Application of implicit function theorems to existence of solutions to ordinary differential equations with nonlocal boundary conditions (Q1856800)

The authors study classical solutions to the boundary value problem \[ y''= a(x,y)(y- b),\quad y(0)= 0,\quad y'(1)= g(y(\xi), y'(\xi)), \] where \(y(x)\) for \(0\leq x\leq 1\) is a scalar unknown function, the functions \(a\), \(g\) and numbers \(b\), \(\xi\) are given. This nonlocal problem models steady-state temperature distributions along a rod when the heat flux at the end \(x= 1\) is regulated depending on the observation of the temperature and of the heat flux at some interior point \(\xi\). Sufficient conditions are given for the existence, uniqueness, and positivity of solutions. Some lemmas proved by the authors have the form of implicit function theorems.