Some fixed point results in ordered $G_p$-metric spaces

Abstract

In this paper, first we present some coincidence point results for six mappings satisfying the generalized $(\psi ,\phi )$-weakly contractive condition in the framework of partially ordered $G_p$-metric spaces. Secondly, we consider α-admissible mappings in the setup of $G_p$-metric spaces. An example is also provided to support our results.

MSC:47H10, 54H25.

1 Introduction and mathematical preliminaries

Recently, Zand and Nezhad [1] have introduced a new generalized metric space, a $G_p$-metric space, as a generalization of both partial metric spaces [2] and G-metric spaces [3].

We will use the following definition of a $G_p$-metric space.

Definition 1.1 [4]

Let X be a nonempty set. Suppose that a mapping $G_p:X\times X\times X\to {\mathbb{R}}^{+}$ satisfies:

($G_{p}1$) $x=y=z$ if $G_p(x,y,z)=G_p(z,z,z)=G_p(y,y,y)=G_p(x,x,x)$;

($G_{p}2$) $G_p(x,x,x)\le G_p(x,x,y)\le G_p(x,y,z)$ for all $x,y,z\in X$ with $z\ne y$;

($G_{p}3$) $G_p(x,y,z)=G_p(p\{x,y,z\})$, where p is any permutation of x, y, z (symmetry in all three variables);

($G_{p}4$) $G_p(x,y,z)\le G_p(x,a,a)+G_p(a,y,z)-G_p(a,a,a)$ for all $x,y,z,a\in X$ (rectangle inequality).

Then $G_p$ is called a $G_p$-metric and $(X,G_p)$ is called a $G_p$-metric space.

The $G_p$-metric $G_p$ is called symmetric if

$$G_p(x,x,y)=G_p(x,y,y)$$

(1)

holds for all $x,y\in X$. Otherwise, $G_p$ is an asymmetric $G_p$-metric.

Remark 1 In [1] (see also [5]), instead of ($G_{p}2$), the following condition was used:

($G_p{2}^{\prime }$) $G_p(x,x,x)\le G_p(x,x,y)\le G_p(x,y,z)$ for all $x,y,z\in X$.

However, with this assumption, it is very easy to obtain that (1) holds for all $x,y\in X$, i.e., the respective space is symmetric. On the other hand, there are a lot of examples of non-symmetric G-metric spaces. Hence, the conclusion stated in [1, 5] that each G-metric space is a $G_p$-metric space (satisfying ($G_p{2}^{\prime }$)) does not hold. With our assumption ($G_{p}2$), this conclusion holds true.

The following are some easy examples of $G_p$-metric spaces.

Example 1.1 Let $X=[0,+\mathrm{\infty })$, and let $G_p:X^3\to {\mathbb{R}}^{+}$ be given by $G_p(x,y,z)=max\{x,y,z\}$. Obviously, $(X,G_p)$ is a symmetric $G_p$-metric space which is not a G-metric space.

Example 1.2 Let $X=\{0,1,2,3,\dots \}$. Define $G_p:X^3\to X$ by

$$G_p(x,y,z)=\{\begin{array}{cc}x+y+z+1,\hfill & x\ne y\ne z,\hfill \\ x+z+1,\hfill & y=z\ne x,\hfill \\ y+z+1,\hfill & x=z\ne y,\hfill \\ x+z+1,\hfill & x=y\ne z,\hfill \\ 1,\hfill & x=y=z.\hfill \end{array}$$

It is easy to see that $(X,G_p)$ is a symmetric $G_p$-metric space.

Example 1.3 [4]

Let $X=\{0,1,2,3\}$. Let

$$\begin{array}{rl}A=& \{(1,0,0),(0,1,0),(0,0,1),(2,0,0),(0,2,0),(0,0,2),(3,0,0),(0,3,0),(0,0,3),\\ (1,2,2),(2,1,2),(2,2,1),(2,3,3),(3,2,3),(3,3,2)\},\\ B=& \{(0,1,1),(1,0,1),(1,1,0),(0,2,2),(2,0,2),(2,2,0),(0,3,3),(3,0,3),(3,3,0),\\ (2,1,1),(1,2,1),(1,1,2),(3,2,2),(2,3,2),(2,2,3)\}.\end{array}$$

Define $G_p:X^3\to {\mathbb{R}}^{+}$ by

$$G(x,y,z)=\{\begin{array}{cc}1\hfill & \text{if }x=y=z\ne 2,\hfill \\ 0\hfill & \text{if }x=y=z=2,\hfill \\ 2\hfill & \text{if }(x,y,z)\in A,\hfill \\ \frac{5}{2}\hfill & \text{if }(x,y,z)\in B,\hfill \\ 3\hfill & \text{if }x\ne y\ne z.\hfill \end{array}$$

It is easy to see that $(X,G_p)$ is an asymmetric $G_p$-metric space.

Proposition 1.1 [1]

Every $G_p$-metric space $(X,G_p)$ defines a metric space $(X,d_{G_p})$ where

$$d_{G_p}(x,y)=G_p(x,y,y)+G_p(y,x,x)-G_p(x,x,x)-G_p(y,y,y)$$

for all $x,y\in X$.

Proposition 1.2 [1]

Let X be a $G_p$-metric space. Then, for each $x,y,z,a\in X$, it follows that:

• (1) $G_p(x,y,z)\le G_p(x,a,a)+G_p(y,a,a)+G_p(z,a,a)-2G_p(a,a,a)$;

• (2) $G_p(x,y,z)\le G_p(x,x,y)+G_p(x,x,z)-G_p(x,x,x)$;

• (3) $G_p(x,y,y)\le 2G_p(x,x,y)-G_p(x,x,x)$;

• (4) $G_p(x,y,z)\le G_p(x,a,z)+G_p(a,y,z)-G_p(a,a,a)$, $a\ne z$.

Definition 1.2 [1]

Let $(X,G_p)$ be a $G_p$-metric space. Let $\{x_n\}$ be a sequence of points of X.

• 1. A point $x\in X$ is said to be a limit of the sequence $\{x_n\}$, denoted by $x_n\to x$, if ${lim}_{n,m\to \mathrm{\infty }}{G}_p(x,x_n,x_m)=G_p(x,x,x)$.

• 2. $\{x_n\}$ is said to be a $G_p$-Cauchy sequence if ${lim}_{n,m\to \mathrm{\infty }}{G}_p(x_n,x_m,x_m)$ exists (and is finite).

• 3. $(X,G_p)$ is said to be $G_p$-complete if every $G_p$-Cauchy sequence in X is $G_p$-convergent to $x\in X$.

Using the above definitions, one can easily prove the following proposition.

Proposition 1.3 [1]

Let $(X,G_p)$ be a $G_p$-metric space. Then, for any sequence $\{x_n\}$ in X and a point $x\in X$, the following are equivalent:

• (1) $\{x_n\}$ is $G_p$-convergent to x.

• (2) $G_p(x_n,x_n,x)\to G_p(x,x,x)$ as $n\to \mathrm{\infty }$.

• (3) $G_p(x_n,x,x)\to G_p(x,x,x)$ as $n\to \mathrm{\infty }$.

Lemma 1.1 [4]

If $G_p$ is a $G_p$-metric on X, then the mappings $d_{G_p},d_{G_p}^{\prime }:X\times X\to R^{+}$, given by

$$d_{G_p}(x,y)=G_p(x,y,y)+G_p(y,x,x)-G_p(x,x,x)-G_p(y,y,y)$$

and

$$d_{G_p}^{\prime }(x,y)=max\{G_p(x,y,y)-G_p(x,x,x),G_p(y,x,x)-G_p(y,y,y)\},$$

define equivalent metrics on X.

Proof $\frac{a+b}{2}\le max\{a,b\}\le a+b$ for all nonnegative real numbers a, b. □

Based on Lemma 2.2 of [6], Parvaneh et al. have proved the following essential lemma.

Lemma 1.2 [4]

• (1) A sequence $\{x_n\}$ is a $G_p$-Cauchy sequence in a $G_p$-metric space $(X,G_p)$ if and only if it is a Cauchy sequence in the metric space $(X,d_{G_p})$.

• (2) A $G_p$-metric space $(X,G_p)$ is $G_p$-complete if and only if the metric space $(X,d_{G_p})$ is complete. Moreover, ${lim}_{n\to \mathrm{\infty }}{d}_{G_p}(x,x_n)=0$ if and only if

  $$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}{G}_p(x,x_n,x_n)& =\underset{n\to \mathrm{\infty }}{lim}{G}_p(x_n,x,x)=\underset{n,m\to \mathrm{\infty }}{lim}{G}_p(x_n,x_n,x_m)\\ =\underset{n,m\to \mathrm{\infty }}{lim}{G}_p(x_n,x_m,x_m)=G_p(x,x,x).\end{array}$$

Lemma 1.3 [4]

Assume that $x_n\to x$ as $n\to \mathrm{\infty }$ in a $G_p$-metric space $(X,G_p)$ such that $G_p(x,x,x)=0$. Then, for every $y\in X$,

• (i) ${lim}_{n\to \mathrm{\infty }}{G}_p(x_n,y,y)=G_p(x,y,y)$,

• (ii) ${lim}_{n\to \mathrm{\infty }}{G}_p(x_n,x_n,y)=G_p(x,x,y)$.

