Multidimensional fixed points for generalized ψ-quasi-contractions in quasi-metric-like spaces

Abstract

In this paper, we introduce the concept of a quasi-metric-like space, and by defining the w-compatibility of two mappings, we obtain multidimensional coincidence point and multidimensional fixed point theorems for generalized ψ-quasi-contractions in quasi-metric-like spaces. Our results extend the fixed point theorems in Vetro and Radenović (Appl. Math. Comput. 219:1594-1600, 2012) and references therein.

MSC:47H10, 54H25.

1 Introduction and preliminaries

In 1987, Guo and Lakshmikantham [1] initiated the study of the coupled fixed point. In 2010, Samet and Vetro [2] presented the concept of a fixed point of N-order as an extension of the coupled fixed point.

Definition 1.1 ([2])

Let X be a non-empty set and let $F:X^N\to X$ ($N\ge 2$) be a given mapping. An element $(x_1,x_2,\dots ,x_N)\in X^N$ is called a fixed point of N-order of the mapping F if

$$\begin{array}{c}F(x_1,x_2,\dots ,x_{N-1},x_N)=x_1,\hfill \\ F(x_2,x_3,\dots ,x_N,x_1)=x_2,\hfill \\ ⋮\hfill \\ F(x_N,x_1,x_2,\dots ,x_{N-1})=x_N.\hfill \end{array}$$

Subsequently, a number of papers occurred on tripled fixed point and quadruple fixed point theory (see, e.g., [3–10]). Berzig and Samet [11] discussed the existence of the fixed point of N-order for m-mixed monotone mappings in complete ordered metric spaces. Very recently, Roldán et al. [12] extended the notion of the fixed point of N-order to the Φ-fixed point and obtained some Φ-fixed point theorems for a mixed monotone mapping in partially ordered complete metric spaces. Afterward, many results on multidimensional fixed points have been established (see, e.g., [13–18]).

Matthews [19] introduced the notion of a partial metric space where the self-distance does not need to be zero. By generalizing the partial metric, Hitzler and Seda [20] presented the concept of a dislocated metric which was redefined as a metric-like by Amini-Harandi [21]. The existence of fixed points in dislocated metric (metric-like) spaces has been discussed by many authors (see, e.g., [22–30]).

Definition 1.2 ([20, 21])

A mapping $\sigma :X\times X\to [0,+\mathrm{\infty })$, where X is a nonempty set, is said to be a dislocated metric (metric-like) on X if, for any $x,y,z\in X$, the following three conditions hold true:

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

(σ 2) $\sigma (x,y)=\sigma (y,x)$;

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

The pair $(X,\sigma )$ is then called a dislocated metric (metric-like) space.

Karapınar et al. [31] introduced the notion of quasi-partial metric spaces and studied some fixed point theorems on quasi-partial metric spaces.

Definition 1.3 ([31])

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

(QPM₁) If $0\le q(x,x)=q(x,y)=q(y,y)$, then $x=y$,

(QPM₂) $q(x,x)\le q(x,y)$,

(QPM₃) $q(x,x)\le q(y,x)$, and

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

for all $x,y,z\in X$. The pair $(X,q)$ is called a quasi-partial metric space.

In this paper, similar to the notation of Amini-Harandi [21], we define a quasi-metric-like space generalizing the metric-like space and the quasi-partial metric space. Furthermore, we discuss the existence and uniqueness of a multidimensional fixed point for a generalized g-ψ-quasi-contractive mapping in quasi-metric-like spaces using the new w-compatibility of two mappings.

2 A quasi-metric-like space

Definition 2.1 A mapping $\rho :X\times X\to [0,+\mathrm{\infty })$, where X is a nonempty set, is said to be a quasi-metric-like on X if, for any $x,y,z\in X$, the following conditions hold:

(ρ 1) $\rho (x,y)=0\Rightarrow x=y$;

(ρ 2) $\rho (x,z)\le \rho (x,y)+\rho (y,z)$.

The pair $(X,\rho )$ is called a quasi-metric-like space.

Definition 2.2 Let $(X,\rho )$ be a quasi-metric-like space. Then

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

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

In this case, x is called a ρ-limit of $\{x_n\}$.

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

• (3) The quasi-metric-like-like space $(X,\rho )$ is called complete if, for every Cauchy sequence $\{x_n\}$ in X, there is some $x\in X$ such that

  $$\begin{array}{rcl}\underset{n\to +\mathrm{\infty }}{lim}\rho (x_n,x)& =& \underset{n\to +\mathrm{\infty }}{lim}\rho (x,x_n)=\rho (x,x)\\ =& \underset{m,n\to +\mathrm{\infty }}{lim}\rho (x_m,x_n)=\underset{m,n\to +\mathrm{\infty }}{lim}\rho (x_n,x_m).\end{array}$$

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

Example 2.3 Let $X=\{0,1\}$, and let

$$\rho (x,y)=\{\begin{array}{ll}2,& \text{if }x=y=0;\\ 1,& \text{if }x=0,y=1;\\ \frac{3}{2},& \text{if }x=1,y=0;\\ 0,& \text{if }x=y=1.\end{array}$$

Then $(X,\rho )$ is a quasi-metric-like space, but $\rho (0,0)\nleqq \rho (1,0)$, so $(X,\rho )$ is not a quasi-partial metric space.

