New existence results for nonlinear functional hybrid differential equations involving the $\psi-$Caputo fractional derivative

Abstract

In this manuscript, we are concerned with the existence result of nonlinear hybrid differential equations involving $\psi-$Caputo fractional derivatives of an arbitrary order $\alpha\in(0,1)$. By applying Krasnoselskii fixed point theorem and some fractional analysis techniques, we prove our main result. As application, a nontrivial example is given to demonstrate the effectiveness of our theoretical result.

Keywords

• $\psi-$fractional integral
• $\psi-$Caputo fractional derivative
• Carathéodory function

Kaynakça

1. [1] R.P. Agarwal, Y. Zhou, J. Wang and X. Luo. Fractional functional differential equations with causal operators in Banach spaces. Mathematical and Compututer Modelling. 54 (2011) 1440–1452.
2. [2] R.P. Agarwal, S.K. Ntouyas, B. Ahmad and A.K. Alzahrani. Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments. Advances in Difference Equations. 1 (2016) 1–15.
3. [3] R.P. Agarwal, S. Hristova and D. O’Regan. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19 (2016) 290–318.
4. [4] R. Almeida. Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul.44 (2017) 460–481.
5. [5] R. Almeida, A.B. Malinowska, and M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathematical Methods in the Applied Sciences. 41(1) (2018) 336–352.
6. [6] D. Baleanu, H. Jafari, H. Khan and S.J Johnston. Results for mild solution of fractional coupled hybrid boundary value problems. Open Math.13 (2015) 601–608.
7. [7] D. Baleanu, H. Khan, H. Jafari and R.A. Alipour. On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions. Advances in Difference Equations . 1(2015) 1–14.
8. [8] A. Belarbi, M. Benchohra, A. Ouahab. Uniqueness results for fractional functional differential equations with infinite delay in Frechet spaces. Appl. Anal. 58 (2006) 1459–1470.

9. [9] M. Caputo. Linear models of dissipation whose Q is almost frequency independent. International Journal of Geographical Information Science. 13(5)(1967) 529–539.
10. [10] B.C. Dhage. On α−condensing mappings in Banach algebras. Math. Student 63(1994)146–152.
11. [11] B.C. Dhage. On a fixed point theorem in Banach algebras with applications. Appl. Math. Lett. 18(2005) 273–280.
12. [12] B.C. Dhage and V. Lakshmikantham. Basic results on hybrid differential equations. Nonlinear Anal. Hybrid 4(2010) 414–424.
13. [13] K. Diethelm. The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin, 2010.
14. [14] T.L. Guo and M. Jiang. Impulsive fractional functional differential equations. Comput. Math. Appl.64 (2012)3414–3424.
15. [15] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematical studies 204. Ed van Mill. Amsterdam. 2006). Elsevier Science B.V. Amsterdam, 2006.
16. [16] R. Hilfer. Applications of Fractional Calculus in Physics .Singapore. 2000.
17. [17] Y. Luchko and J.J. Trujillo. Caputo-type modification of the Erdelyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10( 2007) 249–267. [18] F. Mainardi. Fractals and Fractional Calculus Continuum Mechanics Springer Verlag.1997.
18. [19] I. Podlubny. Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press. New York. 1999.
19. [20] J. Schauder. Der Fixpunktsatz in Functionalraiumen. Studia Math. 2(1930)171–180.