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Progress in Fractional Differentiation & Applications

Author Country (or Countries)

Uzbekistan

Abstract

This paper is devoted to study the existence and uniqueness of a solution for the following fractional hybrid differential equations defined by Riemann-Liouville differential operator order of 0 < α < 1 http://dx.doi.org/10.18576/pfda/090109 􏰕Dα 􏰍x(t)−f(t,x(t))􏰎=f􏰄t,x(t)􏰅, a.et∈J, t0+ 1 2 (1.0) whereDα istheRiemann-Liouvillefractionalderivativeorderof0<α<1,J=[t ,t +a],forsomet ∈R,a>0, f (·,x)∈Cα(J,R) x(t0) = x0 ∈ R, t0+ α 00 0 1 for all x ∈ R and f2 ∈ L (J × R, R). We prove the existence and uniqueness of a solution of the equation (1.0) by using a coupled fixed point theorem. This result extends the existence theorems of [1,2,3,4]. Moreover, we investigate Picard iterations of an operator T defined on a space of continuous functions under two different weak construction conditions. It is shown that Picard iterations of T converge to the unique fixed point if the weak contraction function is a tangent hyperbolic function. If the weak contraction is a fractional linear function then Picard iterations of T converge to the unique fixed point with an algebraic rate. Finally, we investigate approximate solutions of fractional hybrid differential equations via the homotopy analysis method.

Digital Object Identifier (DOI)

http://dx.doi.org/10.18576/pfda/090109

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