Remark 2.4 Every metric-like space is a quasi-metric-like space. Because the limit of a convergent sequence in metric-like space is not necessarily unique [25], the ρ-limit of a convergent sequence in quasi-metric-like spaces is not necessarily unique.

3 Main results

In this section, we establish the coincidence point and fixed point of r-order theorems, and an illustrative example is employed to show the validity of our results.

Definition 3.1 Let X be a nonempty set, and let $g:X\to X$ and let $F:X^r\to X$ ($r\ge 2$) be two given mappings. An element $(x_1,x_2,\dots ,x_r)\in X^r$ is called a coincidence point of r-order of $F:X^r\to X$ and $g:X\to X$ if

$$\begin{array}{c}g(x_1)=F(x_1,x_2,\dots ,x_{r-1},x_r),\hfill \\ g(x_2)=F(x_2,x_3,\dots ,x_r,x_1),\hfill \\ ⋮\hfill \\ g(x_r)=F(x_r,x_1,x_2,\dots ,x_{r-1}).\hfill \end{array}$$

If g is the identity mapping on X, then $(x_1,x_2,\dots ,x_r)\in X^r$ is a fixed point of r-order of the mapping F.

Throughout this paper, we denote all of the coincidence points of r-order of $F:X^r\to X$ and $g:X\to X$ by $C(F,g,r)$.

By Ψ we denote the set of real functions $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ which have the following properties:

• (i) ψ is nondecreasing;

• (ii) $\psi (0)=0$;

• (iii) ${lim}_{t\to +\mathrm{\infty }}(t-\psi (t))=+\mathrm{\infty }$;

• (iv) ${lim}_{s\to t^{+}}\psi (s)<t$ for all $t>0$.

From (iv) and $\psi (t)\le {lim}_{s\to t^{+}}\psi (s)<t$, we deduce that $\psi (t)<t$ for all $t>0$ [32].

Vetro and Radenović [32] introduced the concept of a g-ψ-quasi-contraction. We present the following definition as a generalization of the g-ψ-quasi-contraction.

Definition 3.2 Let $(X,\rho )$ be a quasi-metric-like space, $g:X\to X$ and let $F:X^r\to X$ ($r\ge 2$). F is called a generalized g-ψ-quasi-contraction if there exists $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ such that

$$\rho (F(x_1,x_2,\dots ,x_r),F(y_1,y_2,\dots ,y_r))\le \psi (M(x_1,x_2,\dots ,x_r;y_1,y_2,\dots ,y_r)),$$

(1)

where

$$\begin{array}{r}M(x_1,x_2,\dots ,x_r;y_1,y_2,\dots ,y_r)\\ =max\{\rho (g{x}_1,g{y}_1),\rho (g{x}_2,g{y}_2),\dots ,\\ \rho (g{x}_r,g{y}_r),\rho (g{x}_1,F(x_1,x_2,\dots ,x_r)),\rho (g{x}_2,F(x_2,x_3,\dots ,x_r,x_1)),\dots ,\\ \rho (g{x}_r,F(x_r,x_1,\dots ,x_{r-1})),\rho (g{y}_1,F(y_1,y_2,\dots ,y_r)),\\ \rho (g{y}_2,F(y_2,y_3,\dots ,y_r,y_1)),\dots ,\rho (g{y}_r,F(y_r,y_1,\dots ,y_{r-1})),\\ \rho (g{x}_1,F(y_1,y_2,\dots ,y_r)),\rho (g{x}_2,F(y_2,y_3,\dots ,y_r,y_1)),\dots ,\\ \rho (g{x}_r,F(y_r,y_1,\dots ,y_{r-1})),\rho (g{y}_1,F(x_1,x_2,\dots ,x_r)),\\ \rho (g{y}_2,F(x_2,x_3,\dots ,x_r,x_1)),\dots ,\rho (g{y}_r,F(x_r,x_1,\dots ,x_{r-1}))\},\end{array}$$

(2)

for any $(x_1,x_2,\dots ,x_r),(y_1,y_2,\dots ,y_r)\in X^r$.

If g is the identity mapping, then F is a generalized ψ-quasi-contraction.

Definition 3.3 Let X be a nonempty set. The mappings $g:X\to X$ and $F:X^r\to X$ ($r\ge 2$) are called w-compatible if

$$F(g(x_1),g(x_2),\dots ,g(x_r))=g(F(x_1,x_2,\dots ,x_r)),$$

whenever $(x_1,x_2,\dots ,x_r)\in C(F,g,r)$.

