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 ')

摘要

The paper deals with the approximation of functions belonging to the Sobolev spaces W-infinity(1) and W-2(1) by functions of the form phi = Sigma(n)(k=1) a(k)X([xk,xk+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(xk)(u) = (1)/(d)integral(xk+d)(xk)u(t)dt, k = 1,..., n, defined on the solutions.