Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 215, pp. 1-27.
Title: Positive solutions of fractional differential
equations with derivative terms
Authors: Cuiping Cheng (Hangzhou Dianzi Univ., China)
Zhaosheng Feng (Univ. of Texas-Pan American, Edinburg, TX, USA)
Youhui Su (Xuzhou Institute of Tech., Xuzhou, China)
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.