# Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces

## Abstract

In this article, we establish some common fixed point theorems for a hybrid pair {g, T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Our results unify, generalize and complement various known comparable results from the current literature.

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

## 1. Introduction and preliminaries

Nadler [1] initiated the study of fixed points for multi-valued contraction mappings and generalized the well known Banach fixed point theorem. Then after, many authors studied many fixed point results for multi-valued contraction mappings see [213].

Mustafa and Sims [14] introduced the G-metric spaces as a generalization of the notion of metric spaces. Mustafa et al. [1519] obtained some fixed point theorems for mappings satisfying different contractive conditions. Abbas and Rhoades [20] initiated the study of common fixed point in G-metric spaces. While Saadati et al. [21] studied some fixed point theorems in generalized partially ordered G-metric spaces. Gajić and Crvenković [22, 23] proved some fixed point results for mappings with contractive iterate at a point in G-metric spaces. For other studies in G-metric spaces, we refer the reader to [2438]. Consistent with Mustafa and Sims [14], the following definitions and results will be needed in the sequel.

Definition 1.1. (See [14]). Let X be a non-empty set, G : X × X × X → + be a function satisfying the following properties

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

(G2) 0 < G(x, x, y) for all x, y X with x ≠ y,

(G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z X with y ≠ z,

(G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = ... (symmetry in all three variables),

(G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, aX (rectangle inequality).

Then the function G is called a generalized metric, or, more specially, a G-metric on X, and the pair (X, G) is called a G-metric space.

Definition 1.2. (See [14]). Let (X, G) be a G-metric space, and let (x n ) be a sequence of points of X, therefore, we say that (x n ) is G-convergent to x X if $\underset{n,m\to +\infty }{\text{lim}}G\left(x,{x}_{n},{x}_{m}\right)=0$, that is, for any ε > 0, there exists N such that G(x, x n , x m ) < ε, for all n, m ≥ N. We call x the limit of the sequence and write x n → x or $\underset{n\to +\infty }{\text{lim}}{x}_{n}=x.$

Proposition 1.1. (See [14]). Let (X, G) be a G-metric space. The following statements are equivalent:

1. (1)

(x n ) is G-convergent to x,

2. (2)

G(x n , x n , x) 0 as n → +∞,

3. (3)

G(x n , x, x) 0 as n → +∞,

4. (4)

G(x n , x m , x) 0 as n, m → +.

Definition 1.3. (See [14]). Let (X, G) be a G-metric space. A sequence (x n ) is called a G-Cauchy sequence if for any ε > 0, there is N such that G(x n , x m , x l ) < ε for all m, n, l ≥ N, that is, G(x n , x m , x l ) → 0 as n, m, l → +.

Proposition 1.2. (See [14]). Let (X, G) be a G-metric space. Then the following statements are equivalent:

1. (1)

the sequence (x n ) is G-Cauchy,

2. (2)

for any ε > 0, there exists N such that G(x n , x m , x m ) < ε, for all m, n ≥ N.

Definition 1.4. (See [14]). A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X,G).

Every G-metric on X defines a metric d G on X given by

(1)

Recently, Kaewcharoen and Kaewkhao [34] introduced the following concepts. Let X be a G-metric space. We shall denote CB(X) the family of all nonempty closed bounded subsets of X. Let H(.,.,.) be the Hausdorff G-distance on CB(X), i.e.,

${H}_{G}\left(A,B,C\right)=\text{max}\left\{\underset{x\in A}{\text{sup}}G\left(x,B,C\right),\underset{x\in B}{\text{sup}}G\left(x,C,A\right),\underset{x\in C}{\text{sup}}G\left(x,A,B\right)\right\},$

where

$\begin{array}{l}G\left(x,B,C\right)={d}_{G}\left(x,B\right)+{d}_{G}\left(B,C\right)+{d}_{G}\left(x,C\right),\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}{d}_{G}\left(x,B\right)=\text{inf}\left\{{d}_{G}\left(x,y\right),\phantom{\rule{0.3em}{0ex}}y\in B\right\},\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}{d}_{G}\left(A,B\right)=\text{inf}\left\{{d}_{G}\left(a,b\right),a\in A,b\in B\right\}.\phantom{\rule{2em}{0ex}}\end{array}$

Recall that G(x, y, C) = inf {G(x, y, z), z C}. A mapping T : X → 2X is called a multi-valued mapping. A point x X is called a fixed point of T if x Tx.

