 Research
 Open Access
 Published:
On inclusion of Fcontractions in $(\delta ,k)$weak contractions
Fixed Point Theory and Applications volume 2014, Article number: 65 (2014)
Abstract
We obtain a certain property of Fcontractions, which enables us to generalize and extend Wardowski’s (Fixed Point Theory and Applications 2012:94, 2012) result and other fixed point theorems to the class of nonexpansive operators in real Banach spaces. We give illustrative examples to demonstrate nontrivial applicability of the property and use it to prove that Fcontractions are closely related to $(\delta ,k)$weak contractions introduced by Berinde (Carpath. J. Math. 19(1):722, 2003; Nonlinear Anal. Forum 9(1):4353, 2004).
MSC:47H09, 47H10, 54H25.
1 Introduction and preliminaries
This work is concerned with contractive maps defined on real Banach spaces. Let $(X,d)$ be a metric space and let $T:X\to X$ be a mapping such that there exists at least a constant $L>0$ with $d(Tx,Ty)\le Ld(x,y)$ for all $x,y\in X$, then T is called LLipschitzian operator. T is called a contraction if $L\in [0,1)$ and for $L=1$, T is called a nonexpansive operator. T is called contractive if $d(Tx,Ty)<d(x,y)$, while T is referred to as expansive if $L>1$. A point $p\in X$ is called a fixed point of an operator T if $p=Tp$ and the collection of all fixed points of an operator T is denoted by $Fix(T)$. The study of fixed points of contractive and expansive maps still attracts attention of numerous researchers studying extensions and generalizations of the Banach fixed point result. The famous Banach fixed point theorem, called contraction mapping principle, remains a basic result in fixed point theory because of the simplicity in the conditional requirements of the theorem on one hand and because of the simple nature of the Picard iteration scheme called successive approximations, which gives approximate fixed points with error estimates, on the other hand. The Banach contraction principle in a Banach space $(E,\parallel \cdot \parallel )$ asserts that if $T:E\u27f6E$ is a contraction (i.e., for all $x,y\in E$, $\parallel TxTy\parallel \le L\parallel xy\parallel $ for some $L\in (0,1)$), then T has a unique fixed point given by ${lim}_{n\to \mathrm{\infty}}{T}^{n}{x}_{0}$, where ${x}_{0}$ is any initial point in E. The contraction condition in a complete metric space $(X,d)$ is given by $d(Tx,Ty)\le Ld(x,y)$. There have been many generalizations and extensions of the Banach fixed point theorem, and the approaches used in these generalizations and extensions include (a) replacing the contraction constant L by a suitable function $\varphi :{\mathbb{R}}_{+}\cup \{0\}\u27f6\mathbb{R}$ of $d(x,y)$ to obtain contractive maps [1, 2]; (b) modifying $d(x,y)$ on the $RHS$ with displacements of the forms $d(x,Tx)$ and $d(y,Tx)$ [3–6].
Recently, Wardowski [7] introduced a new type of contraction called Fcontraction in his studies of contractive maps and proved a new fixed point theorem concerning Fcontractions, for which the Banach contraction principle and some other known contractive conditions in the literature can be obtained as special cases.
Definition 1 [7]
Let $F:{\mathbb{R}}_{+}\u27f6\mathbb{R}$ be a mapping satisfying:
F_{1}. F is strictly increasing, i.e., for all $s,t\in {\mathbb{R}}_{+}$ such that $s<t$, $F(s)<F(t)$;
F_{2}. For each sequence ${\{{t}_{n}\}}_{n\in \mathbb{N}}$ of positive numbers ${lim}_{n\to \mathrm{\infty}}{t}_{n}=0$ if and only if ${lim}_{n\to \mathrm{\infty}}F({t}_{n})=\mathrm{\infty}$.
F_{3}. There exists $k\in (0,1)$ such that ${lim}_{t\to {0}^{+}}{t}^{k}F(t)=0$.
A mapping $T:X\u27f6X$ is said to be an Fcontraction if there exists $\tau >0$ such that
The aim of this article is to prove that all Fcontractions satisfy condition (2) as follows:
