Goodrich, Christopher S. An existence result for systems of second-order boundary value problems with nonlinear boundary conditions. (English) Zbl 1310.34035 Dyn. Syst. Appl. 23, No. 4, 601-618 (2014). Summary: We consider the system \[ \begin{aligned} x''(t) &= -\lambda_1 f(t,x(t), y(t)),\qquad t\in (0,1),\\ y''(t) &= -\lambda_2 g(t,x(t), y(t)),\qquad t\in (0,1),\\ x(0) &= \varphi(x),\;y(0)= \psi(y),\\ x(1) &= 0= y(1),\end{aligned} \] and demonstrate that under suitable conditions on the functions \(f,\,g: [0,1]\times\mathbb{R}\times \mathbb{R}\to [0,+\infty)\) and the functionals \(\varphi,\,\psi:{\mathcal C}([0,1])\to \mathbb{R}\) this problem admits at least one positive solution. Since the functionals \(\varphi\) and \(\psi\) can be nonlinear, the boundary condition at \(t=0\) can be quite general. Our approach is based on supposing that each of \(x\mapsto\varphi(x)\) and \(y\mapsto\psi(y)\) behaves, in some sense, like a linear functional as \(\|(x,y)\to +\infty\), and to this end we utilize the concept of the Fréchet derivative at \(+\infty\) in our existence proof. We conclude by providing an explicit example of and discussion regarding our existence theorem. Cited in 4 Documents MSC: 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B09 Boundary eigenvalue problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47G10 Integral operators 47H10 Fixed-point theorems Keywords:positive solution PDFBibTeX XMLCite \textit{C. S. Goodrich}, Dyn. Syst. Appl. 23, No. 4, 601--618 (2014; Zbl 1310.34035)