Approximation of the integral funnel of the Urysohn type integral operator

Abstract

The integral funnel of the closed ball of the space L-p, p > 1, with radius r and centered at the origin under Urysohn type integral operator is defined as the set of graphs of the images of all functions from given ball. Approximation of the integral funnel is considered. The closed ball of the space L-p, p > 1, with radius r is replaced by the set consisting of a finite number of functions. The Hausdorffdistance between integral funnel and the set consisting of the sections of the graphs of images of a finite number of functions is evaluated. It is proved that in the case of appropriate choosing of the discretization parameters the approximating sets converges to the integral funnel. (c) 2018 Elsevier Inc. All rights reserved.