## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2015, Special Issue (2014), Article ID 263748, 11 pages.

### Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

Dongyuan Liu, Zigen Ouyang, and Huilan Wang

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#### Abstract

We consider the following state dependent boundary-value problem ${D}_{0+}^{\alpha}y(t)-p{D}_{0+}^{\beta}g(t,y(\sigma (t)))+f(t,y(\tau (t)))=0$, $$; $y(0)=0$, $\eta y(\sigma (1))=y(1),$ where ${D}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $$, $$, $p\le 0$, $$, $\beta +1-\alpha \ge 0$ the function $g$ is defined as $g(t,u):[\mathrm{0,1}]\times [0,\mathrm{\infty})\to [0,\mathrm{\infty})$, and $g(\mathrm{0,0})=0$ the function $f$ is defined as $f(t,u):[\mathrm{0,1}]\times [0,\mathrm{\infty})\to [0,\mathrm{\infty})\sigma (t)$, $\tau (t)$ are continuous on $t$ and $0\le \sigma (t)$, $\tau (t)\le t$. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2015, Special Issue (2014), Article ID 263748, 11 pages.

**Dates**

First available in Project Euclid: 15 April 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1429105004

**Digital Object Identifier**

doi:10.1155/2015/263748

**Mathematical Reviews number (MathSciNet)**

MR3303263

#### Citation

Liu, Dongyuan; Ouyang, Zigen; Wang, Huilan. Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators. Abstr. Appl. Anal. 2015, Special Issue (2014), Article ID 263748, 11 pages. doi:10.1155/2015/263748. https://projecteuclid.org/euclid.aaa/1429105004

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