Some coupled fixed-point theorems in two quasi-partial metric spaces

Abstract

The purpose of this paper is to prove some new coupled common fixed-point theorems for mappings defined on a set equipped with two quasi-partial metrics. We also provide illustrative examples in support of our new results.

MSC:47H10, 54H25.

1 Introduction and preliminaries

In 1994, Matthews [1] introduced the notion of partial metric spaces as follows.

Definition 1.1 [1]

A partial metric on a nonempty set X is a function $p:X\times X⟶{\mathbb{R}}^{+}$ such that for all $x,y,z\in X$:

(p1) $x=y\iff p(x,x)=p(x,y)=p(y,y)$,

(p2) $p(x,x)\le p(x,y)$,

(p3) $p(x,y)=p(y,x)$,

(p4) $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$.

A partial metric space is a pair $(X,p)$ such that X is a nonempty set and p is a partial metric on X.

In [1], Matthews extended the Banach contraction principle from metric spaces to partial metric spaces. Based on the notion of partial metric spaces, several authors (for example, [2–32]) obtained some fixed-point results for mappings satisfying different contractive conditions. Very recently, Haghi et al. [33] showed in their interesting paper that some fixed-point theorems in partial metric spaces can be obtained from metric spaces.

Karapınar et al. [34] introduced the concept of quasi-partial metric spaces and studied some fixed-point problems on quasi-partial metric spaces. The notion of a quasi-partial metric space is defined as follows.

Definition 1.2 [34]

A quasi-partial metric on nonempty set X is a function $q:X\times X\to {\mathbb{R}}^{+}$ which satisfies:

(QPM₁) If $q(x,x)=q(x,y)=q(y,y)$, then $x=y$,

(QPM₂) $q(x,x)\le q(x,y)$,

(QPM₃) $q(x,x)\le q(y,x)$, and

(QPM₄) $q(x,y)+q(z,z)\le q(x,z)+q(z,y)$

for all $x,y,z\in X$.

A quasi-partial metric space is a pair $(X,q)$ such that X is a nonempty set and q is a quasi-partial metric on X.

Let q be a quasi-partial metric on set X. Then

$$d_q(x,y)=q(x,y)+q(y,x)-q(x,x)-q(y,y)$$

is a metric on X.

Definition 1.3 [34]

Let $(X,q)$ be a quasi-partial metric space. Then

• (i) A sequence $\{x_n\}$ converges to a point $x\in X$ if and only if

  $$q(x,x)=\underset{n\to \mathrm{\infty }}{lim}q(x,x_n)=\underset{n\to \mathrm{\infty }}{lim}q(x_n,x).$$

• (ii) A sequence $\{x_n\}$ is called a Cauchy sequence if ${lim}_{n,m\to \mathrm{\infty }}q(x_n,x_m)$ and ${lim}_{n,m\to \mathrm{\infty }}q(x_m,x_n)$ exist (and are finite).

• (iii) The quasi-partial metric space $(X,q)$ is said to be complete if every Cauchy sequence $\{x_n\}$ in X converges, with respect to ${\tau }_q$, to a point $x\in X$ such that

  $$q(x,x)=\underset{n,m\to \mathrm{\infty }}{lim}q(x_n,x_m)=\underset{n,m\to \mathrm{\infty }}{lim}q(x_n,x_m).$$

Bhaskar and Lakshmikantham [35] introduced the concept of a coupled fixed point and studied some nice coupled fixed-point theorems. Later, Lakshmikantham and Ćirić [36] introduced the notion of a coupled coincidence point of mappings. For some works on a coupled fixed point, we refer the reader to [37–62].

Definition 1.4 [35]

Let X be a nonempty set. We call an element $(x,y)\in X\times X$ a coupled fixed point of the mapping $F:X\times X\to X$ if $F(x,y)=x$ and $F(y,x)=y$.

Definition 1.5 [36]

An element $(x,y)\in X\times X$ is called

• (i) a coupled coincidence point of the mapping $F:X\times X\to X$ and $g:X\to X$ if $F(x,y)=gx$ and $F(y,x)=gy$; in this case $(gx,gy)$ is called coupled point of coincidence of mappings F and g;

• (ii) a common coupled fixed point of mappings $F:X\times X\to X$ and $g:X\to X$ if $F(x,y)=gx=x$ and $F(y,x)=gy=y$;

• (iii) a common coupled fixed point of mappings $F:X\times X\to X$ and $g:X\to X$ if $F(x,y)=gx=x$ and $F(y,x)=gy=y$.

Abbas et al. [37] introduced the concept of w-compatible mappings as follows.

Definition 1.6 [37]

Let X be a nonempty set. We say that the mappings $F:X\times X\to X$ and $g:X\to X$ are w-compatible if $gF(x,y)=F(gx,gy)$ whenever $gx=F(x,y)$ and $gy=F(y,x)$.

