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A fixed point theorem for generalized contractions involving wdistances on complete quasimetric spaces
Fixed Point Theory and Applications volume 2014, Article number: 40 (2014)
Abstract
We obtain a fixed point theorem for generalized contractions on complete quasimetric spaces, which involves wdistances and functions of MeirKeeler and Jachymski type. Our result generalizes in various directions the celebrated fixed point theorems of Boyd and Wong, and Matkowski. Some illustrative examples are also given.
MSC:47H10, 54H25, 54E50.
1 Introduction and preliminaries
In their celebrated paper [1], Kada, Suzuki and Takahashi introduced and studied the notion of a wdistance on a metric space. By using that notion they obtained, among other results, generalizations of the nonconvex minimization theorem of Takahashi [2], of Caristi’s fixed point theorem [3] and of Ekeland’s variational principle [4], as well as a general fixed point theorem that improves fixed point theorems of Subrahmanyam [5], Kannan [6] and Ćirić [7]. This study was continued by Suzuki and Takahashi [8], and by Park [9] who extended several results from [1] to quasimetric spaces. Park’s approach was successful continued by AlHomidan, Ansari and Yao [10], who obtained, among other interesting results, quasimetric versions of CaristiKirk’s fixed point theorem and Nadler’s fixed point theorem by using Qfunctions (a slight generalization of wdistances). More recently, Latif and AlMezel [11], and Marín et al. [12–14] have proved some fixed point theorems both for singlevalued and multivalued mappings in complete quasimetric spaces and preordered quasimetric spaces by using Qfunctions and wdistances, and generalizing in this way wellknown fixed point theorems of Mizoguchi and Takahashi [15], Bianchini and Grandolfi [16], and Boyd and Wong [17], respectively.
In this paper we shall obtain a fixed point theorem for generalized contractions with respect to wdistances on complete quasimetric spaces from which we deduce wdistance versions of Boyd and Wong’s fixed point theorem [17] and of Matkowski’s fixed point theorem [18]. Our approach uses a kind of functions considered by Jachymski in [[19], Corollary of Theorem 2] and that generalizes the notion of a function of MeirKeeler type.
In the sequel the letters ${\mathbb{R}}^{+}$, ℕ and ω will denote the set of nonnegative real numbers, the set of positive integer numbers and the set of nonnegative integer numbers, respectively.
By a quasimetric on a set X we mean a function $d:X\times X\to {\mathbb{R}}^{+}$ such that for all $x,y,z\in X$:

(i)
$d(x,y)=d(y,x)=0\iff x=y$, and

(ii)
$d(x,y)\le d(x,z)+d(z,y)$.
A quasimetric space is a pair $(X,d)$ such that X is a set and d is a quasimetric on X.
Each quasimetric d on a set X induces a topology ${\tau}_{d}$ on X which has as a base the family of open balls $\{{B}_{d}(x,r):x\in X,\epsilon >0\}$, where ${B}_{d}(x,\epsilon )=\{y\in X:d(x,y)<\epsilon \}$ for all $x\in X$ and $\epsilon >0$.
Given a quasimetric d on X, the function ${d}^{1}$ defined by ${d}^{1}(x,y)=d(y,x)$ for all $x,y\in X$, is also a quasimetric on X, and the function ${d}^{s}$ defined by ${d}^{s}(x,y)=max\{d(x,y),d(y,x)\}$ for all $x,y\in X$, is a metric on X.
There exist several different notions of Cauchy sequence and of complete quasimetric space in the literature (see e.g. [20]). In this paper we shall use the following general notion.
A quasimetric space $(X,d)$ is called complete if every Cauchy sequence ${({x}_{n})}_{n\in \omega}$ in the metric space $(X,{d}^{s})$ converges with respect to the topology ${\tau}_{{d}^{1}}$ (i.e., there exists $z\in X$ such that $d({x}_{n},z)\to 0$).
A wdistance on a quasimetric space $(X,d)$ is a function $q:X\times X\to {\mathbb{R}}^{+}$ satisfying the following three conditions:
(W1) $q(x,y)\le q(x,z)+q(z,y)$ for all $x,y,z\in X$;
(W2) $q(x,\cdot ):X\to {\mathbb{R}}^{+}$ is lower semicontinuous on $(X,{\tau}_{{d}^{1}})$ for all $x\in X$;
(W3) for each $\epsilon >0$ there exists $\delta >0$ such that $q(x,y)\le \delta $ and $q(x,z)\le \delta $ imply $d(y,z)\le \epsilon $.