Lemma 1.4 [4]

Assume that $\{x_n\}$, $\{y_n\}$ and $\{z_n\}$ are three sequences in a $G_p$-metric space X such that

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim}{G}_p(x_n,x,x)=\underset{n\to \mathrm{\infty }}{lim}{G}_p(x_n,x_n,x_n)=G_p(x,x,x),\hfill \\ \underset{n\to \mathrm{\infty }}{lim}{G}_p(y_n,y,y)=\underset{n\to \mathrm{\infty }}{lim}{G}_p(y_n,y_n,y_n)=G_p(y,y,y)\hfill \end{array}$$

and

$$\underset{n\to \mathrm{\infty }}{lim}{G}_p(z_n,z,z)=\underset{n\to \mathrm{\infty }}{lim}{G}_p(z_n,z_n,z_n)=G_p(z,z,z).$$

Then

• (i) ${lim}_{n\to \mathrm{\infty }}{G}_p(x_n,y_n,z_n)=G_p(x,y,z)$ and

• (ii) ${lim}_{n\to \mathrm{\infty }}{G}_p(x_n,x_n,y)=G_p(x,x,y)$

for every $y,z\in X$.

Lemma 1.5 [5]

Let $(X,G_p)$ be a $G_p$-metric space. Then

• (A) If $G_p(x,y,z)=0$, then $x=y=z$.

• (B) If $x\ne y$, then $G_p(x,y,y)>0$.

Definition 1.3 [1]

Let $(X_1,G_1)$ and $(X_2,G_2)$ be two $G_p$-metric spaces, and let $f:(X_1,G_1)\to (X_2,G_2)$ be a mapping. Then f is said to be $G_p$-continuous at a point $a\in X_1$ if for a given $\epsilon >0$, there exists $\delta >0$ such that $x,y\in X_1$ and $G_1(a,x,y)<\delta +G_1(a,a,a)$ imply that $G_2(f(a),f(x),f(y))<\epsilon +G_2(f(a),f(a),f(a))$. The mapping f is $G_p$-continuous on $X_1$ if it is $G_p$-continuous at all $a\in X_1$.

Proposition 1.4 [1]

Let $(X_1,G_1)$ and $(X_2,G_2)$ be two $G_p$-metric spaces. Then a mapping $f:X_1\to X_2$ is $G_p$-continuous at a point $x\in X_1$ if and only if it is $G_p$-sequentially continuous at x; that is, whenever $\{x_n\}$ is $G_p$-convergent to x, $\{f(x_n)\}$ is $G_p$-convergent to $f(x)$.

The concept of an altering distance function was introduced by Khan et al. [7] as follows.

Definition 1.4 The function $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is called an altering distance function if the following properties are satisfied:

• 1. ψ is continuous and nondecreasing.

• 2. $\psi (t)=0$ if and only if $t=0$.

A self-mapping f on X is called a weak contraction if the following contractive condition is satisfied:

$$d(fx,fy)\le d(x,y)-\phi (d(x,y)),$$

for all $x,y\in X$, where φ is an altering distance function.

The concept of a weakly contractive mapping was introduced by Alber and Guerre-Delabrere [8] in the setup of Hilbert spaces. Rhoades [9] considered this class of mappings in the setup of metric spaces and proved that a weakly contractive mapping defined on a complete metric space has a unique fixed point.

Zhang and Song [10] introduced the concept of a generalized φ-weakly contractive mapping as follows.

Definition 1.5 Self-mappings f and g on a metric space X are called generalized φ-weak contractions if there exists a lower semicontinuous function $\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\phi (0)=0$ and $\phi (t)>0$ for all $t>0$ such that for all $x,y\in X$,

$$d(fx,gy)\le N(x,y)-\phi (N(x,y)),$$

where

$$N(x,y)=max\{d(x,y),d(x,fx),d(y,gy),\frac{1}{2}[d(x,gy)+d(y,fx)]\}.$$

Based on the above definition, they proved the following common fixed point result.

Theorem 1.1 [10]

Let $(X,d)$ be a complete metric space. If $f,g:X\to X$ are generalized φ-weakly contractive mappings, then there exists a unique point $u\in X$ such that $u=fu=gu$.

So far, many authors extended Theorem 1.1 (see [11–13] and [14]). Moreover, Ðorić [12] generalized it by the definition of generalized $(\psi ,\phi )$-weak contractions.

Definition 1.6 Two mappings $f,g:X\to X$ are called generalized $(\psi ,\phi )$-weakly contractive if there exist two maps $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

$$\psi (d(fx,gy))\le \psi (N(x,y))-\phi (N(x,y)),$$

for all $x,y\in X$, where N and φ are as in Definition 1.5 and $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is an altering distance function.

Theorem 1.2 [12]

Let $(X,d)$ be a complete metric space, and let $f,g:X\to X$ be generalized $(\psi ,\phi )$-weakly contractive self-mappings. Then there exists a unique point $u\in X$ such that $u=fu=gu$.

Recently, many researchers have focused on different contractive conditions in various metric spaces endowed with a partial order and studied fixed point theory in the so-called bi-structured spaces. For more details on fixed point results, their applications, comparison of different contractive conditions and related results in ordered various metric spaces, we refer the reader to [15–29] and the references mentioned therein.

Let X be a nonempty set and $f:X\to X$ be a given mapping. For every $x\in X$, let $f^{-1}(x)=\{u\in X:fu=x\}$.

Definition 1.7 [24]

Let $(X,⪯)$ be a partially ordered set, and let $f,g,h:X\to X$ be given mappings such that $fX\subseteq hX$ and $gX\subseteq hX$. We say that f and g are weakly increasing with respect to h if for all $x\in X$, we have

$$fx⪯gy\text{for all }y\in h^{-1}(fx)$$

and

$$gx⪯fy\text{for all }y\in h^{-1}(gx).$$

If $f=g$, we say that f is weakly increasing with respect to h.

If $h=I$ (the identity mapping on X), then the above definition reduces to that of a weakly increasing mapping [30] (see also [24, 31]).

Definition 1.8 A partially ordered $G_p$-metric space $(X,⪯,G_p)$ is said to have the sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) $\{x_n\}$ in X, $x_n\to x$ implies that $x_n⪯x$ ($x⪯x_n$).

The aim of this paper is to prove some coincidence and common fixed point theorems for weakly $(\psi ,\phi )$-contractive mappings in partially ordered $G_p$-metric spaces.

2 Main results

Let $(X,⪯,G_p)$ be an ordered $G_p$-metric space and $f,g,h,R,S,T:X\to X$ be six self-mappings. Throughout this paper, unless otherwise stated, for all $x,y,z\in X$, let

$$\begin{array}{rl}M(x,y,z)=& max\{G_p(Tx,Ry,Sz),\\ G_p(Tx,fx,fx),G_p(Ry,gy,gy),G_p(Sz,hz,hz),\\ \frac{G_p(Tx,Tx,fx)+G_p(Ry,Ry,gy)+G_p(Sz,Sz,hz)}{3}\}.\end{array}$$

Theorem 2.1 Let $(X,⪯,G_p)$ be a partially ordered $G_p$-metric space with the sequential limit comparison property. Let $f,g,h,R,S,T:X\to X$ be six mappings such that $f(X)\subseteq R(X)$, $g(X)\subseteq S(X)$ and $h(X)\subseteq T(X)$, and RX, SX and TX are $G_p$-complete subsets of X. Suppose that for comparable elements $Tx,Ry,Sz\in X$, we have

$$\psi (2G_p(fx,gy,hz))\le \psi (M(x,y,z))-\phi (M(x,y,z)),$$

(2)

where $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are altering distance functions. Then the pairs $(f,T)$, $(g,R)$ and $(h,S)$ have a coincidence point $z^{\ast }$ in X provided that the pairs $(f,T)$, $(g,R)$ and $(h,S)$ are weakly compatible and the pairs $(f,g)$, $(g,h)$ and $(h,f)$ are partially weakly increasing with respect to R, S and T, respectively. Moreover, if $R{z}^{\ast }$, $S{z}^{\ast }$ and $T{z}^{\ast }$ are comparable, then $z^{\ast }\in X$ is a coincidence point of f, g, h, R, S and T.

Proof Let $x_0$ be an arbitrary point of X. Choose $x_1\in X$ such that $f{x}_0=R{x}_1$, $x_2\in X$ such that $g{x}_1=S{x}_2$ and $x_3\in X$ such that $h{x}_2=T{x}_3$. This can be done as $f(X)\subseteq R(X)$, $g(X)\subseteq S(X)$ and $h(X)\subseteq T(X)$.

