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Coupled systems of boundary value problems with nonlocal boundary conditions. (English) Zbl 1312.34050

Summary: We consider the coupled system \[ -x''= \lambda_1f(t, y(t)),\quad -y''= \lambda_2 g(t,x(t)),\quad t\in (0,1), \] subject to the coupled boundary conditions \[ x(0)= H_1(\varphi_1(y)),\quad x(1)= 0\text{ and }y(0)= H_2(\varphi_2(x)),\quad y(1)= 0. \] Since \(H_1\) and \(H_2\) are nonlinear functions and \(\varphi_1\) and \(\varphi_2\) are linear functionals realized as Stieltjes integrals, the boundary conditions may be nonlocal and nonlinear in character. By assuming that \(\varphi_1\) and \(\varphi_2\) satisfy a particular decomposition hypothesis together with some growth assumptions on \(H_1\) and \(H_2\) at \(0\) and \(+\infty\), we show that this system can possess at least one positive solution even if no growth conditions are imposed on \(f\) and \(g\).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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