Abstract and Applied Analysis

Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

Abstract

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations $({\mathcal{D}}^{\alpha }-\rho t{\mathcal{D}}^{\beta })x(t)=f(t,x(t),{\mathcal{D}}^{\gamma }x(t))$, $t\in (0,1)$ with boundary conditions $x(0)={x}_{0},\mathrm{ }x(1)={x}_{1}$ or satisfying the initial conditions $x(0)=0,\mathrm{ }{x}^{\prime }(0)=1$, where ${\mathcal{D}}^{\alpha }$ denotes Caputo fractional derivative, $\rho$ is constant, $1<\alpha <2,$ and $0<\beta +\gamma \le \alpha$. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions on $f$.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 632681, 14 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168369

Digital Object Identifier
doi:10.1155/2012/632681

Mathematical Reviews number (MathSciNet)
MR2947672

Zentralblatt MATH identifier
1251.34010

Citation

Babakhani, Azizollah; Baleanu, Dumitru. Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 632681, 14 pages. doi:10.1155/2012/632681. https://projecteuclid.org/euclid.aaa/1365168369

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