This paper investigates the following Katugampola fractional differential equation with Erdélyi–Kober fractional integral boundary conditions:
\[
\begin{cases}
D_{\rho}^{\alpha}u(t) + h(t, u(t)) = 0, & 0 < t < T,\\[6pt]
u(0) = 0, & \\[6pt]
u'(T) = \lambda I_{\eta}^{\gamma, \delta} u'(\xi), & 0 < \xi < T,
\end{cases}
\]
where $D_{\rho}^{\alpha}$ is the Katugampola derivative of order $1 < \alpha < 2$, $\rho > 0$ and
$h: [0, T] \times \mathbb{R} \to \mathbb{R}$ is a continuous function,
$I_{\eta}^{\gamma, \delta}$ denotes Erdélyi–Kober fractional integral of order $\delta > 0$, $\eta > 0$,
$\lambda, \gamma \in \mathbb{R}$.
Some new existence and uniqueness results are obtained using nonlinear’s contraction principle and Krasnoselskii’s and Leray–Schauder’s fixed point theorems.
Four examples are given in the last section to illustrate the obtained results.