Theorem 3.4 Let $(X,\rho )$ be a quasi-metric-like space, $g:X\to X$ and $F:X^r\to X$ ($r\ge 2$). Suppose that F is a generalized g-ψ-quasi-contraction with $\psi \in \mathrm{\Psi }$. If $F(X^r)\subseteq g(X)$ and $g(X)$ is a complete subspace of X, then $C(F,g,r)$ is nonempty.

Proof Let $(x_1^0,x_2^0,\dots ,x_r^0)\in X^r$. Since $F(X^r)\subseteq g(X)$, we can construct a sequence $\{(x_1^n,x_2^n,\dots ,x_r^n)\}$ such that

$$g\left(x_i^n\right)=F(x_i^{n-1},x_{i+1}^{n-1},\dots ,x_r^{n-1},x_1^{n-1},\dots ,x_{i-1}^{n-1})\text{for }i=1,2,\dots ,r.$$

Define

$$\begin{array}{c}{O}_n(x_1^0,x_2^0,\dots ,x_r^0)=\{g{x}_1^0,g{x}_2^0,\dots ,g{x}_r^0,g{x}_1^1,g{x}_2^1,\dots ,g{x}_r^1,\dots ,g{x}_1^n,g{x}_2^n,\dots ,g{x}_r^n\},\hfill \\ O(x_1^0,x_2^0,\dots ,x_r^0)=\{g{x}_1^0,g{x}_2^0,\dots ,g{x}_r^0,g{x}_1^1,g{x}_2^1,\dots ,g{x}_r^1,\dots ,g{x}_1^n,g{x}_2^n,\dots ,g{x}_r^n,\dots \},\hfill \\ {\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)=diam\left(O_n(x_1^0,x_2^0,\dots ,x_r^0)\right)=sup\{\rho (x,y):x,y\in O_n(x_1^0,x_2^0,\dots ,x_r^0)\}.\hfill \end{array}$$

If there exists $n_0\in N$ such that ${\delta }_{n_0}(x_1^0,x_2^0,\dots ,x_r^0)=0$, then for any $0\le k\le n_0-1$, $(x_1^k,x_2^k,\dots ,x_r^k)\in C(F,g,r)$.

We suppose that ${\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)>0$, for all $n\in N$.

Step 1. We shall prove that for each $n\in N$,

$${\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)=max\{\underset{1\le i,l\le r,0\le s\le n}{sup}\rho (g{x}_i^0,g{x}_l^s),\underset{1\le i,l\le r,0\le s\le n}{sup}\rho (g{x}_l^s,g{x}_i^0)\}.$$

(3)

In fact, for any $1\le i,l\le r$, $1\le j,s\le n$, we have

$$\begin{array}{rcl}\rho (g{x}_i^j,g{x}_l^s)& =& \rho (F(x_i^{j-1},x_{i+1}^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-1}^{j-1}),\\ F(x_l^{s-1},x_{l+1}^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-1}^{s-1}))\\ \le & \psi \left(M\right(x_i^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-1}^{j-1};\\ x_l^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-1}^{s-1}\left)\right),\end{array}$$

(4)

where

$$\begin{array}{r}M(x_i^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-1}^{j-1};x_l^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-1}^{s-1})\\ =max\{\rho (g{x}_i^{j-1},g{x}_l^{s-1}),\rho (g{x}_{i+1}^{j-1},g{x}_{l+1}^{s-1}),\dots ,\rho (g{x}_{i-1}^{j-1},g{x}_{l-1}^{s-1}),\\ \rho (g{x}_i^{j-1},F(x_i^{j-1},x_{i+1}^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-1}^{j-1})),\\ \rho (g{x}_l^{s-1},F(x_l^{s-1},x_{l+1}^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-1}^{s-1})),\\ \rho (g{x}_{i+1}^{j-1},F(x_{i+1}^{j-1},x_{i+2}^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_i^{j-1})),\\ \rho (g{x}_{l+1}^{s-1},F(x_{l+1}^{s-1},x_{l+2}^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_l^{s-1})),\dots ,\\ \rho (g{x}_{i-1}^{j-1},F(x_{i-1}^{j-1},x_i^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-2}^{j-1})),\\ \rho (g{x}_{l-1}^{s-1},F(x_{l-1}^{s-1},x_l^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-2}^{s-1})),\\ \rho (g{x}_l^{s-1},F(x_i^{j-1},x_{i+1}^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-1}^{j-1})),\\ \rho (g{x}_i^{j-1},F(x_l^{s-1},x_{l+1}^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-1}^{s-1})),\\ \rho (g{x}_{l+1}^{s-1},F(x_{i+1}^{j-1},x_{i+2}^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_i^{j-1})),\\ \rho (g{x}_{i+1}^{j-1},F(x_{l+1}^{s-1},x_{l+2}^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_l^{s-1})),\dots ,\\ \rho (g{x}_{l-1}^{s-1},F(x_{i-1}^{j-1},x_i^{j-1},\dots ,x_r^{j-1},x_1^{j-1},\dots ,x_{i-2}^{j-1})),\\ \rho (g{x}_{i-1}^{j-1},F(x_{l-1}^{s-1},x_l^{s-1},\dots ,x_r^{s-1},x_1^{s-1},\dots ,x_{l-2}^{s-1}))\}\\ =max\{\rho (g{x}_i^{j-1},g{x}_l^{s-1}),\rho (g{x}_{i+1}^{j-1},g{x}_{l+1}^{s-1}),\dots ,\rho (g{x}_{i-1}^{j-1},g{x}_{l-1}^{s-1}),\\ \rho (g{x}_i^{j-1},g{x}_i^j),\rho (g{x}_l^{s-1},g{x}_l^s),\rho (g{x}_{i+1}^{j-1},g{x}_{i+1}^j),\rho (g{x}_{l+1}^{s-1},g{x}_{l+1}^s),\dots ,\\ \rho (g{x}_{i-1}^{j-1},g{x}_{i-1}^j),\rho (g{x}_{l-1}^{s-1},g{x}_{l-1}^s),\rho (g{x}_l^{s-1},g{x}_i^j),\rho (g{x}_i^{j-1},g{x}_l^s),\\ \rho (g{x}_{l+1}^{s-1},g{x}_{i+1}^j),\rho (g{x}_{i+1}^{j-1},g{x}_{l+1}^s),\dots ,\rho (g{x}_{l-1}^{s-1},g{x}_{i-1}^j),\rho (g{x}_{i-1}^{j-1},g{x}_{l-1}^s)\}.\end{array}$$

