# Common fixed points of R-weakly commuting maps in generalized metric spaces

## Abstract

In this paper, using the setting of a generalized metric space, a unique common fixed point of four R-weakly commuting maps satisfying a generalized contractive condition is obtained. We also present example in support of our result.

2000 MSC: 54H25; 47H10; 54E50.

## 1 Introduction and preliminaries

The study of unique common fixed points of mappings satisfying certain contractive conditions has been at the center of rigorous research activity. Mustafa and Sims  generalized the concept of a metric, in which the real number is assigned to every triplet of an arbitrary set. Based on the notion of generalized metric spaces, Mustafa et al.  obtained some fixed point theorems for mappings satisfying different contractive conditions. Study of common fixed point theorems in generalized metric spaces was initiated by Abbas and Rhoades . Abbas et al.  obtained some periodic point results in generalized metric spaces. While, Chugh et al.  obtained some fixed point results for maps satisfying property p in G-metric spaces. Saadati et al.  studied some fixed point results for contractive mappings in partially ordered G-metric spaces. Recently, Shatanawi  obtained fixed points of Φ-maps in G-metric spaces. Abbas et al.  gave some new results of coupled common fixed point results in two generalized metric spaces (see also ).

The aim of this paper is to initiate the study of unique common fixed point of four R-weakly commuting maps satisfying a generalized contractive condition in G-metric spaces.

Consistent with Mustafa and Sims , the following definitions and results will be needed in the sequel.

Definition 1.1. Let X be a nonempty set. Suppose that a mapping G :

X × X × XR+ satisfies:

G1 : G(x, y, z) = 0 if x = y = z;

G2 : 0 < G(x, y, z) for all x, y, z X, with xy;

G3 : G(x, x, y) ≤ G(x, y, z) for all x, y, z X, with yz;

G4 : G(x, y, z) = G(x, z, y) = G(y, z, x) = ··· (symmetry in all three variables); and

G5 : G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a X.

Then G is called a G-metric on X and (X, G) is called a G-metric space.

Definition 1.2. A sequence {x n } in a G-metric space X is:

1. (i)

a G-Cauchy sequence if, for any ε > 0, there is an n 0 N (the set of natural numbers) such that for all n, m, l ≥ n 0, G(x n , x m , x l ) < ε,

2. (ii)

a G-convergent sequence if, for any ε > 0, there is an x X and an n 0 N, such that for all n, mn 0, G(x, x n , x m ) < ε.

A G-metric space on X is said to be G-complete if every G-Cauchy sequence in X is G-convergent in X. It is known that → 0 as n, m → ∞.

Proposition 1.3. Let X be a G-metric space. Then the following are equivalent:

1. (1)

{x n } is G-convergent to x.

2. (2)

G(x n , x m , x) → 0 as n, m → ∞.

3. (3)

G(x n , x n , x) → 0 as n → ∞.

4. (4)

G(x n , x, x) → 0 as n → ∞.

Definition 1.4. A G-metric on X is said to be symmetric if G(x, y, y) = G(y, x, x) for all x, y X.

Proposition 1.5. Every G-metric on X will define a metric d G on X by

${d}_{G}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)=G\left(x,y,\phantom{\rule{2.77695pt}{0ex}}y\right)+G\left(y,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}x\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X.$
(1.1)

For a symmetric G-metric,

${d}_{G}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)=2G\left(x,y,\phantom{\rule{2.77695pt}{0ex}}y\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X.$
(1.2)

However, if G is non-symmetric, then the following inequality holds:

$\frac{3}{2}G\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}y\right)\le {d}_{G}\left(x,\phantom{\rule{2.77695pt}{0ex}}y\right)\le 3G\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}y\right),\phantom{\rule{1em}{0ex}}\forall x,\phantom{\rule{2.77695pt}{0ex}}y\in X.$
(1.3)

It is also obvious that

$G\left(x,\phantom{\rule{2.77695pt}{0ex}}x,\phantom{\rule{2.77695pt}{0ex}}y\right)\le 2G\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}y\right).$

Now, we give an example of a non-symmetric G-metric.

Example 1.6. Let X = {1, 2} and a mapping G : X × X × XR+ be defined as

$\begin{array}{cc}\hfill \left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}z\right)\hfill & \hfill G\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}z\right)\hfill \\ \hfill \left(1,1,1\right),\left(2,2,2\right)\hfill & \hfill 0\hfill \\ \hfill \left(1,1,2\right),\left(1,2,1\right),\left(2,1,1\right)\hfill & \hfill 0.5\hfill \\ \hfill \left(1,2,2\right),\left(2,1,2\right),\left(2,2,1\right)\hfill & \hfill 1.\hfill \end{array}$

Note that G satisfies all the axioms of a generalized metric but G(x, x, y) ≠ G(x, y, y) for distinct x, y in X. Therefore, G is a non-symmetric G-metric on X.

In 1999, Pant  introduced the concept of weakly commuting maps in metric spaces. We shall study R-weakly commuting and compatible mappings in the frame work of G-metric spaces.

Definition 1.7. Let X be a G-metric space and f and g be two self-mappings of X. Then f and g are called R-weakly commuting if there exists a positive real number R such that G(fgx, fgx, gfx) ≤ RG(fx, fx, gx) holds for each x X.

Two maps f and g are said to be compatible if, whenever {x n } in X such that {fx n } and {gx n } are G-convergent to some t X, then limn→∞G(fgx n , fgx n , gfx n ) = 0.

