Existence and uniqueness of mild solutions for nonlinear hybrid Caputo fractional integro-differential equations via fixed point theorems

Abstract

We prove the existence and uniqueness of mild solutions for initial value
problems of nonlinear hybrid first order Caputo fractional
integro-differential equations. The main tool employed here is the
Krasnoselskii and Banach fixed point theorems. An example is also given to
illustrate the main results. In addition, the case of the Higher order
Caputo fractional integro-differential equations is studied.

Keywords

• Hybrid fractional integro-differential equations
• Fixed point theorems
• Existence
• Uniqueness

Kaynakça

1. 1) A. Ardjouni, A. Djoudi, Approximating solutions of nonlinear hybrid Caputo fractional integro-differential equations via Dhage iteration principle, Ural Mathematical Journal 5(1) (2019), 3--12.
2. 2) A. Ardjouni, A. Djoudi, Initial-value problems for nonlinear hybrid implicit Caputo fractional differential equations, Malaya Journal of Matematik 7 (2019), 314--317.
3. 3) M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surveys in Mathematics & its Applications 3 (2008), 1--12.
4. 4) B. C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional integral equations, Differ. Equ. Appl. 5 (2013), 155--184.
5. 5) B. C. Dhage, Basic results in the theory of hybrid differential equations with mixed perturbations of second type, Funct. Differ. Equ. 19 (2012), 87--106.
6. 6) B. C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J. 44 (2004), 145--155.
7. 7) B. C. Dhage, S B. Dhage, S. K. Ntouyas, Approximating solutions of nonlinear second order ordinary differential equations via Dhage iteration principle. Malaya J. Mat. 4(1) (2016), 8--18.
8. 8) B. C. Dhage, G. T. Khurpe, A. Y. Shete, J. N. Salunke, Existence and approximate solutions for nonlinear hybrid fractional integro-differential equations, International Journal of Analysis and Applications 11(2) (2016), 157--167.

9. 9) B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equations, Nonlinear Anal. Hybrid Syst. 4 (2010), 414--424.
10. 10) M. Haoues, A. Ardjouni, A. Djoudi, Existence, uniqueness and monotonicity of positive solutions for hybrid fractional integro-differential equations, Asia Mathematika 4(3) (2020), 1--13.
11. 11) A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B. V., Amsterdam, 2006.
12. 12) H. Lu, S. Sun, D. Yang, Theory of fractional hybrid differential equations with linear perturbations of second type, Bound. Value Probl. 2013(23) (2013).
13. 13) D. R. Smart, Fixed Point Theorems, Cambridge Tracts in Mathematics, Cambridge University Press, London-New York, 1974.
14. 14) S. Sun, Y. Zhao, Z. Han, The existence of solutions for boundary value problem of fractional hybrid differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 4961--4967.
15. 15) Y. Zhao, S. Sun, Z. Han, Theory of fractional hybrid differential equations, Comput. Math. Appl. 62 (2011), 1312--1324.
16. 16) Y. Zhao, S. Sun, Z. Han, Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput. 217 (2011), 6950--6958.
17. 17) Y. Zhao, Y. Sun, Z. Liu, Basic theory of differential equations with mixed perturbations of the second type on time scales, Adv. Differ. Equa. 2019(268) (2019).
18. 18) Y. Zhao, Y. Sun, Z. Liu, Y. Wang, Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type, AIMS Mathematics 5(1) (2019), 557--567.