A fixed point operator for a nonlinear boundary value problem
The authors study the solutions to the following boundary value problem \[ u''+g(t,u,u')=0,\qquad u'(0)=-f(u(0)),\quad u'(\pi)=f(u(\pi)), \tag{1} \] with \(g:[0,\pi]\times\mathbb{R}^2\to\mathbb{R}\) and \(f:\mathbb{R}\to\mathbb{R}\) continuous. Two existence and localization results are provided: one finite-dimensional based on the Brouwer degree and one infinite-dimensional, valid for \(f\) invertible, based on Schauder's theorem. Furthermore, when \(f\) is invertible, by embedding of (1) into the family of problems \[ u''+\lambda g(t,u,u')=0, \qquad u(0)=f^{-1}(-u'(0)),\quad u(\pi)=f^{-1}(u'(\pi)), \tag{\(1_\lambda\)} \] the authors give an iterative construction of a solution to (1). Namely, for \(\varepsilon>0\) small enough, taking a solution \(u_0\) to (\(1_{\lambda_0}\)) as ``seed'', they define a sequence converging in \(C^1([0,\pi])\) to a solution to (\(1_{\lambda_0+\varepsilon}\)). Then, since (\(1_0\)) is solvable, they show, under suitable assumptions, the existence of a sequence \(0=\lambda_0<\lambda_1<\ldots <\lambda_n=1\) with the property that a solution to (\(1_{\lambda_i}\)) can be constructed recursively.
scientific article; zbMATH DE number 1763700