Continuing this way, construct a sequence $\{z_n\}$ defined by $z_{3n+1}=R{x}_{3n+1}=f{x}_{3n}$, $z_{3n+2}=S{x}_{3n+2}=g{x}_{3n+1}$ and $z_{3n+3}=T{x}_{3n+3}=h{x}_{3n+2}$ for all $n\ge 0$. The sequence $\{z_n\}$ in X is said to be a Jungck-type iterative sequence with initial guess $x_0$.

As $x_1\in R^{-1}(f{x}_0)$, $x_2\in S^{-1}(g{x}_1)$ and $x_3\in T^{-1}(h{x}_2)$ and the pairs $(f,g)$, $(g,h)$ and $(h,f)$ are partially weakly increasing with respect to R, S and T, respectively, we have

$$R{x}_1=f{x}_0⪯g{x}_1=S{x}_2⪯h{x}_2=T{x}_3⪯f{x}_3=R{x}_4.$$

Continuing this process, we obtain $R{x}_{3n+1}⪯S{x}_{3n+2}⪯T{x}_{3n+3}$ for all $n\ge 0$.

We will complete the proof in three steps.

Step I. We will prove that $\{z_n\}$ is a $G_p$-Cauchy sequence. First, we show that ${lim}_{k\to \mathrm{\infty }}{G}_p(z_k,z_{k+1},z_{k+2})=0$.

Define $G_{p_k}=G_p(z_k,z_{k+1},z_{k+2})$. Suppose $G_{p_{k_0}}=0$ for some $k_0$. Then $z_{k_0}=z_{k_0+1}=z_{k_0+2}$. In the case that $k_0=3n$, then $z_{3n}=z_{3n+1}=z_{3n+2}$ gives $z_{3n+1}=z_{3n+2}=z_{3n+3}$. Indeed,

$$\begin{array}{rl}\psi (2G_p(z_{3n+1},z_{3n+2},z_{3n+3}))& =\psi (2G_p(f{x}_{3n},g{x}_{3n+1},h{x}_{3n+2}))\\ \le \psi (M(x_{3n},x_{3n+1},x_{3n+2}))-\phi (M(x_{3n},x_{3n+1},x_{3n+2})),\end{array}$$

where

$$\begin{array}{r}M(x_{3n},x_{3n+1},x_{3n+2})\\ =max\{G_p(T{x}_{3n},R{x}_{3n+1},S{x}_{3n+2}),G_p(T{x}_{3n},f{x}_{3n},f{x}_{3n}),\\ G_p(R{x}_{3n+1},g{x}_{3n+1},g{x}_{3n+1}),G_p(S{x}_{3n+2},h{x}_{3n+2},h{x}_{3n+2}),\\ \frac{G_p(T{x}_{3n},T{x}_{3n},f{x}_{3n})+G_p(R{x}_{3n+1},R{x}_{3n+1},g{x}_{3n+1})+G_p(S{x}_{3n+2},S{x}_{3n+2},h{x}_{3n+2})}{3}\}\\ =max\{G_p(z_{3n},z_{3n+1},z_{3n+2}),G_p(z_{3n},z_{3n+1},z_{3n+1}),\\ G_p(z_{3n+1},z_{3n+2},z_{3n+2}),G_p(z_{3n+2},z_{3n+3},z_{3n+3}),\\ \frac{G_p(z_{3n},z_{3n},z_{3n+1})+G_p(z_{3n+1},z_{3n+1},z_{3n+2})+G_p(z_{3n+2},z_{3n+2},z_{3n+3})}{3}\}\\ =max\{0,0,0,G_p(z_{3n+2},z_{3n+3},z_{3n+3}),\frac{0+0+G_p(z_{3n+2},z_{3n+2},z_{3n+3})}{3}\}\\ =G_p(z_{3n+2},z_{3n+3},z_{3n+3})\\ \le 2G_p(z_{3n+2},z_{3n+2},z_{3n+3})\\ =2G_p(z_{3n+1},z_{3n+2},z_{3n+3}).\end{array}$$

Thus

$$\psi (2G_p(z_{3n+1},z_{3n+2},z_{3n+3}))\le \psi (2G_p(z_{3n+1},z_{3n+2},z_{3n+3}))-\phi (G_p(z_{3n+2},z_{3n+3},z_{3n+3}))$$

implies that $\phi (G_p(z_{3n+2},z_{3n+3},z_{3n+3}))=0$, that is, $z_{3n+1}=z_{3n+2}=z_{3n+3}$. Similarly, if $k_0=3n+1$, then $z_{3n+1}=z_{3n+2}=z_{3n+3}$ gives $z_{3n+2}=z_{3n+3}=z_{3n+4}$. Also, if $k_0=3n+2$, then $z_{3n+2}=z_{3n+3}=z_{3n+4}$ implies that $z_{3n+3}=z_{3n+4}=z_{3n+5}$. Consequently, the sequence $\{z_k\}$ becomes constant for $k\ge k_0$, hence $\{z_k\}$ is $G_p$-Cauchy.

Suppose that

$$z_k\ne z_{k+1}\ne z_{k+2}$$

(3)

for each k. We now claim that the following inequality holds:

$$G_p(z_{k+1},z_{k+2},z_{k+3})\le G_p(z_k,z_{k+1},z_{k+2})=M(x_k,x_{k+1},x_{k+2})$$

(4)

for each $k=1,2,3,\dots$ .

Let $k=3n$ and for $n\ge 0$, $G_p(z_{3n+1},z_{3n+2},z_{3n+3})>G_p(z_{3n},z_{3n+1},z_{3n+2})>0$. Then, as $T{x}_{3n}⪯R{x}_{3n+1}⪯S{x}_{3n+2}$, using (2) we obtain that

$$\begin{array}{rcl}\psi (G_p(z_{3n+1},z_{3n+2},z_{3n+3}))& \le & \psi (2G_p(z_{3n+1},z_{3n+2},z_{3n+3}))\\ =& \psi (2G_p(f{x}_{3n},g{x}_{3n+1},h{x}_{3n+2}))\\ \le & \psi (M(x_{3n},x_{3n+1},x_{3n+2}))-\phi (M(x_{3n},x_{3n+1},x_{3n+2})),\end{array}$$

(5)

where

$$\begin{array}{r}M(x_{3n},x_{3n+1},x_{3n+2})\\ =max\{G_p(T{x}_{3n},R{x}_{3n+1},S{x}_{3n+2}),\\ G_p(T{x}_{3n},f{x}_{3n},f{x}_{3n}),G_p(R{x}_{3n+1},g{x}_{3n+1},g{x}_{3n+1}),G_p(S{x}_{3n+2},h{x}_{3n+2},h{x}_{3n+2}),\\ \frac{G_p(T{x}_{3n},T{x}_{3n},f{x}_{3n})+G_p(R{x}_{3n+1},R{x}_{3n+1},g{x}_{3n+1})+G_p(S{x}_{3n+2},S{x}_{3n+2},h{x}_{3n+2})}{3}\}\\ =max\{G_p(z_{3n},z_{3n+1},z_{3n+2}),\\ G_p(z_{3n},z_{3n+1},z_{3n+1}),G_p(z_{3n+1},z_{3n+2},z_{3n+2}),G_p(z_{3n+2},z_{3n+3},z_{3n+3}),\\ \frac{G_p(z_{3n},z_{3n},z_{3n+1})+G_p(z_{3n+1},z_{3n+1},z_{3n+2})+G_p(z_{3n+2},z_{3n+2},z_{3n+3})}{3}\}\\ \le max\{G_p(z_{3n},z_{3n+1},z_{3n+2}),G_p(z_{3n+1},z_{3n+2},z_{3n+3}),\\ \frac{2G_p(z_{3n},z_{3n+1},z_{3n+2})+G_p(z_{3n+1},z_{3n+2},z_{3n+3})}{3}\}\\ =G_p(z_{3n+1},z_{3n+2},z_{3n+3}).\end{array}$$

Hence (5) implies that

$$\psi (G_p(z_{3n+1},z_{3n+2},z_{3n+3}))\le \psi (G_p(z_{3n+1},z_{3n+2},z_{3n+3}))-\phi (M(x_{3n},x_{3n+1},x_{3n+2})),$$

which is possible only if $M(x_{3n},x_{3n+1},x_{3n+2})=0$, that is, $G_p(z_{3n},z_{3n+1},z_{3n+2})=0$. A contradiction to (3). Hence, $G_p(z_{3n+1},z_{3n+2},z_{3n+3})\le G_p(z_{3n},z_{3n+1},z_{3n+2})$ and

$$M(x_{3n},x_{3n+1},x_{3n+2})=G_p(z_{3n},z_{3n+1},z_{3n+2}).$$

Therefore, (4) is proved for $k=3n$.

