Arbitrary Binary Relations, Contraction Mappings, and b -Metric Spaces
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ARBITRARY BINARY RELATIONS, CONTRACTION MAPPINGS, AND b-METRIC SPACES S. Chandok
UDC 517.54
We prove some results on the existence and uniqueness of fixed points defined on a b-metric space endowed with an arbitrary binary relation. As applications, we obtain some statements on the coincidence of points involving a pair of mappings. Our results generalize, extend, modify and unify several well-known results and, especially, the results obtained by Alam and Imdad [J. Fixed Point Theory Appl., 17, 693– 702 (2015); Fixed Point Theory, 18, 415–432 (2017), and Filomat, 31, 4421–4439 (2017)] and Berzig [J. Fixed Point Theory Appl., 12, 221–238 (2012)]. In addition, we provide an example to illustrate the suitability of the obtained results.
1. Relation-Theoretic Notions, and Preliminary Results We begin with some preliminary definitions and notation used in what follows. Definition 1.1 [7, 11]. Let X be a (nonempty) set and let k ≥ 1 be a given real number. A function d : X ⇥ X ! [0, +1) is a b-metric if and only if, for all x, y, z 2 X, the following conditions are satisfied: (b1 ) d(x, y) = 0 if and only if x = y,
(b2 ) d(x, y) = d(y, x), � � (b3 ) d(x, z) k d(x, y) + d(y, z) .
The pair (X, d) is called a b-metric space.
It should be noted that the class of b-metric spaces is effectively larger than the class of metric spaces because a b-metric is a metric when k = 1. An example presented below shows that, in general, a b-metric need not necessarily to be a metric (see also [1, 14, 17]). Example 1.1. Let (X, d) be a metric space and let ⇢(x, y) = (d(x, y))p ,
p > 1,
be a real number. Then ⇢ is a b-metric with k = 2p−1 but ⇢ is not a metric on X. For more concepts, such as b-convergence, b-completeness, and b-Cauchy sequences and b-closed sets in b-metric spaces, we refer the reader to [1, 13, 14, 17] and the references therein. At the same time, for the other concepts, such as partial ordering, comparability, well-ordered, nondecreasing, increasing, dominated, dominating, etc., we refer the reader to [1, 9, 12, 14, 17, 18, 21]. In what follows, by N and R we denote the set of all nonnegative integers and the set of all real numbers, respectively. Throughout the paper, R stands for a nonempty binary relation but, for the sake of simplicity, we often write binary relation instead of nonempty binary relation. School of Mathematics, Thapar University, Patiala, India; e-mail: [email protected]; [email protected]. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 565–574, April, 2020. Original article submitted December 8, 2016; revision submitted January 31, 2020. 0041-5995/20/7204–0651
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Springer Science+Business Media, LLC
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Definition 1.2. Let R be a binary relation defined on a nonempty set X and let x, y 2 X. We say that x and y are R-comparative if either (x, y) 2 R or (y, x) 2 R. This is denoted by [x, y] 2 R. Definition 1.3. A binary relation R on a nonempty set X is called: (1) reflexive if (x, x) 2 R for every x 2 X; (2) symmetric if (y, x) 2 R whe
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