Positive solutions of nonlinear second-order \(m\)-point boundary value problem
The author presents, under suitable conditions, results concerning the existence of single and multiple positive solutions to the nonlinear second-order \(m\)-point boundary value problem \[ u''(t)+\lambda a(t)f(u(t))=0,\;0<t<1,\quad u(0)=0,\;u(1)=\sum_{i=1}^{m-2}a_iu(\xi_i), \tag{*} \] where \(\lambda>0\) is a parameter, \(a_i \geq 0\) for \(1\leq i \leq m-3\) and \(a_{m-2}>0,\) \(\xi_i\) satisfy \(0<\xi_1<\dots<\xi_{m-2}<1\) and \(\sum_{i=1}^{m-2}a_i\xi_i<1\). An explicit interval is determined such that the existence of at least one positive solution to the boundary value problem (\(*\)) is guaranteed in that interval. The existence of at least two positive solutions of problem (\(*\)) is discussed for an appropriate interval of \(\lambda\). Krasnoselskii's fixed-point theorem of cone expansion-compression type is used to obtain these results.
scientific article; zbMATH DE number 2173631