Similarly, it can be shown that

$$G_p(z_{3n+2},z_{3n+3},z_{3n+4})\le G_p(z_{3n+1},z_{3n+2},z_{3n+3})=M(x_{3n+1},x_{3n+2},x_{3n+3})$$

(6)

and

$$G_p(z_{3n+3},z_{3n+4},z_{3n+5})\le G_p(z_{3n+2},z_{3n+3},z_{3n+4})=M(x_{3n+2},x_{3n+3},x_{3n+4}).$$

(7)

Hence, $\{G_p(z_k,z_{k+1},z_{k+2})\}$ is a nonincreasing sequence of nonnegative real numbers. Therefore, there is $r\ge 0$ such that

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_k,z_{k+1},z_{k+2})=r.$$

(8)

Since

$$G_p(z_{k+1},z_{k+2},z_{k+3})\le M(x_k,x_{k+1},x_{k+2})\le G_p(z_k,z_{k+1},z_{k+2}),$$

(9)

taking the limit as $k\to \mathrm{\infty }$ in (9), we obtain

$$\underset{k\to \mathrm{\infty }}{lim}M(x_k,x_{k+1},x_{k+2})=r.$$

(10)

Taking the limit as $n\to \mathrm{\infty }$ in (5), using (8), (10) and the continuity of ψ and φ, we have $\psi (r)\le \psi (r)-\phi (r)$. Therefore, $\phi (r)=0$. Hence

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_k,z_{k+1},z_{k+2})=0$$

(11)

from our assumptions about φ. Also, from Definition 1.1, part ($G_{p}2$), we have

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_k,z_{k+1},z_{k+1})=0,$$

(12)

and since $G_p(x,y,y)\le 2G_p(x,x,y)$ for all $x,y\in X$, we have

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_k,z_k,z_{k+1})=0.$$

(13)

Step II. We now show that $\{z_n\}$ is a $G_p$-Cauchy sequence in X. Therefore, we will show that

$$\underset{m,n\to \mathrm{\infty }}{lim}{G}_p(z_m,z_n,z_n)=0.$$

Because of (11), (12) and (13), it is sufficient to show that

$$\underset{m,n\to \mathrm{\infty }}{lim}{G}_p(z_{3m},z_{3n},z_{3n})=0,$$

i.e., we prove that $\{z_{3n}\}$ is $G_p$-Cauchy.

Suppose the opposite. Then there exists $\epsilon >0$ for which we can find subsequences $\{z_{3m(k)}\}$ and $\{z_{3n(k)}\}$ of $\{z_{3n}\}$ such that $n(k)>m(k)\ge k$ and

$$G_p(z_{3m(k)},z_{3n(k)},z_{3n(k)})\ge \epsilon ,$$

(14)

and $n(k)$ is the smallest number such that the above statement holds; i.e.,

$$G_p(z_{3m(k)},z_{3n(k)-3},z_{3n(k)-3})<\epsilon .$$

(15)

From the rectangle inequality and (15), we have

$$\begin{array}{r}{G}_p(z_{3m(k)},z_{3n(k)},z_{3n(k)})\\ \le G_p(z_{3m(k)},z_{3n(k)-3},z_{3n(k)-3})+G_p(z_{3n(k)-3},z_{3n(k)},z_{3n(k)})\\ <\epsilon +G_p(z_{3n(k)-3},z_{3n(k)},z_{3n(k)})\\ <\epsilon +G_p(z_{3n(k)-3},z_{3n(k)-2},z_{3n(k)-2})+G_p(z_{3n(k)-2},z_{3n(k)-1},z_{3n(k)-1})\\ +G_p(z_{3n(k)-1},z_{3n(k)},z_{3n(k)}).\end{array}$$

(16)

Taking limit as $k\to \mathrm{\infty }$ in (16), from (12) and (14) we obtain that

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_{3m(k)},z_{3n(k)},z_{3n(k)})=\epsilon .$$

(17)

Using the rectangle inequality and ($G_{p}2$), we have

$$\begin{array}{r}{G}_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+2})\\ \le G_p(z_{3m(k)},z_{3n(k)},z_{3n(k)})+G_p(z_{3n(k)},z_{3n(k)+1},z_{3n(k)+2})\\ \le G_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+1})+G_p(z_{3n(k)+1},z_{3n(k)},z_{3n(k)})\\ +G_p(z_{3n(k)},z_{3n(k)+1},z_{3n(k)+2})\\ \le G_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+2})+G_p(z_{3n(k)+1},z_{3n(k)+2},z_{3n(k)+2})\\ +G_p(z_{3n(k)+1},z_{3n(k)},z_{3n(k)})+G_p(z_{3n(k)},z_{3n(k)+1},z_{3n(k)+2}).\end{array}$$

(18)

Taking limit as $k\to \mathrm{\infty }$, we have

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+2})\le \epsilon \le \underset{k\to \mathrm{\infty }}{lim}{G}_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+2}).$$

Finally,

$$\begin{array}{r}{G}_p(z_{3m(k)+1},z_{3n(k)+2},z_{3n(k)+3})\\ \le G_p(z_{3m(k)+1},z_{3m(k)},z_{3m(k)})+G_p(z_{3m(k)},z_{3n(k)+2},z_{3n(k)+3})\\ \le G_p(z_{3m(k)+1},z_{3m(k)},z_{3m(k)})+G_p(z_{3m(k)},z_{3n(k)},z_{3n(k)})\\ +G_p(z_{3n(k)},z_{3n(k)+2},z_{3n(k)+3}).\end{array}$$

(19)

Taking limit as $k\to \mathrm{\infty }$ and using (17), we have

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_{3m(k)+1},z_{3n(k)+2},z_{3n(k)+3})\le \epsilon .$$

Consider,

$$\begin{array}{r}{G}_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+2})\\ \le G_p(z_{3m(k)},z_{3m(k)+1},z_{3m(k)+1})+G_p(z_{3m(k)+1},z_{3n(k)+1},z_{3n(k)+2})\\ \le G_p(z_{3m(k)},z_{3m(k)+1},z_{3m(k)+1})+G_p(z_{3m(k)+1},z_{3n(k)+3},z_{3n(k)+3})\\ +G_p(z_{3n(k)+3},z_{3n(k)+1},z_{3n(k)+2})\\ \le G_p(z_{3m(k)},z_{3m(k)+1},z_{3m(k)+1})+G_p(z_{3m(k)+1},z_{3n(k)+2},z_{3n(k)+3})\\ +G_p(z_{3n(k)+1},z_{3n(k)+2},z_{3n(k)+3}).\end{array}$$

(20)

Taking limit as $k\to \mathrm{\infty }$ and using (11), (12) and (13), we have

$$\epsilon \le \underset{k\to \mathrm{\infty }}{lim}{G}_p(z_{3m(k)+1},z_{3n(k)+2},z_{3n(k)+3}).$$

Therefore,

$$\underset{k\to \mathrm{\infty }}{lim}{G}_p(z_{3m(k)+1},z_{3n(k)+2},z_{3n(k)+3})=\epsilon .$$

(21)

As $T{x}_{m(k)}⪯R{x}_{n(k)+1}⪯S{x}_{n(k)+2}$, so from (2) we have

$$\begin{array}{r}\psi (G_p(z_{3m(k)+1},z_{3n(k)+2},z_{3n(k)+3}))\\ =\psi (G_p(f{x}_{3m(k)},g{x}_{3n(k)+1},h{x}_{3n(k)+2}))\\ \le \psi (M(x_{3m(k)},x_{3n(k)+1},x_{3n(k)+2}))-\phi (M(x_{3m(k)},x_{3n(k)+1},x_{3n(k)+2})),\end{array}$$

(22)

where

$$\begin{array}{r}M(x_{3m(k)},x_{3n(k)+1},x_{3n(k)+2})\\ =max\{G_p(T{x}_{3m(k)},R{x}_{3n(k)+1},S{x}_{3n(k)+2}),G_p(T{x}_{3m(k)},f{x}_{3m(k)},f{x}_{3m(k)}),\\ G_p(R{x}_{3n(k)+1},g{x}_{3n(k)+1},g{x}_{3n(k)+1}),G_p(S{x}_{3n(k)+2},h{x}_{3n(k)+2},h{x}_{3n(k)+2}),\\ \frac{\begin{array}{c}{G}_p(T{x}_{3m(k)},T{x}_{3m(k)},f{x}_{3m(k)})+G_p(R{x}_{3n(k)+1},R{x}_{3n(k)+1},g{x}_{3n(k)+1})\\ +G_p(S{x}_{3n(k)+2},S{x}_{3n(k)+2},h{x}_{3n(k)+2})\end{array}}{3}\}\\ =max\{G_p(z_{3m(k)},z_{3n(k)+1},z_{3n(k)+2}),G_p(z_{3m(k)},z_{3m(k)+1},z_{3m(k)+1}),\\ G_p(z_{3n(k)+1},z_{3n(k)+2},z_{3n(k)+2}),G_p(z_{3n(k)+2},z_{3n(k)+3},z_{3n(k)+3}),\\ \frac{\begin{array}{c}{G}_p(z_{3m(k)},z_{3m(k)},z_{3m(k)+1})+G_p(z_{3n(k)+1},z_{3n(k)+1},z_{3n(k)+2})\\ +G_p(z_{3n(k)+2},z_{3n(k)+2},z_{3n(k)+3})\end{array}}{3}\}.\end{array}$$

Taking limit as $k\to \mathrm{\infty }$ and using (12), (13), (17), (21) in (22), we have

$$\psi (\epsilon )\le \psi (\epsilon )-\phi (\epsilon )<\psi (\epsilon ),$$

a contradiction. Hence, $\{z_n\}$ is a $G_p$-Cauchy sequence.

