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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2007 Volume 71, Issue 1, Pages 155–186 (Mi im609)

This article is cited in 1 paper

Approximation by step functions of functions belonging to Sobolev spaces and uniqueness of solutions of differential equations of the form $u''=F(x,u,u')$

T. Yu. Semenova

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The paper deals with the approximation of functions belonging to the Sobolev spaces $W^1_\infty$ and $W^1_2$ by functions of the form $\varphi=\sum_{k=1}^n a_k \chi_{[x_k,x_k+d]}$. The results obtained are applied to the study of the stability of solutions of non-linear second-order differential equations of a special form. We consider the problem of whether two solutions can coincide given supplementary information in terms of the values of the functionals $l_{x_k}(u)=\frac{1}{d}\int_{x_k}^{x_k+d}u(t)\,dt$, $k=1,\dots,n$, defined on the solutions.

UDC: 517.9

MSC: 41A25, 41A30, 34B05

Received: 06.10.2005
Revised: 23.03.2006

DOI: 10.4213/im609


 English version:
Izvestiya: Mathematics, 2007, 71:1, 149–180

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