Document Type : Research Paper

Authors

1 Department of mathematics, Alexandria University, Alexandria, Egypt.

2 Department of mathematics, College of Science, Qassim University, P.O. Box 6644 Buraidah 51452 , Saudi Arabia.

3 Department of mathematics, Lebanese International University, Lebanon, Saida.

Abstract

In this paper, we discuss the existence results for a class of hybrid initial value problems of Riemann-Liouville fractional differential equations. Our investigation is based on the Dhage hybrid fixed point theorem, remarks and some special cases will be discussed. The continuous dependence of the unique solution on one of its functions will be proved.

Keywords

[1] B. Ahmad, S.K. Ntouyas and J. Tariboon, A nonlocal hybrid boundary value problem of Caputo fractional integro-differential equations, Acta Math. Sci., 36(6) (2016),pp. 1631-1640.
[2] S. Abbas, Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differ. Equ., 2011(9) (2011), pp. 1-11.
 
[3] Sh.M Al-Issa and N.M. Mawed, Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra, J. Nonlinear Sci. Appl.,14(4) (2021), pp. 181-195.
 
[4] J. Banas and B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl., 332(2) (2007), pp. 1371-1379.
 
[5] B.C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Math. J., 44(1) (2004), pp. 145–155.
 
[6]B.C. Dhage andB.D. Karande, First order integro-differential equations in Banach algebras involving Caratheodory and discontinuous nonlinearities, Electron. J. Qual. Theory Differ. Equ., 2005(21) (2005), pp. 1-16.
 
[7] B.C. Dhage and V. Lakshmikantham, Basic results on hybrid differential equation, Nonlinear Anal. Hybrid Syst.,4(3) (2010), pp. 414-424.
 
[8] M.A. Darwish and K. Sadarangani, Existence of solutions for hybrid fractional pantograph equations, Appl. Anal. Discrete Math.,9 (2015), pp. 150-167.
 
[9] A.M.A. El-Sayed, F.M. Gaafar and H.H.G. Hashem, On the maximal and minimal solutions of arbitrary-orders nonlinear functional integral and differenbtial equations, Math. Sci. Res. J.,8(11), (2004), pp. 336-348.
 
[10] A.M.A El-Sayed and H.H.G. Hashem, Integrable and continuous solutions of a nonlinear quadratic integral equation, Electron. J. Qual. Theory Differ. Equ., 2008(25) (2008), pp. 1-10.
[11] A.M.A El-Sayed and H.H.G. Hashem, Monotonic solutions of functional integral and differential equations of fractional order, Electron. J. Qual. Theory Differ. Equ., 2009(2009), pp. 1-8.
 
[12] A.M.A El-Sayed and H.H.G. Hashem, Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra, Fract. Calc. Appl. Anal., 16(4) (2013), pp. 816-826.
 
[13] A.M.A. El-Sayed, Sh.M Al-Issa and N.M. Mawed, On A Coupled System of Hybrid Fractional-order Differential Equations in Banach Algebras, Adv. Dyn. Syst. Appl., 16(1) (2021),pp. 91-112.
 
[14]A.R Elsonbatyand A.M.A El-Sayed, Further nonlinear dynamical analysis of simple jerk system with multiple attractors,
Nonlinear Dynamics,87(2) (2017), pp. 1169-1186.
[15] F.M. Gaafar, Positive solutions of a quadratic integro-differential equation, J. Egyptian Math. Soc., 22(2) ( 2014), pp. 162-166.
[16] M.A.E. Herzallah and D. Baleanu,On Fractional Order Hybrid Differential Equations, Abstr. Appl. Anal., 2014 (2014), 7 pages.
[17] H. Lu, S. Sun, D. Yang and H. Teng, Theory of fractional hybrid differential equations with linear perturbations of second type, Bound. Value Probl.,2013(23)(2013), 16 pages.
[18] E. H. Rothe, Topological Methods in the Theory of Nonlinear Integral Equations, Amer. Math. Monthly, 75(3) ( 1968), pp. 318-319.
 
[19] S. Sitho, S. K. Ntouyas and J. Tariboon, Existence results for hybrid fractional integro-differential equations, Bound. Value Probl., 2015(113) (2015), 13 pages.
 
[20] L. Zheng andX. Zhang, Modeling and analysis of modern fluid problems, Math. Sci. Eng, Academic Press, London, 2017.
 
[21] Y. Zhao, S. Suna, Z. Hana andQ. Li, Theory of fractional hybrid differential equations, Comput. Math. Appl., 62(3) (2011), pp. 1312-1324.