## 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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### References:

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