Monotone positive solutions of second-order multi-point boundary value problems
The following second-order multi-point boundary value problem \[ \begin{cases} u''(t) + a(t)f(u(t)) = 0, \quad 0<t<1, \\ u(0) - \displaystyle\sum_{i=1}^m a_i u(\xi_i) = 0, \quad u'(1) - \sum_{i=1}^m b_i u'(\xi_i) = b \end{cases} \] is considered, where \(0<\xi_1<\xi_2<\cdots<\xi_m<1\), \(a_i\), \(b_i \in [0,\infty)\), \(i=1,2,\dots,m\) and \(b>0\) is a parameter. It is assumed that \(0 < \sum_{i=1}^m a_i < 1\), \(\sum_{i=1}^m b_i < 1\), \(a \in C[0,1]\), \(a(t) \geq 0\), \(a(t) \not\equiv 0\) on \([0,1]\), \(f \in C[0,\infty)\) and \(f(x) \geq 0\) on \([0,\infty)\). Existence and nonexistence results for positive solutions are obtained in the case \[ f_0:=\lim_{x\to0+} f(x)/x=0\text{ and }f_\infty:=\lim_{x\to\infty} f(x)/x=\infty \] and the case \[ f_0=\infty\text{ and }f_\infty=0. \]
scientific article; zbMATH DE number 5518807
scientific article