Abstract
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.
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1 Introduction and preliminaries
The Banach contraction principle [1] 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 [10]. 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 [24] introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle [1]. 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 [22].
Let () denote the set of all coincidence points (the set of all common fixed points) of self-mappings f and g.
Definition 2 ([25])
Let be a metric space and . The mapping f is called a g-contraction if there exists such that
holds for all .
In 1976, Jungck [25] 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 .
Let Ϝ be the collection of all mappings that satisfy the following conditions:
-
(C1)
F is strictly increasing, that is, for all such that implies that .
-
(C2)
For every sequence of positive real numbers, and are equivalent.
-
(C3)
There exists such that
Definition 3 ([24])
Let be a metric space and . A mapping is said to be an F-contraction on X if there exists such that
for all .
Note that every F-contraction is continuous (see [24]). We extend the above definition to two mappings.
Definition 4 Let be a metric space, and . The mapping f is said to be an F-contraction with respect to g on X if there exists such that
for all satisfying .
By different choices of mappings F in (1) and (2), one obtains a variety of contractions [24].
Example 1 Let be given by . It is clear that . Suppose that is an F-contraction with respect to a self-mapping g on X. From (2) we have
which implies that
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.
Example 2 Consider the following sequence of partial sums [[24], Example 2.5]:
Let and d be the usual metric on X. Let and be defined as
Let be given by . As
so f is not a g-contraction. If we take , then and f is an -contraction with respect to a mapping g (taking ). Indeed, the following holds:
for all . For all with , we have
Definition 5 ([26], 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.
Example 3 Let be endowed with the usual ordering and defined by for some and for some real number . Note that
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 [27]).
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 8 Let be a partial ordered set, then x, y in X are called comparable elements if either or holds true. Moreover, we define by
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
We present the following theorem as a generalization of results in [25] and [[24], Theorem 2.1].
Theorem 2 Let be a partially ordered set such that there exists a metric d on X, and let be an F-contraction with respect to on Δ with . Assume that f is dominating and g is dominated. Then
-
(a)
f and g have a coincidence point in X provided that is complete and has the sequential limit comparison property.
-
(b)
is well ordered if and only if is a singleton.
-
(c)
f and g have a unique common fixed point if f and g are weakly compatible and is well ordered.
Proof (a) Let be an arbitrary point of X. Since the range of g contains the range of f, there exists a point in X such that . As f is dominating and g is dominated, so we have
Hence . Continuing this process, having chosen in X, we obtain in X such that
So, we obtain for every . For the sake of simplicity, take
for all . If there exists for which , then implies that , that is, . Now we assume that for all . As f is an F-contraction with respect to g on Δ, so we obtain
That is,
On taking limit as , we obtain . Hence by (C2). Now, by (C3), there exists such that . Note that
Taking limit as in (4), we have . Consequently, . Thus there exists in ℕ such that for all , that is, for all . Now, for integers , we obtain
This shows that is a Cauchy sequence in . As is complete, so there exists q in such that . Let be such that . The sequential limit comparison property implies that . As so . Hence from (2) we have
Since , therefore by (C2) we have . Hence implies that . That is, . Uniqueness of limit implies , that is, .
(b) Now suppose that is well ordered. We prove that is a singleton. Assume on the contrary that there exists another point w in X such that with . Since is well ordered, so . Now from (2) we have
a contradiction. Therefore . Hence f and g have a unique coincidence point p in X. The converse follows immediately.
-
(c)
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. □
Example 4 Let be endowed with usual metric and usual order. Define mappings by
Clearly, g is dominated and f is dominating. Define as . If and , then
Hence, for , inequality (2) is satisfied. Similarly, for and , we have
Hence, for , inequality (2) is satisfied. We can take a so that
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.
Theorem 3 Let be a partially ordered set such that there exists a complete metric d on X. If self-mappings f and g on X are weakly increasing and for some satisfy
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.
Proof Let be an arbitrary point of X. Define a sequence in X as follows: and . Since f and g are weakly increasing, we have and . Hence and for every . Now define
for all . Using (5) the following holds for every :
Similarly,
Therefore, for all , we have
Thus
Taking limit as in (7), we get
By (C2) and (C3) we get and such that . Note that
By taking limit as in (8), we get . This implies that there exists such that for all . Consequently, we obtain for all . Now, for integers , we have
This shows that is a Cauchy sequence in X, so there exists p in X such that . As X has the sequential limit comparison property, so . Therefore
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.
Example 5 Let be given by
Then f has a unique fixed point . Note that holds for every even natural number n and x in . On the other hand, define a mapping as
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 .
For further details on these properties, we refer to [20, 28].
Let be a metric space and be a self-mapping. The set is called the orbit of x [29]. 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.
Theorem 4 Let X be a nonempty set such that there exists a complete metric d on X. Suppose that satisfies
for some and for all x in X such that . Then f has the property P provided that f is orbitally continuous on X.
Proof First we show that . Let . Define a sequence in X, such that , for all . Denote for all . If there exists for which , then and the proof is finished. Suppose that for all . Using (9), we obtain
for every . By taking limit as in the above inequality, we obtain that , which together with (C2) gives . From (C3), there exists such that . Note that
On taking limit as , we get . Hence there exists such that for all . Consequently for all . Now, for integers such that
This shows that is a Cauchy sequence. Since and X is complete, which implies that there exists x in X such that . Since f is orbitally continuous at x, so . Hence f has a fixed point and is true for . Now assume . Suppose on the contrary that but , then . Now consider
Thus . Hence . By (C2) , a contradiction. So . □
Theorem 5 Let be a partially ordered set such that there exists a complete metric d on X and f, g self-mappings on X. Further assume that f, g are weakly increasing and satisfy
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.
Proof By Theorem 3, f and g have a common fixed point. Suppose on the contrary that
but , then there are three possibilities (a) , (b) , (c) and . Without loss of generality, let , that is, , so we get
As , so we have . By (C2) , a contradiction. Hence . □
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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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Abbas, M., Ali, B. & Romaguera, S. Fixed and periodic points of generalized contractions in metric spaces. Fixed Point Theory Appl 2013, 243 (2013). https://doi.org/10.1186/1687-1812-2013-243
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DOI: https://doi.org/10.1186/1687-1812-2013-243