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

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.