Definition 1.5. Let X be a given non empty set. Assume that g : XX and T : X → 2X.

If w = gx Tx for some x X, then x is called a coincidence point of g and T and w is a point of coincidence of g and T.

Mappings g and T are called weakly compatible if gx Tx for some x X implies gT(x) Tg(x).

Proposition 1.3. (see [34]). Let X be a given non empty set. Assume that g : XX and T : X → 2X are weakly compatible mappings. If g and T have a unique point of coincidence w = gx Tx, then w is the unique common fixed point of g and T.

In this article, we establish some common fixed point theorems for a hybrid pair {g,T} of single valued and multi-valued maps satisfying a generalized contractive condition defined on G-metric spaces. Also, an example is presented.

## 2. Main results

We start this section with the following lemma, which is the variant of the one given in Nadler [1] or Assad and Kirk [4]. Its proof is a simple consequence of the definition of the Hausdorff G-distance H G (A, B, B).

Lemma 2.1. If A, B CB(X) and a A, then for each ε > 0, there exists b B such that G(a,b,b) ≤ H G (A, B, B) + ε.

The main result of the article is the following.

Theorem 2.1. Let (X, G) be a G-metric space. Set g : XX and T : XCB(X). Assume that there exists a function α : [0,+∞) → [0,1) satisfying $\underset{r\to {t}^{+}}{\text{lim sup}}\alpha \left(r\right)<1$ for every t ≥ 0 such that

${H}_{G}\left(Tx,Ty,Tz\right)\le \alpha \left(G\left(gx,gy,gz\right)\right)G\left(gx,gy,gz\right),$
(2)

for all x, y, z X. If for any x X, Tx g(X) and g(X) is a G-complete subspace of X, then g and T have a point of coincidence in X. Furthermore, if we assume that gp Tp and gq Tq implies G(gq, gp, gp) ≤ H G (Tq, Tp, Tp), then

(i) g and T have a unique point of coincidence.

(ii) If in addition g and T are weakly compatible, then g and T have a unique common fixed point.

Proof. Let x0 be arbitrary in X. Since Tx0 g(X), choose x1 X such that gx1 Tx0. If gx1= gx0, we finished. Assume that gx0 ≠ gx1, so G(gx0, gx1, gx1) > 0. We can choose a positive integer n1 such that

${\alpha }^{{n}_{1}}\left(G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\right)\le \left[1-\alpha \left(G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\right)\right]G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right).$

By Lemma 2.1 and the fact that Tx1 g(X), there exists gx2 Tx1 such that

$G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\le {H}_{G}\left(T{x}_{0},T{x}_{1},T{x}_{1}\right)+{\alpha }^{{n}_{1}}\left(G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\right).$

Using the two above inequalities and (2), it follows that

$\begin{array}{ll}\hfill G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)& \le {H}_{G}\left(T{x}_{0},T{x}_{1},T{x}_{1}\right)+{\alpha }^{{n}_{1}}\left(G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \alpha \left(G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\right)G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)+\left[1-\alpha \left(G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\right)\right]G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right)\phantom{\rule{2em}{0ex}}\\ =G\left(g{x}_{0},g{x}_{1},g{x}_{1}\right).\phantom{\rule{2em}{0ex}}\end{array}$

If gx1 = gx2, we finished. Assume that gx1 ≠ gx2. Now we choose a positive integer n2> n1 such that

${\alpha }^{{n}_{2}}\left(G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\right)\le \left[1-\alpha \left(G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\right)\right]G\left(g{x}_{2},g{x}_{2},g{x}_{2}\right).$

Since Tx2 CB(X) and the fact that Tx2 g(X), we may select gx3 Tx2 such that from Lemma 2.1

$G\left(g{x}_{2},g{x}_{3},g{x}_{3}\right)\le {H}_{G}\left(T{x}_{1},T{x}_{2},T{x}_{2}\right)+{\alpha }^{{n}_{2}}\left(G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\right),$

and then, similarly to the previous case, we have

$\begin{array}{ll}\hfill G\left(g{x}_{2},g{x}_{3},g{x}_{3}\right)& \le {H}_{G}\left(T{x}_{1},T{x}_{2},T{x}_{2}\right)+{\alpha }^{{n}_{2}}\left(G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \alpha \left(G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\right)G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)+\left[1-\alpha \left(G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\right)\right]G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right)\phantom{\rule{2em}{0ex}}\\ =G\left(g{x}_{1},g{x}_{2},g{x}_{2}\right).\phantom{\rule{2em}{0ex}}\end{array}$

