Best proximity point theorems for cyclic contractions mappings
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Publication Details
Author list: Mongkolkeha C., Kumam P.
Publisher: Hindawi
Publication year: 2017
Start page: 201
End page: 228
Number of pages: 28
ISBN: 9781351243360; 9780815369455
ISSN: 0146-9428
eISSN: 1745-4557
Languages: English-Great Britain (EN-GB)
Abstract
Many problems in mathematics such as equilibrium and variational inequalities, mathematical economics, game theory and optimization can be formulated as equations of the form T x = x, where T is a self mapping in some suitable framework, because of its ability to solve ordinary differential equations, integral equations, matrix equations and others. However, given nonempty subsets A and B of X. A mapping T: A → B (or T: A∪B → A∪B) using the equation T x = x does not necessarily have a solution. It is worthwhile to find an approximate solution x under mapping T so that the error d(x, T x) is minimal. This is the basis of the optimal approximation theory that includes a generalized fixed point. Whenever A coincides with B, the optimization problem known as a best proximity point of the mapping T reduces to a fixed point problem. Fan [7] introduced a classical best approximation theorem: if A is a nonempty compact convex subset of a Hausdorff locally convex topological vector space B and T: A ? B is a continuous mapping, there exists an element x ε A such that d(T x, x) = d(T x, A):= inf{d(T x, y): y ε A}. Several authors, including Prolla [15], Reich [16], Sehgal [20, 21], derived extensions of Fan’s theorem in many directions. © 2018 by Taylor & Francis Group, LLC.
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