On the existence of two solutions of the four-point problem
The author considers the four-point boundary value problem (1) \(u'' = f(t, u', u'',s)\), \(u \in R\), \(t \in [a,b]\), (2) \(u(a) = u(c)\), \(u(d) = u(b)\), \(a < c \leq d < b\), where \(s \in R\) is a bifurcation parameter and the continuous function \(f(t,x,y,s)\) is increasing in \(s\). Using the techniques of lower and upper solutions as well as the degree theory, the author proves an Ambrosetti-Prodi-like result for (1), (2) considered in some domain \(D = \{x \in C^2(J) : - \alpha^2 \geq x(t) \leq \beta^2 \forall t \in J\}\): namely, under certain conditions imposed on \(f\) there exists \(s_0 \in (p,s_1)\) such that 1) for \(s < s_0\) BVP (1), (2) has no solution in \(D\); 2) for \(s = s_0\) BVP (1), (2) has at least one solution; 3) for \(s \in (0; s_1]\) has at least two solutions.
scientific article; zbMATH DE number 811584