By repeating this process, for each k *, we may choose a positive integer n k such that

${\alpha }^{{n}_{k}}\left(G\left(g{x}_{k-1},g{x}_{k},g{x}_{k}\right)\right)\le \left[1-\alpha \left(G\left(g{x}_{k-1},g{x}_{k},g{x}_{k}\right)\right)\right]G\left(g{x}_{k-1},g{x}_{k},g{x}_{k}\right).$

Again, we may select gx k+1 Tx k such that

$G\left(g{x}_{k},g{x}_{k+1},g{x}_{k+1}\right)\le {H}_{G}\left(T{x}_{k-1},T{x}_{k},T{x}_{k}\right)+{\alpha }^{{n}_{k}}\left(G\left(g{x}_{k-1},g{x}_{k},g{x}_{k}\right)\right).$
(3)

The last two inequalities together imply that

$G\left(g{x}_{k},g{x}_{k+1},g{x}_{k+1}\right)\le G\left(g{x}_{k-1},g{x}_{k},g{x}_{k}\right),$

which shows that the sequence of nonnegative numbers {d k }, given by d k = G(gx k-1 , gx k , gx k ), k = 1, 2,. . ., is non-increasing. This means that there exists d ≥ 0 such that

$\underset{k\to +\infty }{\text{lim}}{d}_{k}=d.$

Let now prove that the {gx k } is a G-Cauchy sequence.

Using the fact that, by hypothesis for t = d, $\underset{r\to {d}^{+}}{\text{lim sup}}\alpha \left(t\right)<1$, it results that there exists a rank k0 such that for k ≥ k0, we have α(d k ) < h, where

$\underset{t\to {d}^{+}}{\text{lim sup}}\alpha \left(t\right)

Now, by (3) we deduce that the sequence {d k } satisfies the following recurrence inequality

${d}_{k+1}\le {H}_{G}\left(T{x}_{k-1},T{x}_{k},T{x}_{k}\right)+{\alpha }^{{n}_{k}}\left({d}_{k}\right)\le \alpha \left({d}_{k}\right){d}_{k}+{\alpha }^{{n}_{k}}\left({d}_{k}\right),\phantom{\rule{1em}{0ex}}k\ge 1.$
(4)

By induction, from (4), we get

${d}_{k+1}\le \prod _{i=1}^{k}\alpha \left({d}_{i}\right){d}_{1}+\sum _{m=1}^{k-1}\prod _{i=m+1}^{k}\alpha \left({d}_{i}\right){\alpha }^{{n}_{m}}\left({d}_{m}\right)+{\alpha }^{{n}_{k}}\left({d}_{k}\right),\phantom{\rule{1em}{0ex}}k\ge 1,$

which, by using the fact that α < 1, can be simplified to

${d}_{k+1}\le \prod _{i=1}^{k}\alpha \left({d}_{i}\right){d}_{1}+\sum _{m=1}^{k-1}\prod _{i=\text{max}\left\{{k}_{0},m+1\right\}}^{k}\alpha \left({d}_{i}\right){\alpha }^{{n}_{m}}\left({d}_{m}\right)+{\alpha }^{{n}_{k}}\left({d}_{k}\right),\phantom{\rule{1em}{0ex}}k\ge 1,$

Referring to the proof of Theorem 2.1 in [11] or Lemma 3.2 in [12], we may obtain

$\prod _{i=1}^{k}\alpha \left({d}_{i}\right){d}_{1}+\sum _{m=1}^{k-1}\prod _{i=\text{max}\left\{{k}_{0},m+1\right\}}^{k}\alpha \left({d}_{i}\right){\alpha }^{{n}_{m}}\left({d}_{m}\right)+{\alpha }^{{n}_{k}}\left({d}_{k}\right)\le c{h}^{k},$

where c is a positive constant. We deduce that

${d}_{k+1}=G\left(g{x}_{k},g{x}_{k+1},g{x}_{k+1}\right)\le c{h}^{k}.$

Now for k ≥ k0 and m is a positive arbitrary integer, we have using the property (G4)

