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Existence theorems for a second order $$m$$-point boundary value problem. (English) Zbl 0884.34024
The paper is devoted to the study of the $$m$$-point boundary value problem $x''= f(t,x(t),x'(t))+ e(t),\;t\in(0,t),\;x'(0)= 0,\;x(1)= \sum^{m-2}_{i=1} a_ix(\xi_i),\tag{1}$ where $$f: [0,1]\times \mathbb{R}^2\to\mathbb{R}$$ and $$e:[0, 1]\to\mathbb{R}$$ are continuous, $$0<\xi_1<\xi_2<\cdots< \xi_{m-2}<1,$$ and the real numbers $$a_i$$ have the same sign.
The author considers both the case $$a= \sum^{m-2}_{i=1} a_i\neq 1$$ and the resonance case $$a= 1$$, where the associated linear homogeneous BVP has nontrivial solutions. The existence of a solution for (1) is proved in both cases. The sign conditions are required for $$f$$ only, and no growth restrictions. The proof is based on the nonlinear alternative of Leray-Schauder type.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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##### References:
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