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.


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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