Some higher-order multi-point boundary value problem at resonance. (English) Zbl 1059.34010

Du, Zengji; Lin, Xiaojie; Ge, Weigao

J. Comput. Appl. Math. 177, No. 1, 55-65 (2005).

Summary: This paper deals with the existence of solutions for the following \(n\)th-order multi-point boundary value problem at resonance \[ \begin{gathered} x^{(n)}(t)= f(t, x(t), x'(t),\dots, x^{(n-1)}(t))+ e(t),\quad t\in (0,1),\\ x(0)= \sum^{m-2}_{i=1} \alpha_i x(\xi_i),\;x'(0)=\cdots= x^{n-2)}(0)= 0,\quad x(1)= x(\eta),\end{gathered} \] where \(f: [0, 1]\times\mathbb{R}^n\to\mathbb{R}\) is a continuous function, \(e\in L^1[0,1]\), \(\alpha_i\in \mathbb{R}\), \(1\leq i\leq m-2\), \(0< \xi_1< \xi_2<\cdots< \xi_{m-2}< 1\) and \(0<\eta< 1\). An existence theorem is obtained by using the coincidence degree theory of Mawhin.

MSC:

34B10

Nonlocal and multipoint boundary value problems for ordinary differential equations

Keywords:

Multipoint boundary value problem; Resonance; Fredholm operators; Coincidence degree theory

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