EXISTENCE RESULTS IN THE THEORY OF HYBRID FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

Öz

We study in this paper, the existence results for initial value problems for hybrid fractional integro-dierential equations. By using fixed point theorems for the sum of three operators are used for proving the main results.An example is also given to demonstrate the applications of our main results.

Anahtar Kelimeler

• Fractional derivative,differential equations,fixed point theorem

Kaynakça

1. Lakshmikantham, V., Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal. 69(8), 2677-2682 (2008).
2. Podlubny, I., Fractional Differential Equations, Academic Press, San Diego (1999).
3. Sabatier, J. Agrawal, O. P. Machado, JAT (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007).
4. Tariboon, J., Ntouyas, S. K., Sudsutad, W Fractional integral problems for fractional differential equations via Caputo derivative. Adv. Deffer. Equ. 181 (2014).
5. Ahmad, B., Ntouyas, S. K., A four-point nonlocal, integral boundary value problem for fractional differential equations of arbitrary order. Electron. J. Qual. Theory Differ. Equ. 2011, 22 (2011).
6. Ahmad, B., Sivasundaram, S., Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions. Commun. Appl. Anal. 13, 121-228 (2009).
7. Ahmad, B., Sivasundaram, S., On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl. Math. Comput. 217, 480-487 (2010).
8. Ahmad, B., Ntouyas, S. K., Alsaedi, A., Existence theorems for nonlocal multi-valued Hadamard fractional integro-differential boundary value problems. J. Inequal. Appl. 2014, 454 (2014).

9. Chen, W., Zhao, Y: Solvability of boundary value problems of nonlinear fractional differential equations. Adv. Differ. Equ. 2015, 36 (2015).
10. Zhao, Y., Sun, S., Han, Z., Li, Q., The existence of multiple positive solutions for boundary value problems of nonlinear fractional dierential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 2086-2097 (2011).
11. Zhao, Y, Sun, S, Han, Z, Li, Q: Theory of fractional hybrid differential equations. Comput. Math. Appl. 62, 1312-1324 (2011).
12. Sun, S., Zhao, Y., Han, Z., Li, Y., The existence of solutions for boundary value problem of fractional hybrid differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 4961-4967 (2012).
13. Ahmad, B., Ntouyas, S. K., An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions. Abstr. Appl. Anal. 2014, Article ID 705809 (2014).
14. Dhage, B. C., Ntouyas, S. K., Existence results for boundary value problems for fractional hybrid differential inclusions.Topol. Methods Nonlinear Anal. 44, 229-238 (2014).
15. Zhao, Y., Wang, Y., Existence of solutions to boundary value problem of a class of nonlinear fractional differential equations. Adv. Differ. Equ. 2014, 174 (2014).
16. Ahmad, B., Ntouyas, S. K., Alsaedi, A., Existence results for a system of coupled hybrid fractional differential equations.Sci. World J. 2014, Article ID 426438 (2014).
17. Dhage, B. C., Lakshmikantham, V., Basic results on hybrid differential equations, Nonlinear Anal. Hybrid 4 (2010) 414-424.
18. Dhage, B. C., Basic results in the theory of hybrid differential equations with mixed perturbations of second type. Funct. Differ. Equ. 19, 1-20 (2012).
19. Dhage, B. C., A fixed point theorem in Banach algebras with applications to functional integral equations. Kyungpook Math. J. 44, 145-155 (2004).
20. M. Benchohra, S. Hamani, S.K. Ntouyas, Boundary value problems for differential equations with fractional order, Surveys in Mathematics and its Appl. 3, 1-12 (2008).
21. Surang Sitho, Sotiris K Ntouyas and Jessada Tariboon: Existence results for hybrid fractional integro-differential equations ,Boundary Value Problems (2015).
22. Y. Zhao, S. Suna, Z. Han, Q. Li, Theory of fractional hybrid differential equations, Computers and Mathematics with Appl. 62, 1312-1324 (2011).