Very recently, Shatanawi and Pitea [38] obtained some common coupled fixed-point results for a pair of mappings in quasi-partial metric space.

Theorem 1.1 (see [[38], Theorem 2.1])

Let $(X,q)$ be a quasi-partial metric space, $g:X\to X$ and $F:X\times X\to X$ be two mappings. Suppose that there exist $k_1$, $k_2$, and $k_3$ in $[0,1)$ with $k_1+k_2+k_3<1$ such that the condition

$$\begin{array}{r}q(F(x,y),F(u,v))+q(F(y,x),F(v,u))\\ \le k_1[q(gx,gu)+q(gy,gv)]+k_2[q(gx,F(x,y))+q(gy,F(y,x))]\\ +k_3[q(gu,F(u,v))+q(gv,F(v,u))]\end{array}$$

(1.1)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric q.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)$ and $gy=F(y,x)$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(x,x)$.

The aim of this article is to prove some new coupled common fixed-point theorems for mappings defined on a set equipped with two quasi-partial metrics.

The following lemma is crucial in our work.

Lemma 1.1 [38]

Let $(X,q)$ be a quasi-partial metric space. Then the following statements hold true:

• (i) If $q(x,y)=0$, then $x=y$.

• (ii) If $x\ne y$, then $q(x,y)>0$ and $q(y,x)>0$.

In this manuscript, we generalize, improve, enrich, and extend the above coupled common fixed-point results. We also state some examples to illustrate our results. This paper can be considered as a continuation of the remarkable works of Aydi [12], Karapınar et al. [34], and Shatanawi and Pitea [38].

2 Main results

Now we shall prove our main results.

Theorem 2.1 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and let $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exist $k_1$, $k_2$, $k_3$, $k_4$, and $k_5$ in $[0,1)$ with

$$k_1+k_2+k_3+2k_4+k_5<1$$

(2.1)

such that the condition

$$\begin{array}{r}{q}_1(F(x,y),F(u,v))+q_1(F(y,x),F(v,u))\\ \le k_1[q_2(gx,gu)+q_2(gy,gv)]+k_2[q_2(gx,F(x,y))+q_2(gy,F(y,x))]\\ +k_3[q_2(gu,F(u,v))+q_2(gv,F(v,u))]+k_4[q_2(gx,F(u,v))+q_2(gy,F(v,u))]\\ +k_5[q_2(gu,F(x,y))+q_2(gv,F(y,x))]\end{array}$$

(2.2)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Proof Let $x_0,y_0\in X$. Since $F(X\times X)\subset g(X)$, we can choose $x_1,y_1\in X$ such that $g{x}_1=F(x_0,y_0)$ and $g{y}_1=F(y_0,x_0)$. Similarly, we can choose $x_2,y_2\in X$ such that $g{x}_2=F(x_1,y_1)$ and $g{y}_2=F(y_1,x_1)$. Continuing in this way we construct two sequences $\{x_n\}$ and $\{y_n\}$ in X such that

$$g{x}_{n+1}=F(x_n,y_n)\text{and}g{y}_{n+1}=F(y_n,x_n),\mathrm{\forall }n\ge 0.$$

(2.3)

It follows from (2.2) and (QPM₄) that

$$\begin{array}{r}{q}_1(g{x}_n,g{x}_{n+1})+q_1(g{y}_n,g{y}_{n+1})\\ =q_1(F(x_{n-1},y_{n-1}),F(x_n,y_n))+q_1(F(y_{n-1},x_{n-1}),F(y_n,x_n))\\ \le k_1[q_2(g{x}_{n-1},g{x}_n)+q_2(g{y}_{n-1},g{y}_n)]\\ +k_2\left[q_2(g{x}_{n-1},F(x_{n-1},y_{n-1})+q_2(g{y}_{n-1},F(y_{n-1},x_{n-1})))\right]\\ +k_3[q_2(g{x}_n,F(x_n,y_n))+q_2(g{y}_n,F(y_n,x_n))]\\ +k_4[q_2(g{x}_{n-1},F(x_n,y_n))+q_2(g{y}_{n-1},F(y_n,x_n))]\\ +k_5[q_2(g{x}_n,F(x_{n-1},y_{n-1}))+q_2(g{y}_n,F(y_{n-1},x_{n-1}))]\\ =(k_1+k_2)[q_2(g{x}_{n-1},g{x}_n)+q_2(g{y}_{n-1},g{y}_n)]+k_3[q_2(g{x}_n,g{x}_{n+1})+q_2(g{y}_n,g{y}_{n+1})]\\ +k_4[q_2(g{x}_{n-1},g{x}_{n+1})+q_2(g{y}_{n-1},g{y}_{n+1})]+k_5[q_2(g{x}_n,g{x}_n)+q_2(g{y}_n,g{y}_n)]\\ \le (k_1+k_2)[q_2(g{x}_{n-1},g{x}_n)+q_2(g{y}_{n-1},g{y}_n)]+k_3[q_2(g{x}_n,g{x}_{n+1})+q_2(g{y}_n,g{y}_{n+1})]\\ +k_4[q_2(g{x}_{n-1},g{x}_n)+q_2(g{x}_n,g{x}_{n+1})-q_2(g{x}_n,g{x}_n)+q_2(g{y}_{n-1},g{y}_n)+q_2(g{y}_n,g{y}_{n+1})\\ -q_2(g{y}_n,g{y}_n)]+k_5[q_2(g{x}_n,g{x}_{n+1})+q_2(g{y}_n,g{y}_{n+1})]\\ \le (k_1+k_2+k_4)[q_2(g{x}_{n-1},g{x}_n)+q_2(g{y}_{n-1},g{y}_n)]\\ +(k_3+k_4+k_5)[q_2(g{x}_n,g{x}_{n+1})+q_2(g{y}_n,g{y}_{n+1})]\\ \le (k_1+k_2+k_4)[q_1(g{x}_{n-1},g{x}_n)+q_1(g{y}_{n-1},g{y}_n)]\\ +(k_3+k_4+k_5)[q_1(g{x}_n,g{x}_{n+1})+q_1(g{y}_n,g{y}_{n+1})],\end{array}$$

