Pata Zamfirescu Type Fixed-Disc Results with a Proximal Application
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Pata Zamfirescu Type Fixed-Disc Results with a Proximal Application Nihal Özgür1
· Nihal Ta¸s1
Received: 21 April 2020 / Revised: 1 November 2020 / Accepted: 4 November 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract This paper concerns with the geometric study of fixed points of a self-mapping on a metric space. We establish new generalized contractive conditions which ensure that a self-mapping has a fixed disc or a fixed circle. We introduce the notion of a best proximity circle and explore some proximal contractions for a non-self-mapping as an application. Necessary illustrative examples are presented to highlight the importance of the obtained results. Keywords Fixed disc · Pata Zamfirescu type x0 -mapping · Proximity point · Proximity circle Mathematics Subject Classification Primary 54H25; Secondary 47H09 · 47H10
1 Introduction and Motivation Fixed-point theory has an important role due to solutions of the equation T x = x where T is a self-mapping on a metric (resp. some generalized metric) space. This theory has been extensively studied with various applications in diverse research areas such as integral equations, differential equations, engineering, statistics, and economics. Some questions have been arisen for the existence and uniqueness of fixed points. Some fixed-point problems are as follows: (1) Is there always a solution of the equation T x = x? (2) What are the existence conditions for a fixed point of a self-mapping?
Communicated by Rosihan M. Ali.
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Nihal Özgür [email protected] Nihal Ta¸s [email protected]
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Department of Mathematics, Balıkesir University, 10145 Balıkesir, Turkey
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N. Özgür, N. Ta¸s
(3) What are the uniqueness conditions if there is a fixed point of a self-mapping? (4) Can the number of fixed points be more than one? (5) If the number of fixed points is more than one, is there a geometric interpretation of these points? Considering the above questions, many researchers have been studied on fixed-point theory with different aspects. Some generalized contractive conditions have been investigated to guarantee the existence and uniqueness of a fixed point of a self-mapping. For example, in [22], an existence theorem was given for a generalized contraction mapping. In [16], a refinement of the classical Banach contraction principle was obtained. A new generalization of these results was derived by using both of the above contractive conditions in [3]. In [2], a survey of various variants of fixed point results for single- and multivalued mappings under the Pata-type conditions was given (for more results, see [3–5,17] and the references therein). In the cases in which the fixed-point equation T x = x has no solution, the notion of “best proximity point” has been appeared as an approximate solution x such that the error d (x, T x) is minimum. For example, the existence of best proximity point was investigated using the Pata-type proximal mappings in [3]. These results are the generalizations of ones obt
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