On a class of \(m\)-point boundary value problems. (Q2918596)

The paper studies the existence and nonexistence of positive solutions of the nonlinear second-order differential system NEWLINE\[NEWLINE \begin{cases} u''(t) + b(t) f(v(t)) = 0,\\ v''(t) + c(t) g(u(t)) = 0, & t\in (0,T), \end{cases} NEWLINE\]NEWLINE with \(m\)-point boundary conditions NEWLINE\[NEWLINE\begin{aligned} \beta u(0)- \gamma u'(0)&=0, \quad u(T)=\sum _{i=1}^{m-2} a_i u(\xi _i) + b_0,\\ \beta v(0)-\gamma v'(0)&=0, \quad v(T)=\sum _{i=1}^{m-2} a_i v(\xi _i)+b_0, \quad m\in{\mathbb N}, \;m\geq 3. \end{aligned}NEWLINE\]NEWLINE Using a fixed point theorem and the corresponding Green function, the author proves that there exists at least one positive solution provided \(b_0 >0\) is sufficiently small. On the other hand, no positive solution exists for \(b_0\) sufficiently large.