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A new a priori estimate for multi-point boundary-value problem. (English) Zbl 0980.34009

Summary: Let \(f:[0,1]\times \mathbb{R}^2\to \mathbb{R}\) be a function satisfying Carathéodory’s conditions and \(e(t)\in L^{1}[0,1]\). Let \(0<\xi _1<\xi_2<\dots <\xi_{m-2}<1\) and \(a_i\in \mathbb{R}\) for \(i=1,2,\dots ,m-2\), be given. A priori estimates of the form \[ \|x\|_{\infty }\leq C\|x''\|_1, \quad \|x^\prime\|_{\infty }\leq C\|x''\|_1 \] are needed to obtain the existence of a solution to the multipoint boundary value problem \[ x''(t)=f(t,x(t),x^\prime(t))+e(t),\quad 0<t<1, \qquad x(0)=0,\quad x(1)=\sum_{i=1}^{m-2}a_ix(\xi_i), \] using Leray Schauder continuation theorem. The purpose of this paper is to obtain a new a priori estimate of the form \(\|x\|_{\infty}\leq C\|x''\|_1\). This new estimate then enables the author to obtain a new existence theorem. Further, he obtains a new a priori estimate of the form \(\|x\|_{\infty }\leq C\|x''\|_1\) for the three-point boundary value problem \[ x''(t)=f(t,x(t),x^\prime(t))+e(t),\quad 0<t<1, \qquad x^\prime(0)=0,\quad x(1)=\alpha x(\eta), \] where \(\eta \in (0,1)\) and \(\alpha \in \mathbb{R}\) are given. The estimate obtained for the three-point boundary value problem turns out to be sharper than the one obtained by particularizing the \(m\)-point boundary value estimate to the three-point case.

MSC:

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