Dislocated metric space to metric spaces with some fixed point theorems
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Dislocated metric space to metric spaces with some fixed point theorems Erdal Karapınar1* and Peyman Salimi2 *
Correspondence: [email protected]; [email protected] 1 Department of Mathematics, Atilim University, Incek, Ankara 06836, Turkey Full list of author information is available at the end of the article
Abstract In this paper, we notice the notions metric-like space and dislocated metric space are exactly the same. After this historical remark, we discuss the existence and uniqueness of a fixed point of a cyclic mapping in the context of metric-like spaces. We consider some examples to illustrate the validity of the derived results of this paper. MSC: 47H10; 54H25 Keywords: dislocated metric spaces; metric-like spaces; fixed point
1 Introduction and preliminaries Fixed point theory is one of the most dynamic research subjects in nonlinear sciences. Regarding the feasibility of application of it to the various disciplines, a number of authors have contributed to this theory with a number of publications. The most impressing result in this direction was given by Banach, called the Banach contraction mapping principle: Every contraction in a complete metric space has a unique fixed point. In fact, Banach demonstrated how to find the desired fixed point by offering a smart and plain technique. This elementary technique leads to increasing of the possibility of solving various problems in different research fields. This celebrated result has been generalized in many abstract spaces for distinct operators. In particular, Hitzler [] obtained one of interesting characterizations of the Banach contraction mapping principle by introducing dislocated metric spaces, which is rediscovered by Amini-Harandi []. Definition . A dislocated (metric-like) on a nonempty set X is a function σ : X × X → [, +∞) such that for all x, y, z ∈ X: (σ ) if σ (x, y) = then x = y, (σ ) σ (x, y) = σ (y, x), (σ ) σ (x, y) ≤ σ (x, z) + σ (z, y), and the pair (X, σ ) is called a dislocated (metric-like) space. The motivation of defining this new notion is to get better results in logic programming semantics (see, e.g., [, ]). Following these initial reports, many authors paid attention to the subject and have published several papers (see, e.g., [–]). Another interesting generalization of the Banach contraction mapping principle was given by Kirk et al. [] via a cyclic mapping (see, e.g., [–]). In this remarkable paper, the mappings, for which the existence and uniqueness of a fixed point were discussed, do not need to be continuous. © 2013 Karapınar and Salimi; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Karapınar and Salimi Fixed Point Theory and Applications 2013, 2013:222 http://www.fixedpointtheoryandapplications.com/content/2013/1/222
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