So, for $1\le i,l\le r$, $1\le j,s\le n$, we have

$$\rho (g{x}_i^j,g{x}_l^s)\le \psi \left({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\right)<{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0).$$

(5)

Hence, equation (3) is true.

Step 2. Now, we claim that for each $n\in N$, ${lim}_{n\to +\mathrm{\infty }}{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)<+\mathrm{\infty }$. For this, we distinguish three cases.

Since the sequence $\{{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\}$ is nondecreasing, there exists ${lim}_{n\to +\mathrm{\infty }}{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)$.

Case 1. If, for all $n\in N$, ${\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)=diam\{g{x}_1^0,g{x}_2^0,\dots ,g{x}_r^0\}$, then the claim holds.

Case 2. Suppose that there exist $n_0\in N$, $1\le i_0,l_0\le r$, and $1\le s_0\le n_0$ such that

$${\delta }_{n_0}(x_1^0,x_2^0,\dots ,x_r^0)=\rho (g{x}_{i_0}^0,g{x}_{l_0}^{s_0}),$$

then, for any $n\ge n_0$, there exist $1\le i,l\le r$, and $1\le s\le n$ such that

$${\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\le \rho (g{x}_i^0,g{x}_l^s).$$

By equation (5), we obtain

$$\begin{array}{rcl}{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)& \le & \rho (g{x}_i^0,g{x}_l^1)+\rho (g{x}_l^1,g{x}_l^s)\\ \le & \rho (g{x}_i^0,g{x}_l^1)+\psi \left({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\right),\end{array}$$

which implies that

$${\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)-\psi \left({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\right)\le \rho (g{x}_i^0,g{x}_l^1).$$

(6)

Suppose that ${lim}_{n\to +\mathrm{\infty }}{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)=+\mathrm{\infty }$, from the property (iii) of ψ, we have

$$\underset{n\to +\mathrm{\infty }}{lim}({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)-\psi \left({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\right))=+\mathrm{\infty }.$$

Nevertheless, by equation (6), we get

$$\underset{n\to +\mathrm{\infty }}{lim}({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)-\psi \left({\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)\right))\le \rho (g{x}_i^0,g{x}_l^1),$$

which is a contradiction. Thus, ${lim}_{n\to +\mathrm{\infty }}{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)<+\mathrm{\infty }$.

Case 3. If there exist $n_1\in N$, $1\le i_1,l_1\le r$, and $1\le s_1\le n_1$ such that

$${\delta }_{n_1}(x_1^0,x_2^0,\dots ,x_r^0)=\rho (g{x}_{l_1}^{s_1},g{x}_{i_1}^0),$$

the proof is similar to Case 2.

Step 3. Next, we prove that, for every $1\le i\le r$, $\{g{x}_i^n\}$ is a Cauchy sequence in $(X,\rho )$.

