. Utilizing advanced hybrid fixed-point theory, we derive results that not only establish conditions for the existence of solutions but also demonstrate their uniform local attractiveness. Our findings offer valuable insights into the behavior of these fractional differential equations and provide a solid theoretical foundation for future research and applications in this field.

Keywords

• $\psi$-Hilfer fractional derivative
• uniformly locally attractive
• hybrid fixed-point theory
• fractional differential equation

References

1. Abdo, M.S., Panchal, S.K., Wahash, H.A., Ulam-Hyers-Mittag-Leffler stability for a ψ-Hilfer problem with fractional order and infinite delay, Results in Applied Mathematics, 7(2020), 100–115.
2. Ahmad, B., Ntouyas, S.K., An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstract and Applied Analysis, 2014(2014).
3. Ahmad, M., Zada, A., Wang, X., Existence, uniqueness and stability of implicit switched coupled fractional differential equations of ψ-Hilfer type, International Journal of Nonlinear Sciences and Numerical Simulation, 21(3-4)(2020), 327–337.
4. Atshan, S.M., Hamoud, A.A., Qualitative analysis of ABR-fractional Volterra-Fredholm system, Nonlinear Functional Analysis and Applications, (2024), 113–130.
5. Dhage, B.C., Ntouyas, S.K., Existence results for boundary value problems for fractional hybrid differential inclusions, Topological Methods in Nonlinear Analysis, 44(2014), 229–238.
6. Dhage, B.C., Lakshmikantham, V., Basic results on hybrid differential equations, Nonlinear Analysis: Hybrid Systems, 4(2010), 414–424.
7. Dhage, B.C., On α-condensing mappings in Banach algebras, Math. Stud., 63(1994), 146–152.
8. Dhage, B.C., Existence and attractivity theorems for nonlinear hybrid fractional integrodifferential equations with anticipation and retardation, Cubo (Temuco), 22(3)(2020), 325–350.

9. Hamoud, A.A., Mohammed, N.M., Existence and uniqueness of solutions for the neutral fractional integro differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 29(2022), 49–61.
10. Hamoud, A.A., Khandagale, A.D., Shah, R., Ghadle, K.P., Some new results on Hadamard neutral fractional nonlinear Volterra-Fredholm integro-differential equations, Discontinuity, Nonlinearity, and Complexity, 12(04)(2023), 893–903.
11. Herrmann, R., Fractional Calculus for Physicist, World Scientific, Singapore, 2014.
12. Hilal, K., Kajouni, A., Lmou, H.,Existence and stability results for a coupled system of Hilfer fractional Langevin equation with non local integral boundary value conditions, Filomat, 37(2023), 1241–1259.
13. Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific Publishing Co. Inc, River Edge, NJ, Singapore, 2000.
14. Jameel, S.A.M., Rahman, S.A., Hamoud, A.A., Analysis of Hilfer fractional Volterra-Fredholm system, Nonlinear Functional Analysis and Applications, (2024), 259–273.
15. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and aplications of fractional differential equations, in NorthHoland Mathematics Studies, vol. 24, Elsevier, Amsterdam, Netherlands, 2006.
16. Lmou, H., Hilal, K., Kajouni, A., Existence and uniqueness results for Hilfer Langevin fractional pantograph differential equations and inclusions, International Journal of Difference Equations, 18(1)(2023), 145–162.
17. Lmou, H., Hilal, K., Kajouni, A., A new result for ψ-Hilfer fractional pantograph-type Langevin equation and inclusions, Journal of Mathematics, (2022).
18. Louakar, A., Kajouni, A., Hilal, K., Lmou, H., A class of implicit fractional ψ-Hilfer Langevin equation with time delay and impulse in the weighted space, Communications in Advanced Mathematical Sciences, 7(2)(2024), 88–103.
19. Mahmudov, N., Matar, M.M., Existence of mild solution for hybrid differential equations with arbitrary fractional order, TWMS J. Pure Appl. Math., 8(2)(2017), 160–169.
20. Mali, A.D., Kucche, K.D., Vanterler da Costa Sousa, J., On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 24(4)(2023), 1425–1445.
21. More, P.M., Karande, B.D., Amrutrao, P.S., Locally attractivity solution of coupled fractional integral equation in Banach algebras, Asian Research Journal of Mathematics, 19(1)(2023), 45–55.
22. Shabna, M.S., Ranjin, M.C., On existence of ψ-Hilfer hybrid fractional differential equations, South East Asian International Journal of Mathematics and Mathematical Sciences, 16(2020).
23. Shah, K., Ali, A., Bushnaq, S., HyersUlam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Mathematical Methods in the Applied Sciences, 41(17)(2018), 8329–8343.
24. Si Bachir, F., Abbas, S., Benbachir, M., Benchohra, M., N’Gu´er´ekata, G.M., Existence and attractivity results for ψ-Hilfer hybrid fractional differential equations, Cubo., 23(2021), 145–159.
25. Sousa, J.V.C., Capelas Oliveira, E., A Gronwall inequality and the Cauchy type problem by means of Hilfer operator, Diff. Equ. and Appl., 11(1)(2019), 87–106.
26. Sousa, J.V.D.C., Capelas de Oliveira, E.C., On the ψ-Hilfer fractional derivative, Commun Nonlinear Sci Numer Simulat, 60(2018), 72–91.
27. Zhao, Y., Sun, S., Han, Z., Li, Q., Theory of fractional hybrid differential equations, Computers and Mathematcs with Applications, 62(3)(2011), 1312–1324.