$\begin{array}{ll}\hfill G\left(g{x}_{k},g{x}_{k+m},g{x}_{k+m}\right)& \le G\left(g{x}_{k},g{x}_{k+1},g{x}_{k+1}\right)+G\left(g{x}_{k+1},g{x}_{k+2},g{x}_{k+2}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\cdots +G\left(g{x}_{k+m-2},g{x}_{k+m-1},g{x}_{k+m-1}\right)+G\left(g{x}_{k+m-1},g{x}_{k+m},g{x}_{k+m}\right)\phantom{\rule{2em}{0ex}}\\ \le c\left[{h}^{k}+{h}^{k+1}+\cdots +{h}^{k+m-1}\right]\phantom{\rule{2em}{0ex}}\\ \le c\frac{{h}^{k}}{1-h}\to 0\phantom{\rule{2.77695pt}{0ex}}\text{as}\phantom{\rule{2.77695pt}{0ex}}k\to +\infty ,\phantom{\rule{2em}{0ex}}\end{array}$

since 0 < h < 1. This shows that the sequence {gx n } is G-Cauchy in the complete subspace g(X). Thus, there exists q g(X) such that, from Proposition 1.1

$\underset{n\to +\infty }{\text{lim}}G\left(g{x}_{n},g{x}_{n},q\right)=\underset{n\to +\infty }{\text{lim}}G\left(g{x}_{n},q,q\right)=0.$
(5)

Since q g(X), then there exists p X such that q = gp. From (5), we have

$\underset{n\to +\infty }{\text{lim}}G\left(g{x}_{n},g{x}_{n},gp\right)=\underset{n\to +\infty }{\text{lim}}G\left(g{x}_{n},gp,gp\right)=0.$
(6)

We claim that gp Tp. Indeed, from (2), we have

$\begin{array}{ll}\hfill G\left(g{x}_{n+1},Tp,Tp\right)& \le {H}_{G}\left(T{x}_{n},Tp,Tp\right)\phantom{\rule{2em}{0ex}}\\ \le \alpha \left(G\left(g{x}_{n},gp,gp\right)\right)G\left(g{x}_{n},gp,gp\right).\phantom{\rule{2em}{0ex}}\end{array}$
(7)

Letting n → +∞ in (7) and using (6), we get

$G\left(gp,Tp,Tp\right)=\underset{n\to +\infty }{\text{lim}}G\left(g{x}_{n+1},Tp,Tp\right)=0,$

that is, gp Tp. That is T and g have a point of coincidence. Now, assume that if gp Tp and gq Tq, then G(gq, gp, gp) ≤ H G (Tq, Tp, Tp). We will prove the uniqueness of a point of coincidence of g and T. Suppose that gp Tp and gq Tq. By (2) and this assumption, we have

$\begin{array}{ll}\hfill G\left(gq,gp,gp\right)& \le {H}_{G}\left(Tq,Tp,Tp\right)\phantom{\rule{2em}{0ex}}\\ \le \alpha \left(G\left(gq,gp,gp\right)\right)G\left(gq,gp,gp\right),\phantom{\rule{2em}{0ex}}\end{array}$
(8)

and since α(G(gq, gp, gp)) < G(gq, gp, gp), so necessarily from (8), we have G(gq, gp, gp) = 0, i.e., gp = gq. In view of

${H}_{G}\left(Tq,Tp,Tp\right)\le \alpha \left(G\left(gq,gp,gp\right)\right)G\left(gq,gp,gp\right)=0,$

we get Tq = Tp. Thus, T and g have a unique point of coincidence. Suppose that g and T are weakly compatible. By applying Proposition 1.3, we obtain that g and T have a unique common fixed point.

Corollary 2.1. Let (X,G) be a complete G-metric space. Assume that T : XCB(X) satisfies the following condition

${H}_{G}\left(Tx,Ty,Tz\right)\le \alpha \left(G\left(x,y,z\right)\right)G\left(x,y,z\right),$
(9)

for all x, y, z X, where α : [0,+∞) → [0,1) satisfies $\underset{r\to {t}^{+}}{\text{lim sup}}\alpha \left(r\right)<1$ for every t ≥ 0. Then T has a fixed point in X. Furthermore, if we assume that p Tp and q Tq implies G(q, p, p) ≤ H G (Tq, Tp, Tp), then T has a unique fixed point.