for some $M\ge 1$ and for all x and y in a certain subset of a closed convex and bounded subset of a Banach space where $x\ne y$. Examples of Fcontractions and some particular types of the function F, which yield various known contractive conditions, are given in [7] where it is also proved that Fcontractions have unique fixed points given by the limit of the Picard successive approximation ${lim}_{n\to \mathrm{\infty}}{T}^{n}{x}_{0}$ mentioned earlier. The main result proved by Wardowski in [7] is stated below.
Theorem 1.1 [7]
Let $(X,d)$ be a complete metric space and let $T:X\to X$ be an Fcontraction. Then T has a unique fixed point ${x}^{\ast}\in X$ and for every ${x}_{0}\in X$ a sequence ${\{{T}^{n}{x}_{0}\}}_{n\in \mathbb{N}}$ is convergent to ${x}^{\ast}$.
Examples of contractive conditions making use of the displacements $d(x,Tx)$ and $d(y,Tx)$ is the class of $(\delta ,k)$weak contractions introduced and used by Berinde [4] to obtain fixed point and uniqueness theorems for a large class of weakly Picard operators. We give the definition of weak contraction below.
Definition 2 [4]
Let X be a metric space, $\delta \in (0,1)$ and $k\ge 0$, then a mapping $T:X\u27f6X$ is called $(\delta ,k)$weak contraction (or a weak contraction) if and only if
The purpose of this article is to prove that if $T:E\u27f6E$ is an Fcontraction on a Banach space E, then T satisfies condition (2) which enables us to prove that T is a $(\delta ,k)$weak contraction. In addition we prove as an extension that the averaging operator ${S}_{\lambda}=\lambda I+(1\lambda )T$ for a nonexpansive mapping T is an example of $(\delta ,k)$weak contraction. It should be recalled that $Fix(T)=Fix({S}_{\lambda})$ for $\lambda \in [0,1)$. We give illustrative examples below to give an insight concerning property (2).
Example 1.1 Let E be real numbers ℝ, let $K\subset \mathbb{R}$ be a closed interval $[0,1]$ and define $T:K\to K$ by $Tx={e}^{x}$. Clearly, T is contractive since, by a mean value theorem, ${e}^{x}{e}^{y}={e}^{{c}_{xy}}xy$ for some constant ${c}_{xy}\in (0,1)$ for each $x,y\in K$. This yields ${e}^{x}{e}^{y}<{e}^{0}xy$ giving the desired result ${e}^{x}{e}^{y}<xy$. But T does not satisfy condition (2) as shown below: For all $x,y\in K=[0,1]$, we have $xy\le y{e}^{x}$ and, given any constant $M\ge 1$, we can find an open neighborhood of zero ${K}_{0}$ such that $\frac{y{e}^{x}}{xy}>M$ for some distinct $x,y\in {K}_{0}$. By our hypothesis, T may not have a fixed point in ${K}_{0}$ which is in concrete agreement with reality.
On the other hand, let X and K be, respectively, ℝ and $[0,1]$, while $T:K\to K$ is defined by $Tx=x{e}^{x}$. Similarly, T is contractive since, by a mean value theorem, $x{e}^{x}y{e}^{y}=(1{c}_{xy}){e}^{{c}_{xy}}xy<xy$. In this case, T satisfies our hypothesis and also has a fixed point, as shown below, in an appropriate open neighborhood ${K}_{0}$ of zero: $xy\le yx{e}^{x}$ for all $x,y\in K$, but we can find $M\ge 1$ such that $yx{e}^{x}\le Mxy$. We observe that since $xy>0$, $\frac{yx{e}^{x}}{xy}\le \frac{xy}{xy}+\frac{x(1{e}^{x})}{xy}<M$ for all x and y in an appropriate neighborhood of zero ${K}_{0}$. It is remarkable that in agreement with our hypothesis, T has a unique fixed point $0\in {K}_{0}$. We shall prove in this work that this holds for all contractive maps, i.e., all contractive operators that satisfy condition (2) have unique fixed points, by proving that it is true for nonexpansive operators.
Other examples of $(\delta ,k)$weak contractions are given in [4, 8]. It is shown in [3, 4] that a lot of wellknown contractive conditions in literature are special cases of weak contraction condition (3) as it does not require that $\delta +k$ be less than 1, which is assumed in almost all fixed point theorems based on the contractive condition which involves displacements of the forms $d(x,y)$, $d(x,Tx)$, $d(y,Ty)$, $d(x,Ty)$, $d(y,Tx)$ (see, for example, Berinde [3], Kannan [5] and Zamfirescu [6]). In [4] Berinde proved the theorem below.
Theorem 1.2 Let $(X,d)$ be a complete metric space and let $T:X\to X$ be a $(\delta ,k)$weak contraction. Then