Example 1.8. Let X = [0, 2] with complete G-metric defined by

$G\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}z\right)=max\left\{|x-y|,\phantom{\rule{2.77695pt}{0ex}}|x-z|,\phantom{\rule{2.77695pt}{0ex}}|y-z|\right\}.$

Let f, g, S, T : XX defined by

and

Then note that the pairs {f, S} and {g, T} are R-weakly commuting as they commute at their coincidence points. The pair {f, S} is continuous compatible while the pair {g, T} is non-compatible. To see that g and T are non-compatible, consider a decreasing sequence {x n } in X such that x n → 1. Then $g{x}_{n}\to \frac{1}{2}$, $T{x}_{n}\to \frac{1}{2}$. $gT{x}_{n}=\frac{4-{x}_{n}}{4}\to \frac{3}{4}$ and $Tg{x}_{n}=\frac{2-{x}_{n}}{4}\to \frac{1}{4}$. □

## 2 Common fixed point theorems

In this section, we obtain some unique common fixed point results for four mappings satisfying certain generalized contractive conditions in the framework of a generalized metric space. We start with the following result.

Theorem 2.1. Let X be a complete G-metric space. Suppose that {f, S} and {g, T} be pointwise R-weakly commuting pairs of self-mappings on X satisfying

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}gy\right)\le h\mathrm{max}\left\{G\left(Sx,Sx,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,fx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,gy,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}\end{array}$
(2.1)

and

$\begin{array}{c}G\left(fx,gy,gy\right)\le h\phantom{\rule{0.1em}{0ex}}\mathrm{max}\left\{G\left(Sx,Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,Sx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}\end{array}$
(2.2)

for all x, y X, where h [0, 1). Suppose that fX TX, gX SX, and one of the pair {f, S} or {g, T} is compatible. If the mappings in the compatible pair are continuous, then f, g, S and T have a unique common fixed point.

Proof. Suppose that f and g satisfy the conditions (2.1) and (2.2). If G is symmetric, then by adding these, we have

$\begin{array}{c}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}gy\right)\\ \phantom{\rule{1em}{0ex}}\le \frac{h}{2}max\left\{{d}_{G}\left(Sx,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Sx\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}\left[{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)+{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Sx\right)\right]∕2\right\}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{h}{2}max\left\{{d}_{G}\left(Sx,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Sx\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}\left[{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)+{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Sx\right)\right]∕2\right\}\\ \phantom{\rule{1em}{0ex}}=hmax\left\{{d}_{G}\phantom{\rule{2.77695pt}{0ex}}\left(Sx,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Sx\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}\left[{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)+{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Sx\right)\right]∕2\right\},\end{array}$

for all x, y X with 0 ≤ h < 1, the existence and uniqueness of a common fixed point follows from . However, if X is non-symmetric G-metric space, then by the definition of metric d G on X and (1.3), we obtain

$\begin{array}{c}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}gy\right)\\ \phantom{\rule{1em}{0ex}}=G\left(fx,\phantom{\rule{2.77695pt}{0ex}}fx,\phantom{\rule{2.77695pt}{0ex}}gy\right)+G\left(fx,\phantom{\rule{2.77695pt}{0ex}}gy,\phantom{\rule{2.77695pt}{0ex}}gy\right)\\ \phantom{\rule{1em}{0ex}}\le \frac{2h}{3}max\left\{{d}_{G}\left(Sx,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Sx\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}\left[{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)+{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Sx\right)\right]∕2\right\}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{2h}{3}max\left\{{d}_{G}\left(Sx,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Sx\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}\left[{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)+{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Sx\right)\right]∕2\right\}\\ \phantom{\rule{1em}{0ex}}=\frac{4h}{3}max\left\{{d}_{G}\left(Sx,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Sx\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}Ty\right),\phantom{\rule{2.77695pt}{0ex}}\left[{d}_{G}\left(fx,\phantom{\rule{2.77695pt}{0ex}}Ty\right)+{d}_{G}\left(gy,\phantom{\rule{2.77695pt}{0ex}}{S}_{X}\right)\right]∕2\right\},\end{array}$

for all x, y X. Here, the contractivity factor $\frac{4h}{3}$ needs not be less than 1. Therefore, metric d G gives no information. In this case, let x0 be an arbitrary point in X. Choose x1 and x2 in X such that gx0 = Sx1 and fx1 = Tx2. This can be done, since the ranges of S and T contain those of g and f, respectively. Again choose x3 and x4 in X such that gx2 = Sx 3 and fx3 = Tx4. Continuing this process, having chosen x n in X such that gx2n= Sx2n+1and fx2n+1= Tx2n+2, n = 0, 1, 2, .... Let

${y}_{2n}=S{x}_{2n+1}=g{x}_{2n}\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}{y}_{2n+1}=T{x}_{2n+2}=f{x}_{2n+1}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}n=0,1,2,\phantom{\rule{2.77695pt}{0ex}}\dots \phantom{\rule{2.77695pt}{0ex}}.$

For a given n N, if n is even, so n = 2k for some k N. Then from (2.1)