Several examples of wdistances on quasimetric spaces may be found in [9–12].
Note that if d is a metric on X then it is a wdistance on $(X,d)$. Unfortunately, this does not hold for quasimetric spaces, in general. Indeed, in [[12], Lemma 2.2] there was observed the following.
Lemma 1 If q is a wdistance on a quasimetric space $(X,d)$, then for each $\epsilon >0$ there exists $\delta >0$ such that $q(x,y)\le \delta $ and $q(x,z)\le \delta $ imply ${d}^{s}(y,z)\le \epsilon $.
It follows from Lemma 1 (see [[12], Proposition 2.3]) that if a quasimetric d on X is also a wdistance on $(X,d)$, then the topologies induced by d and by the metric ${d}^{s}$ coincide, so $(X,{\tau}_{d})$ is a metrizable topological space.
2 Results and examples
Meir and Keeler proved in [21] that if f is a selfmap of a complete metric space $(X,d)$ satisfying the condition that for each $\epsilon >0$ there is $\delta >0$ such that, for any $x,y\in X$, with $\epsilon \le d(x,y)<\epsilon +\delta $ we have $d(fx,fy)<\epsilon $, then f has a unique fixed point $z\in X$ and ${f}^{n}x\to z$ for all $x\in X$.
This wellknown result suggests the notion of a MeirKeeler function:
A function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is said to be a MeirKeeler function if $\varphi (0)=0$, and satisfies the following condition:
(MK) For each $\epsilon >0$ there exists $\delta >0$ such that
Remark 1 It is obvious that if ϕ is a MeirKeeler function then $\varphi (t)<t$ for all $t>0$.
Later on, Jachymski proved in [19] the following interesting result and showed that both Boyd and Wong’s fixed point theorem and Matkowski’s fixed point theorem are easy consequences of it.
Theorem 1 ([[19], Corollary of Theorem 2])
Let f be a selfmap of a complete metric space $(X,d)$ such that $d(fx,fy)<d(x,y)$ for $x\ne y$, and $d(fx,fy)\le \varphi (d(x,y))$ for all $x,y\in X$, where $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfies the condition
(Ja) for each $\epsilon >0$ there exists $\delta >0$ such that for any $t\in {\mathbb{R}}^{+}$,
Then f has a unique fixed point $z\in X$ and ${f}^{n}x\to z$ for all $x\in X$.
Theorem 1 suggests the following notion:
A function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is said to be a Jachymski function if $\varphi (0)=0$ and it satisfies condition (Ja) of Theorem 1.
Remark 2 Obviously, each MeirKeeler function is a Jachymski function. However, the converse does not follow even in the case that $\varphi (t)<t$ for all $t>0$: Indeed, let $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ defined as $\varphi (t)=0$ for all $t\in [0,1]$ and $\varphi (t)=1$ otherwise. Clearly ϕ is a Jachymski function such that $\varphi (t)<t$ for all $t>0$. Finally, for $\epsilon =1$ and any $\delta >0$ we have $\varphi (\epsilon +\delta /2)=\epsilon $, so ϕ is not a MeirKeeler function.
Now we establish the main result of this paper.
Theorem 2 Let f be a selfmap of a complete quasimetric space $(X,d)$. If there exist a wdistance q on $(X,d)$ and a Jachymski function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (t)<t$ for all $t>0$, and
for all $x,y\in X$, then f has a unique fixed point $z\in X$. Moreover $q(z,z)=0$.
Proof Fix ${x}_{0}\in X$. For each $n\in \omega $ let ${x}_{n}={f}^{n}{x}_{0}$. Then
for all $n\in \omega $.
First, we shall prove that ${\{{x}_{n}\}}_{n\in \omega}$ is a Cauchy sequence in $(X,{d}^{s})$.
To this end put ${r}_{n}=q({x}_{n},{x}_{n+1})$ for all $n\in \omega $.
If there is ${n}_{0}\in \omega $ such that ${r}_{{n}_{0}}=0$, then ${r}_{n}=0$ for all $n\ge {n}_{0}$ by (2) and our assumption that $\varphi (0)=0$. Therefore $q({x}_{n},{x}_{m})=0$ whenever $m>n\ge {n}_{0}$ by condition (W1), and consequently, ${d}^{s}({x}_{n},{x}_{m})=0$ by Lemma 1. Thus ${x}_{n}={x}_{{n}_{0}+1}$ for all $n\ge {n}_{0}+1$.
