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A second order \(m\)-point boundary value problem at resonance. (English) Zbl 0824.34023

The problem of existence of solutions for the boundary value problem \(x''= f(t, x, x')+ e(t)\), \(t\in (0, 1)\), \(x(0)= 0\), \(x'(1)= \sum^{m- 2}_{i= 1} a_ i x'(\xi_ i)\) is studied. Here \(f: [0, 1]\times \mathbb{R}^ 2\to \mathbb{R}\) is a Carathéodory function of sublinear growth, \(e\in L_ 1[0, 1]\), \(a_ i\geq 0\); \(0< \xi_ 1<\cdots< \xi_{m- 2}< 1\), \(m\geq 3\). Using Mawhin’s version of the Leray-Schauder continuation theorem, the author investigates this problem in the case of resonance, when \(\sum^{m- 2}_{i= 1} a_ i= 1\).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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