Positive solutions for fractional differential equations with p-Laplacian

In this paper, we study the existence of positive solution to boundary value problem for fractional differential equation with a one-dimensional $p$-Laplacian operator {begin{equation*} \begin{cases D_{0^+}^\sigma (\phi_p ( u (t))) - g (t) f (u (t)) = 0, ,\\( t \in (0, 1 \\ phi_p ( u (0)) = \phi_p ( u (1)) = 0 \\( a u (0) - b u (0) = \sum_{i = 1}^{m - 2} a_i u (\xi_i ,(c u (1) + d u (1) = \sum_{i = 1}^{m - 2} b_i u (\xi_i {* end{cases} \end{equationwhere $D_{0^+}^\alpha$ is the Riemann-Liouville fractional derivative of order $1 \sigma \leq 2$, $\phi_p (s) = |s|^{p - 2}s$, $p 1$ and $f$ is a lower semi-continuous function. By using Krasnoselskii s fixed point theorems in a cone, the existence of one positive solution and multiple positive solutions for nonlinear singular boundary value problems is obtained