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Journal of Operators
Volume 2013 (2013), Article ID 368501, 5 pages
A Generalization of Prešić Type Mappings in Metric-Like Spaces
1Department of Applied Mathematics, Shri Vaishnav Institute of Technology & Science, Gram Baroli, Sanwer Road, Indore, Madhya Pradesh 453331, India
2Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
Received 20 March 2013; Accepted 1 April 2013
Academic Editor: Ram U. Verma
Copyright © 2013 Satish Shukla and Brian Fisher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We generalize the result of Prešić in metric-like spaces by proving some common fixed point theorems for Prešić type mappings in metric-like spaces. An example is given which shows that the generalization is proper.
1. Introduction and Preliminaries
Let be any metric space, and let be any mapping; then is said to be a contraction on if there exists such that
A point is called a fixed point of if . Banach  proved that every contraction on a complete metric space has a unique fixed point and this result is known as the Banach contraction principle. There are several generalizations of this famous principle. One such generalization is given by Prešić [2, 3]. When studying the convergence of some particular sequences, Prešić [2, 3] proved the following theorem.
Theorem 1. Let be a complete metric space, a positive integer, and a mapping satisfying the following contractive type condition: for every , where are nonnegative constants such that . Then there exists a unique point such that . Moreover, if are arbitrary points in and for , then the sequence is convergent and .
Note that the -step iterative sequence given by (3) represents a nonlinear difference equation. In view of Prešić theorem, it is obvious that if this sequence is convergent (which is ensured by the Prešić theorem) then the limit of the sequence is a fixed point of . The result of Prešić is generalized by several authors, and some generalizations and applications of Prešić theorem can be seen in [4–15].
On the other hand, Matthews  introduced the notion of a partial metric space as a part of the study of denotational semantics of a dataflow network. In this space, the usual metric is replaced by a partial metric with an interesting property that the self-distance of any point of space may not be zero. Further, Matthews showed that the Banach contraction principle is valid in a partial metric space and can be applied in program verifications. O'Neill  generalized the concept of a partial metric space a bit further by admitting negative distances. The partial metric defined by O'Neill is called the dualistic partial metric. Heckmann  generalized it by omitting the small self-distance axiom. The partial metric defined by Heckmann is called a weak partial metric.
Recently, Amini-Harandi  generalized the partial metric spaces by introducing the metric-like spaces. Amini-Harandi introduced the notion of a -Cauchy sequence and completeness of metric-like spaces and proved some fixed point theorems in such spaces. In this paper, we prove some common fixed point theorems for Prešić type mappings in metric-like spaces. Our results generalize and extend the result of Prešić from complete spaces into -complete metric-like spaces. An example is given which shows that the generalization is proper.
First we recall some definitions about partial metric and metric-like spaces.
Definition 2 (see ). A partial metric on a nonempty set is a function ( stands for nonnegative reals) such that for all :1) if and only if ; (2); (3); (4).A partial metric space is a pair such that is a nonempty set and is a partial metric on . A sequence in converges to a point if and only if . A sequence in is called a -Cauchy sequence if exists and is finite. is said to be complete if every -Cauchy sequence in converges to a point such that .
Definition 3 (see ). A metric-like on a nonempty set is a function such that for all :(1) implies ; (2); (3).A metric-like space is a pair such that is a nonempty set and is a metric-like on . Note that a metric-like satisfies all the conditions of metric except that may be positive for some . Each metric-like on generates a topology on whose base is the family of open -balls
A sequence in converges to a point if and only if . A sequence is said to be -Cauchy if exists and is finite. The metric-like space is called complete if for each -Cauchy sequence , there exists such that
Every partial metric space is a metric-like space but the converse may not be true.
Example 4 (see ). Let and be defined by
Then is a metric-like space, but it is not a partial metric space, as .
Example 5. Let , , and be defined by
Then is a metric-like space, but for , it is not a partial metric space, as .
Example 6. Let and be defined by
Then is a metric-like space, but it is not a partial metric space, as .
For the following definition, we refer to  and the references therein.
Definition 7. Let be a nonempty set, a positive integer, and a mapping. If , then is called a fixed point of .
Definition 8. Let be a nonempty set, a positive integer, and and mappings.(a)An element is said to be a coincidence point of and if .(b)If , then is called a point of coincidence of and .(c)If , then is called a common fixed point of and .(d)Mappings and are said to be commuting if for all .(e)Mappings and are said to be weakly compatible if they commute at their coincidence points.
Let be a metric-like space, a positive integer, and a mapping. is said to be Prešić type if for all , where are nonnegative constants such that .
The following lemma shows that the self-distance of a point of coincidence of two maps of a metric-like space satisfying Prešić type contractive condition is minimum (zero), and it will be used in the sequel.
Lemma 9. Let be a metric-like space, a positive integer, and , two mappings. Suppose that the following condition holds: for all , where are nonnegative constants such that . If and have a point of coincidence , then .
Proof. Let be any point of coincidence of and . Then there exists such that .
Now suppose that . Then it follows from (10) that a contradiction. Therefore we must have .
Now we can state our main results.
2. Main Results
The following theorem extends and generalizes the result of Prešić in metric-like spaces.
Theorem 10. Let be a metric-like space, a positive integer, and , two mappings such that and is a complete subspace of . Suppose that the following condition holds: for every , where are nonnegative constants such that . Then and have a unique point of coincidence and . Moreover, if and are weakly compatible, then is the unique common fixed point of and .
Proof. Let be arbitrary points. As , we can define a sequence in by for and for .
For simplicity put for and , where .
By mathematical induction, we will show that According to the definition of , it is clear that (13) is true for . Let the following inequalities: be the induction hypothesis.
Now using (12), we obtain
Thus, the inductive proof of (13) is complete.
Now let with . Then from (13), we have As, , it follows from the previous inequality that Thus, is a -Cauchy sequence in . By the completeness of , there exists such that and
We will now show that is a point of coincidence of and . For any we have Using (12) and writing , in the previous inequality we obtain Letting and using (18) in the previous inequality, we obtain Therefore, is a point of coincidence of and .
We will now show that it is unique. Suppose that is another point of coincidence of and . Then there exists such that . By Lemma 9, we have . Now it follows from (12) that As , it follows from previous inequality that a contradiction. Therefore, we must have , that is, . Thus the point of coincidence of and is unique.
Now suppose that and are weakly compatible. Put . Then , so is a point of coincidence of and . However, by the uniqueness of , we have . Thus, is the unique common fixed point of and .
Taking (i.e., the identity mapping of ) we obtain following fixed point result for Prešić type mapping in a metric-like space.
Corollary 11. Let be a complete metric-like space, a positive integer, and a Prešić type mapping on . Then has a unique fixed point and .
Example 12. Let , and define by
Then, is a complete metric-like space. Note that is not a partial metric space, as .
Note that is not a Prešić type mapping in the usual metric space , where for all . Indeed, for the points we have and . Therefore, the condition (2) of Theorem 1 is not satisfied with . Thus the result of Prešić is not applicable here.
On the other hand, it is easy to see that satisfies all the conditions of Corollary 11, with and ; that is, is the unique fixed point of .
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