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An existence result for systems of second-order boundary value problems with nonlinear boundary conditions. (English) Zbl 1310.34035

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.

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
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