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 \((0,1).\)