Positive solutions of fractional differential equations with derivative terms

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, 
\n$$\\displaylines{
\n D_{0^+}^{\\alpha}u(t)+f(t,u(t),u'(t))=0,\\quad t\\in (0,1),\\;
\n n-1<\\alpha\\leq n,\\cr
\n u^{(i)}(0)=0, \\quad i=0,1,2,\\dots,n-2,\\cr
\n [D_{0^+}^{\\beta}u(t)]_{t=1}=0, \\quad 2\\leq\\beta\\leq n-2,
\n }$$
\nwhere $n>4 $ $(n\\in\\mathbb{N})$, $D_{0^+}^{\\alpha}$ is the standard Riemann-Liouville fractional derivative of order $\\alpha$ and $f(t,u,u'):[0,1] \\times [0,\\infty)\\times(-\\infty,+\\infty)
\n \\to [0,\\infty)$ 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.