Step III. We will show that f, g, h, R, S and T have a coincidence point.

Since $\{z_n\}$ is a $G_p$-Cauchy sequence in the complete $G_p$-metric space X, from Lemma 1.2, $\{z_n\}$ is a Cauchy sequence in the metric space $(X,d_{G_p})$. Completeness of $(X,G_p)$ yields that $(X,d_{G_p})$ is also complete. Then there exists $z^{\ast }\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}{d}_{G_p}(z_n,z^{\ast })=0.$$

(23)

Now, since ${lim}_{m,n\to \mathrm{\infty }}{G}_p(z_m,z_n,z_n)=0$, (23) and part (2) of Lemma 1.2 yield that $G_p(z^{\ast },z^{\ast },z^{\ast })=0$.

Since $R(X)$ is $G_p$-complete and $\{z_{3n+1}\}\subseteq R(X)$, there exists $u\in X$ such that $z^{\ast }=Ru$ and

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}{G}_p(z_{3n+1},z_{3n+1},Ru)\\ =\underset{n\to \mathrm{\infty }}{lim}{G}_p(R{x}_{3n+1},R{x}_{3n+1},Ru)=\underset{n\to \mathrm{\infty }}{lim}{G}_p(f{x}_{3n},f{x}_{3n},Ru)=G(Ru,Ru,Ru)=0.\end{array}$$

(24)

By similar arguments, there exist $v,w\in X$ such that $z^{\ast }=Sv=Tw$ and

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}{G}_p(z_{3n+2},z_{3n+2},z^{\ast })\\ =\underset{n\to \mathrm{\infty }}{lim}{G}_p(S{x}_{3n+2},S{x}_{3n+2},z^{\ast })=\underset{n\to \mathrm{\infty }}{lim}{G}_p(g{x}_{3n+1},g{x}_{3n+1},z^{\ast })=G(z^{\ast },z^{\ast },z^{\ast })=0\end{array}$$

(25)

and

$$\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim}{G}_p(z_{3n+3},z_{3n+3},z^{\ast })\\ =\underset{n\to \mathrm{\infty }}{lim}{G}_p(T{x}_{3n+3},T{x}_{3n+3},z^{\ast })=\underset{n\to \mathrm{\infty }}{lim}{G}_p(h{x}_{3n+2},h{x}_{3n+2},z^{\ast })=G(z^{\ast },z^{\ast },z^{\ast })=0.\end{array}$$

(26)

Now, we prove that w is a coincidence point of f and T.

Since $S{x}_{3n+2}\to z^{\ast }=Tw=Ru$ as $n\to \mathrm{\infty }$, so $S{x}_{3n+2}⪯Tw=Ru$. Therefore, from (2), we have

$$\begin{array}{r}\psi (G_p(fw,gu,h{x}_{3n+2}))\le \psi (M(w,u,x_{3n+2}))-\phi (M(w,u,x_{3n+2})),\end{array}$$

(27)

where

$$\begin{array}{r}M(w,u,x_{3n+2})\\ =max\{G_p(Tw,Ru,S{x}_{3n+2}),G(Tw,fw,fw),\\ G_p(Ru,gu,gu),G(S{x}_{3n+2},h{x}_{3n+2},h{x}_{3n+2}),\\ \frac{G_p(Tw,Tw,fw)+G(Ru,Ru,gu)+G_p(S{x}_{3n+2},S{x}_{3n+2},h{x}_{3n+2})}{3}\}.\end{array}$$

Taking limit as $n\to \mathrm{\infty }$ in (27), as $G(z^{\ast },z^{\ast },z^{\ast })=0$, from Lemma 1.3, we obtain that

$$\begin{array}{r}\psi \left(G_p(fw,gu,z^{\ast })\right)\\ \le \psi \left(G_p(fw,gu,z^{\ast })\right)\\ -\phi (max\{G_p(z^{\ast },fw,fw),G_p(z^{\ast },gu,gu),\frac{G_p(z^{\ast },z^{\ast },fw)+G_p(z^{\ast },z^{\ast },gu)}{3}\}),\end{array}$$

which implies that $gu=fw=z^{\ast }=Tw=Ru$.

As f and T are weakly compatible, we have $f{z}^{\ast }=fTw=Tfw=T{z}^{\ast }$. Thus $z^{\ast }$ is a coincidence point of f and T.

Similarly it can be shown that $z^{\ast }$ is a coincidence point of the pairs $(g,R)$ and $(h,S)$.

Now, let $R{z}^{\ast }$, $S{z}^{\ast }$ and $T{z}^{\ast }$ be comparable. By (2) we have

$$\psi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)\le \psi \left(M(z^{\ast },z^{\ast },z^{\ast })\right)-\phi \left(M(z^{\ast },z^{\ast },z^{\ast })\right),$$

(28)

where

$$\begin{array}{rl}M(z^{\ast },z^{\ast },z^{\ast })=& max\{G_p(T{z}^{\ast },R{z}^{\ast },S{z}^{\ast }),\\ G_p(T{z}^{\ast },f{z}^{\ast },f{z}^{\ast }),G_p(R{z}^{\ast },g{z}^{\ast },g{z}^{\ast }),G_p(S{z}^{\ast },h{z}^{\ast },h{z}^{\ast }),\\ \frac{G_p(T{z}^{\ast },T{z}^{\ast },f{z}^{\ast })+G_p(R{z}^{\ast },R{z}^{\ast },g{z}^{\ast })+G_p(S{z}^{\ast },S{z}^{\ast },h{z}^{\ast })}{3}\}\\ =& G_p(T{z}^{\ast },R{z}^{\ast },S{z}^{\ast })=G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast }).\end{array}$$

Hence (28) gives

$$\psi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)\le \psi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)-\phi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)=0.$$

Therefore $f{z}^{\ast }=g{z}^{\ast }=h{z}^{\ast }=T{z}^{\ast }=R{z}^{\ast }=S{z}^{\ast }$. □

Theorem 2.2 Let $(X,⪯,G_p)$ be a partially ordered complete $G_p$-metric space. Let $f,g,h:X\to X$ be three mappings. Suppose that for every three comparable elements $x,y,z\in X$, we have

$$\psi (2G_p(fx,gy,hz))\le \psi (M(x,y,z))-\phi (M(x,y,z)),$$

(29)

where

$$\begin{array}{rl}M(x,y,z)=& max\{G_p(x,y,z),\\ G_p(x,fx,fx),G_p(y,gy,gy),G_p(z,hz,hz),\\ \frac{G_p(x,x,fx)+G_p(y,y,gy)+G_p(z,z,hz)}{3}\}\end{array}$$

and $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are altering distance functions. Let f, g, h be continuous and the pairs $(f,g)$, $(g,h)$ and $(h,f)$ be partially weakly increasing. Then f, g and h have a common fixed point $z^{\ast }$ in X.

Proof Let $x_0$ be an arbitrary point and $x_{3n+1}=f{x}_{3n}$, $x_{3n+2}=g{x}_{3n+1}$ and $x_{3n+3}=h{x}_{3n+2}$ for all $n\ge 0$.

Following the proof of the previous theorem, we can show that there exists $z^{\ast }\in X$ such that

$$G_p(z^{\ast },z^{\ast },z^{\ast })=0$$

(30)

and

$$\underset{n\to \mathrm{\infty }}{lim}{G}_p(x_{3n},x_{3n},z^{\ast })=0.$$

(31)

Continuity of f yields that

$$\underset{n\to \mathrm{\infty }}{lim}{G}_p(f{x}_{3n},f{x}_{3n},f{z}^{\ast })=G_p(f{z}^{\ast },f{z}^{\ast },f{z}^{\ast }).$$

(32)

By the rectangle inequality, we have

$$G_p(f{z}^{\ast },z^{\ast },z^{\ast })\le G_p(f{z}^{\ast },f{x}_{3n},f{x}_{3n})+G_p(x_{3n+1},z^{\ast },z^{\ast })$$

(33)

and

$$G_p(f{z}^{\ast },f{z}^{\ast },z^{\ast })\le G_p(z^{\ast },f{x}_{3n},f{x}_{3n})+G_p(f{x}_{3n},f{z}^{\ast },f{z}^{\ast }).$$

(34)

Taking limit as $n\to \mathrm{\infty }$ in (33) and (34), from (30) we obtain

$$G_p(f{z}^{\ast },z^{\ast },z^{\ast })\le G_p(f{z}^{\ast },f{z}^{\ast },f{z}^{\ast })$$

and

$$G_p(f{z}^{\ast },f{z}^{\ast },z^{\ast })\le G_p(f{z}^{\ast },f{z}^{\ast },f{z}^{\ast }).$$

Similar inequalities are obtained for g and h.