$\begin{array}{l}G\left({y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n}\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(f{x}_{2k+1},f{x}_{2k+1},g{x}_{2k}\right)\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{G\left(S{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}S{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}T{x}_{2k}\right),\phantom{\rule{0.1em}{0ex}}G\left(f{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}f{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}S{x}_{2k+1}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}G\left(g{x}_{2k},g{x}_{2k},T{x}_{2k}\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(f{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}f{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}T{x}_{2k}\right)+G\left(g{x}_{2k},g{x}_{2k},S{x}_{2k+1}\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\phantom{\rule{0.1em}{0ex}}\mathrm{max}\left\{G\left({y}_{2}k,\phantom{\rule{0.1em}{0ex}}{y}_{2}k,\phantom{\rule{0.1em}{0ex}}{y}_{2}k-1\right),\phantom{\rule{0.1em}{0ex}}G\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}G\left({y}_{2}k,\phantom{\rule{0.1em}{0ex}}y2{\phantom{\rule{0.1em}{0ex}}}_{k},\phantom{\rule{0.1em}{0ex}}{y}_{2}k-1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left({y}_{2k+1},{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k-1}\right)+G\left({y}_{2k},{y}_{2k},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}\le h\phantom{\rule{0.1em}{0ex}}\mathrm{max}\left\{G\left({y}_{2}{\phantom{\rule{0.1em}{0ex}}}_{k},\phantom{\rule{0.1em}{0ex}}{y}_{2}{\phantom{\rule{0.1em}{0ex}}}_{k},\phantom{\rule{0.1em}{0ex}}{y}_{2}{\phantom{\rule{0.1em}{0ex}}}_{k-1}\right),\phantom{\rule{0.1em}{0ex}}G\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right)+G\left({y}_{2}{\phantom{\rule{0.1em}{0ex}}}_{k},\phantom{\rule{0.1em}{0ex}}{y}_{2}{\phantom{\rule{0.1em}{0ex}}}_{k},\phantom{\rule{0.1em}{0ex}}{y}_{2}{\phantom{\rule{0.1em}{0ex}}}_{k-1}\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left({y}_{n},\phantom{\rule{0.1em}{0ex}}{y}_{n},\phantom{\rule{0.1em}{0ex}}{y}_{n-1}\right),\phantom{\rule{0.1em}{0ex}}G\left({y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n}\right)\right\}.\end{array}$

This implies that

$G\left({y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n}\right)\le hG\left({y}_{n},\phantom{\rule{2.77695pt}{0ex}}{y}_{n},\phantom{\rule{2.77695pt}{0ex}}{y}_{n-1}\right).$

If n is odd, then n = 2k + 1 for some k N. In this case (2.1) gives

$\begin{array}{l}G\left({y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n}\right)\\ \phantom{\rule{0.5em}{0ex}}=G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1}\right)\\ \phantom{\rule{0.5em}{0ex}}=G\phantom{\rule{0.1em}{0ex}}\left(f{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}f{x}_{2k+2}+g{x}_{2k+1}\right)\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{G\phantom{\rule{0.1em}{0ex}}\left(S{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}S{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}T{x}_{2k+1}\right),\phantom{\rule{0.1em}{0ex}}G\left(f{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}f{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}S{x}_{2k+2}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}G\left(g{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}g{x}_{2k+1},T{x}_{2k+1}\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(f{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}f{x}_{2k+2},\phantom{\rule{0.1em}{0ex}}T{x}_{2k+1}\right)+G\left(g{x}_{2k+1},g{x}_{2k+1},\phantom{\rule{0.1em}{0ex}}S{x}_{2k+2}\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\phantom{\rule{0.1em}{0ex}}\mathrm{max}\left\{G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right),\phantom{\rule{0.1em}{0ex}}G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right),\phantom{\rule{0.1em}{0ex}}\left[G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right)+G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1}\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}\le h\phantom{\rule{0.1em}{0ex}}\mathrm{max}\left\{G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right),\phantom{\rule{0.1em}{0ex}}G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1}\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1}\right)+G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\phantom{\rule{0.1em}{0ex}}\mathrm{max}\left\{G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1},\phantom{\rule{0.1em}{0ex}}{y}_{2k}\right),\phantom{\rule{0.1em}{0ex}}G\phantom{\rule{0.1em}{0ex}}\left({y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+2},\phantom{\rule{0.1em}{0ex}}{y}_{2k+1}\right)\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left({y}_{n},\phantom{\rule{0.1em}{0ex}}{y}_{n},\phantom{\rule{0.1em}{0ex}}{y}_{n-1}\right),\phantom{\rule{0.1em}{0ex}}G\left({y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n+1},\phantom{\rule{0.1em}{0ex}}{y}_{n}\right)\right\},\end{array}$

that is,

$G\left({y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n}\right)\le hG\left({y}_{n},\phantom{\rule{2.77695pt}{0ex}}{y}_{n},\phantom{\rule{2.77695pt}{0ex}}{y}_{n-1}\right).$

Continuing the above process, we have

$G\left({y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n}\right)\le {h}^{n}G\left({y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{0}\right).$

Thus, if y0 = y1, we get G(y n , yn+1, yn+1) = 0 for each n N. Hence, y n = yn+1for each n N. Therefore, {y n } is G-Cauchy. So we may assume that y0y1.

