A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces

Abstract

We obtain a fixed point theorem for generalized contractions on complete quasi-metric spaces, which involves w-distances and functions of Meir-Keeler 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 w-distance 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 quasi-metric spaces. Park’s approach was successful continued by Al-Homidan, Ansari and Yao [10], who obtained, among other interesting results, quasi-metric versions of Caristi-Kirk’s fixed point theorem and Nadler’s fixed point theorem by using Q-functions (a slight generalization of w-distances). More recently, Latif and Al-Mezel [11], and Marín et al. [12–14] have proved some fixed point theorems both for single-valued and multi-valued mappings in complete quasi-metric spaces and preordered quasi-metric spaces by using Q-functions and w-distances, and generalizing in this way well-known 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 w-distances on complete quasi-metric spaces from which we deduce w-distance 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 Meir-Keeler type.

In the sequel the letters ${\mathbb{R}}^{+}$, ℕ and ω will denote the set of non-negative real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively.

By a quasi-metric 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 quasi-metric space is a pair $(X,d)$ such that X is a set and d is a quasi-metric on X.

Each quasi-metric 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 quasi-metric 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 quasi-metric 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 quasi-metric space in the literature (see e.g. [20]). In this paper we shall use the following general notion.

A quasi-metric 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 $d(x_n,z)\to 0$).

Definition 1 ([9, 10])

A w-distance on a quasi-metric 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 w-distances on quasi-metric spaces may be found in [9–12].

Note that if d is a metric on X then it is a w-distance on $(X,d)$. Unfortunately, this does not hold for quasi-metric spaces, in general. Indeed, in [[12], Lemma 2.2] there was observed the following.

Lemma 1 If q is a w-distance on a quasi-metric 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 quasi-metric d on X is also a w-distance 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 self-map 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 well-known result suggests the notion of a Meir-Keeler function:

A function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is said to be a Meir-Keeler function if $\varphi (0)=0$, and satisfies the following condition:

(MK) For each $\epsilon >0$ there exists $\delta >0$ such that

$$\epsilon \le t<\epsilon +\delta \text{implies}\varphi (t)<\epsilon ,\text{for all }t\in {\mathbb{R}}^{+}.$$

Remark 1 It is obvious that if ϕ is a Meir-Keeler 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 self-map 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 $t\in {\mathbb{R}}^{+}$,

$$\epsilon <t<\epsilon +\delta \mathit{\text{implies}}\varphi (t)\le \epsilon .$$

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 Meir-Keeler 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 Meir-Keeler function.

Now we establish the main result of this paper.

Theorem 2 Let f be a self-map of a complete quasi-metric space $(X,d)$. If there exist a w-distance q on $(X,d)$ and a Jachymski function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (t)<t$ for all $t>0$, and

$$q(fx,fy)\le \varphi (q(x,y)),$$

(1)

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

$$q(x_{n+1},x_{n+2})\le \varphi (q(x_n,x_{n+1})),$$

(2)

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

$$r<t<r+\delta \Rightarrow \varphi (t)\le r.$$

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

$$q(x_k,x_{n+k})<\frac{\delta }{2}+\mu .$$

(3)

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

$$q(x_k,x_{n+k+1})\le q(x_k,x_{k+1})+q(x_{k+1},x_{n+k+1})<\mu +\frac{\delta }{2}.$$

• 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),

$$q(x_k,x_{n+k+1})\le q(x_k,x_{k+1})<\mu <\mu +\frac{\delta }{2}.$$

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

$$\begin{array}{rcl}q(x_k,x_{n+k+1})& \le & q(x_k,x_{k+1})+q(x_{k+1},x_{n+k+1})\\ \le & q(x_k,x_{k+1})+\varphi (q(x_k,x_{n+k}))<\mu +\frac{\delta }{2}.\end{array}$$

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),

$$q(x_k,x_i)=q(x_k,x_{n+k})<\frac{\delta }{2}+\mu <\delta \text{and}q(x_k,x_j)=q(x_k,x_{m+k})<\frac{\delta }{2}+\mu <\delta .$$

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,

$$q(x_k,z)<q(x_k,x_{n+k})+\epsilon .$$

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

$$q(z,z)=q(fz,fz)\le \varphi (q(z,z))<q(z,z),$$

a contradiction.

Finally, let $u\in X$ such that $u=fu$ and $u\ne z$. If $q(u,z)>0$ we deduce that

$$q(u,z)=q(fu,fz)\le \varphi (q(u,z))<q(u,z),$$

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 self-map of a complete metric space $(X,d)$. If there exist a w-distance q on $(X,d)$ and a Jachymski function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (t)<t$ for all $t>0$, and

$$q(fx,fy)\le \varphi (q(x,y)),$$

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 self-map of a complete quasi-metric space $(X,d)$. If there exist a w-distance q on $(X,d)$ and a Meir-Keeler function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that

$$q(fx,fy)\le \varphi (q(x,y)),$$

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 self-map of a complete quasi-metric space $(X,d)$. If there exist a w-distance 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

$$q(fx,fy)\le \varphi (q(x,y)),$$

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 Meir-Keeler 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 self-map of a complete quasi-metric space $(X,d)$. If there exist a w-distance q on $(X,d)$ and a non-decreasing function $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $\varphi (0)=0$, ${\varphi }^n(t)\to 0$ for all $t>0$, and

$$q(fx,fy)\le \varphi (q(x,y)),$$

(4)

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 Meir-Keeler 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 non-decreasing 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

$$d(fx,fy)\le \varphi (d(x,y)),$$

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 w-distance q on a complete quasi-metric space $(X,d)$ but not for d. Moreover, Corollary 1 cannot be applied for any w-distance on the metric space $(X,d^s)$.

Example 2 Let $X=\omega$ and let d be the quasi-metric on X defined as

$$\begin{array}{c}d(x,x)=0\text{for all }x\in X;\hfill \\ d(n,0)=1/n\text{for all }n\in \mathbb{N};\hfill \\ d(0,n)=1\text{for all }n\in \mathbb{N};\hfill \\ d(n,m)=|1/n-1/m|\text{for all }n,m\in \mathbb{N}.\hfill \end{array}$$

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 w-distance 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 $fn=n-1$ for all $n\in \mathbb{N}$, and $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ as $\varphi (0)=0$ and $\varphi (t)=n-1$ where $t\in (n-1,n]$, $n\in \mathbb{N}$.

It is routine to check that ϕ is a Jachymski function satisfying $\varphi (t)<t$ for all $t>0$ (in fact, it is a Meir-Keeler function).

Since $q(fx,f0)=0$ for all $x\in X$, and for each $n,m\in X$ with $m\ne 0$, we have

$$q(fn,fm)=fm=m-1=\varphi (m)=\varphi (q(n,m)),$$

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

$$d(f0,fn)=d(0,n-1)=1>0=\varphi (1)=\varphi (d(0,n)).$$

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 quasi-metric on ${\mathbb{R}}^{+}$ given by $d(x,y)=max\{y-x,0\}$ 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 w-distance 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 Meir-Keeler 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

$$q(fx,fy)=fy=\varphi (y)=\varphi (q(x,y)).$$

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 non-decreasing function.

Observe also that the w-distance q cannot be replaced by the quasi-metric d because for $1<y\le 2$ we have

$$d(f1,fy)=d(0,\frac{y}{2})=\frac{y}{2}>0=\varphi (y-1)=\varphi (d(1,y)).$$