Let

$$O(g{x}_1^p,g{x}_2^p,\dots ,g{x}_r^p)=\{g{x}_1^p,g{x}_2^p,\dots ,g{x}_r^p,g{x}_1^{p+1},g{x}_2^{p+1},\dots ,g{x}_r^{p+1},\dots \},$$

and let

$$\delta (x_1^p,x_2^p,\dots ,x_r^p)=diam\left(O(g{x}_1^p,g{x}_2^p,\dots ,g{x}_r^p)\right),p=0,1,2,\dots .$$

Then,

$$\begin{array}{rcl}\delta (x_1^p,x_2^p,\dots ,x_r^p)& \le & \delta (x_1^0,x_2^0,\dots ,x_r^0)\\ =& \underset{n\to +\mathrm{\infty }}{lim}{\delta }_n(x_1^0,x_2^0,\dots ,x_r^0)<+\mathrm{\infty },p=0,1,2,\dots .\end{array}$$

Since

$$0\le \delta (x_1^{p+1},x_2^{p+1},\dots ,x_r^{p+1})\le \delta (x_1^p,x_2^p,\dots ,x_r^p),p=0,1,2,\dots ,$$

there exists $\delta \ge 0$ such that

$$\underset{p\to +\mathrm{\infty }}{lim}\delta (x_1^p,x_2^p,\dots ,x_r^p)=\delta .$$

If $\delta >0$, using the monotonicity of $\{\delta (x_1^p,x_2^p,\dots ,x_r^p)\}$ and the property (iv) of ψ, we conclude that

$$\underset{p\to +\mathrm{\infty }}{lim}\psi \left(\delta (x_1^p,x_2^p,\dots ,x_r^p)\right)=\underset{\delta (x_1^p,x_2^p,\dots ,x_r^p)\to {\delta }^{+}}{lim}\psi \left(\delta (x_1^p,x_2^p,\dots ,x_r^p)\right)<\delta .$$

(7)

However, by equation (4), we have, for any $p\ge 0$,

$$\delta (x_1^{p+1},x_2^{p+1},\dots ,x_r^{p+1})\le \psi \left(\delta (x_1^p,x_2^p,\dots ,x_r^p)\right),$$

which implies that

$$\delta =\underset{p\to +\mathrm{\infty }}{lim}\delta (x_1^{p+1},x_2^{p+1},\dots ,x_r^{p+1})\le \underset{p\to +\mathrm{\infty }}{lim}\psi \left(\delta (x_1^p,x_2^p,\dots ,x_r^p)\right),$$

which contradicts equation (7). Therefore, ${lim}_{p\to +\mathrm{\infty }}\delta (x_1^p,x_2^p,\dots ,x_r^p)=\delta =0$, that is, for every $1\le i\le r$, $\{g{x}_i^n\}$ is a Cauchy sequence in $(X,\rho )$.

Step 4. Finally, we prove that $C(F,g,r)$ is nonempty.

Since $g(X)$ is a complete subspace of X, there exist $u_i=g{x}_i^{\ast }$, $i=1,2,\dots ,r$, such that

$$\begin{array}{rcl}\underset{n\to +\mathrm{\infty }}{lim}\rho (g{x}_i^{\ast },g{x}_i^n)& =& \underset{n\to +\mathrm{\infty }}{lim}\rho (g{x}_i^n,g{x}_i^{\ast })=\underset{m,n\to +\mathrm{\infty }}{lim}\rho (g{x}_i^n,g{x}_i^m)\\ =& \underset{m,n\to +\mathrm{\infty }}{lim}\rho (g{x}_i^m,g{x}_i^n)=\rho (g{x}_i^{\ast },g{x}_i^{\ast })=\rho (u_i,u_i)=0.\end{array}$$

(8)

For $1\le i\le r$, $n\in N$, from

$$\begin{array}{r}\rho (g{x}_i^{n+1},F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\\ \le \rho (g{x}_i^{n+1},g{x}_i^{\ast })+\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\end{array}$$

and

$$\begin{array}{r}\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))-\rho (g{x}_i^{\ast },g_i^{n+1})\\ \le \rho (g{x}_i^{n+1},F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\end{array}$$

we get

$$\begin{array}{r}\underset{n\to +\mathrm{\infty }}{lim}\rho (g{x}_i^{n+1},F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\\ =\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })).\end{array}$$

(9)

For any $1\le i\le r$, $n\in N$, we have

$$\begin{array}{r}\rho (g{x}_i^{n+1},F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\\ =\rho (F(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n),F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\\ \le \psi \left(M(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n;x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })\right),\end{array}$$

(10)