(1)
$Fix(T)=\{x\in X:Tx=x\}\ne \mathrm{\varnothing}$.

(2)
For any ${x}_{0}\in X$, the Picard iteration ${\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}}$ given by ${x}_{n+1}=T{x}_{n}$, $n=0,1,2,\dots $ converges to some ${x}^{\ast}\in Fix(T)$.

(3)
The estimates
$$d({x}_{n},{x}^{\ast})\le \frac{{\delta}^{n}}{1\delta}d({x}_{0},{x}_{1}),\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$$(4)
hold, where δ is the constant appearing in (3).

(4)
Under the additional condition that there exist $\theta \in (0,1)$ and some ${k}_{1}\ge 0$ such that
$$d(Tx,Ty)\le \theta .d(x,y)+{k}_{1}.d(x,Tx)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X,$$(6)
the fixed point ${x}^{\ast}$ is unique and the Picard iteration converges at the rate $d({x}_{n},{x}^{\ast})\le \theta d({x}_{n1},{x}^{\ast})$, $n\in \mathbb{N}$.
In the sequel we shall make use of the following result in connection with metric and normed linear spaces.
Proposition 1.1 Let Y be a subset of a complete metric space $(X,d)$. Then $(Y,d)$ is a metric space and $(Y,d)$ is complete if and only if Y is closed in X.
2 Main results
We now present our main results. It should be mentioned here that in our extension of Wardowski result on Fcontractions we no longer guarantee the uniqueness of fixed points. Our first result is concerned with the inclusion of Fcontractions in $(\delta ,k)$weak contractions, while the second result is an extension of the Wardowski result. First, we prove the following important and supporting lemma.
Lemma 2.1 Let V be a real normed linear space and let $T:V\u27f6V$ be any map. If $\parallel xy\parallel \le \parallel yTx\parallel $, then $Tx\ne Ty$ for any distinct $x,y\in V$ satisfying $x,y\notin Fix(T)$.
Proof Given a selfmapping T of a real normed linear space V with $\parallel xy\parallel \le \parallel yTx\parallel $, where $x,y\in V$ satisfy $x\ne y$; $x,y\notin Fix(T)$, let $z={\lambda}_{0}x+(1{\lambda}_{0})Ty$ be a projection of the point y onto the line segment $[x,Ty]$ for some ${\lambda}_{0}\in [0,1]$, then $\parallel yz\parallel \le \parallel yTy\parallel $. This yields $\parallel y({\lambda}_{0}x+(1{\lambda}_{0})Ty)\parallel \le \parallel yTy\parallel $. When y, z and Ty are colinear (in particular when V is onedimensional), we have
It is clear that $Tx=Ty$ in (7) implies $x\in Fix(T)$, which is a contradiction; therefore $Tx\ne Ty$. The end of the proof. □
Theorem 2.1 Let E be a real Banach space, let K be a bounded closed and convex subset of E and $T:K\to K$. If T is an Fcontraction, then

i.
There exists an open subset ${K}_{1}\subset K$ such that T satisfies the following condition:
$$\parallel yTx\parallel \le M\parallel xy\parallel \phantom{\rule{1em}{0ex}}\mathit{\text{whenever}}\parallel xy\parallel \le \parallel yTx\parallel $$
for some $M\ge 1$ and for all $x,y\in {K}_{1}$; $x\ne y$, $x,y\notin Fix(T)$.

ii.
The Fcontraction T is a $(\delta ,k)$weak contraction.