Let n, m N with m > n,

$\begin{array}{c}G\left({y}_{n},\phantom{\rule{2.77695pt}{0ex}}{y}_{m},\phantom{\rule{2.77695pt}{0ex}}{y}_{m}\right)\\ \phantom{\rule{1em}{0ex}}\le G\left({y}_{n},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+1}\right)+G\left({y}_{n+1},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+2},\phantom{\rule{2.77695pt}{0ex}}{y}_{n+2}\right)+\cdots +G\left({y}_{m-1},\phantom{\rule{2.77695pt}{0ex}}{y}_{m},\phantom{\rule{2.77695pt}{0ex}}{y}_{m}\right)\\ \phantom{\rule{1em}{0ex}}\le {h}^{n}G\left({y}_{0},\phantom{\rule{2.77695pt}{0ex}}{y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\right)+{h}^{n+1}G\left({y}_{0},\phantom{\rule{2.77695pt}{0ex}}{y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\right)+\cdots +{h}^{m-1}G\left({y}_{0},\phantom{\rule{2.77695pt}{0ex}}{y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\right)\\ \phantom{\rule{1em}{0ex}}={h}^{n}G\left({y}_{0},\phantom{\rule{2.77695pt}{0ex}}{y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\right)\sum _{i=0}^{m-n-1}{h}^{i}\\ \phantom{\rule{1em}{0ex}}\le \frac{{h}^{n}}{1-h}G\left({y}_{0},\phantom{\rule{2.77695pt}{0ex}}{y}_{1},\phantom{\rule{2.77695pt}{0ex}}{y}_{1}\right),\end{array}$

and so G(y n , y m , y m ) → 0 as m, n → ∞. Hence {y n } is a Cauchy sequence in X. Since X is G-complete, there exists a point z X such that limn→∞y n = z.

Consequently

$\underset{n\to \infty }{lim}{y}_{2n}=\underset{n\to \infty }{lim}S{x}_{2n+1}=\underset{n\to \infty }{lim}g{x}_{2n}=z$

and

$\underset{n\to \infty }{lim}{y}_{2n+1}=\underset{n\to \infty }{lim}T{x}_{2n+2}=\underset{n\to \infty }{lim}f{x}_{2n+1}=z.$

Let f and S be continuous compatible mappings. Compatibility of f and S implies that limn→∞G(fSx2n+1, fSx2n+1, Sfx2n+1) = 0, that is G(fz, fz, Sz) = 0 which implies that fz = Sz. Since fX TX, there exists some u X such that fz = Tu. Now from (2.1), we have

(2.3)

Also, from (2.2)

(2.4)

Combining above two inequalities, we get

$G\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}gu\right)\le {h}^{2}G\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}gu\right).$

Since h < 1, so that fz = gu. Hence, fz = Sz = gu = Tu. As the pair {g, T} is R-weakly commuting, there exists R > 0 such that

$G\left(gTu,\phantom{\rule{2.77695pt}{0ex}}gTu,\phantom{\rule{2.77695pt}{0ex}}Tgu\right)\le RG\left(gu,\phantom{\rule{2.77695pt}{0ex}}gu,\phantom{\rule{2.77695pt}{0ex}}Tu\phantom{\rule{2.77695pt}{0ex}}\right)=0,$

that is, gTu = Tgu. Moreover, ggu = gTu = Tgu = TTu. Similarly, the pair {f, S} is R-weakly commuting, there exists some R > 0 such that

$G\left(fSz,\phantom{\rule{2.77695pt}{0ex}}fSz,\phantom{\rule{2.77695pt}{0ex}}Sfz\right)\le RG\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}Sz\right)=0,$

so that fSz = Sfz and ffz = fSz = Sfz = SSz.

Now by (2.1)

$\begin{array}{c}G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}fz\right)=G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}gu\right)\\ \le h\mathrm{max}\left\{G\left(Sfz,\phantom{\rule{0.1em}{0ex}}Sfz,\phantom{\rule{0.1em}{0ex}}Tu\right),\phantom{\rule{0.1em}{0ex}}G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}Sfz\right),\phantom{\rule{0.1em}{0ex}}G\left(gu,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}Tu\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}Tu\right)+G\left(gu,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}Sfz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}gu\right),\phantom{\rule{0.1em}{0ex}}G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}ffz\right),\phantom{\rule{0.1em}{0ex}}G\left(gu,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}gu\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}gu\right)+G\left(gu,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}ffz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}fz\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}fz\right)+G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}ffz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}\frac{h}{2}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}fz\right)+G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}ffz\right)\right],\end{array}$

so that

$G\left(ffz,\phantom{\rule{2.77695pt}{0ex}}ffz,\phantom{\rule{2.77695pt}{0ex}}fz\right)\le hG\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}ffz\right).$
(2.5)

Again from (2.2), we have

$\begin{array}{c}G\left(ffz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right)=G\left(ffz,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}gu\right)\\ \le h\mathrm{max}\left\{G\left(Sfz,Tu,Tu\right),G\left(ffz,\phantom{\rule{0.1em}{0ex}}Sfz,\phantom{\rule{0.1em}{0ex}}Sfz\right),\phantom{\rule{0.1em}{0ex}}G\left(gu,Tu,Tu\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(f{f}_{Z},\phantom{\rule{0.1em}{0ex}}Tu,\phantom{\rule{0.1em}{0ex}}Tu\right)+G\left(gu,\phantom{\rule{0.1em}{0ex}}Sfz,\phantom{\rule{0.1em}{0ex}}Sfz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(Sfz,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}gu\right),\phantom{\rule{0.1em}{0ex}}G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}ffz\right),\phantom{\rule{0.1em}{0ex}}G\left(gu,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}gu\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}gu,\phantom{\rule{0.1em}{0ex}}gu\right)+G\left(gu,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}ffz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(ffz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right)+G\left(fz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}ffz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}\frac{h}{2}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right)+G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}fz\right)\right],\end{array}$

which implies

$G\left(ffz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}fz\right)\le hG\left(ffz,\phantom{\rule{2.77695pt}{0ex}}ffz,\phantom{\rule{2.77695pt}{0ex}}fz\right).$
(2.6)