Proof. It follows by taking g the identity on X in Theorem 2.1.

Corollary 2.2. Let (X, G) be a G-metric space. Assume that g : XX and T : XCB(X) satisfy the following condition

${H}_{G}\left(Tx,Ty,Tz\right)\le kG\left(gx,gy,gz\right),$
(10)

for all x, y, z X, where k [0,1). If for any x X, Tx g(X) and g(X) is a G-complete subspace of X, then g and T have a point of coincidence in X. Furthermore, if we assume that gp Tp and gq Tq implies G(gq, gp, gp) ≤ H G (Tq, Tp, Tp), then

(i) g and T have a unique point of coincidence.

(ii) If in addition g and T are weakly compatible, then g and T have a unique common fixed point.

Proof. It follows by taking α(t) = k, k [0,1), in Theorem 2.1.

In the case of single-valued mappings, that is, if T : XX, (i.e., Tx = {Tx} for any x X), it is obviously that

${H}_{G}\left(Tx,Ty,Tz\right)=G\left(Tx,Ty,Tz\right),\phantom{\rule{1em}{0ex}}\forall \phantom{\rule{0.3em}{0ex}}x,y,z\in X.$

Furthermore, if gp Tp (i.e., gp = Tp) and gq Tq (i.e., gq = Tq), then clearly,

$G\left(gq,gp,gp\right)=G\left(Tq,Tp,Tp\right)={H}_{G}\left(Tq,Tp,Tp\right),$

that is, the assumption given in Theorem 2.1 is verified.

Also, the single-valued mappings T, g : XX are said weakly compatible if Tgx = gTx whenever Tx = gx for some x X.

Now, we may state the following corollaries from Theorem 2.1 and the precedent corollaries:

Corollary 2.3. Let (X, G) be a complete G-metric space. Assume that T : XX satisfies the following condition

$G\left(Tx,Ty,Tz\right)\le \alpha \left(G\left(x,y,z\right)\right)G\left(x,y,z\right)$
(11)

for all x, y, z X, where α : [0, +∞) → [0, 1) satisfies $\underset{r\to {t}^{+}}{\text{lim sup}}\alpha \left(r\right)<1$ for every t ≥ 0. Then, T has a unique fixed point.

Corollary 2.4. Let (X, G) be a G-metric space. Assume that g : XX and T : XX satisfy the following condition

$G\left(Tx,Ty,Tz\right)\le \alpha \left(G\left(gx,gy,gz\right)\right)G\left(gx,gy,gz\right)$
(12)

for all x, y, z X, where α : [0, +∞) → [0, 1) satisfies $\underset{r\to {t}^{+}}{\text{lim sup}}\alpha \left(r\right)<1$ for every t ≥ 0. If T(X) g(X) and g(X) is a G-complete subspace of X, then

(i) g and T have a unique point of coincidence.

(ii) Furthermore, if g and T are weakly compatible, then g and T have a unique common fixed point.

Now, we introduce an example to support the useability of our results.

Example 2.1. Let X = [0, 1]. Define T : XCB(X) by $Tx=\left[0,\frac{1}{16}x\right]$ and define g : X → X by $gx=\sqrt{x}$. Define a G-metric on X by G(x, y, z) = max{|x-y|, |x-z|, |y-z|}. Also, define α : [0, +∞) → [0, 1) by $\alpha \left(t\right)=\frac{1}{2}$ Then:

1. (1)

Tx g(X) for all x X.

2. (2)

g(X) is a G-complete subspace of X.

3. (3)

g and T are weakly compatible.

4. (4)

H G (Tx, Ty, Tz) ≤ α(G(gx, gy, gz))G(gx, gy, gz) for all x, y, z X.