which implies that

$$q_1(g{x}_n,g{x}_{n+1})+q_1(g{y}_n,g{y}_{n+1})\le \frac{k_1+k_2+k_4}{1-k_3-k_4-k_5}[q_1(g{x}_{n-1},g{x}_n)+q_1(g{y}_{n-1},g{y}_n)].$$

(2.4)

Put $k=\frac{k_1+k_2+k_4}{1-k_3-k_4-k_5}$. Obviously, $0\le k<1$. By repetition of the above inequality (2.4) n times, we get

$$q_1(g{x}_n,g{x}_{n+1})+q_1(g{y}_n,g{y}_{n+1})\le k^n[q_1(g{x}_0,g{x}_1)+q_1(g{y}_0,g{y}_1)].$$

(2.5)

Next, we shall prove that $\{g{x}_n\}$ and $\{g{y}_n\}$ are Cauchy sequences in $g(X)$.

In fact, for each $n,m\in \mathbb{N}$, $m>n$, from (QPM₄) and (2.5) we have

$$\begin{array}{rcl}{q}_1(g{x}_n,g{x}_m)+q_1(g{y}_n,g{y}_m)& \le & \sum _{i=n}^{m-1}[q_1(g{x}_i,g{x}_{i+1})+q_1(g{y}_i,g{y}_{i+1})]\\ \le & \sum _{i=n}^{m-1}{k}^i[q_1(g{x}_0,g{x}_1)+q_1(g{y}_0,g{y}_1)]\\ \le & \frac{k^n}{1-k}[q_1(g{x}_0,g{x}_1)+q_1(g{y}_0,g{y}_1)].\end{array}$$

(2.6)

This implies that

$$\underset{n,m\to \mathrm{\infty }}{lim}[q_1(g{x}_n,g{x}_m)+q_1(g{y}_n,g{y}_m)]=0,$$

and so

$$\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{x}_n,g{x}_m)=0\text{and}\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{y}_n,g{y}_m)=0.$$

(2.7)

By similar arguments as above, we can show that

$$\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{x}_m,g{x}_n)=0\text{and}\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{y}_m,g{y}_n)=0.$$

(2.8)

Hence $\{g{x}_n\}$ and $\{g{y}_n\}$ are Cauchy sequences in $(gX,q_1)$. Since $(gX,q_1)$ is complete, there exist $gx,gy\in g(X)$ such that $\{g{x}_n\}$ and $\{g{y}_n\}$ converge to gx and gy with respect to ${\tau }_{q_1}$, that is,

$$\begin{array}{rcl}{q}_1(gx,gx)& =& \underset{n\to \mathrm{\infty }}{lim}{q}_1(gx,g{x}_n)=\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{x}_n,gx)\\ =& \underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{x}_m,g{x}_n)=\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{x}_n,g{x}_m)\end{array}$$

(2.9)

and

$$\begin{array}{rcl}{q}_1(gy,gy)& =& \underset{n\to \mathrm{\infty }}{lim}{q}_1(gy,g{y}_n)=\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{y}_n,gy)\\ =& \underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{y}_m,g{y}_n)=\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{y}_n,g{y}_m).\end{array}$$

(2.10)

Combining (2.7)-(2.10), we have

$$\begin{array}{rcl}{q}_1(gx,gx)& =& \underset{n\to \mathrm{\infty }}{lim}{q}_1(gx,g{x}_n)=\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{x}_n,gx)\\ =& \underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{x}_m,g{x}_n)=\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{x}_n,g{x}_m)=0\end{array}$$