where

$$\begin{array}{r}M(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n;x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })\\ =max\{\rho (g{x}_i^n,g{x}_i^{\ast }),\rho (g{x}_{i+1}^n,g{x}_{i+1}^{\ast }),\dots ,\rho (g{x}_{i-1}^n,g{x}_{i-1}^{\ast }),\\ \rho (g{x}_i^n,F(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n)),\\ \rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\\ \rho (g{x}_{i+1}^n,F(x_{i+1}^n,x_{i+2}^n,\dots ,x_r^n,x_1^n,\dots ,x_i^n)),\\ \rho (g{x}_{i+1}^{\ast },F(x_{i+1}^{\ast },x_{i+2}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_i^{\ast })),\dots ,\\ \rho (g{x}_r^n,F(x_r^n,x_1^n,\dots ,x_{r-1}^n)),\rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\\ \rho (g{x}_1^n,F(x_1^n,x_2^n,\dots ,x_r^n)),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^n,F(x_{i-1}^n,x_i^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-2}^n)),\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast })),\\ \rho (g{x}_i^{\ast },F(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n)),\\ \rho (g{x}_i^n,F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\\ \rho (g{x}_{i+1}^{\ast },F(x_{i+1}^n,x_{i+2}^n,\dots ,x_r^n,x_1^n,\dots ,x_i^n)),\\ \rho (g{x}_{i+1}^n,F(x_{i+1}^{\ast },x_{i+2}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_i^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^n,x_1^n,\dots ,x_{r-1}^n)),\rho (g{x}_r^n,F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\\ \rho (g{x}_1^{\ast },F(x_1^n,x_2^n,\dots ,x_r^n)),\rho (g{x}_1^n,F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^n,x_i^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-2}^n)),\\ \rho (g{x}_{i-1}^n,F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}.\end{array}$$

By equations (8) and (9), for any $\epsilon >0$, there exists $n_0\in N$, and, for every $n>n_0$ and $1\le i\le r$, we have

$$\begin{array}{r}max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}\\ \le M(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n;x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })\\ \le max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}+\epsilon .\end{array}$$

(11)

Thus, for each $1\le i\le r$,

$$\begin{array}{r}\underset{n\to +\mathrm{\infty }}{lim}M(x_i^n,x_{i+1}^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n;x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })\\ =max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}.\end{array}$$

(12)

If

$$\begin{array}{r}max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}>0,\end{array}$$

using equations (9), (10), (11), and (12) and the property (iv) of ψ, we obtain, for every $1\le i\le r$,

$$\begin{array}{r}\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\\ =\underset{n\to +\mathrm{\infty }}{lim}\rho (g{x}_i^{n+1},F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))\\ \le \underset{n\to +\mathrm{\infty }}{lim}\psi \left(M(x_i^n,\dots ,x_r^n,x_1^n,\dots ,x_{i-1}^n;x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })\right)\\ <max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}.\end{array}$$

(13)

By the arbitrariness of i in equation (13), we deduce that

$$\begin{array}{r}max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}\\ <max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\},\end{array}$$

which is a contradiction. Therefore,

$$\begin{array}{r}max\{\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })),\dots ,\\ \rho (g{x}_r^{\ast },F(x_r^{\ast },x_1^{\ast },\dots ,x_{r-1}^{\ast })),\rho (g{x}_1^{\ast },F(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })),\dots ,\\ \rho (g{x}_{i-1}^{\ast },F(x_{i-1}^{\ast },x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-2}^{\ast }))\}=0,\end{array}$$

which implies that $\rho (g{x}_i^{\ast },F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }))=0$, $i=1,2,\dots ,r$.

Thus,

$$g{x}_i^{\ast }=F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }),i=1,2,\dots ,r,$$

that is, $(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast })\in C(F,g,r)$. □

Theorem 3.5 Let $(X,\rho )$ be a quasi-metric-like space. Let $g:X\to X$ and let $F:X^r\to X$ ($r\ge 2$) be mappings satisfying all the conditions of Theorem  3.4. If F and g are w-compatible, then F and g have a unique coincidence point of r-order, which is a fixed point of g and a fixed point of r-order of F. Moreover, the coincidence point of r-order is of the form $(u^{\ast },u^{\ast },\dots ,u^{\ast })$ for some $u^{\ast }\in X$.

Proof Suppose that there exist $(x_1^{\ast },x_2^{\ast },\dots ,x_r^{\ast }),(x_1^{\ast \ast },x_2^{\ast \ast },\dots ,x_r^{\ast \ast })\in C(F,g,r)$, that is, for each $1\le i\le r$,

$$g{x}_i^{\ast }=F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }),$$

(14)

$$g{x}_i^{\ast \ast }=F(x_i^{\ast \ast },x_{i+1}^{\ast \ast },\dots ,x_r^{\ast \ast },x_1^{\ast \ast },\dots ,x_{i-1}^{\ast \ast }).$$

(15)

First, we prove that, for any $1\le i,j,k\le r$,

$$g{x}_i^{\ast }=g{x}_j^{\ast }=g{x}_k^{\ast \ast }.$$

(16)

By equations (1), (14), and (15), for $1\le i\le r-1$, we have

$$\begin{array}{rl}\rho (g{x}_i^{\ast },g{x}_{i+1}^{\ast \ast })=& \rho (F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast }),\\ F(x_{i+1}^{\ast \ast },x_{i+2}^{\ast \ast },\dots ,x_r^{\ast \ast },x_1^{\ast \ast },\dots ,x_i^{\ast \ast }))\\ \le & \psi \left(M(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast };x_{i+1}^{\ast \ast },x_{i+2}^{\ast \ast },\dots ,x_r^{\ast \ast },x_1^{\ast \ast },\dots ,x_i^{\ast \ast })\right),\end{array}$$