iii.
The averaged operator ${S}_{\lambda}=\lambda I+(1\lambda )T$ is a $(\delta ,k)$weak contraction and T and ${S}_{\lambda}$ have a common unique fixed point in K for $\lambda \in (0,1)$.
Proof Given a bounded closed convex subset K of a real Banach space E. Let $T:K\u27f6K$ be an Fcontraction and let Z denote the collection of elements of K such that if $x,y\in Z$ then $\parallel xy\parallel \le \parallel yTx\parallel $, $x\ne y$, $x,y\notin Fix(T)$. We shall show that (2) is satisfied whenever and $\parallel xy\parallel \le \parallel yTx\parallel $ in an open set ${K}_{1}\subset K$ to be derived shortly. Combining these inequalities, we obtain the following:
Adding (8) and (9) yields
Set ${K}_{o}=Z\cap {\{{T}^{n}{x}_{0}\}}_{n\ge 1}$, where ${\{{T}^{n}{x}_{0}\}}_{n\ge 1}$ is a sequence of successive approximations starting from an arbitrary ${x}_{0}\in K$. Clearly, ${K}_{o}\ne \mathrm{\varnothing}$ since we can always find $n,m\in \mathbb{N}$ such that $\parallel {T}^{n}{x}_{0}{T}^{m}{x}_{0}\parallel \le \parallel {T}^{m}{x}_{0}{T}^{n+1}{x}_{0}\parallel $ for any ${x}_{0}\in K$ and by Lemma 2.1 ${T}^{m}{x}_{0}\ne {T}^{n+1}{x}_{0}$ if ${x}_{0},T{x}_{0}\notin Fix(T)$. It follows, in this case, that $\parallel yTy\parallel $ takes the form $\parallel {x}_{m}{x}_{m+1}\parallel $, while $\parallel xy\parallel \le \parallel yTx\parallel $ takes the form $\parallel {x}_{n}{x}_{m}\parallel \le \parallel {x}_{m}{x}_{n+1}\parallel $, where ${x}_{k}={T}^{k}{x}_{0}$, $k=1,2,\dots $ . Clearly, $\parallel {x}_{m}{x}_{m+1}\parallel \le \parallel {x}_{n}{x}_{m}\parallel $ since the condition $\parallel {x}_{n}{x}_{m}\parallel \le \parallel {x}_{m}{x}_{n+1}\parallel $ implies that $n\ge m$. This means that $\parallel yTy\parallel \le \parallel xy\parallel $ in ${K}_{o}$ so, in ${K}_{o}$, (10) yields $\parallel yTx\parallel \le 3\parallel xy\parallel $. Let ${K}_{1}$ be the smallest open set in K containing ${K}_{o}$. Using the continuity of T, we conclude that if T is an Fcontraction, then condition (2) is implied in ${K}_{1}$. This proves i.