(2.11)

and

$$\begin{array}{rcl}{q}_1(gy,gy)& =& \underset{n\to \mathrm{\infty }}{lim}{q}_1(gy,g{y}_n)=\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{y}_n,gy)\\ =& \underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{y}_m,g{y}_n)=\underset{n,m\to \mathrm{\infty }}{lim}{q}_1(g{y}_n,g{y}_m)=0.\end{array}$$

(2.12)

By (QPM₄) we obtain

$$\begin{array}{rcl}{q}_1(g{x}_{n+1},F(x,y))& \le & q_1(g{x}_{n+1},gx)+q_1(gx,F(x,y))-q_1(gx,gx)\\ \le & q_1(g{x}_{n+1},gx)+q_1(gx,F(x,y))\\ \le & q_1(g{x}_{n+1},gx)+q_1(gx,g{x}_{n+1})+q_1(g{x}_{n+1},F(x,y))-q_1(g{x}_{n+1},g{x}_{n+1})\\ \le & q_1(g{x}_{n+1},gx)+q_1(gx,g{x}_{n+1})+q_1(g{x}_{n+1},F(x,y)).\end{array}$$

Letting $n\to \mathrm{\infty }$ in the above inequalities and using (2.11), we have

$$\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{x}_{n+1},F(x,y))\le q_1(gx,F(x,y))\le \underset{n\to \mathrm{\infty }}{lim}{q}_1(g{x}_{n+1},F(x,y)).$$

That is,

$$\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{x}_{n+1},F(x,y))=q_1(gx,F(x,y)).$$

(2.13)

Similarly, using (2.12) we have

$$\underset{n\to \mathrm{\infty }}{lim}{q}_1(g{y}_{n+1},F(y,x))=q_1(gy,F(y,x)).$$

(2.14)

Now we prove that $F(x,y)=gx$ and $F(y,x)=gy$. In fact, it follows from (2.2) and (2.3) that

$$\begin{array}{r}{q}_1(g{x}_{n+1},F(x,y))+q_1(g{y}_{n+1},F(y,x))\\ =q_1(F(x_n,y_n),F(x,y))+q_1(F(y_n,x_n))\\ \le k_1[q_2(g{x}_n,gx)+q_2(g{y}_n,gy)]+k_2[q_2(g{x}_n,F(x_n,y_n))+q_2(g{y}_n,F(y_n,x_n))]\\ +k_3[q_2(gx,F(x,y))+q_2(gy,F(y,x))]+k_4[q_2(g{x}_n,F(x,y))+q_2(g{y}_n,F(y,x))]\\ +k_5[q_2(gx,F(x_n,y_n))+q_2(gy,F(y_n,x_n))]\\ =k_1[q_2(g{x}_n,gx)+q_2(g{y}_n,gy)]+k_2[q_2(g{x}_n,g{x}_{n+1})+q_2(g{y}_n,g{y}_{n+1})]\\ +k_3[q_2(gx,F(x,y))+q_2(gy,F(y,x))]+k_4[q_2(g{x}_n,F(x,y))+q_2(g{y}_n,F(y,x))]\\ +k_5[q_2(gx,g{x}_{n+1})+q_2(gy,g{y}_{n+1})]\\ \le k_1[q_1(g{x}_n,gx)+q_1(g{y}_n,gy)]+k_2[q_1(g{x}_n,g{x}_{n+1})+q_1(g{y}_n,g{y}_{n+1})]\\ +k_3[q_1(gx,F(x,y))+q_1(gy,F(y,x))]+k_4[q_1(g{x}_n,F(x,y))+q_1(g{y}_n,F(y,x))]\\ +k_5[q_1(gx,g{x}_{n+1})+q_1(gy,g{y}_{n+1})].\end{array}$$

Letting $n\to \mathrm{\infty }$ in the above inequality, using (2.11)-(2.14), we obtain

$$q_1(gx,F(x,y))+q_1(gy,F(y,x))\le (k_3+k_4)[q_1(gx,F(x,y))+q_1(gy,F(y,x))].$$

(2.15)

By (2.1) we have $k_3+k_4<1$. Hence, it follows from (2.15) that $q_1(gx,F(x,y))=q_1(gy,F(y,x))=0$. By Lemma 1.1, we get $F(x,y)=gx$ and $F(y,x)=gy$. Hence, $(gx,gy)$ is a coupled point of coincidence of mappings F and g.

