On a nonlocal BVP with nonlinear boundary conditions (Q351143)

The author considers the existence of a positive solution to a boundary value problem where one of the boundary conditions is allowed to be nonlocal and nonlinear, namely \[ \begin{gathered} u''(t)+f(t,u(t))=0,\;t \in (0,1),\\ u(0) =H_{1}(\varphi (u))+\int_E H_{2}(s,u(s))\,ds,\;u(1) =0.\\ \end{gathered} \] Here, \(E\) is a measurable subset of \((0,1)\), \(\varphi\) is a linear functional given by a Stieltjes integral, \(H_{1}\), \( H_{2}\) are continuous functions. The methodology relies on the Guo-Krasnosel'skii fixed point theorem on cone compressions and cone expansions. The author also provides some examples to illustrate the theoretical results.