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In this article, we investigate the boundary-value problem \begin{equation*} \begin{cases}x''(t)+h(t)f(x(t))=0,\quad t\in[0,1],\\ x(0)=\beta x'(0),\quad x(1)=x(\eta),\end{cases} \end{equation*} where $\beta\ge0$, $\eta\in(0,1)$, $f\in C([0,\infty), [0,\infty))$ is nondecreasing, and importantly $h$ changes sign on $[0,1]$. By the Guo-Krasnosel'skii fixed-point theorem in a cone, the existence of positive solutions is obtained via a special cone in terms of superlinear or sublinear behavior of $f$.