From (2.5) and (2.6), we obtain

$G\left(ffz,\phantom{\rule{2.77695pt}{0ex}}ffz,\phantom{\rule{2.77695pt}{0ex}}fz\right)\le {h}^{2}G\left(ffz,\phantom{\rule{2.77695pt}{0ex}}ffz,\phantom{\rule{2.77695pt}{0ex}}fz\right),$

and since h2 < 1 so that ffz = fz. Hence, ffz = Sfz = fz, and fz is the common fixed point of f and S. Since gu = fz, following arguments similar to those given above we conclude that fz is a common fixed point of g and T as well. Now we show the uniqueness of fixed point. For this, assume that there exists another point w in X which is the common fixed point of f, g, S and T. From (2.1), we obtain

$\begin{array}{c}G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}w\right)=G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}gw\right)\\ \le \phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(Sfz,\phantom{\rule{0.1em}{0ex}}Sfz,\phantom{\rule{0.1em}{0ex}}Tw\right),\phantom{\rule{0.1em}{0ex}}G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}Sfz\right),\phantom{\rule{0.1em}{0ex}}G\left(gw,\phantom{\rule{0.1em}{0ex}}gw,\phantom{\rule{0.1em}{0ex}}Tw\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}ffz,\phantom{\rule{0.1em}{0ex}}Tw\right)+G\left(gw,\phantom{\rule{0.1em}{0ex}}gw,\phantom{\rule{0.1em}{0ex}}Sfz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}w\right),\phantom{\rule{0.1em}{0ex}}G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right),\phantom{\rule{0.1em}{0ex}}G\left(w,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}w\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}w\right)+G\left(w,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}fz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}\frac{h}{2}\left[G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}w\right)+G\left(w,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}fz\right)\right],\end{array}$

which implies that

$G\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}w\right)\le hG\left(w,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}fz\right).$
(2.7)

From (2.2), we get

$\begin{array}{c}G\left(fz,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}w\right)=G\left(ffz,\phantom{\rule{0.1em}{0ex}}gw,\phantom{\rule{0.1em}{0ex}}gw\right)\\ \le \phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(Sfz,\phantom{\rule{0.1em}{0ex}}Tw,\phantom{\rule{0.1em}{0ex}}Tw\right),\phantom{\rule{0.1em}{0ex}}G\left(ffz,\phantom{\rule{0.1em}{0ex}}Sfz,\phantom{\rule{0.1em}{0ex}}Sfz\right),\phantom{\rule{0.1em}{0ex}}G\left(gw,\phantom{\rule{0.1em}{0ex}}Tw,\phantom{\rule{0.1em}{0ex}}Tw\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(ffz,\phantom{\rule{0.1em}{0ex}}Tw,\phantom{\rule{0.1em}{0ex}}Tw\right)+G\left(gw,\phantom{\rule{0.1em}{0ex}}Sfz,\phantom{\rule{0.1em}{0ex}}Sfz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(fz,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}w\right),\phantom{\rule{0.1em}{0ex}}G\left(fz,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right),\phantom{\rule{0.1em}{0ex}}G\left(w,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}w\right),\\ \phantom{\rule{0.5em}{0ex}}\left[G\left(fz,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}w\right)+G\left(w,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right)\right]/2\right\}\\ =\phantom{\rule{0.1em}{0ex}}\frac{h}{2}\left[G\left(fz,\phantom{\rule{0.1em}{0ex}}w,\phantom{\rule{0.1em}{0ex}}w\right)+G\left(w,\phantom{\rule{0.1em}{0ex}}fz,\phantom{\rule{0.1em}{0ex}}fz\right)\right],\end{array}$

which implies

$G\left(fz,\phantom{\rule{2.77695pt}{0ex}}w,\phantom{\rule{2.77695pt}{0ex}}w\right)\le hG\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}w\right).$
(2.8)

Now (2.7) and (2.8) give

$G\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}w\right)\le {h}^{2}G\left(fz,\phantom{\rule{2.77695pt}{0ex}}fz,\phantom{\rule{2.77695pt}{0ex}}w\right),$

and fz = w. This completes the proof.

Example 2.2. Let X = {0, 1, 2} with G-metric defined by

is a non-symmetric G-metric on X because G(0, 0, 1) ≠ G(0, 1, 1).

Let f, g, S, T : XX defined by

$\begin{array}{ccccc}\hfill x\hfill & \hfill f\left(x\right)\hfill & \hfill g\left(x\right)\hfill & \hfill S\left(x\right)\hfill & \hfill T\left(x\right)\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}$

Then fX TX and gX SX, with the pairs {f, S} and {g, T} are R-weakly commuting as they commute at their coincidence points.

Now to get (2.1) and (2.2) satisfied, we have the following nine cases: (I) x, y = 0, (II) x = 0, y = 2, (III) x = 1, y = 0, (IV) x = 1, y = 2, (V) x = 2, y = 0, (VI) x = 2, y = 2. For all these cases, f(x) = g(y) = 0 implies G(fx, fx, gy) = 0 and (2.1) and (2.2) hold.

