Positive solutions of second-order three-point boundary value problems with sign-changing coefficients

• Main Authors: Ye Xue, Guowei Zhang
• Format: Article
• Language: English
• Published: University of Szeged 2016-10-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: positive solution; fixed point theorem; cone; sign-changing coefficient
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=5158

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 $f$.