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