Otherwise, we assume, without loss of generality, that ${r}_{n+1}<{r}_{n}$ for all $n\in \omega $. Then ${\{{r}_{n}\}}_{n\in \omega}$ converges to some $r\in {\mathbb{R}}^{+}$. Of course, $r<{r}_{n}$ for all $n\in \omega $.
If $r>0$ there exists $\delta =\delta (r)$ such that
Take ${n}_{\delta}\in \mathbb{N}$ such that ${r}_{n}<r+\delta $ for all $n\ge {n}_{\delta}$. Therefore $\varphi ({r}_{n})\le r$, so by condition (2), ${r}_{n+1}\le r$ for all $n\ge {n}_{\delta}$, a contradiction. Consequently $r=0$.
Now choose an arbitrary $\epsilon >0$. There exists $\delta =\delta (\epsilon )$, with $\delta \in (0,\epsilon )$, for which conditions (W3) and (Ja) hold. Similarly, for $\delta /2$ there exists $\mu =\mu (\delta /2)$, with $\mu \in (0,\delta /2)$ for which conditions (W3) and (Ja) also hold, i.e.,
$q(x,y)\le \mu $ and $q(x,z)\le \mu $, imply $d(y,z)\le \delta /2$, and for any $t>0$, $\delta /2<t<\delta /2+\mu $ implies $\varphi (t)\le \delta /2$.
Since ${r}_{n}\to 0$, there exists ${k}_{0}\in \mathbb{N}$ such that ${r}_{n}<\mu $ for all $n\ge {k}_{0}$.
By using a similar technique to the one given by Jachymski in [[19], Theorem 2] we shall prove, by induction, that for each $k\ge {k}_{0}$ and each $n\in \mathbb{N}$, we have
Indeed, fix $k\ge {k}_{0}$. Since $q({x}_{k},{x}_{k+1})<\mu $, condition (3) follows for $n=1$.
Assume that (3) holds for some $n\in \mathbb{N}$. We shall distinguish two cases.

Case 1: $q({x}_{k},{x}_{n+k})>\delta /2$. Then we deduce from the induction hypothesis and condition (Ja) that
$$\varphi (q({x}_{k},{x}_{n+k}))\le \delta /2,$$
so by (1), $q({x}_{k+1},{x}_{n+k+1})\le \delta /2$. Therefore

Case 2: $q({x}_{k},{x}_{n+k})\le \delta /2$.
If $q({x}_{k},{x}_{n+k})=0$, we deduce that $q({x}_{k+1},{x}_{n+k+1})=0$ by (1). So, by (W1),
If $q({x}_{k},{x}_{n+k})>0$, we deduce that $\varphi (q({x}_{k},{x}_{n+k}))<q({x}_{k},{x}_{n+k})\le \delta /2$, so
Now take $i,j\in \mathbb{N}$ with $i,j>k$. Then $i=n+k$ and $j=m+k$ for some $n,m\in \mathbb{N}$. Hence, by (3),
Now, from Lemma 1 it follows that ${d}^{s}({x}_{i},{x}_{j})\le \epsilon $ whenever $i,j>k$. We conclude that ${\{{x}_{n}\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $(X,{d}^{s})$.
Since $(X,d)$ is complete, there exists $z\in X$ such that $d({x}_{n},z)\to 0$.
Next we show that $q({x}_{n},z)\to 0$: Indeed, choose an arbitrary $\epsilon >0$. We have proved (see (3)) that there is ${k}_{0}\in \mathbb{N}$ such that $q({x}_{k},{x}_{n+k})<\epsilon $ for all $k\ge {k}_{0}$ and $n\in \mathbb{N}$. Fix $k\ge {k}_{0}$. Since $d({x}_{n},z)\to 0$ it follows from condition (W2) that, for n sufficiently large,
Hence $q({x}_{k},z)<2\epsilon $ for all $k\ge {k}_{0}$. We deduce that $q({x}_{n},z)\to 0$.
From (1) it follows that $q({x}_{n+1},fz)\to 0$. So ${d}^{s}(z,fz)=0$ by Lemma 1. Consequently $z=fz$, i.e., is a fixed point of f. Furthermore $q(z,z)=0$. In fact, otherwise we have
a contradiction.
