Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations

Abstract

We investigate the existence and multiplicity of positive solutions for the nonlinear fractional differential equation initial value problem $D_{0+}^{\alpha }u\left(t\right)+D_{0+}^{\beta }u\left(t\right)=f(t,u\left(t\right)),\hspace{0.17em}$ , where is the standard Riemann-Liouville differentiation and $f:\left[\mathrm{0,1}\right]\times \left[0,\infty \right)\to \left[0,\infty \right)$ is continuous. By using some fixed-point results on cones, some existence and multiplicity results of positive solutions are obtained.