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Positive solutions to singular system with four-point coupled boundary conditions. (English) Zbl 1232.34034

This paper is devoted to the solvability of the system \[ -x''(t)=f\bigl(t,x(t),y(t)\bigr),\;t\in(0,1), \]
\[ -y''(t)=g\bigl(t,x(t),y(t)\bigr),\;t\in(0,1), \]
\[ x(0)=0,\;x(1)=\alpha y(\xi),\;\;y(0)=0,\;y(1)=\beta x(\eta), \] where \(f,g:(0,1)\times[0,\infty)\times[0,\infty)\to[0,\infty)\) are continuous and unbounded at \(t=0\) and \(t=1\), and the parameters \(\alpha, \beta, \xi, \eta\) are such that \(\xi,\eta\in(0,1)\) and \(0<\alpha\beta\xi\eta<1.\)
It is assumed in addition that the function \(f\) is such that \[ f(t,1,1)\in C((0,1),(0,\infty)),\;\; \int_0^1t(1-t)f(t,1,1)dt<+\infty \] and there exist constants \(\alpha_i,\beta_i,\,i=1,2,\) and \(c\) with the properties \(0\leq\alpha_i\leq\beta_i<1,\,i=1,2,\;\beta_1+\beta_2<1\) and for \(t\in(0,1)\) and \(x,y\in[0,\infty)\) \[ \begin{aligned}&c^{\beta_1}f(t,x,y)\leq f(t,cx,y)\leq c^{\alpha_1}f(t,x,y),\;0<c\leq1,\\ &c^{\alpha_1}f(t,x,y)\leq f(t,cx,y)\leq c^{\beta_1}f(t,x,y),\;c\geq1,\\ &c^{\beta_2}f(t,x,y)\leq f(t,x,cy)\leq c^{\alpha_2}f(t,x,y),\;0<c\leq1,\\ &c^{\alpha_2}f(t,x,y)\leq f(t,x,cy)\leq c^{\beta_2}f(t,x,y),\;c\geq1.\end{aligned} \] It is assumed also that similar conditions hold for \(g\).
Using the Guo-Krasnosel’skii fixed point theorem, the authors establish the existence of a solution \((x,y)\) such that \(x,y\in C[0,1]\cap C^2(0,1)\) and \(x\) and \(y\) are positive on \((0,1).\)

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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