Existence of positive solutions for nonlinear fractional functional differential equation

Abstract

In this paper, the existence of positive solutions for the nonlinear Caputo fractional functional differential equation in the form

$$\{\begin{array}{c}{D}_{0+}^{q}y\left(t\right)+r\left(t\right)f\left(y_t\right)=0\text{,}\forall t\in (0,1),q\in (n-1,n]\text{,}\hfill \\ y^{\left(i\right)}\left(0\right)=0\text{,}0\le i\le n-3\text{,}\hfill \\ \alpha y^{(n-2)}\left(t\right)-\beta y^{(n-1)}\left(t\right)=\eta \left(t\right)\text{,}t\in [-\tau ,0]\text{,}\hfill \\ \gamma y^{(n-2)}\left(t\right)+\delta y^{(n-1)}\left(t\right)=\xi \left(t\right)\text{,}t\in [1,1+a]\hfill \end{array}$$

is studied. By constructing a special cone and using Krasnosel’skii’s fixed point theorem, various results on the existence of at least one or two positive solutions to the fractional functional differential equation are established. The main results improve and generalize the existing results.