## Abstract and Applied Analysis

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

#### Abstract

We consider the following state dependent boundary-value problem ${D}_{0+}^{\alpha }y(t)-p{D}_{0+}^{\beta }g(t,y(\sigma (t)))+f(t,y(\tau (t)))=0$, $0; $y(0)=0$, $\eta y(\sigma (1))=y(1),$ where ${D}^{\alpha }$ is the standard Riemann-Liouville fractional derivative of order $1<\alpha <2$, $0<\eta <1$, $p\le 0$, $0<\beta <1$, $\beta +1-\alpha \ge 0$ the function $g$ is defined as $g(t,u):[0,1]\times[0,\mathrm{\infty })\to [0,\mathrm{\infty })$, and $g(0,0)=0$ the function $f$ is defined as $f(t,u):[0,1]\times[0,\mathrm{\infty })\to [0,\mathrm{\infty })\sigma (t)$, $\tau (t)$ 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

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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