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Positive solutions to a system of second-order nonlocal boundary value problems. (English) Zbl 1089.34022

The authors consider a second-order system of the type \[ -u''=f(t,u,v),\;-v''=g(t,u,v), \] with two-point nonlocal boundary condition in \([0,1]\) of type \[ u(0)=v(0)=0,\; u(1)=H_1 \biggl(\int_0^1u(s)\,d\alpha(s)\biggr),\quad v(1)=H_2\biggl(\int_0^1u(s)\,d\beta(s)\biggr). \] \(f\) and \(g\) are positive, continuous and defined for \(u,\,v\geq0\); \(\alpha,\,\beta\) are increasing functions. They give several sets of conditions that involve the behaviour of \(H_i(u)/u\) at \(u=0\) and at \(u=+\infty\) and also the behaviour of \(f(t,u,v)/p(v)\), \(g(t,u,v)/q(u)\), \(f(t,u,v)/(u+v)\), \(g(t,u,v,)/(u+v)\) (\(p\), \(q\) are adequate positive functions). Based on index theorems of Guo and Lakshmikantham, three existence results for positive solutions are given, one of them being a two-solution theorem. The proof relies essentially on estimates with repect to the cone of positive functions and its subcone characterized by \( \int_0^1u(s)\,\sin\pi s\,ds\geq\frac{2}{\pi^2}\| u\| \).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
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