Some fixed point theorems concerning F-contraction in complete metric spaces
© Piri and Kumam; licensee Springer. 2014
Received: 11 April 2014
Accepted: 10 September 2014
Published: 13 October 2014
In this paper, we extend the result of Wardowski (Fixed Point Theory Appl. 2012:94, 2012) by applying some weaker conditions on the self map of a complete metric space and on the mapping F, concerning the contractions defined by Wardowski. With these weaker conditions, we prove a fixed point result for F-Suzuki contractions which generalizes the result of Wardowski.
1 Introduction and preliminaries
Throughout this article, we denote by ℝ the set of all real numbers, by the set of all positive real numbers, and by ℕ the set of all natural numbers.
In 1922, Polish mathematician Banach  proved a very important result regarding a contraction mapping, known as the Banach contraction principle. It is one of the fundamental results in fixed point theory. Due to its importance and simplicity, several authors have obtained many interesting extensions and generalizations of the Banach contraction principle (see [2–9] and references therein). Subsequently, in 1962, M Edelstein proved the following version of the Banach contraction principle.
Theorem 1.1 
Let (X, d) be a compact metric space and let be a self-mapping. Assume that holds for all with . Then T has a unique fixed point in X.
In 2008, Suzuki  proved generalized versions of Edelstein’s results in compact metric space as follows.
Theorem 1.2 
Then T has a unique fixed point in X.
In 2012, Wardowski  introduce a new type of contractions called F-contraction and prove a new fixed point theorem concerning F-contractions. In this way, Wardowski  generalized the Banach contraction principle in a different manner from the well-known results from the literature. Wardowski defined the F-contraction as follows.
F is strictly increasing, i.e. for all such that , ;
For each sequence of positive numbers, if and only if ;
There exists such that .
Remark 1.4 From (F1) and (1) it is easy to conclude that every F-contraction is necessarily continuous.
Wardowski  stated a modified version of the Banach contraction principle as follows.
Theorem 1.5 
Let be a complete metric space and let be an F-contraction. Then T has a unique fixed point and for every the sequence converges to .
Very recently, Secelean  proved the following lemma.
Lemma 1.6 
if , then ;
if , and , then .
By proving Lemma 1.6, Secelean showed that the condition (F2) in Definition 1.3 can be replaced by an equivalent but a more simple condition,
or, also, by
(F2″) there exists a sequence of positive real numbers such that .
Remark 1.7 Define by , then . Note that with the F-contraction reduces to a Banach contraction. Therefore, the Banach contractions are a particular case of F-contractions. Meanwhile there exist F-contractions which are not Banach contractions (see [11, 12]).
In this paper, we use the following condition instead of the condition (F3) in Definition 1.3:
(F3′) F is continuous on .
We denote by the set of all functions satisfying the conditions (F1), (F2′), and (F3′).
Example 1.8 Let , , , . Then .
Remark 1.9 Note that the conditions (F3) and (F3′) are independent of each other. Indeed, for , satisfies the conditions (F1) and (F2) but it does not satisfy (F3), while it satisfies the condition (F3′). Therefore, . Again, for , , , where denotes the integral part of α, satisfies the conditions (F1) and (F2) but it does not satisfy (F3′), while it satisfies the condition (F3) for any . Therefore, . Also, if we take , then and . Therefore, .
In view of Remark 1.9, it is meaningful to consider the result of Wardowski  with the mappings instead . Also, we define the F-Suzuki contraction as follows and we give a new version of Theorem 1.5.
2 Main results
Then T has a unique fixed point and for every the sequence converges to .
which is a contradiction. Therefore, the fixed point is unique. □
Theorem 2.2 Let be a complete metric space and be an F-Suzuki contraction. Then T has a unique fixed point and for every the sequence converges to .
which is a contradiction. Thus, the fixed point is unique. □
Therefore for all . Hence T is an F-contraction and .
The generated iterations start from a point . denotes
3 × 103
tends to 0
≥τ = 1
≥τ = 1
≥τ = 1
The second author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (Under NUR Project ‘Theoretical and Computational fixed points for Optimization problems’ No. 57000621).
- Banach B: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
- Suzuki T: A new type of fixed point theorem in metric spaces. Nonlinear Anal. 2009, 71: 5313–5317. 10.1016/j.na.2009.04.017View ArticleMathSciNetGoogle Scholar
- Suzuki T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 2001, 253: 440–458. 10.1006/jmaa.2000.7151View ArticleMathSciNetGoogle Scholar
- Suzuki T: Several fixed point theorems concerning τ -distance. Fixed Point Theory Appl. 2004, 2004: 195–209.View ArticleGoogle Scholar
- Tataru D: Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. J. Math. Anal. Appl. 1992, 163: 345–392. 10.1016/0022-247X(92)90256-DView ArticleMathSciNetGoogle Scholar
- Vályi I: A general maximality principle and a fixed point theorem in uniform space. Period. Math. Hung. 1985, 16: 127–134. 10.1007/BF01857592View ArticleGoogle Scholar
- Włodarczyk K, Plebaniak R: Quasigauge spaces with generalized quasipseudodistances and periodic points of dissipative set-valued dynamic systems. Fixed Point Theory Appl. 2011., 2011: Article ID 712706Google Scholar
- Włodarczyk K, Plebaniak P: Kannan-type contractions and fixed points in uniform spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 90Google Scholar
- Włodarczyk K, Plebaniak R: Contractivity of Leader type and fixed points in uniform spaces with generalized pseudodistances. J. Math. Anal. Appl. 2012, 387: 533–541. 10.1016/j.jmaa.2011.09.006View ArticleMathSciNetGoogle Scholar
- Edelstein M: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 1962, 37: 74–79.View ArticleMathSciNetGoogle Scholar
- Wardowski D: Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 94Google Scholar
- Secelean NA: Iterated function systems consisting of F -contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 277 10.1186/1687-1812-2013-277Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.