## International Journal of Differential Equations

### Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

#### Abstract

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: $({}^{C}{D}_{\mathrm{0}+}^{q}u)(t)=f(t,u(t))$,  $t\in (\mathrm{0,1})$, $u(\mathrm{0})={u}^{\mathrm{\prime }\mathrm{\prime }}(\mathrm{0})=\mathrm{0}, ({}^{C}{D}_{\mathrm{0}+}^{{\sigma }_{\mathrm{1}}}u)(\mathrm{1})=\lambda ({I}_{\mathrm{0}+}^{{\sigma }_{\mathrm{2}}}u)(\mathrm{1})$, where $\mathrm{2}, $\mathrm{0}<{\sigma }_{\mathrm{1}}\le \mathrm{1}$, ${\sigma }_{\mathrm{2}}>\mathrm{0}$, and $\lambda \ne \mathrm{\Gamma }(\mathrm{2}+{\sigma }_{\mathrm{2}})/\mathrm{\Gamma }(\mathrm{2}-{\sigma }_{\mathrm{1}})$. The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

#### Article information

Source
Int. J. Differ. Equ., Volume 2017 (2017), Article ID 4683581, 7 pages.

Dates
Received: 11 October 2016
Revised: 14 December 2016
Accepted: 15 December 2016
First available in Project Euclid: 24 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1487905255

Digital Object Identifier
doi:10.1155/2017/4683581

Mathematical Reviews number (MathSciNet)
MR3607978

Zentralblatt MATH identifier
06915935

#### Citation

Su, Cheng-Min; Sun, Jian-Ping; Zhao, Ya-Hong. Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation. Int. J. Differ. Equ. 2017 (2017), Article ID 4683581, 7 pages. doi:10.1155/2017/4683581. https://projecteuclid.org/euclid.ijde/1487905255

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