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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.

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text: DOI
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