Next, we will show that the coupled point of coincidence is unique. Suppose that $(x^{\ast },y^{\ast })\in X\times X$ with $F(x^{\ast },y^{\ast })=g{x}^{\ast }$ and $F(y^{\ast },x^{\ast })=g{y}^{\ast }$. Using (2.2), (2.11), (2.12), and (QPM₃), we obtain

$$\begin{array}{r}{q}_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })\\ =q_1(F(x,y),F(x^{\ast },y^{\ast }))+q_1(F(y,x),F(y^{\ast },x^{\ast }))\\ \le k_1[q_2(gx,g{x}^{\ast })+q_2(gy,g{y}^{\ast })]+k_2[q_2(gx,F(x,y))+q_2(gy,F(y,x))]\\ +k_3[q_2(g{x}^{\ast },F(x^{\ast },y^{\ast }))+q_2(g{y}^{\ast },F(y^{\ast },x^{\ast }))]\\ +k_4[q_2(gx,F(x^{\ast },y^{\ast }))+q_2(gy,F(y^{\ast },x^{\ast }))]\\ +k_5[q_2(g{x}^{\ast },F(x,y))+q_2(g{y}^{\ast },F(y,x))]\\ =k_1[q_2(gx,g{x}^{\ast })+q_2(gy,g{y}^{\ast })]+k_2[q_2(gx,gx)+q_2(gy,gy)]\\ +k_3[q_2(g{x}^{\ast },g{x}^{\ast })+q_2(g{y}^{\ast },g{y}^{\ast })]+k_4[q_2(gx,g{x}^{\ast })+q_2(gy,g{y}^{\ast })]\\ +k_5[q_2(g{x}^{\ast },gx)+q_2(g{y}^{\ast },gy)]\\ \le (k_1+k_4)[q_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })]+k_2[q_1(gx,gx)+q_1(gy,gy)]\\ +k_3[q_1(g{x}^{\ast },g{x}^{\ast })+q_1(g{y}^{\ast },g{y}^{\ast })]+k_5[q_1(g{x}^{\ast },gx)+q_1(g{y}^{\ast },gy)]\\ \le (k_1+k_3+k_4)[q_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })]\\ +k_5[q_1(g{x}^{\ast },gx)+q_1(g{y}^{\ast },gy)].\end{array}$$

This implies that

$$q_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })\le \frac{k_5}{1-k_1-k_3-k_4}\cdot [q_1(g{x}^{\ast },gx)+q_1(g{y}^{\ast },gy)].$$

(2.16)

Similarly, we have

$$q_1(g{x}^{\ast },gx)+q_1(g{y}^{\ast },gy)\le \frac{k_5}{1-k_1-k_3-k_4}\cdot [q_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })].$$

(2.17)

Substituting (2.17) into (2.16), we obtain

$$q_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })\le {\left(\frac{k_5}{1-k_1-k_3-k_4}\right)}^2\cdot [q_1(gx,g{x}^{\ast })+q_1(gy,g{y}^{\ast })].$$

(2.18)

Since $\frac{k_5}{1-k_1-k_3-k_4}<1$, from (2.18), we must have $q_1(gx,g{x}^{\ast })=q_1(gy,g{y}^{\ast })=0$. By Lemma 1.1, we get $gx=g{x}^{\ast }$ and $gy=g{y}^{\ast }$, which implies the uniqueness of the coupled point of coincidence of F and g, that is, $(gx,gy)$.

Next, we will show that $gx=gy$. In fact, from (2.2), (2.11), and (2.12) we have

$$\begin{array}{r}{q}_1(gx,gy)+q_1(gy,gx)\\ =q_1(F(x,y),F(y,x))+q_1(F(y,x),F(x,y))\\ \le k_1[q_2(gx,gy)+q_2(gy,gx)]+k_2[q_2(gx,F(x,y))+q_2(gy,F(y,x))]\\ +k_3[q_2(gy,F(y,x))+q_2(gx,F(x,y))]+k_4[q_2(gx,F(y,x))+q_2(gy,F(x,y))]\\ +k_5[q_2(gy,F(x,y))+q_2(gx,F(y,x))]\\ =k_1[q_2(gx,gy)+q_2(gy,gx)]+k_2[q_2(gx,gx)+q_2(gy,gy)]\\ +k_3[q_2(gy,gy)+q_2(gx,gx)]+k_4[q_2(gx,gy)+q_2(gy,gx)]\\ +k_5[q_2(gy,gx)+q_2(gx,gy)]\\ \le k_1[q_1(gx,gy)+q_1(gy,gx)]+k_2[q_1(gx,gx)+q_1(gy,gy)]\\ +k_3[q_1(gy,gy)+q_1(gx,gx)]+k_4[q_1(gx,gy)+q_1(gy,gx)]\\ +k_5[q_1(gy,gx)+q_1(gx,gy)]\\ =(k_1+k_4+k_5)[q_1(gx,gy)+q_1(gy,gx)].\end{array}$$

(2.19)

Since $k_1+k_4+k_5<1$, we have $q_1(gx,gy)=q_1(gy,gx)=0$. By Lemma 1.1, we get $gx=gy$.

