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Zbl 1168.34310
Meng, Fanchao; Du, Zengji
Solvability of a second-order multi-point boundary value problem at resonance. (English)
[J] Appl. Math. Comput. 208, No. 1, 23-30 (2009). ISSN 0096-3003

Summary: Based on the coincidence degree theory of Mawhin, we get a general existence result for the following second-order multi-point boundary value problem at resonance $$x''(t)= f(t,x(t),x'(t))+e(t), \quad t\in(0,1),$$ $$x(0)= \sum_{i=1}^m \alpha_ix(\xi_i), \qquad x'(1)= \sum_{j=1}^n \beta_jx'(\eta_j),$$ where $f:[0,1]\times\Bbb R^2\to\Bbb R$ is a Carathéodory function, $e\in L^1[0,1]$, $0<\xi_1<\xi_2<\cdots< \xi_m<1$, $\alpha_i\in\Bbb R$, $i=1,2,\dots,m$, $m\ge 2$ and $0<\eta_1<\cdots<\eta_n<1$, $\beta_j\in\Bbb R$, $j=1,\dots,n$, $n\ge 1$. In this paper, both of the boundary value conditions are responsible for resonance.

MSC 2000:: *34B10 Multipoint boundary value problems
47N20 Appl. of operator theory to differential and integral equations

Keywords: multi-point boundary value problem; coincidence degree theory; resonance

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