Metric-like spaces, partial metric spaces and fixed points

There exist many generalizations of the concept of metric spaces in the literature. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. After that, fixed point results in partial metric spaces were studied by many other authors [2–11]. In this paper, we first introduce a new generalization of a partial metric space which is called a metric-like space. Then, we give some fixed point results in such spaces. Our fixed point theorems, even in the case of partial metric spaces, generalize and improve some well-known results in the literature.

In the rest of this section, we recall some definitions and facts which will be used throughout the paper.

Definition 1.1 A mapping $p:X\times X\to R^{+}$, where X is a nonempty set, is said to be a partial metric on X if for any $x,y,z\in X$, the following four conditions hold true:

(P1) $x=y$ if and only if $p(x,x)=p(y,y)=p(x,y)$;

(P2) $p(x,x)\le p(x,y)$;

(P3) $p(x,y)=p(y,x)$;

(P4) $p(x,z)\le p(x,y)+p(y,z)-p(y,y)$.

A basic example of a partial metric space is the pair $({\mathbb{R}}^{+},p)$, where $p(x,y)=max\{x,y\}$ for all $x,y\in {\mathbb{R}}^{+}$. For some other examples of partial metric spaces see [1–11] and references therein.

We first introduce the concept of a metric-like space.

Definition 2.1 A mapping $\sigma :X\times X\to R^{+}$, where X is a nonempty set, is said to be metric-like on X if for any $x,y,z\in X$, the following three conditions hold true:

(σ 1) $\sigma (x,y)=0\Rightarrow x=y$;

(σ 2) $\sigma (x,y)=\sigma (y,x)$;

(σ 3) $\sigma (x,z)\le \sigma (x,y)+\sigma (y,z)$.

Then a sequence $\{x_n\}$ in the metric-like space $(X,\sigma )$ converges to a point $x\in X$ if and only if ${lim}_{n\to \mathrm{\infty }}\sigma (x_n,x)=\sigma (x,x)$.

Every partial metric space is a metric-like space. Below we give another example of a metric-like space.

Then $(X,\sigma )$ is a metric-like space, but since $\sigma (0,0)\nleqq \sigma (0,1)$, then $(X,\sigma )$ is not a partial metric space.

Remark 2.3 Let $X=\{0,1\}$, let $\sigma (x,y)=1$ for each $x,y\in X$, and let $x_n=1$ for each $n\in \mathbb{N}$. Then it is easy to see that $x_n\to 0$ and $x_n\to 1$, and so in metric-like spaces the limit of a convergent sequence is not necessarily unique.

Some slight modifications of the proof of Theorem 2.1 in [12] yield the following result which is a generalization of the well-known fixed point theorem of Ćirić [13].

Then T has a fixed point.

From our assumption, we have

where

and so $\sigma (T{x}_n,Tu)\to \sigma {(Tu,u)}^{+}$. Then from (5) and our assumptions on ψ, we get $\sigma (Tu,u)<\sigma (Tu,u)$, a contradiction. Thus, $\sigma (Tu,u)=0$ and so $Tu=u$. □

Example 2.5 Let ${\psi }_1(t)=kt$ for each $t\in [0,\mathrm{\infty })$, where $k\in [0,1)$, and let ${\psi }_2(t)=t-ln(1+t)$ for each $t\in [0,\mathrm{\infty })$. Then ${\psi }_1$ and ${\psi }_2$ satisfy the conditions of Theorem 2.4.

Now we illustrate our previous result by the following example.

for each $x,y\in X$. Then all the required hypotheses of Theorem 2.4 are satisfied. Then T has a unique fixed point.

for all $x,y\in X$, where $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a nondecreasing continuous function such that $\phi (t)=0$ if and only if $t=0$. Then T has a unique fixed point.

which gives $\phi (\sigma (u,v))=0$ and so $u=v$. □

Then T satisfies the hypothesis of Theorem 2.7 and so T has a fixed point ($x=0$ is the unique fixed point of T). Now since ${lim}_{t\to \mathrm{\infty }}\phi (t)=1<\mathrm{\infty }$, we cannot invoke Theorem 2.1 of [9] to show the existence of fixed point of T.

The following corollary improves Theorem 1 in [2].

Then T has a unique fixed point.

Thus, $p(x,y)=0$ and $x=y$. □

The following corollary improves Corollary 1 and Theorem 2 in [2] and the main fixed point result of Matthews [1].

for all $x,y\in X$, where $\lambda \in [0,1)$. Then T has a unique fixed point.

Proof Let $\psi (t)=\lambda t$ for each $t\in [0,\mathrm{\infty })$ and apply Corollary 2.9. □

Now, we present the following version of Rakotch’s fixed point theorem [14] in metric-like spaces.

for each $x,y\in X$ with $x\ne y$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is nonincreasing. Then T has a unique fixed point.

and so $\sigma (x_0,T{x}_0)=0$, that is, $T{x}_0=x_0$. The uniqueness easily follows from our contractive condition on T. □

The following corollary is another new extension of Matthews’s fixed point result [1].

for each $x,y\in X$ with $x\ne y$, where $\alpha :[0,\mathrm{\infty })\to [0,1)$ is nonincreasing. Then T has a unique fixed point.