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

Abstract:
In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative,
$$\displaylines{ D_{0^+}^{\alpha}u(t)+f(t,u(t),u'(t))=0,\quad t\in (0,1),\; n-1<\alpha\leq n,\cr u^{(i)}(0)=0, \quad i=0,1,2,\dots,n-2,\cr [D_{0^+}^{\beta}u(t)]_{t=1}=0, \quad 2\leq\beta\leq n-2, }$$
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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