Finally, assume that g and F are w-compatible. Let $u=gx$, then we have $u=gx=F(x,y)=gy=F(y,x)$, so that

$$gu=ggx=g(F(x,y))=F(gx,gy)=F(u,u).$$

(2.20)

Consequently, $(u,u)$ is a coupled coincidence point of F and g, and therefore $(gu,gu)$ is a coupled point of coincidence of F and g, and by its uniqueness, we get $gu=gx$. Thus, we obtain $F(u,u)=gu=u$. Therefore, $(u,u)$ is the unique common coupled fixed point of F and g. This completes the proof of Theorem 2.1. □

In Theorem 2.1, if we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then we get the following.

Corollary 2.1 Let $(X,q)$ be a quasi-partial metric space, $F:X\times X\to X$ and $g:X\to X$ be two mappings. Suppose that there exist $k_1$, $k_2$, $k_3$, $k_4$ and $k_5$ in $[0,1)$ with $k_1+k_2+k_3+2k_4+k_5<1$ such that the condition

$$\begin{array}{r}q(F(x,y),F(u,v))+q(F(y,x),F(v,u))\\ \le k_1[q(gx,gu)+q(gy,gv)]+k_2[q(gx,F(x,y))+q(gy,F(y,x))]\\ +k_3[q(gu,F(u,v))+q(gv,F(v,u))]+k_4[q(gx,F(u,v))+q(gy,F(v,u))]\\ +k_5[q(gu,F(x,y))+q(gv,F(y,x))]\end{array}$$

(2.21)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric q.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Remark 2.1 Corollary 2.1 improve and extend Theorem 2.1 of Shatanawi and Pitea [38]; the contractive condition defined by (1.1) is replaced by the new contractive condition defined by (2.23).

Corollary 2.2 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exist $a_i\in [0,1)$ ($i=1,2,3,\dots ,10$) with

$$a_1+a_2+a_3+a_4+a_5+a_6+2(a_7+a_8)+a_9+a_{10}<1$$

(2.22)

such that the condition

$$\begin{array}{r}{q}_1(F(x,y),F(u,v))\\ \le a_1{q}_2(gx,gu)+a_2{q}_2(gy,gv)+a_3{q}_2(gx,F(x,y))+a_4{q}_2(gy,F(y,x))\\ +a_5{q}_2(gu,F(u,v))+a_6{q}_2(gv,F(v,u))+a_7{q}_2(gx,F(u,v))+a_8{q}_2(gy,F(v,u))\\ +a_9{q}_2(gu,F(x,y))+a_{10}{q}_2(gv,F(y,x))\end{array}$$

(2.23)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Proof Given $x,y,u,v\in X$. It follows from (2.23) that

$$\begin{array}{r}{q}_1(F(x,y),F(u,v))\\ \le a_1{q}_2(gx,gu)+a_2{q}_2(gy,gv)+a_3{q}_2(gx,F(x,y))+a_4{q}_2(gy,F(y,x))\\ +a_5{q}_2(gu,F(u,v))+a_6{q}_2(gv,F(v,u))+a_7{q}_2(gx,F(u,v))+a_8{q}_2(gy,F(v,u))\\ +a_9{q}_2(gu,F(x,y))+a_{10}{q}_2(gv,F(y,x))\end{array}$$

(2.24)

and

$$\begin{array}{r}{q}_1(F(y,x),F(v,u))\\ \le a_1{q}_2(gy,gv)+a_2{q}_2(gx,gu)+a_3{q}_2(gy,F(y,x))+a_4{q}_2(gx,F(x,y))\\ +a_5{q}_2(gv,F(v,u))+a_6{q}_2(gu,F(u,v))\\ +a_7{q}_2(gy,F(v,u))+a_8{q}_2(gx,F(u,v))\\ +a_9{q}_2(gv,F(y,x))+a_{10}{q}_2(gu,F(x,y)).\end{array}$$

(2.25)

Adding inequality (2.24) to inequality (2.25), we get

$$\begin{array}{r}{q}_1(q_1(F(x,y),F(u,v))+F(y,x),F(v,u))\\ \le (a_1+a_2)[q_2(gx,gu)+q_2(gy,gv)]+(a_3+a_4)[q_2(gx,F(x,y))+q_2(gy,F(y,x))]\\ +(a_5+a_6)[q_2(gu,F(u,v))+q_2(gv,F(v,u))]\\ +(a_7+a_8)[q_2(gx,F(u,v))+q_2(gy,F(v,u))]\\ +(a_9+a_{10})[q_2(gu,F(x,y))+q_2(gv,F(y,x))].\end{array}$$

(2.26)

Therefore, the result follows from Theorem 2.1. □

Remark 2.2 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$ and $a_7=a_8=a_9=a_{10}=0$, then Corollary 2.2 is reduced to Corollary 2.1 of Shatanawi and Pitea [38].