On the other hand, as $z^{\ast }⪯z^{\ast }⪯z^{\ast }$, using (29) we obtain that

$$\begin{array}{rl}\psi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)& \le \psi \left(2G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)\\ \le \psi \left(M(z^{\ast },z^{\ast },z^{\ast })\right)-\phi \left(M(z^{\ast },z^{\ast },z^{\ast })\right),\end{array}$$

(35)

where

$$\begin{array}{rl}M(z^{\ast },z^{\ast },z^{\ast })=& max\{G_p(z^{\ast },z^{\ast },z^{\ast }),\\ G_p(z^{\ast },f{z}^{\ast },f{z}^{\ast }),G_p(z^{\ast },g{z}^{\ast },g{z}^{\ast }),G_p(z^{\ast },h{z}^{\ast },h{z}^{\ast }),\\ \frac{G_p(z^{\ast },z^{\ast },f{z}^{\ast })+G_p(z^{\ast },z^{\ast },g{z}^{\ast })+G_p(z^{\ast },z^{\ast },h{z}^{\ast })}{3}\}\\ \le & max\{G_p(f{z}^{\ast },f{z}^{\ast },f{z}^{\ast }),G_p(g{z}^{\ast },g{z}^{\ast },g{z}^{\ast }),G_p(h{z}^{\ast },h{z}^{\ast },h{z}^{\ast })\}.\end{array}$$

(36)

We consider three cases as follows:

• 1. $f{z}^{\ast }=g{z}^{\ast }=h{z}^{\ast }$.

• 2. $f{z}^{\ast }\ne g{z}^{\ast }\ne h{z}^{\ast }$.

• 3. a. $f{z}^{\ast }=g{z}^{\ast }\ne h{z}^{\ast }$, or b. $f{z}^{\ast }\ne g{z}^{\ast }=h{z}^{\ast }$.

For case 1, by (36), $M(z^{\ast },z^{\ast },z^{\ast })\le G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })$.

For case 2, by ($G_{p}2$), $M(z^{\ast },z^{\ast },z^{\ast })\le G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })$.

Now, from (35),

$$\psi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)\le \psi \left(G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)-\phi \left(M(z^{\ast },z^{\ast },z^{\ast })\right),$$

(37)

hence $M(z^{\ast },z^{\ast },z^{\ast })=0$. Therefore, $z^{\ast }=f{z}^{\ast }=g{z}^{\ast }=h{z}^{\ast }$.

On the other hand, for case 3, part a, by ($G_{p}2$), $M(z^{\ast },z^{\ast },z^{\ast })\le 2G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })$ and hence from (35), we have

$$\psi \left(2G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)\le \psi \left(2G_p(f{z}^{\ast },g{z}^{\ast },h{z}^{\ast })\right)-\phi \left(M(z^{\ast },z^{\ast },z^{\ast })\right),$$

(38)

hence $M(z^{\ast },z^{\ast },z^{\ast })=0$. Therefore, $z^{\ast }=f{z}^{\ast }=g{z}^{\ast }=h{z}^{\ast }$.

Now, let $x^{\ast }$ and $y^{\ast }$ as two fixed points of f, g and h be comparable. So, from (29) we have

$$\begin{array}{rl}\psi \left(2G_p(x^{\ast },x^{\ast },y^{\ast })\right)& =\psi \left(2G_p(f{x}^{\ast },g{x}^{\ast },h{y}^{\ast })\right)\\ \le \psi \left(M(x^{\ast },x^{\ast },y^{\ast })\right)-\phi \left(M(x^{\ast },x^{\ast },y^{\ast })\right),\end{array}$$

(39)

where

$$\begin{array}{rl}M(x^{\ast },x^{\ast },y^{\ast })=& max\{G_p(x^{\ast },x^{\ast },y^{\ast }),\\ G_p(x^{\ast },f{x}^{\ast },f{x}^{\ast }),G_p(x^{\ast },g{x}^{\ast },g{x}^{\ast }),G_p(y^{\ast },h{y}^{\ast },h{y}^{\ast }),\\ \frac{G_p(x^{\ast },x^{\ast },f{x}^{\ast })+G_p(x^{\ast },x^{\ast },g{x}^{\ast })+G_p(y^{\ast },y^{\ast },h{y}^{\ast })}{3}\}\\ \le & 2G_p(x^{\ast },x^{\ast },y^{\ast }).\end{array}$$

Hence (39) gives

$$\psi \left(2G_p(x^{\ast },x^{\ast },y^{\ast })\right)\le \psi \left(2G_p(x^{\ast },x^{\ast },y^{\ast })\right)-\phi \left(M(x^{\ast },x^{\ast },y^{\ast })\right).$$

Therefore, $\phi (M(x^{\ast },x^{\ast },y^{\ast }))=0$ and hence $x^{\ast }=y^{\ast }$. □

The following corollaries are special cases of the above results.

Corollary 2.1 Let $(X,⪯,G_p)$ be a partially ordered complete $G_p$-metric space. Let $f:X\to X$ be a mapping such that for every three comparable elements $x,y,z\in X$, we have

$$\psi (2G_p(fx,fy,fz))\le \psi (M(x,y,z))-\phi (M(x,y,z)),$$

(40)

where

$$\begin{array}{rl}M(x,y,z)=& max\{G_p(x,y,z),\\ G_p(x,fx,fx),G_p(y,fy,fy),G_p(z,fz,fz),\\ \frac{G_p(x,x,fx)+G_p(y,y,fy)+G_p(z,z,fz)}{3}\}\end{array}$$

and $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are altering distance functions. Then f has a fixed point in X provided that $fx⪯f(fx)$ for all $x\in X$ and either

• a. f is continuous, or

• b. X has the sequential limit comparison property.

Moreover, f has a unique fixed point provided that the fixed points of f are comparable.

Taking $y=z$ in Corollary 2.1, we obtain the following common fixed point result.

Corollary 2.2 Let $(X,⪯,G_p)$ be a partially ordered complete $G_p$-metric space, and let f be a self-mapping on X such that for every comparable elements $x,y\in X$,

$$\psi (2G_p(fx,fy,fy))\le \psi (M(x,y,y))-\phi (M(x,y,y)),$$

(41)

where

$$M(x,y,y)=max\{G_p(x,y,y),G(x,fx,fx),G_p(y,fy,fy),\frac{G_p(x,x,fx)+2G_p(y,y,fy)}{3}\},$$

and $\psi ,\phi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ are altering distance functions. Then f has a fixed point in X provided that $fx⪯f(fx)$ for all $x\in X$ and either

• a. f is continuous, or

• b. X has the sequential limit comparison property.

3 Fixed point results via an α-admissible mapping with respect to η in $G_p$-metric spaces

Samet et al. [32] defined the notion of α-admissible mappings and proved the following result.

Definition 3.1 Let T be a self-mapping on X and $\alpha :X\times X\to [0,+\mathrm{\infty })$ be a function. We say that T is an α-admissible mapping if

$$x,y\in X,\alpha (x,y)\ge 1⟹\alpha (Tx,Ty)\ge 1.$$

Denote with Ψ the family of all nondecreasing functions $\psi :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ such that ${\sum }_{n=1}^{\mathrm{\infty }}{\psi }^n(t)<+\mathrm{\infty }$ for all $t>0$, where ${\psi }^n$ is the n th iterate of ψ.

Theorem 3.1 Let $(X,d)$ be a complete metric space and T be an α-admissible mapping. Assume that

$$\alpha (x,y)d(Tx,Ty)\le \psi (d(x,y)),$$

(42)

where $\psi \in \mathrm{\Psi }$. Also suppose that the following assertions hold:

• (i) there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge 1$;

• (ii) either T is continuous or for any sequence $\{x_n\}$ in X with $\alpha (x_n,x_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and $x_n\to x$ as $n\to +\mathrm{\infty }$, we have $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$.

Then T has a fixed point.

For more details on α-admissible mappings, we refer the reader to [33–37].

Very recently, Salimi et al. [38] modified and generalized the notions of α-ψ-contractive mappings and α-admissible mappings as follows.

