The Criterions of the Extremality of a Sequence for the Problem of Finding the Distance Between Two Convex Sets of the Linear Normed Space

• Видавець: Кам'янець-Подільський національний університет імені Івана Огієнка
• Дата: 2019
• Автори: Гудима, Уляна Василівна, Гнатюк, Василь Олексійович
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2019
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/188966

The research and construction of effective methods for decisioning special classes of extreme problems are actual problems of modern mathematics.In the writings of P. L. Chebyshev investigated a class of problems with non-differentiable functions. These works became the basis of approximation theory.Many problems of this theory are partial cases of the problem of the best approximation of an element of linear normed space by convex set of this space. Important questions of the study of this problem are establishment of the criterion of the extremality of an element. M. P. Korniichuk and V. M. Tikhomirov established the criterion of the extremality of an element. The basis of thes criterion was relation of duality.An important problem, the partial case of which is the problem of the best approximation of an element of a linear normed space by a convex set of this space, is the problem of finding the distance between two convex sets of linear normed space. In the article [1] established relation of duality and the criterion of the extremality of an element for this problem.However, the set of extremal elements for some problems is an empty set. But any extremal problem has an extremal sequence. So research into this sequence is relevant.In this article established the criterions of the extremal sequence of the basis of which is relation of duality and criterion of the Kolmogorov type for the problem of finding the distance between two convex sets of linear normed space. These results were used for the problem of the find the distance between two closed hyperplanes of the linear normed space.