1. (VII)

For x = 0, y = 1, then fx = 0, gy = 2, Sx = 0, Ty = 1.

$\begin{array}{l}G\left(fx,fx,\phantom{\rule{0.1em}{0ex}}gy\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(0,0,2\right)=1\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{1,0,2,1\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(0,0,1\right),\phantom{\rule{0.1em}{0ex}}G\left(0,\phantom{\rule{0.1em}{0ex}}0,0\right),\phantom{\rule{0.1em}{0ex}}G\left(2,2,1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(0,\phantom{\rule{0.1em}{0ex}}0,1\right)+G\left(2,2,0\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}.\end{array}$

Thus, (2.1) is satisfied where $h=\frac{4}{5}$.

Also

$\begin{array}{l}G\left(fx,gy,\phantom{\rule{0.1em}{0ex}}gy\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(0,2,2\right)=1\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{2,0,2,1.5\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(0,1,1\right),\phantom{\rule{0.1em}{0ex}}G\left(0,\phantom{\rule{0.1em}{0ex}}0,0\right),\phantom{\rule{0.1em}{0ex}}G\left(2,1,1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(0,\phantom{\rule{0.1em}{0ex}}1,1\right)+G\left(2,0,0\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left(fx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}.\end{array}$

Thus, (2.2) is satisfied where $h=\frac{4}{5}$.

1. (VIII)

Now when x = 1, y = 1, then fx = 0, gy = 2, Sx = 2, Ty = 1.

$\begin{array}{l}G\left(fx,fx,\phantom{\rule{0.1em}{0ex}}gy\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(0,0,2\right)=1\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{2,1,2,0.5\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(2,2,1\right),\phantom{\rule{0.1em}{0ex}}G\left(0,\phantom{\rule{0.1em}{0ex}}0,2\right),\phantom{\rule{0.1em}{0ex}}G\left(2,2,1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(0,\phantom{\rule{0.1em}{0ex}}0,1\right)+G\left(2,2,2\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}.\end{array}$

Thus, (2.1) is satisfied where $h=\frac{4}{5}$.

And

$\begin{array}{l}G\left(fx,gy,\phantom{\rule{0.1em}{0ex}}gy\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(0,2,2\right)=1\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{2,1,2,1\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(2,1,1\right),\phantom{\rule{0.1em}{0ex}}G\left(0,\phantom{\rule{0.1em}{0ex}}2,2\right),\phantom{\rule{0.1em}{0ex}}G\left(2,1,1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(0,\phantom{\rule{0.1em}{0ex}}1,1\right)+G\left(2,2,2\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left(fx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}.\end{array}$

Thus, (2.2) is satisfied where $h=\frac{4}{5}$.

1. (IX)

If x = 2, y = 1, then fx = 0, gy = 2, Sx = 1, Ty = 1 and

$\begin{array}{l}G\left(fx,fx,\phantom{\rule{0.1em}{0ex}}gy\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(0,0,2\right)=1\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{0,1,2,1.5\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(1,1,1\right),\phantom{\rule{0.1em}{0ex}}G\left(0,\phantom{\rule{0.1em}{0ex}}0,1\right),\phantom{\rule{0.1em}{0ex}}G\left(2,2,1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(0,\phantom{\rule{0.1em}{0ex}}0,1\right)+G\left(2,2,1\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}.\end{array}$

Thus, (2.1) is satisfied where $h=\frac{4}{5}$.

Also

$\begin{array}{l}G\left(fx,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}gy\right)\\ \phantom{\rule{0.5em}{0ex}}=G\left(0,2,2\right)=1\\ \phantom{\rule{0.5em}{0ex}}\le h\mathrm{max}\left\{0,2,2,2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(1,1,1\right),\phantom{\rule{0.1em}{0ex}}G\left(0,\phantom{\rule{0.1em}{0ex}}1,1\right),\phantom{\rule{0.1em}{0ex}}G\left(2,1,1\right),\phantom{\rule{0.1em}{0ex}}\left[G\left(0,\phantom{\rule{0.1em}{0ex}}1,1\right)+G\left(2,1,1\right)\right]/2\right\}\\ \phantom{\rule{0.5em}{0ex}}=h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \phantom{\rule{0.5em}{0ex}}\phantom{\rule{0.5em}{0ex}}\left[G\left(fx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}.\end{array}$

Thus, (2.2) is satisfied where $h=\frac{4}{5}$.

Hence, for all x, y X, (2.1) and (2.2) are satisfied for $h=\frac{4}{5}<1$ so that all the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed point for all of the mappings f, g, S and T.

In Theorem 2.1, if we take f = g, then we have the following corollary.