Finally, let $u\in X$ such that $u=fu$ and $u\ne z$. If $q(u,z)>0$ we deduce that
a contradiction. So $q(u,z)=0$. Similarly we check that $q(u,u)=0$. Since $q(z,z)=0$, we deduce from Lemma 1 that ${d}^{s}(u,z)=0$, i.e., $u=z$. We conclude that z is the unique fixed point of f. □
Corollary 1 Let f be a selfmap of a complete metric space $(X,d)$. If there exist a wdistance q on $(X,d)$ and a Jachymski function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (t)<t$ for all $t>0$, and
for all $x,y\in X$, then f has a unique fixed point $z\in X$. Moreover $q(z,z)=0$.
Corollary 2 Let f be a selfmap of a complete quasimetric space $(X,d)$. If there exist a wdistance q on $(X,d)$ and a MeirKeeler function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that
for all $x,y\in X$, then f has a unique fixed point $z\in X$. Moreover $q(z,z)=0$.
Proof Apply Remarks 1 and 2, and Theorem 2. □
Corollary 3 [13]
Let f be a selfmap of a complete quasimetric space $(X,d)$. If there exist a wdistance q on $(X,d)$ and a right upper semicontinuous function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (0)=0$, $\varphi (t)<t$ for all $t>0$, and
for all $x,y\in X$, then f has a unique fixed point $z\in X$. Moreover $q(z,z)=0$.
Proof It suffices to show that ϕ is a MeirKeeler function. Assume the contrary. Then there exist $\epsilon >0$ and a sequence ${\{{t}_{n}\}}_{n\in \mathbb{N}}$ of positive real numbers such that $\epsilon \le {t}_{n}<\epsilon +1/n$ but $\varphi ({t}_{n})\ge \epsilon $ for all $n\in \mathbb{N}$. Since $\epsilon \varphi (\epsilon )>0$, it follows from right upper semicontinuity of ϕ that $\varphi ({t}_{n})\varphi (\epsilon )<\epsilon \varphi (\epsilon )$ eventually, i.e., $\varphi ({t}_{n})<\epsilon $, a contradiction. We conclude that f has a unique fixed point by Corollary 2. □
Corollary 4 Let f be a selfmap of a complete quasimetric space $(X,d)$. If there exist a wdistance q on $(X,d)$ and a nondecreasing function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (0)=0$, ${\varphi}^{n}(t)\to 0$ for all $t>0$, and
for all $x,y\in X$, then f has a unique fixed point $z\in X$. Moreover $q(z,z)=0$.
Proof Again it suffices to show that ϕ is a MeirKeeler function. Assume the contrary. Then there exist $\epsilon >0$ and a sequence ${\{{t}_{n}\}}_{n\in \mathbb{N}}$ of positive real numbers such that $\epsilon \le {t}_{n}<\epsilon +1/n$ but $\varphi ({t}_{n})\ge \epsilon $ for all $n\in \mathbb{N}$. Since ϕ is nondecreasing we deduce that $\varphi (t)\ge \epsilon $ whenever $t\ge \epsilon $. Hence ${\varphi}^{n}(t)\ge \epsilon $ whenever $t\ge \epsilon $, which contradicts the hypothesis that ${\varphi}^{n}(t)\to 0$ for all $t>0$. We conclude that f has a unique fixed point by Corollary 2. □
Remark 3 In [22] the authors proved Corollary 2 for the case that $(X,d)$ is a complete metric space. Note also that Boyd and Wong’s fixed point theorem [17] and Matkowski’s fixed point theorem [18] are special cases of Corollaries 3 and 4, respectively, when $(X,d)$ is a complete metric space and q is the metric d.
We conclude the paper with some examples that illustrate and validate the obtained results.
The first example shows that condition ‘$\varphi (t)<t$ for all $t>0$’ in Theorem 2 cannot be omitted.
Example 1 Let $X=\{0,1\}$ and let d be the discrete metric on X, i.e., $d(x,x)=0$ for all $x\in X$ and $d(x,y)=1$ whenever $x\ne y$. Let $f:X\to X$ defined as $f0=1$ and $f1=0$, and $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ defined as $\varphi (1)=1$ and $\varphi (t)=0$ for all $x\in {\mathbb{R}}^{+}\mathrm{\setminus}\{1\}$. It is clear that ϕ is a Jachysmki function such that
for all $x,y\in X$. However, f has no fixed point.
