Positive Solutions for a Kind of Singular Nonlinear Fractional Differential Equations with Integral Boundary Conditions

Abstract

In this paper, we study the following fractional differential equation:

$$\displaystyle{ D_{0+}^{\alpha }u(t) + f(t,u(t),(\phi u)(t),(\psi u)(t)) = 0,\ \ 0 < t < 1, }$$

with integral boundary condition:

$$\displaystyle{ u(0) = 0,\ \ u(1) =\int _{ 0}^{1}g(s)u(s)\mbox{ d}s, }$$

where $1 < \alpha < 2;(\phi u)(t) =\int _{ 0}^{t}\gamma (t,s)u(s)\mbox{ d}s;(\psi u)(t) =\int _{ 0}^{t}\delta (t,s)u(s)\mbox{ d}s;0 <\int _{ 0}^{1}g(s)$ $u(s)\mbox{ d}s < 1$; f satisfies the Carathéodory conditions on $[0,1] \times \mathcal{B}$, $\mathcal{B} = (0,+\infty ) \times [0,+\infty {)}^{2}$; f is positive; f(t,x,y,z) is singular at x = 0; and D ^α is the standard Riemann–Liouville fractional derivative. By using the fixed point theorems on cones, we get the existence of positive solution.