In the following, we use the same definitions, notations and structures given in [1]. We start first with Caristi’s [2] fixed point theorem.

Theorem 1.1 [2]

Let $\left(X,d\right)$ be a complete metric space. Let $f:X\to X$ and let ϕ be a lower semi-continuous function from X into $\left[0,\mathrm{\infty }\right)$. Assume that $d\left(x,f\left(x\right)\right)\le \varphi \left(x\right)-\varphi \left(f\left(x\right)\right)$ for all $x\in X$. Then f has a fixed point in X.

Lemma 2.3 [3]

Let $\left(X,p\right)$ be a partial metric space and let ${p}^{s}:X×X\to \left[0,\mathrm{\infty }\right)$ be defined by

${p}^{s}\left(x,y\right)=2p\left(x,y\right)-p\left(x,x\right)-p\left(y,y\right)$
(1)

for all $x,y\in X$. Then $\left(X,{p}^{s}\right)$ is a metric space.

To emphasize that the function given in (1) is a metric, we use the notation ${d}_{p}$ instead of ${p}^{s}$, that is,

(2)

Let $\left(X,p\right)$ be a partial metric space. Following [1], consider $\varphi :X\to \left[0,\mathrm{\infty }\right)$ and $g:X\to X$ not necessarily a continuous function such that

$2p\left(x,g\left(x\right)\right)-p\left(x,x\right)-p\left(g\left(x\right),g\left(x\right)\right)\le \varphi \left(x\right)-\varphi \left(g\left(x\right)\right),\phantom{\rule{1em}{0ex}}x\in X.$

By (2), we can write

${d}_{p}\left(x,g\left(x\right)\right)\le \varphi \left(x\right)-\varphi \left(g\left(x\right)\right).$

The author [1] defines the class of mappings Φ and ${\mathrm{\Phi }}_{g}$ as follows:

and

We re-write Φ as

It is well known also that $\left(X,p\right)$ is complete if and only if $\left(X,{d}_{p}\right)$ is complete (see, e.g., [3, 4]).

Under these observations, keeping (2) in mind, we conclude that Lemma 3.1 in [1] remains true without using any properties of a partial metric. On the other hand, in Lemma 3.2 in [1] the completeness assumption is missed. It can be re-formulated correctly as follows.

Updated Lemma 3.2 [1]

Let $\left\{{x}_{n}\right\}$ be a sequence in a complete partial metric space $\left(X,p\right)$ such that

where ϕ is a lower semi-continuous function. Then ${lim}_{n\to \mathrm{\infty }}{x}_{n}=\overline{x}$ and ${d}_{p}\left(\overline{x},{x}_{n}\right)\le \varphi \left({x}_{n}\right)-\varphi \left(\overline{x}\right)$ for each n.

Moreover, in Definition 2.2 in [1], the open and closed balls associated to a partial metric p are not defined correctly, because the term $p\left(x,x\right)$ is missing, that is, we should have

${B}_{\epsilon }\left(x\right)=\left\{y\in X,p\left(x,y\right)

It is clear that there is nothing in this paper [1] to prove. Indeed, the main result of [1] is a consequence of Theorem 1.1.

The following definition already exists in the literature.

Definition 3.3 (cf. [1])

Let $\left(X,p\right)$ be a partial metric space.

(1) For $A\subset X$, define the diameter of a subset A, written $D\left(A\right)$, by

$\begin{array}{rcl}D\left(A\right)& =& \underset{\left({x}_{i},{x}_{j}\right)\in A}{sup}\left\{2p\left({x}_{i},{x}_{j}\right)-p\left({x}_{i},{x}_{i}\right)-p\left({x}_{j},{x}_{j}\right)\right\}\\ =& \underset{\left({x}_{i},{x}_{j}\right)\in A}{sup}{d}_{p}\left({x}_{i},{x}_{j}\right).\end{array}$

(2) Let $r\left(A\right)={inf}_{x\in A}\left(\varphi \left(x\right)\right)$. Note that $B\subset A$ implies $r\left(B\right)\ge r\left(A\right)$.

(3) Let ${\mathrm{\Phi }}^{\prime }\subset \mathrm{\Phi }$. For each $x\in X$, define ${S}_{x}=\left\{f\left(x\right)|f\in {\mathrm{\Phi }}^{\prime }\right\}$.

Keeping (2) in mind, we conclude easily.

Lemma 3.4 [1]

$D\left({S}_{x}\right)\le 2\left(\varphi \left(x\right)-r\left({S}_{x}\right)\right)$.

Consequently, we derive Theorem 3.5 in [1] without using any property of the partial metric. As a conclusion, this paper is just a repetition of usual results by using equality (2).