Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation

Abstract

In this paper, the following nonlinear fractional ordinary differential boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} D_{0+}^{\alpha } u(t) +\lambda f(t,u(t))+ q(t)=0, &{} 0<t<1, \\ u(0) = u'(1) =u''(0)= \dots = u^{(n-1)}(0)=0, &{} \\ \end{array} \right. \end{aligned}$$

is considered, where $\alpha (n-1 <\alpha \le n)$ is a real number. $\lambda > 0$ is a parameter. $D_{0+}^{\alpha }$ is the standard Caputo differentiation. Some sufficient conditions for the existence of positive solutions to this boundary value problem of nonlinear fractional differential equation are established by nonlinear alternative of Leray–Schauder type and Guo–Krasnoselskii fixed point theorem on cones. As applications, some examples are provided to illustrate our main results.