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Solvability of a second-order multi-point boundary value problem at resonance. (English) Zbl 1168.34310
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 \(f:[0,1]\times\mathbb R^2\to\mathbb R\) is a Carathéodory function, \(e\in L^1[0,1]\), \(0<\xi_1<\xi_2<\cdots< \xi_m<1\), \(\alpha_i\in\mathbb R\), \(i=1,2,\dots,m\), \(m\geq 2\) and \(0<\eta_1<\cdots<\eta_n<1\), \(\beta_j\in\mathbb R\), \(j=1,\dots,n\), \(n\geq 1\). In this paper, both of the boundary value conditions are responsible for resonance.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:
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