Corollary 2.3 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(gx,gu)+q_2(gy,gv)]$$

(2.27)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Remark 2.3 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then Corollary 2.3 is reduced to Corollary 2.2 of Shatanawi and Pitea [38].

Corollary 2.4 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(gx,F(x,y))+q_2(gy,F(y,x))]$$

(2.28)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Remark 2.4 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then Corollary 2.4 is reduced to Corollary 2.3 of Shatanawi and Pitea [38].

Corollary 2.5 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(gu,F(u,v))+q_2(gv,F(v,u))]$$

(2.29)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Remark 2.5 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then Corollary 2.5 is reduced to Corollary 2.4 of Shatanawi and Pitea [38].

Corollary 2.6 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exists $k\in [0,\frac{1}{2})$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(gx,F(u,v))+q_2(gy,F(v,u))]$$

(2.30)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Corollary 2.7 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$, $g:X\to X$ be two mappings. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(gu,F(x,y))+q_2(gv,F(y,x))]$$

(2.31)

holds for all $x,y,u,v\in X$. Also, suppose we have the following hypotheses:

• (i) $F(X\times X)\subset g(X)$.

• (ii) $g(X)$ is a complete subspace of X with respect to the quasi-partial metric $q_1$.

Then the mappings F and g have a coincidence point $(x,y)$ satisfying $gx=F(x,y)=F(y,x)=gy$.

Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point of the form $(u,u)$.

Let $g=I_X$ (the identity mapping) in Theorem 2.1 and Corollaries 2.1-2.7. Then we have the following results.

Corollary 2.8 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exist $k_1$, $k_2$, $k_3$, $k_4$, and $k_5$ in $[0,1)$ with $k_1+k_2+k_3+2k_4+k_5<1$ such that the condition

$$\begin{array}{r}{q}_1(F(x,y),F(u,v))+q_1(F(y,x),F(v,u))\\ \le k_1[q_2(x,u)+q_2(y,v)]+k_2[q_2(x,F(x,y))+q_2(y,F(y,x))]\\ +k_3[q_2(u,F(u,v))+q_2(v,F(v,u))]+k_4[q_2(x,F(u,v))+q_2(y,F(v,u))]\\ +k_5[q_2(u,F(x,y))+q_2(v,F(y,x))]\end{array}$$

(2.32)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Corollary 2.9 Let $(X,q)$ be a complete quasi-partial metric space, $F:X\times X\to X$ be a mapping. Suppose that there exist $k_1$, $k_2$, $k_3$, $k_4$, and $k_5$ in $[0,1)$ with $k_1+k_2+k_3+2k_4+k_5<1$ such that the condition

$$\begin{array}{r}q(F(x,y),F(u,v))+q(F(y,x),F(v,u))\\ \le k_1[q(x,u)+q(y,v)]+k_2[q(x,F(x,y))+q(y,F(y,x))]\\ +k_3[q(u,F(u,v))+q(v,F(v,u))]+k_4[q(x,F(u,v))+q(y,F(v,u))]\\ +k_5[q(u,F(x,y))+q(v,F(y,x))]\end{array}$$

(2.33)

holds for all $x,y,u,v\in X$. Then F has a unique coupled fixed point of the form $(u,u)$.

Remark 2.6 Corollary 2.9 improve and extend Corollary 2.5 of Shatanawi and Pitea [38], the contractive condition is replaced by the new contractive condition defined by (2.35).

Corollary 2.10 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exist $a_i\in [0,1)$ ($i=1,2,3,\dots ,10$) with

$$a_1+a_2+a_3+a_4+a_5+a_6+2(a_7+a_8)+a_9+a_{10}<1$$

(2.34)

such that the condition

$$\begin{array}{r}{q}_1(F(x,y),F(u,v))\\ \le a_1{q}_2(x,u)+a_2{q}_2(y,v)+a_3{q}_2(x,F(x,y))+a_4{q}_2(y,F(y,x))\\ +a_5{q}_2(u,F(u,v))+a_6{q}_2(v,F(v,u))+a_7{q}_2(x,F(u,v))+a_8{q}_2(y,F(v,u))\\ +a_9{q}_2(u,F(x,y))+a_{10}{q}_2(v,F(y,x))\end{array}$$

(2.35)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space. Then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Remark 2.7

• (1) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$ and $a_7=a_8=a_9=a_{10}=0$, then Corollary 2.10 is reduced to Corollary 2.6 of Shatanawi and Pitea [38].

• (2) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$ and $a_i=0$ ($i=3,4,5,\dots ,10$), then Corollary 2.10 extends Theorem 2.1 of Aydi [12] on the class of quasi-partial metric spaces.

