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Solvability of a multi-point boundary value problen of Neumann type. (English) Zbl 0993.34009

This paper is concerned with the existence of a solution to the \(m\)-point boundary value problem \[ x''(t)= f(t,x(t),x'(t))+e(t),\quad 0<t<1, \qquad x(0)= 0,\quad x'(1)= \sum_{i=1}^{m-2}a_ix'(\xi _i), \] where \(f:[0,1]\times \mathbb{R}^2\rightarrow \mathbb{R}\) is a function satisfying Carathéodory’s conditions and \(e\in L^1[0,1]\); \(\xi _i\in (0,1),\) \(a_i\in \mathbb{R}\), \(i=1,2,\dots,m-2\); \(0<\xi _1<\xi _2<\dots<\xi _{m-2}\). The authors give sufficient conditions for a solution to this multipoint boundary value problem and for its particular case of the three-point boundary value problem. The proofs of the existence theorems are based on the Leray-Schauder continuation theorem and use Poincaré-type a priori estimates on \( x'\) provided by the authors. The abstract results are illustrated by examples.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
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