Fixed and periodic points of generalized contractions in metric spaces
© Abbas et al.; licensee Springer. 2013
Received: 1 July 2013
Accepted: 8 October 2013
Published: 7 November 2013
Wardowski (Fixed Point Theory Appl. 2012:94, 2012, doi:10.1186/1687-1812-2012-94) introduced a new type of contraction called F-contraction and proved a fixed point result in complete metric spaces, which in turn generalizes the Banach contraction principle. The aim of this paper is to introduce F-contractions with respect to a self-mapping on a metric space and to obtain common fixed point results. Examples are provided to support results and concepts presented herein. As an application of our results, periodic point results for the F-contractions in metric spaces are proved.
MSC:47H10, 47H07, 54H25.
1 Introduction and preliminaries
The Banach contraction principle  is a popular tool in solving existence problems in many branches of mathematics (see, e.g., [2–4]). Extensions of this principle were obtained either by generalizing the domain of the mapping or by extending the contractive condition on the mappings [5–9]. Initially, existence of fixed points in ordered metric spaces was investigated and applied by Ran and Reurings . Since then, a number of results have been proved in the framework of ordered metric spaces (see [11–18]). Contractive conditions involving a pair of mappings are further additions to the metric fixed point theory and its applications (for details, see [19–23]).
Recently, Wardowski  introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle . In this paper, we introduce an F-contraction with respect to a self-mapping on a metric space and obtain common fixed point results in an ordered metric space. In the last section, we give some results on periodic point properties of a mapping and a pair of mappings in a metric space. We begin with some basic known definitions and results which will be used in the sequel. Throughout this article, ℕ, , ℝ denote the set of natural numbers, the set of positive real numbers and the set of real numbers, respectively.
Definition 1 Let f and g be self-mappings on a set X. If for some x in X, then x is called a coincidence point of f and g and w is called a coincidence point of f and g. Furthermore, if whenever x is a coincidence point of f and g, then f and g are called weakly compatible mappings .
Let () denote the set of all coincidence points (the set of all common fixed points) of self-mappings f and g.
Definition 2 ()
holds for all .
In 1976, Jungck  obtained the following useful generalization of the Banach contraction principle.
Theorem 1 Let g be a continuous self-mapping on a complete metric space . Then g has a fixed point in X if and only if there exists a g-contraction mapping such that f commutes with g and .
F is strictly increasing, that is, for all such that implies that .
For every sequence of positive real numbers, and are equivalent.
- (C3)There exists such that
Definition 3 ()
for all .
Note that every F-contraction is continuous (see ). We extend the above definition to two mappings.
for all satisfying .
By different choices of mappings F in (1) and (2), one obtains a variety of contractions .
Therefore an -contraction map f with respect to g reduces to a g-contraction mapping.
Now we give an example of an F-contraction with respect to a self-mapping g on X which is not a g-contraction on X.
Definition 5 (, Dominance condition)
Let be a partially ordered set. A self-mapping f on X is said to be (i) a dominated map if for each x in X, (ii) a dominating map if for each x in X.
for all x in X. Thus g is dominated and f is a dominating map.
Definition 6 Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all x in X (see ).
Definition 7 Let X be a nonempty set. Then is called an ordered metric space if is a metric space and is a partially ordered set.
Definition 9 An ordered metric space is said to have the sequential limit comparison property if for every non-decreasing sequence (non-increasing sequence) in X such that implies that ().
2 Common fixed point results in ordered metric spaces
f and g have a coincidence point in X provided that is complete and has the sequential limit comparison property.
is well ordered if and only if is a singleton.
f and g have a unique common fixed point if f and g are weakly compatible and is well ordered.
Since , therefore by (C2) we have . Hence implies that . That is, . Uniqueness of limit implies , that is, .
Now if f and g are weakly compatible mappings, then we have , that is, q is the coincidence point of f and g. But q is the only point of coincidence of f and g, so . Hence q is the unique common fixed point of f and g. □
is satisfied for all , whenever . Hence f is an F-contraction with respect to g on . Hence all the conditions of Theorem 2 are satisfied. Moreover, is the coincidence point of f and g. Also note that f and g are weakly compatible and is the common fixed point of g and f as well.
Now we give a common fixed point result without imposing any type of commutativity condition for self-mappings f and g on X. Moreover, we relax the dominance conditions on f and g as well.
for all such that , then , provided that X has the sequential limit comparison property. Further, f and g have a unique common fixed point if and only if is well ordered.
Since , by (C2) we have . This implies , which further implies that . Hence and . Similarly, we obtain . This shows that p is a common fixed point of g and f. Now suppose that is well ordered. We prove that is a singleton. Assume on the contrary that there exists another point q in X such that with . Obviously, . So, from (5) we have , a contradiction. Therefore . Hence g and f have a unique common fixed point p in X. The converse follows immediately. □
3 Periodic point results in metric spaces
If x is a fixed point of the self-mapping f, then x is a fixed point of for every , but the converse is not true. In the sequel, we denote by the set of all fixed points of f.
Then g has the same fixed point as for every n.
Definition 10 The self-mapping f is said to have the property P if for every . A pair of self-mappings is said to have the property Q if .
Let be a metric space and be a self-mapping. The set is called the orbit of x . A mapping f is called orbitally continuous at p if implies that . A mapping f is orbitally continuous on X if f is orbitally continuous for all .
In this section we prove some periodic point results for self-mappings on complete metric spaces.
for some and for all x in X such that . Then f has the property P provided that f is orbitally continuous on X.
Thus . Hence . By (C2) , a contradiction. So . □
for some , for all x, y in X such that . Then f and g have the property Q provided that X has the sequential limit comparison property.
As , so we have . By (C2) , a contradiction. Hence . □
The third author thanks for the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.
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