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A note on multi-point boundary value problems. (English) Zbl 1127.34006
This work is concerned with solvability of the multipoint boundary value problem $$x''(t) = f(t,x(t),x'(t)), \quad t \in (0,1),$$ $$x(0) = \sum_{j=1}^n \alpha_j x(\xi_j), \quad x(1) = \sum_{j=1}^n \beta_j x(\eta_j),$$ where $0 < \eta_j, \xi_j < 1$, $\eta_j, \xi_j \in \Bbb{R}$, $j = 1,2,\dots,n$, $n \geq 2$, $e \in L^1[0,1]$, and $f:[0,1] \times \Bbb{R}^2 \to \Bbb{R}$ is a Carathéodory function. The additional assumptions $$\sum_{j=1}^n \alpha_j = 1 = \sum_{j=1}^n \beta_j, \quad \sum_{j=1}^n \alpha_j \xi_j = 0 = \sum_{j=1}^n \beta_j \eta_j$$ are “critical”, that is, responsible for resonance. This note complements the result by {\it N. Kosmatov} [Nonlinear Anal., Theory Methods Appl. 65, 622--633 (2006; Zbl 1121.34023)] and extends the result by the first author [Appl. Math. Comput. 143, 275--299 (2003; Zbl 1071.34014)]. The existence result follows from the celebrated coincidence degree theorem due to {\it J. Mawhin} [Topological degree methods in nonlinear boundary value problems (Regional Conference Series in Mathematics, No. 40, Providence, R.I., The American Mathematical Society) (1979; Zbl 0414.34025)].

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47H11 Degree theory (nonlinear operators)
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##### References:
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