• (3) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, $a_1=a_2$ and $a_i=0$ ($i=3,4,5,\dots ,10$), then Corollary 2.10 extends the Corollary 2.2 of Aydi [12] on the class of quasi-partial metric spaces.

• (4) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$ and $a_i=0$ ($i=1,2,4,6,7,8,9,10$), then Corollary 2.10 extends Theorem 2.4 of Aydi [12] on the class of quasi-partial metric spaces.

• (5) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$ and $a_i=0$ ($i=1,2,3,4,5,6,8,10$), then Corollary 2.10 extends Theorem 2.5 of Aydi [12] on the class of quasi-partial metric spaces.

• (6) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, $a_3=a_9$ and $a_i=0$ ($i=1,2,4,5,6,7,8,10$), then Corollary 2.10 extends Corollary 2.6 of Aydi [12] on the class of quasi-partial metric spaces.

• (7) If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, $a_7=a_9$ and $a_i=0$ ($i=1,2,3,4,5,6,8,10$), then Corollary 2.10 extends Corollary 2.7 of Aydi [12] on the class of quasi-partial metric spaces.

Corollary 2.11 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(x,u)+q_2(y,v)]$$

(2.36)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space. Then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Remark 2.8 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then Corollary 2.11 is reduced to Corollary 2.7 of Shatanawi and Pitea [38].

Corollary 2.12 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(x,F(x,y))+q_2(y,F(y,x))]$$

(2.37)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Remark 2.9 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then Corollary 2.12 is reduced to Corollary 2.8 of Shatanawi and Pitea [38].

Corollary 2.13 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(u,F(u,v))+q_2(v,F(v,u))]$$

(2.38)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Remark 2.10 If we take $q_1(x,y)=q_2(x,y)$ for all $x,y\in X$, then Corollary 2.13 is reduced to Corollary 2.9 of Shatanawi and Pitea [38].

Corollary 2.14 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exists $k\in [0,\frac{1}{2})$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(x,F(u,v))+q_2(y,F(v,u))]$$

(2.39)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Corollary 2.15 Let $q_1$ and $q_2$ be two quasi-metrics on X such that $q_2(x,y)\le q_1(x,y)$, for all $x,y\in X$, and $F:X\times X\to X$ be a mapping. Suppose that there exists $k\in [0,1)$ such that the condition

$$q_1(F(x,y),F(u,v))+q(F(y,x),F(v,u))\le k[q_2(u,F(x,y))+q_2(v,F(y,x))]$$

(2.40)

holds for all $x,y,u,v\in X$. If $(X,q_1)$ is a complete quasi-partial metric space, then the mapping F has a unique coupled fixed point of the form $(u,u)$.

Now, we introduce an example to support our results.

Example 2.1 Let $X=[0,1]$, and two quasi-partial metrics $q_1$, $q_2$ on X be given as

$$q_1(x,y)=|x-y|+x\text{and}{q}_2(x,y)=\frac{1}{2}(|x-y|+x)$$

for all $x,y\in X$. Also, define $F:X\times X\to X$ and $g:X\to X$ as

$$F(x,y)=\frac{x+y}{16}\text{and}gx=\frac{x}{2}$$

for all $x,y\in X$. Then

• (1) $(X,q_1)$ is a complete quasi-partial metric space.

• (2) $F(X\times X)\subset X$.

• (3) F and g is w-compatible.

• (4) For any $x,y,u,v\in X$, we have

  $$q_1(F(x,y),F(u,v))+q_1(F(y,x)+F(v,u))\le \frac{1}{2}(q_2(gx,gu)+q_2(gy,gv)).$$

Proof The proofs of (1), (2), and (3) are clear. Next we show that (4). In fact, for $x,y,u,v\in X$, we have

$$\begin{array}{r}{q}_1(F(x,y),F(u,v))+q_1(F(y,x)+F(v,u))\\ =q_1(\frac{x+y}{16},\frac{u+v}{16})+q_1(\frac{y+x}{16},\frac{v+u}{16})\\ =\frac{1}{8}(|x+y-(u+v)|+(x+y))\\ =\frac{1}{4}\left(\right|\frac{1}{2}(x+y)-\frac{1}{2}(u+v)|+\frac{1}{2}(x+y))\\ \le \frac{1}{4}\left(\right|\frac{1}{2}x-\frac{1}{2}u|+\frac{1}{2}x+|\frac{1}{2}y-\frac{1}{2}v|+\frac{1}{2}y)\\ =\frac{1}{2}(q_2(gx,gu)+q_2(gy,gv)).\end{array}$$

Thus, F and g satisfy all the hypotheses of Corollary 2.3. So, F and g have a unique common coupled fixed point. Here $(0,0)$ is the unique common coupled fixed point of F and g. □