Electron. J. Diff. Equ., Vol. 2012 (2012), No. 215, pp. 1-27.

Positive solutions of fractional differential equations with derivative terms

Cuiping Cheng, Zhaosheng Feng, Youhui Su

In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative,
 D_{0^+}^{\alpha}u(t)+f(t,u(t),u'(t))=0,\quad t\in (0,1),\;
 n-1<\alpha\leq n,\cr
 u^{(i)}(0)=0, \quad  i=0,1,2,\dots,n-2,\cr
 [D_{0^+}^{\beta}u(t)]_{t=1}=0, \quad 2\leq\beta\leq n-2,
where $n>4 $ $(n\in\mathbb{N})$, $D_{0^+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative of order $\alpha$ and $f(t,u,u'):[0,1] \times [0,\infty)\times(-\infty,+\infty)
 \to [0,\infty)$ satisfies the Caratheodory type condition. Sufficient conditions are obtained for the existence of at least one or two positive solutions by using the nonlinear alternative of the Leray-Schauder type and Krasnosel'skii's fixed point theorem. In addition, several other sufficient conditions are established for the existence of at least triple, n or 2n-1 positive solutions. Two examples are given to illustrate our theoretical results.

Submitted August 15, 2012. Published November 29, 2012.
Math Subject Classifications: 34A08, 34B18, 34K37.
Key Words: Positive solution; equicontinuity; fractional differential equation; fixed point theorem; Caratheodory type condition.

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Cuiping Cheng
Department of Applied Mathematics
Hangzhou Dianzi University
Hangzhou 310018, China
email: chengcp0611@163.com
Zhaosheng Feng
Department of Mathematics
University of Texas-Pan American
Edinburg, TX 78539, USA
email: zsfeng@utpa.edu; fax: (956) 665-5091
Youhui Su
Department of Mathematics
Xuzhou Institute of Technology, Xuzhou 221116, China
Department of Mathematics, Indiana University
Bloomington, IN 47405, USA
email: youhsu@indiana.edu

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