Definition 3.2 [38]

Let T be a self-mapping on X and $\alpha ,\eta :X\times X\to [0,+\mathrm{\infty })$ be two functions. We say that T is an α-admissible mapping with respect to η if

$$x,y\in X,\alpha (x,y)\ge \eta (x,y)⟹\alpha (Tx,Ty)\ge \eta (Tx,Ty).$$

Note that if we take $\eta (x,y)=1$, then this definition reduces to Definition 3.1. Also, if we take $\alpha (x,y)=1$, then we say that T is an η-subadmissible mapping.

The following result properly contains Theorem 3.1 and Theorems 2.3 and 2.4 of [37].

Theorem 3.2 [38]

Let $(X,d)$ be a complete metric space and T be an α-admissible mapping with respect to η. Assume that

$$x,y\in X,\alpha (x,y)\ge \eta (x,y)⟹d(Tx,Ty)\le \psi (M(x,y)),$$

(43)

where $\psi \in \mathrm{\Psi }$ and

$$M(x,y)=max\{d(x,y),\frac{d(x,Tx)+d(y,Ty)}{2},\frac{d(x,Ty)+d(y,Tx)}{2}\}.$$

Also, suppose that the following assertions hold:

• (i) there exists $x_0\in X$ such that $\alpha (x_0,T{x}_0)\ge \eta (x_0,T{x}_0)$;

• (ii) either T is continuous or for any sequence $\{x_n\}$ in X with $\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$ and $x_n\to x$ as $n\to +\mathrm{\infty }$, we have $\alpha (x_n,x)\ge \eta (x_n,x)$ for all $n\in \mathbb{N}\cup \{0\}$.

Then T has a fixed point.

In fact, the Banach contraction principle and Theorem 3.2 hold for the following example, but Theorem 3.1 does not hold.

Example 3.1 [38]

Let $X=[0,\mathrm{\infty })$ be endowed with the usual metric $d(x,y)=|x-y|$ for all $x,y\in X$, and let $T:X\to X$ be defined by $Tx=\frac{1}{4}x$. Also, define $\alpha :X^2\to [0,\mathrm{\infty })$ by $\alpha (x,y)=3$ and $\psi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ by $\psi (t)=\frac{1}{2}t$.

Theorem 3.3 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous α-admissible mapping with respect to η on X, there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge \eta (x_0,f{x}_0)$ and if any sequence $\{x_n\}$ in X converges to a point x, then we have $\alpha (x,x)\ge \eta (x,x)$. Assume that

$$\begin{array}{r}\alpha (x,y)\ge \eta (x,y)\\ ⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}\end{array}$$

(44)

for all $x,y\in X$, where $0\le r<1$. Then f has a fixed point.

Proof Let $x_0\in X$ and define a sequence $\{x_n\}$ by $x_n=f^n{x}_0$ for all $n\in \mathbb{N}$. Since f is an α-admissible mapping with respect to η and $\alpha (x_0,x_1)=\alpha (x_0,f{x}_0)\ge \eta (x_0,f{x}_0)=\eta (x_0,x_1)$, we deduce that $\alpha (x_1,x_2)=\alpha (f{x}_0,f{x}_1)\ge \eta (f{x}_0,f{x}_1)=\eta (x_1,x_2)$. Continuing this process, we get $\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$. Now, from (44) we have

$$\begin{array}{r}{G}_p(f{f}^n{x}_0,f^2{f}^n{x}_0,f^2{f}^n{x}_0)\\ \le rmax\{G_p(f^n{x}_0,f{f}^n{x}_0,f{f}^n{x}_0),G_p(f{f}^n{x}_0,f^2{f}^n{x}_0,f^2{f}^n{x}_0)\},\end{array}$$

which implies

$$G_p(f^{n+1}{x}_0,f^{n+2}{x}_0,f^{n+2}{x}_0)\le r{G}_p(f^n{x}_0,f^{n+1}{x}_0,f^{n+1}{x}_0).$$

(45)

Continuing the above process, we can obtain

$$G_p(f^n{x}_0,f^{n+1}{x}_0,f^{n+1}{x}_0)\le r{G}_p(f^{n-1}{x}_0,f^n{x}_0,f^n{x}_0)\le \cdots \le r^n{G}_p(x_0,f{x}_0,f{x}_0).$$

(46)

Then, for any $m>n$, by (46) we get

$$\begin{array}{rl}{G}_p(f^n{x}_0,f^m{x}_0,f^m{x}_0)\le & G_p(f^n{x}_0,f^{n+1}{x}_0,f^{n+1}{x}_0)+G_p(f^{n+1}{x}_0,f^m{x}_0,f^m{x}_0)\\ \le & G_p(f^n{x}_0,f^{n+1}{x}_0,f^{n+1}{x}_0)+G_p(f^{n+1}{x}_0,f^{n+2}{x}_0,f^{n+2}{x}_0)\\ +G_p(f^{n+2}{x}_0,f^m{x}_0,f^m{x}_0)\\ \le & G(f^n{x}_0,f^{n+1}{x}_0,f^{n+1}{x}_0)+G_p(f^{n+1}{x}_0,f^{n+2}{x}_0,f^{n+2}{x}_0)\\ +G_p(f^{n+2}{x}_0,f^{n+3}{x}_0,f^{n+3}{x}_0)+\cdots +G_p(f^{m-1}{x}_0,f^m{x}_0,f^m{x}_0)\\ \le & \frac{r^n}{1-r}{G}_p(x_0,f{x}_0,f{x}_0).\end{array}$$

This implies that ${lim}_{m,n\to +\mathrm{\infty }}{G}_p(f^n{x}_0,f^m{x}_0,f^m{x}_0)=0$, that is, $\{x_n\}$ is a $G_p$-Cauchy sequence.

Since $\{x_n\}$ is a $G_p$-Cauchy sequence in the complete $G_p$-metric space X, from Lemma 1.2, $\{x_n\}$ is a Cauchy sequence in the metric space $(X,d_{G_p})$. Completeness of $(X,G_p)$ yields that $(X,d_{G_p})$ is also complete. Then there exists $z\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}{d}_{G_p}(x_n,z)=0.$$

(47)

Since ${lim}_{m,n\to +\mathrm{\infty }}{G}_p(x_n,x_m,x_m)=0$, from Lemma 1.2 we get

$$\underset{n\to +\mathrm{\infty }}{lim}{G}_p(x_n,z,z)=\underset{n\to +\mathrm{\infty }}{lim}{G}_p(x_n,x_n,z)=G_p(z,z,z)=0.$$

(48)

From the continuity of f, we have

$$\underset{n\to +\mathrm{\infty }}{lim}{G}_p(x_{n+1},fz,fz)=G_p(fz,fz,fz),$$

and hence we get

$$G_p(z,fz,fz)\le \underset{n\to +\mathrm{\infty }}{lim}G(z,x_{n+1},x_{n+1})+\underset{n\to +\mathrm{\infty }}{lim}G(x_{n+1},fz,fz)=G_p(fz,fz,fz).$$

So, we get that $G_p(z,fz,fz)\le G_p(fz,fz,fz)$. Since the opposite inequality always holds, we get that

$$G_p(z,fz,fz)=G_p(fz,fz,fz).$$

As $\alpha (z,z)\ge \eta (z,z)$ we have

$$G_p(z,fz,fz)=G_p(fz,fz,fz)\le rmax\{G_p(z,z,z),G_p(z,fz,fz),G_p(z,fz,fz)\},$$

(49)

where $0\le r<1$. Hence, $G_p(z,fz,fz)\le r{G}_p(z,fz,fz)$. Thus, $G_p(z,fz,fz)=0$, that is, $z=fz$. □

If in Theorem 3.3 we take $\eta (x,y)=1$, then we deduce the following corollary.

Corollary 3.1 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$$\alpha (x,y)\ge 1⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

for all $x,y\in X$, where $0\le r<1$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $\alpha (x,x)\ge 1$. Then f has a fixed point.

If in Theorem 3.3 we take $\alpha (x,y)=1$, then we deduce the following corollary.

Corollary 3.2 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous η-subadmissible mapping on X, and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$$\eta (x,y)\le 1⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

(50)

for all $x,y\in X$, where $0\le r<1$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $1\ge \eta (x,x)$. Then f has a fixed point.

In the following theorem, we omit the continuity of the mapping f.

Theorem 3.4 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space and f be an α-admissible mapping with respect to η on X such that

$$\begin{array}{r}\alpha (x,y)\ge \eta (x,y)\\ ⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}\end{array}$$

(51)

for all $x,y\in X$, where $0\le r<1$. Assume that the following conditions hold:

• (i) there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge \eta (x_0,f{x}_0)$;

• (ii) if $\{x_n\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\alpha (x_n,x)\ge \eta (x_n,x)$ for all $n\in \mathbb{N}\cup \{0\}$.

Then f has a fixed point.

