POSITIVE SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH P-LAPLACIAN

Abstract

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{equation*}
where $D_{0^+}^\alpha$ 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.