Corollary 2.3. Let X be a complete G-metric space. Suppose that {f, S} and {f, T} be pointwise R-weakly commuting pairs of self-mappings on X satisfying

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}fy\right)\le h\mathrm{max}\left\{G\left(Sx,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(fy,\phantom{\rule{0.1em}{0ex}}fy,\phantom{\rule{0.1em}{0ex}}Ty\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(fy,\phantom{\rule{0.1em}{0ex}}fy,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}\end{array}$
(2.9)

and

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fy,\phantom{\rule{0.1em}{0ex}}fy\right)\le h\mathrm{max}\left\{G\left(Sx,Ty,\phantom{\rule{0.1em}{0ex}}Ty\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,Sx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(fy,Ty,\phantom{\rule{0.1em}{0ex}}Ty\right)\right\}\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}Ty,\phantom{\rule{0.1em}{0ex}}Ty\right)+G\left(fy,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}\end{array}$
(2.10)

for all x, y X, where h [0, 1). Suppose that fX SX TX, and one of the pairs {f, S} or {f, T} is compatible. If the mappings in the compatible pair are continuous, then f, S and T have a unique common fixed point.

Also, if we take S = T in Theorem 2.1, then we get the following.

Corollary 2.4. Let X be a complete G-metric space. Suppose that {f, S} and {g, S} are pointwise R-weakly commuting pairs of self-maps on X and

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}gy\right)\le h\mathrm{max}\left\{G\left(Sx,Sx,\phantom{\rule{0.1em}{0ex}}Sy\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,fx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,gy,\phantom{\rule{0.1em}{0ex}}Sy\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}Sy\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}\end{array}$
(2.11)

and

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}gy\right)\le h\mathrm{max}\left\{G\left(Sx,Sy,\phantom{\rule{0.1em}{0ex}}Sy\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,Sx,\phantom{\rule{0.1em}{0ex}}Sx\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,Sy,\phantom{\rule{0.1em}{0ex}}Sy\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}Sy,\phantom{\rule{0.1em}{0ex}}Sy\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}Sx,\phantom{\rule{0.1em}{0ex}}Sx\right)\right]/2\right\}\end{array}$
(2.12)

hold for all x, y X, where h [0, 1). Suppose that fX gX SX and one of the pairs {f, S} or {g, S} is compatible. If the mappings in the compatible pair are continuous, then f, g and S have a unique common fixed point.

Corollary 2.5. Let X be a complete G-metric space. Suppose that f and g are two self-mappings on X satisfying

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}gy\right)\phantom{\rule{0.1em}{0ex}}\le \phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(x,\phantom{\rule{0.1em}{0ex}}x,\phantom{\rule{0.1em}{0ex}}y\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}x\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}y\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}y\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}x\right)\right]/2\right\}\end{array}$
(2.13)

and

(2.14)

for all x, y X, where h [0, 1). Suppose that one of f or g is continuous, then f and g have a unique common fixed point.

Proof. Taking S and T as identity maps on X, the result follows from Theorem 2.1.

Corollary 2.6. Let X be a complete G-metric space and f be a self-map on X such that

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}fy\right)\phantom{\rule{0.1em}{0ex}}\le \phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(x,\phantom{\rule{0.1em}{0ex}}x,\phantom{\rule{0.1em}{0ex}}y\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}x\right),\phantom{\rule{0.1em}{0ex}}G\left(fy,\phantom{\rule{0.1em}{0ex}}fy,\phantom{\rule{0.1em}{0ex}}y\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}y\right)+G\left(fy,\phantom{\rule{0.1em}{0ex}}fy,\phantom{\rule{0.1em}{0ex}}x\right)\right]/2\right\}\end{array}$
(2.15)

and

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fy,\phantom{\rule{0.1em}{0ex}}fy\right)\phantom{\rule{0.1em}{0ex}}\le \phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(x,\phantom{\rule{0.1em}{0ex}}y,\phantom{\rule{0.1em}{0ex}}y\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}x,\phantom{\rule{0.1em}{0ex}}x\right),\phantom{\rule{0.1em}{0ex}}G\left(fy,\phantom{\rule{0.1em}{0ex}}y,\phantom{\rule{0.1em}{0ex}}y\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}y,\phantom{\rule{0.1em}{0ex}}y\right)+G\left(fy,\phantom{\rule{0.1em}{0ex}}x,\phantom{\rule{0.1em}{0ex}}x\right)\right]/2\right\}\end{array}$
(2.16)

hold for all x, y X, where h [0, 1). Then f has a unique fixed point.

Proof. If we take f = g, and S and T as identity maps on X, then from f has a unique fixed point by Theorem 2.1.

## 3 Application

Let Ω = [0, 1] be bounded open set in , L2(Ω), the set of functions on Ω whose square is integrable on Ω. Consider an integral equation

$p\left(t,\phantom{\rule{2.77695pt}{0ex}}x\left(t\right)\right)=\underset{\Omega }{\int }q\left(t,\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}x\left(s\right)\right)ds$
(3.1)

where p : Ω × and q : Ω × Ω × be two mappings. Define G : X × X × X+ by

$G\left(x,\phantom{\rule{2.77695pt}{0ex}}y,\phantom{\rule{2.77695pt}{0ex}}z\right)=\underset{t\in \Omega }{sup}|x\left(t\right)-y\left(t\right)|+\underset{t\in \Omega }{sup}|y\left(t\right)-z\left(t\right)|+\underset{t\in \Omega }{sup}|z\left(t\right)-x\left(t\right)|.$

Then X is a G-complete metric space. We assume the following that is there exists a function G : Ω × +:

1. (i)

p(s, v(t)) ≥ ∫Ω q(t, s, u(s)) dsG(s, v(t)) for each s, t Ω..

2. (ii)

p(s, v(t)) - G(s, v(t)) ≤ h |p(s, v(t)) - v(t)|.

Then integral equation (3.1) has a solution in L2(Ω).