Proof Let $x_0\in X$ be such that $\alpha (x_0,f{x}_0)\ge \eta (x_0,f{x}_0)$ and define a sequence $\{x_n\}$ in X by $x_n=f^n{x}_0=f{x}_{n-1}$ for all $n\in \mathbb{N}$. Following the proof of Theorem 3.1, we have $\alpha (x_n,x_{n+1})\ge \eta (x_n,x_{n+1})$ for all $n\in \mathbb{N}\cup \{0\}$ and there exists $x\in X$ such that $x_n\to x$ as $n\to +\mathrm{\infty }$. Hence, from (ii) we deduce that $\alpha (x_n,x)\ge \eta (x_n,x)$ for all $n\in \mathbb{N}\cup \{0\}$.

Hence, by (51), it follows that for all n,

$$G_p(x_{n+1},fx,fx)\le rmax\{G_p(x_n,x,x),G_p(x_n,x_{n+1},x_{n+1}),G_p(x,fx,fx)\}.$$

Taking the limit as $n\to +\mathrm{\infty }$ in the above inequality, from Lemma 1.3 we obtain $(1-r)G(x,fx,fx)\le 0$, which implies that $x=fx$. □

Corollary 3.3 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space and f be an α-admissible mapping on X such that

$$\alpha (x,y)\ge 1⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

(52)

for all $x,y\in X$, where $0\le r<1$. Assume that the following conditions hold:

• (i) there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$;

• (ii) if $\{x_n\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$.

Then f has a fixed point.

Example 3.2 Let $X=[0,+\mathrm{\infty })$ and $G_p(x,y,z)=max\{x,y,z\}$ be a $G_p$-metric on X. Define $f:X\to X$ by

$$fx=\{\begin{array}{cc}\frac{x}{24}\hfill & \text{if }x\in [0,1]\cup \{2\}=U,\hfill \\ 37/12\hfill & \text{if }x=3,\hfill \\ {(1+x)}^x\hfill & \text{if }x\in [0,+\mathrm{\infty })\setminus ([0,1]\cup \{2,3\})=V,\hfill \end{array}$$

and $\alpha :X\times X\to [0,+\mathrm{\infty })$ by

$$\alpha (x,y)=\{\begin{array}{cc}1\hfill & \text{if }x,y\in [0,1],\hfill \\ 1/8\hfill & \text{if }x=2\text{ and }y=3,\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

Now, we prove that all the hypotheses of Corollary 3.3 are satisfied and hence f has a fixed point.

Let $x,y\in X$, if $\alpha (x,y)\ge 1$, then $x,y\in [0,1]$. On the other hand, for all $x\in [0,1]$, we have $fx\le 1$ and hence $\alpha (fx,fy)\ge 1$. This implies that f is an α-admissible mapping on X. Obviously, $\alpha (0,f0)\ge 1$.

Now, if $\{x_n\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$ and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\{x_n\}\subseteq [0,1]$ and hence $x\in [0,1]$. This implies that $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$.

If $\alpha (x,y)\ge 1$, then $x,y\in [0,1]$. Hence,

$$\begin{array}{rl}{G}_p(fx,fy,fy)& =max\{fx,fy\}=max\{\frac{x}{24},\frac{y}{24}\}\\ \le \frac{1}{12}max\{x,y\}\\ \le \frac{1}{12}max\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}.\end{array}$$

Thus, all the conditions of Corollary 3.3 are satisfied and therefore f has a fixed point ($x=0$).

Corollary 3.4 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space and f be an η-subadmissible mapping on X such that

$$\eta (x,y)\le 1⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

for all $x,y\in X$, where $0\le r<1$. Assume that the following conditions hold:

• (i) there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$;

• (ii) if $\{x_n\}$ is a sequence in X such that $\eta (x_n,x_{n+1})\le 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\eta (x_n,x)\le 1$ for all $n\in \mathbb{N}\cup \{0\}$.

Then f has a fixed point.

4 Consequences

Theorem 4.1 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$$\alpha (x,y)G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

(53)

for all $x,y\in X$, where $0\le r<1$ and if any sequence $\{x_n\}$ in X converges to a point x, then we have $\alpha (x,x)\ge \eta (x,x)$. Then f has a fixed point.

Proof Assume that $\alpha (x,y)\ge 1$, then from (53) we get

$$G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}.$$

That is,

$$\alpha (x,y)\ge 1⟹G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}.$$

Hence all the conditions of Corollary 3.1 hold and f has a fixed point. □

Similarly, we can deduce the following results.

Theorem 4.2 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$${(G_p(fx,fy,fy)+\ell )}^{\alpha (x,y)}\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}+\ell$$

for all $x,y\in X$, where $0\le r<1$ and $\ell \ge 1$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $\alpha (x,x)\ge 1$. Then f has a fixed point.

Theorem 4.3 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$${(\alpha (x,y)+\ell )}^{G_p(fx,fy,fy)}\le {(1+\ell )}^{rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}}$$

(54)

for all $x,y\in X$, where $0\le r<1$ and $\ell >0$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $\alpha (x,x)\ge 1$. Then f has a fixed point.

Theorem 4.4 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous η-subadmissible mapping on X, and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$$G_p(fx,fy,fy)\le r\eta (x,y)max\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

(55)

for all $x,y\in X$, where $0\le r<1$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $1\ge \eta (x,x)$. Then f has a fixed point.

Theorem 4.5 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous η-subadmissible mapping on X, and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$$G_p(fx,fy,fy)+\ell \le {(rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}+\ell )}^{\eta (x,y)}$$

for all $x,y\in X$, where $0\le r<1$ and $\ell \ge 1$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $1\ge \eta (x,x)$. Then f has a fixed point.

Theorem 4.6 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be a continuous η-subadmissible mapping on X, and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$${(1+\ell )}^{G_p(fx,fy,fy)}\le {(\eta (x,y)+\ell )}^{rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}}$$

(56)

for all $x,y\in X$, where $0\le r<1$ and $\ell >0$, and if any sequence $\{x_n\}$ in X converges to a point x, then we have $1\ge \eta (x,x)$. Then f has a fixed point.

Theorem 4.7 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be an α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$$\alpha (x,y)G_p(fx,fy,fy)\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

for all $x,y\in X$, where $0\le r<1$. If $\{x_n\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$, then f has a fixed point.

Theorem 4.8 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be an α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$${(G_p(fx,fy,fy)+\ell )}^{\alpha (x,y)}\le rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}+\ell$$

for all $x,y\in X$, where $0\le r<1$ and $\ell \ge 1$. If $\{x_n\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$, then f has a fixed point.

Theorem 4.9 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be an α-admissible mapping on X, and there exists $x_0\in X$ such that $\alpha (x_0,f{x}_0)\ge 1$. Assume that

$${(\alpha (x,y)+\ell )}^{G_p(fx,fy,fy)}\le {(1+\ell )}^{rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}}$$

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for all $x,y\in X$, where $0\le r<1$ and $\ell >0$. If $\{x_n\}$ is a sequence in X such that $\alpha (x_n,x_{n+1})\ge 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\alpha (x_n,x)\ge 1$ for all $n\in \mathbb{N}\cup \{0\}$, then f has a fixed point.

Theorem 4.10 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be an η-subadmissible mapping on X, and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$$G_p(fx,fy,fy)\le r\eta (x,y)max\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}$$

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for all $x,y\in X$, where $0\le r<1$. If $\{x_n\}$ is a sequence in X such that $\eta (x_n,x_{n+1})\le 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, we have $\eta (x_n,x)\le 1$ for all $n\in \mathbb{N}\cup \{0\}$, then f has a fixed point.

Theorem 4.11 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be an η-subadmissible mapping on X and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$$G_p(fx,fy,fy)+\ell \le {(rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}+\ell )}^{\eta (x,y)}$$

for all $x,y\in X$, where $0\le r<1$ and $\ell \ge 1$. If $\{x_n\}$ is a sequence in X such that $\eta (x_n,x_{n+1})\le 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, we have $\eta (x_n,x)\le 1$ for all $n\in \mathbb{N}\cup \{0\}$, then f has a fixed point.

Theorem 4.12 Let $(X,G_p)$ be a $G_p$-complete $G_p$-metric space, f be an η-subadmissible mapping on X, and there exists $x_0\in X$ such that $\eta (x_0,f{x}_0)\le 1$. Assume that

$${(1+\ell )}^{G_p(fx,fy,fy)}\le {(\eta (x,y)+\ell )}^{rmax\{G_p(x,y,y),G_p(x,fx,fx),G_p(y,fy,fy)\}}$$

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for all $x,y\in X$, where $0\le r<1$ and $\ell >0$. If $\{x_n\}$ is a sequence in X such that $\eta (x_n,x_{n+1})\le 1$ for all n and $x_n\to x$ as $n\to +\mathrm{\infty }$, then $\eta (x_n,x)\le 1$ for all $n\in \mathbb{N}\cup \{0\}$, then f has a fixed point.