(17)

where

$$\begin{array}{r}M(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast };x_{i+1}^{\ast \ast },x_{i+2}^{\ast \ast },\dots ,x_r^{\ast \ast },x_1^{\ast \ast },\dots ,x_i^{\ast \ast })\\ =max\{\rho (g{x}_1^{\ast },g{x}_2^{\ast \ast }),\dots ,\rho (g{x}_{r-1}^{\ast },g{x}_r^{\ast \ast }),\rho (g{x}_r^{\ast },g{x}_1^{\ast \ast }),\\ \rho (g{x}_2^{\ast \ast },g{x}_1^{\ast }),\rho (g{x}_3^{\ast \ast },g{x}_2^{\ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_{r-1}^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_r^{\ast }),\\ \rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast })\}.\end{array}$$

(18)

Set

$$\begin{array}{rcl}\zeta & =& max\{\rho (g{x}_1^{\ast },g{x}_2^{\ast \ast }),\dots ,\rho (g{x}_{r-1}^{\ast },g{x}_r^{\ast \ast }),\rho (g{x}_r^{\ast },g{x}_1^{\ast \ast }),\\ \rho (g{x}_2^{\ast \ast },g{x}_1^{\ast }),\rho (g{x}_3^{\ast \ast },g{x}_2^{\ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_{r-1}^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_r^{\ast }),\\ \rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast })\}.\end{array}$$

(19)

Similarly, we have

$$\rho (g{x}_r^{\ast },g{x}_1^{\ast \ast })\le \psi (\zeta ),\rho (g{x}_1^{\ast \ast },g_r^{\ast })\le \psi (\zeta )$$

(20)

and

$$\rho (g{x}_{i+1}^{\ast \ast },g{x}_i^{\ast })\le \psi (\zeta ),i=1,2,\dots ,r-1.$$

(21)

By equations (1), (14), (15), and the monotonicity of ψ, for $1\le i\le r$, we also have

$$\rho (g{x}_i^{\ast },g{x}_i^{\ast })\le \psi (max\{\rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast })\})\le \psi (\zeta )$$

(22)

and

$$\rho (g{x}_i^{\ast \ast },g{x}_i^{\ast \ast })\le \psi (max\{\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast })\})\le \psi (\zeta ).$$

(23)

From equations (17) to (23), we can conclude that

$$\zeta \le \psi (\zeta ),$$

which is a contradiction, unless $\zeta =0$. So

$$\begin{array}{r}max\{\rho (g{x}_1^{\ast },g{x}_2^{\ast \ast }),\dots ,\rho (g{x}_{r-1}^{\ast },g{x}_r^{\ast \ast }),\rho (g{x}_r^{\ast },g{x}_1^{\ast \ast }),\\ \rho (g{x}_2^{\ast \ast },g{x}_1^{\ast }),\rho (g{x}_3^{\ast \ast },g{x}_2^{\ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_{r-1}^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_r^{\ast }),\\ \rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast })\}=0,\end{array}$$

that is,

$$g{x}_i^{\ast }=g{x}_{i+1}^{\ast \ast },i=1,2,\dots ,r-1,$$

(24)

$$g{x}_r^{\ast }=g{x}_1^{\ast \ast }.$$

(25)

On the other hand, for any $1\le i\le r$, we obtain

$$\begin{array}{r}\rho (g{x}_i^{\ast },g{x}_i^{\ast \ast })\\ \le \psi \left(M(x_i^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast };x_i^{\ast \ast },\dots ,x_r^{\ast \ast },x_1^{\ast \ast },\dots ,x_{i-1}^{\ast \ast })\right)\\ =\psi (max\{\rho (g{x}_i^{\ast },g{x}_i^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast \ast }),\rho (g{x}_1^{\ast },g{x}_1^{\ast \ast }),\dots ,\\ \rho (g{x}_{i-1}^{\ast },g{x}_{i-1}^{\ast \ast }),\rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\\ \rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast }),\rho (g{x}_i^{\ast \ast },g{x}_i^{\ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast }),\dots ,\\ \rho (g{x}_{i-1}^{\ast \ast },g{x}_{i-1}^{\ast })\left\}\right).\end{array}$$

(26)

Set

$$\begin{array}{rcl}\lambda & =& max\{\rho (g{x}_i^{\ast },g{x}_i^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast \ast }),\rho (g{x}_1^{\ast },g{x}_1^{\ast \ast }),\dots ,\\ \rho (g{x}_{i-1}^{\ast },g{x}_{i-1}^{\ast \ast }),\rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\\ \rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast }),\rho (g{x}_i^{\ast \ast },g{x}_i^{\ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast }),\dots ,\\ \rho (g{x}_{i-1}^{\ast \ast },g{x}_{i-1}^{\ast })\}.\end{array}$$