Proof. Define (fx)(t) = p(t, x(t)) and (gx)(t) = ∫Ωq(t, s, x(s)) ds. Now

$\begin{array}{lll}\hfill G\left(fx,\phantom{\rule{2.77695pt}{0ex}}fx,\phantom{\rule{2.77695pt}{0ex}}gy\right)& =2\underset{t\in \Omega }{sup}|\left(fx\right)\left(t\right)-\left(gy\right)\left(t\right)|\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =2\underset{t\in \Omega }{sup}\left|p\left(t,\phantom{\rule{2.77695pt}{0ex}}x\left(t\right)\right)-\underset{\Omega }{\int }q\left(t,\phantom{\rule{2.77695pt}{0ex}}s,\phantom{\rule{2.77695pt}{0ex}}y\left(t\right)\right)dt\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \le 2\underset{t\in \Omega }{sup}|p\left(t,\phantom{\rule{2.77695pt}{0ex}}x\left(t\right)\right)-G\left(t,\phantom{\rule{2.77695pt}{0ex}}x\left(t\right)\right)|\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \le 2h\underset{t\in \Omega }{sup}|p\left(t,\phantom{\rule{2.77695pt}{0ex}}x\left(t\right)\right)-x\left(t\right)|\phantom{\rule{2em}{0ex}}& \hfill \text{(4)}\\ =\phantom{\rule{2.77695pt}{0ex}}hG\left(fx,\phantom{\rule{2.77695pt}{0ex}}fx,x\right).\phantom{\rule{2em}{0ex}}& \hfill \text{(5)}\\ \hfill \text{(6)}\end{array}$

Thus

$\begin{array}{c}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}gy\right)\phantom{\rule{0.1em}{0ex}}\le \phantom{\rule{0.1em}{0ex}}h\mathrm{max}\left\{G\left(x,\phantom{\rule{0.1em}{0ex}}x,\phantom{\rule{0.1em}{0ex}}y\right),\phantom{\rule{0.1em}{0ex}}G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}x\right),\phantom{\rule{0.1em}{0ex}}G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}y\right),\\ \left[G\left(fx,\phantom{\rule{0.1em}{0ex}}fx,\phantom{\rule{0.1em}{0ex}}y\right)+G\left(gy,\phantom{\rule{0.1em}{0ex}}gy,\phantom{\rule{0.1em}{0ex}}x\right)\right]/2\right\}\end{array}$

is satisfied. Similarly (2.14) is satisfied. Now we can apply Corollary 2.5 to obtain the solution of integral equation (3.1) in L2(Ω).

Remark 1. Theorems 2.8-2.9 in  and Corollaries 2.6-2.8 in  are special cases of our results Theorem 2.1 and Corollaries 2.3-2.6.

Remark 2. A G-metric naturally induces a metric d G given by d G (x, y) = G(x, y, y) + G(x, x, y). If the G-metric is not symmetric, the inequalities (2.1) and (2.2) do not reduce to any metric inequality with the metric d G . Hence, our theorems do not reduce to fixed point problems in the corresponding metric space (X, d G ).

## References

1. Mustafa Z, Sims B: Some remarks concerning D -metric spaces. Proceedings of the International Conference on Fixed Point Theory and Applications, Valencia, Spain 2003, 189–198.

2. Mustafa Z, Sims B: A new approach to generalized metric spaces. J Nonlinear Convex Anal 2006,7(2):289–297.

3. Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theor Appl 2008, 2008: 12. Article ID 189870

4. Mustafa Z, Sims B: Fixed point theorems for contractive mapping in complete G -metric spaces. Fixed Point Theor Appl 2009, 2009: 10. >Article ID 917175

5. Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int J Math Math Sci 2009, 2009: 10. Article ID 283028

6. Mustafa Z, Awawdeh F, Shatanawi W: Fixed point theorem for expansive mappings in G -metric spaces. Int J Contemp Math Sci 2010, 5: 2463–2472.

7. Abbas M, Rhoades BE: Common fixed point results for non-commuting mappings without continuity in generalized metric spaces. Appl Math Comput 2009, 215: 262–269. 10.1016/j.amc.2009.04.085

8. Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Appl Math Comput 2010, 217: 4094–4099. 10.1016/j.amc.2010.10.026

9. Chugh R, Kadian T, Rani A, Rhoades BE: Property p in G -metric spaces. Fixed Point Theor Appl 2010, 2010: 12. Article ID 401684

10. Saadati R, Vaezpour SM, Vetro P, Rhoades BE: Fixed point theorems in generalized partially ordered G -metric spaces. Math Comput Modelling 2010,52(5–6):797–801. 10.1016/j.mcm.2010.05.009

11. Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theor Appl 2010, 2010: 9. Article ID 181650

12. Abbas M, Khan AR, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl Math Comput 2011, 217: 6328–6336. 10.1016/j.amc.2011.01.006

13. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math Comput Modelling 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036

14. Pant RP: R -weak commutativity and common fixed points. Soochow J Math 1999,1(25):37–42.

## Author information

Authors

### Corresponding author

Correspondence to Safeer Hussain Khan.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Abbas, M., Khan, S.H. & Nazir, T. Common fixed points of R-weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl 2011, 41 (2011). https://doi.org/10.1186/1687-1812-2011-41

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2011-41

### Keywords

• R-weakly commuting maps
• compatible maps
• common fixed point
• generalized metric space 