ii.
By Theorem 1.1, Fcontractions have unique fixed points ${x}^{\ast}$. In this case we can take ${K}_{1}$ to be the open ball $B({x}^{\ast};r)$ centered at ${x}^{\ast}$ with radius r. Clearly, $T:B({x}^{\ast};r)\to B({x}^{\ast};r)$ for a suitable value of r. In $B({x}^{\ast};r)$ we have $\parallel TxTy\parallel <\parallel xy\parallel $ since Fcontractions are contractive operators. We shall show that this contractive condition and boundedness of $B({x}^{\ast};r)$ imply that Fcontractions are $(\delta ,k)$weak contractions. This follows from the fact that $\parallel TxTy\parallel <\delta \parallel xy\parallel +(1\delta )(\parallel yTx\parallel +\parallel yTy\parallel )\le \delta \parallel xy\parallel +(1\delta )(M+1)\parallel yTx\parallel $.
The proof of iii. is an application of Theorem 1.2 and Proposition 1.1. Recall that Proposition 1.1 asserts that $(K,\parallel \cdot \parallel )$ is a complete metric space since E is a Banach space and K is a closed subset of E. It suffices to show that the averaging operator ${S}_{\lambda}$ (for a contractive map T) is a $(\delta ,k)$weak contraction on K; the existence of a fixed point in K follows via Theorem 1.2.
Next, let ${S}_{\lambda}$, $\lambda \in (0,1)$ denote an averaging operator, i.e., ${S}_{\lambda}=\lambda I+(1\lambda )T$, we obtain
When $\parallel yTx\parallel \le \parallel xy\parallel $, then (11) yields $\parallel {S}_{\lambda}x{S}_{\lambda}y\parallel \le 2(1\lambda )\parallel xy\parallel +\parallel y{S}_{\lambda}x\parallel $ and we are done. It is easy to show, in ${K}_{1}$ derived above, that $\parallel {x}_{n}{x}_{m}\parallel \u27f60$ as $n,m\u27f6\mathrm{\infty}$. But in ${K}_{1}$, when $\parallel xy\parallel \le \parallel yTx\parallel $, then by (2) equation (11) yields $4(1\lambda )\parallel xy\parallel +\parallel y{S}_{\lambda}x\parallel $. So, choosing $\lambda \in (0,1)$ such that $(1\lambda )<min\{\frac{1}{2},\frac{1}{4}\}=\frac{1}{4}$, we conclude that ${S}_{\lambda}$ is a $(\delta ,k)$weak contraction with $\delta \ge \frac{1\lambda}{4}$ and $k=1$, where λ satisfies $4(1\lambda )<1$. Therefore, by Theorem 1.2 T has a fixed point in K, and successive approximations for ${S}_{\lambda}$ converge to a fixed point of T in K. The end of the proof. □
From the above method of the proof of Theorem 2.1, we obtain a fixed point theorem stated below for contractive mappings satisfying condition (2).
Theorem 2.2 Let E be a real Banach space, let K be a bounded closed and convex subset of E and $T:K\to K$. If T is a nonexpanive map such that there exist a constant $M\ge 1$ and an open subset ${K}_{1}\subset K$ such that T satisfies the following condition:
for all $x,y\in {K}_{1}$; with $x\ne y$, $x,y\notin Fix(T)$. Then the averaged operator ${S}_{\lambda}=\lambda I+(1\lambda )T$ is a $(\delta ,k)$weak contraction and T has a fixed point in K.
Remark 2.1 We conclude with the following observations:

1.
The uniqueness of fixed points of nonexpansive maps (whenever they exist) is guaranteed if condition (6) in Theorem 1.2 is satisfied for a corresponding averaged operator ${S}_{\lambda}$.

2.
Condition (2), Lemma 2.1 and Theorem 2.1 together with other properties of $(\delta ,k)$weak contractions constitute the properties of Fcontractions and the properties of nonexpansive operators with a nonempty fixed point set $Fix(T)$.

3.
The bounded closed convex subset K can be replaced with a closed convex subset K, with emphasis on the boundedness of the open set ${K}_{1}$.
References
 1.
Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S00029939196902395599
 2.
Geraghty MA: On contractive mappings. Proc. Am. Math. Soc. 1973, 40(2):604–608. 10.1090/S00029939197303341765
 3.
Berinde V: On the approximation of fixed point of weak contraction mappings. Carpath. J. Math. 2003, 19(1):7–22.
 4.
Berinde V: Approximating fixed point of weak contractions using Picard’s iteration. Nonlinear Anal. Forum 2004, 9(1):43–53.
 5.
Kannan R: Some result on fixed points. Bull. Calcutta Math. Soc. 1968, 10: 71–76.
 6.
Zamfirescu T: Fixed point theorem in metric spaces. Arch. Math. 1992, 23: 292–298.
 7.
Wardowski D: Fixed points of a new contractive type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 94
 8.
Berinde M, Berinde V: On a general class of multivalued weakly Picard mappings. J. Math. Anal. Appl. 2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016
Acknowledgements
The author is delighted to acknowledge the comments from the reviewers which have enhanced deeper insight. Also acknowledged is the support from the Springer Open editorial team for all their facilities and friendliness. Thank you all.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Udoutun, X. On inclusion of Fcontractions in $(\delta ,k)$weak contractions. Fixed Point Theory Appl 2014, 65 (2014). https://doi.org/10.1186/16871812201465
Received:
Accepted:
Published:
Keywords
 $(\delta ,k)$weak contraction
 Fcontractions
 fixed points
 Kranoselskii iteration
 LLipschitzian maps