The next is an example where we can apply Theorem 2 for an appropriate wdistance q on a complete quasimetric space $(X,d)$ but not for d. Moreover, Corollary 1 cannot be applied for any wdistance on the metric space $(X,{d}^{s})$.
Example 2 Let $X=\omega $ and let d be the quasimetric on X defined as
Clearly $(X,d)$ is complete (observe that ${\{n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $(X,{d}^{s})$ with $d(n,0)\to 0$).
Let q be the wdistance on $(X,d)$ given by $q(x,y)=y$ for all $x,y\in X$.
Now define $f:X\to X$ as $f0=0$ and $fn=n1$ for all $n\in \mathbb{N}$, and $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ as $\varphi (0)=0$ and $\varphi (t)=n1$ where $t\in (n1,n]$, $n\in \mathbb{N}$.
It is routine to check that ϕ is a Jachymski function satisfying $\varphi (t)<t$ for all $t>0$ (in fact, it is a MeirKeeler function).
Since $q(fx,f0)=0$ for all $x\in X$, and for each $n,m\in X$ with $m\ne 0$, we have
it follows that all conditions of Theorem 2 are satisfied. In fact $z=0$ is the unique fixed point of f.
However, the contraction condition (1) is not satisfied for d. Indeed, for any $n>1$ we have
Finally, note that we cannot apply Corollary 1 because $(X,{d}^{s})$ is not complete (observe that ${\{n\}}_{n\in \mathbb{N}}$ is a Cauchy sequence in $(X,{d}^{s})$ that does not converge in $(X,{d}^{s})$).
We conclude with an example where we can apply Corollary 2 but not Corollaries 3 and 4.
Example 3 Let d be the quasimetric on ${\mathbb{R}}^{+}$ given by $d(x,y)=max\{yx,0\}$ for all $x,y\in {\mathbb{R}}^{+}$. Since ${d}^{s}$ is the usual metric on ${\mathbb{R}}^{+}$ it immediately follows that $({\mathbb{R}}^{+},d)$ is complete.
Define $q:{\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ as $q(x,y)=y$. It is clear that q is a wdistance on $({\mathbb{R}}^{+},d)$.
Now let $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, defined by $\varphi (t)=t/2$ if $t\in (1,2]$, and $\varphi (t)=0$ otherwise.
Then ϕ is a MeirKeeler function: Indeed, we first note that $\varphi (0)=0$. Now, given $\epsilon >0$ we distinguish the following cases:

(1)
if $0<\epsilon <1$, we take $\delta =1\epsilon $, and thus, from $\epsilon \le t<\epsilon +\delta =1$, it follows $\varphi (t)=0<\epsilon $;

(2)
if $\epsilon =1$, we take $\delta =1/2$, and thus, from $1<t<3/2$, it follows $\varphi (t)=t/2<3/4<\epsilon $, whereas $\varphi (1)=0<\epsilon $;

(3)
if $1<\epsilon <2$, we take $\delta =2\epsilon $, and thus, from $\epsilon \le t<\epsilon +\delta =2$, it follows $\varphi (t)=t/2<1<\epsilon $;

(4)
if $\epsilon \ge 2$, we fix $\delta >0$, and thus, from $\epsilon \le t<\epsilon +\delta $, it follows $\varphi (t)<\epsilon $ because $\varphi (2)=1$ and $\varphi (t)=0$ for $t>2$.
Finally, taking $f=\varphi $, we obtain $q(fx,fy)\le \varphi (q(x,y))$ for all $x,y\in X$, because
Therefore, all conditions of Corollary 2 are satisfied. In fact, $z=0$ is the unique fixed point of f.
However, ϕ is not right upper semicontinuous at $t=1$, so we cannot apply Corollary 3.
Similarly, we cannot apply Corollary 4 because ϕ is not a nondecreasing function.
Observe also that the wdistance q cannot be replaced by the quasimetric d because for $1<y\le 2$ we have
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Acknowledgements
The authors are grateful to the referees for several useful suggestions. They also thank the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM201237894C0201.
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Alegre, C., Marín, J. & Romaguera, S. A fixed point theorem for generalized contractions involving wdistances on complete quasimetric spaces. Fixed Point Theory Appl 2014, 40 (2014). https://doi.org/10.1186/16871812201440
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Keywords
 fixed point
 generalized contraction
 wdistance
 complete quasimetric space