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On the existence of two solutions of the four-point problem. (English) Zbl 0835.34026
The author considers the four-point boundary value problem (1) $u'' = f(t, u', u'',s)$, $u \in R$, $t \in [a,b]$, (2) $u(a) = u(c)$, $u(d) = u(b)$, $a < c \le d < b$, 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 $D = \{x \in C^2(J) : - \alpha^2 \ge x(t) \le \beta^2 \forall t \in J\}$: 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.

34B10Nonlocal and multipoint boundary value problems for ODE
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