(27)

Similarly, for any

$$\begin{array}{rcl}\xi & \in & \{\rho (g{x}_1^{\ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_r^{\ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast \ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast \ast })\\ \rho (g{x}_i^{\ast \ast },g{x}_i^{\ast }),\dots ,\rho (g{x}_r^{\ast \ast },g{x}_r^{\ast }),\rho (g{x}_1^{\ast \ast },g{x}_1^{\ast }),\dots ,\rho (g{x}_{i-1}^{\ast \ast },g{x}_{i-1}^{\ast })\},\end{array}$$

we have

$$\xi \le \psi (\lambda ).$$

(28)

By equations (26), (27), and (28), we get

$$\lambda \le \psi (\lambda ),$$

which is a contradiction, unless $\lambda =0$. That is,

$$g{x}_i^{\ast }=g{x}_i^{\ast \ast },i=1,2,\dots ,r.$$

(29)

Therefore, equations (24), (25), and (29) imply that equation (16) is true.

Next, we prove that the coincidence point of r-order is unique.

In view of equation (16), let $g{x}_i^{\ast }=u^{\ast }$, $i=1,2,\dots ,r$.

Using the w-compatibility of F and g, we conclude that

$$\begin{array}{rcl}g{u}^{\ast }& =& g\left(g{x}_i^{\ast }\right)=g\left(F(x_i^{\ast },x_{i+1}^{\ast },\dots ,x_r^{\ast },x_1^{\ast },\dots ,x_{i-1}^{\ast })\right)\\ =& F(g{x}_i^{\ast },g{x}_{i+1}^{\ast },\dots ,g{x}_r^{\ast },g{x}_1^{\ast },\dots ,g{x}_{i-1}^{\ast })=F(u^{\ast },u^{\ast },\dots ,u^{\ast }).\end{array}$$

So, $u^{\ast }\in C(F,g,r)$. By equation (16), we can deduce that $g{u}^{\ast }=g{x}_i^{\ast }$, $i=1,2,\dots ,r$.

Thus,

$$u^{\ast }=g{x}_i^{\ast }=g{u}^{\ast }=F(u^{\ast },u^{\ast },\dots ,u^{\ast }).$$

(30)

Moreover, equations (16) and (30) imply that $(u^{\ast },u^{\ast },\dots ,u^{\ast })$ is the unique coincidence point of r-order of F and g, $u^{\ast }$ is a fixed point of g, and $(u^{\ast },u^{\ast },\dots ,u^{\ast })$ is a fixed point of r-order of F. □

For each $a\in (0,1)$, setting $\psi (t)=at$ in Theorem 3.4 and Theorem 3.5, we obtain the following results.

Corollary 3.6 Let $(X,\rho )$ be a quasi-metric-like space, let $g:X\to X$ and let $F:X^r\to X$ ($r\ge 2$). Suppose there exists $a\in (0,1)$ such that F is a generalized g-ψ-quasi-contraction with $\psi (t)=at$. If $F(X^r)\subseteq g(X)$ and $g(X)$ is a complete subspace of X, then $C(F,g,r)$ is nonempty.

Corollary 3.7 Let $(X,\rho )$ be a quasi-metric-like space. Let $g:X\to X$ and let $F:X^r\to X$ ($r\ge 2$) be mappings satisfying all the conditions of Corollary  3.6. If F and g are w-compatible, then F and g have a unique coincidence point of r-order, which is a fixed point of g and a fixed point of r-order of F. Moreover, the coincidence point of r-order is of the form $(u^{\ast },u^{\ast },\dots ,u^{\ast })$ for some $u^{\ast }\in X$.

Example 3.8 Let $X=\{0,1,2\}$, Define $\rho :X\times X\to [0,+\mathrm{\infty })$ as follows:

$$\begin{array}{c}\rho (0,0)=0,\rho (1,1)=3,\rho (2,2)=\frac{1}{2},\hfill \\ \rho (0,1)=3,\rho (0,2)=\frac{3}{2},\rho (1,0)=\frac{5}{2},\hfill \\ \rho (2,0)=3,\rho (1,2)=\frac{4}{5},\rho (2,1)=4.\hfill \end{array}$$

Then $(X,\rho )$ is a complete quasi-metric-like space.

Define $g:X\times X\to X$ by

$$g0=1,g1=2,g2=0,$$

and $F:X^r\to X$ ($r\ge 2$) by

$$F(x_1,x_2,\dots ,x_r)=\{\begin{array}{ll}0,& \text{if }{x}_1=x_2=\cdots =x_r;\\ min\{x_1,x_2,\dots ,x_r\},& \text{otherwise}.\end{array}$$

It is easy to prove that g and F satisfy all conditions of Theorem 3.4 by taking $\psi (t)=\frac{5}{6}t$, and the proof would be lengthy.

Here, $(2,2,\dots ,2)\in C(F,g,r)$.