Proof. The proofs of (1), (2), and (3) are clear. By (1), we have

To prove (4), let x, y, z X. If x = y = z = 0, then

${H}_{G}\left(Tx,Ty,Tz\right)=0\le \alpha \left(G\left(gx,gy,gz\right)\right)G\left(gx,gy,gz\right).$

Thus, we may assume that x, y, and z are not all zero. With out loss of generality, we assume that x ≤ y ≤ z. Then

Since x ≤ y ≤ z, so $\left[0,\frac{1}{16}x\right]\subseteq \left[0,\frac{1}{16}y\right]\subseteq \left[0,\frac{1}{16}z\right]$ This implies that

${d}_{G}\left(\left[0,\frac{1}{16}x\right],\left[0,\frac{1}{16}y\right]\right)={d}_{G}\left(\left[0,\frac{1}{16}y\right],\left[0,\frac{1}{16}z\right]\right)={d}_{G}\left(\left[0,\frac{1}{16}x\right],\left[0,\frac{1}{16}z\right]\right)=0.$

For each $0\le a\le \frac{1}{16}x,$ we have

$G\left(a,\left[0,\frac{1}{16}y\right],\left[0,\frac{1}{16}z\right]\right)={d}_{G}\left(a,\left[0,\frac{1}{16}y\right]\right)+{d}_{G}\left(\left[0,\frac{1}{16}y\right],\left[0,\frac{1}{16}z\right]\right)+{d}_{G}\left(a,\left[0,\frac{1}{16}z\right]\right)=0.$

Also, for each $0\le b\le \frac{1}{16}y$, we have

$\begin{array}{c}G\left(b,\left[0,\frac{1}{16}z\right],\left[0,\frac{1}{16}x\right]\right)={d}_{G}\left(b,\left[0,\frac{1}{16}z\right]\right)+{d}_{G}\left(\left[0,\frac{1}{16}z\right],\left[0,\frac{1}{16}x\right]\right)+{d}_{G}\left(b,\left[0,\frac{1}{16}x\right]\right)\\ =\left\{\underset{2b-\frac{x}{8}\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}b\ge \frac{x}{16}.}{0\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}b\le \frac{x}{16}}\end{array}$

This yields that

$\underset{0\le b\le \frac{1}{16}y}{\text{sup}}G\left(b,\left[0,\frac{1}{16}z\right],\left[0,\frac{1}{16}x\right]\right)=\frac{y}{8}-\frac{x}{8}.$

Moreover, for each $0\le c\le \frac{1}{16}z$, we have

$\begin{array}{c}G\left(c,\left[0,\frac{1}{16}x\right],\left[0,\frac{1}{16}y\right]\right)={d}_{G}\left(c,\left[0,\frac{1}{16}x\right]\right)+{d}_{G}\left(\left[0,\frac{1}{16}x\right],\left[0,\frac{1}{16}y\right]\right)+{d}_{G}\left(c,\left[0,\frac{1}{16}y\right]\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=\left\{\begin{array}{c}0\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}c\le \frac{x}{16}\hfill \\ 2c-\frac{x}{8}\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}\frac{x}{16}\le c\le \frac{y}{16}\hfill \\ 4c-\frac{x}{8}-\frac{y}{8}\phantom{\rule{2.77695pt}{0ex}}\text{if}\phantom{\rule{2.77695pt}{0ex}}c\ge \frac{y}{16}.\hfill \end{array}\right\\end{array}$

This yields that

$\underset{0\le c\le \frac{1}{16}z}{\text{sup}}G\left(c,\left[\frac{1}{16}c\right],\left[0,\frac{1}{16}y\right]\right)=\frac{z}{4}-\frac{x}{8}-\frac{y}{8}.$

We deduce that

$\begin{array}{ll}\hfill {H}_{G}\left(Tx,Ty,Tz\right)& =\frac{z}{4}-\frac{x}{8}-\frac{y}{8}\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{4}\left(z-x\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2}\left(\frac{1}{2}\left(z-x\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{2}\left(\frac{z-x}{\sqrt{x}+\sqrt{z}}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{2}\left(\sqrt{z}-\sqrt{x}\right)\phantom{\rule{2em}{0ex}}\end{array}$

On the other hand, it is obvious that all other hypotheses of Theorem 2.1 are satisfied and so g and T have a unique common fixed point, which is u = 0.

Remark 1 • Theorem 2.1 improves Kaewcharoen and Kaewkhao [[34], Theorem 3.3] (in case b = c = d = 0).

• Corollary 2.3 generalizes Mustafa [[15], Theorem 5.1.7] and Shatanawi [[35], Corollary 3.4].

## References

1. Nadler SB: Multi-valued contraction mappings. Pacific J Math 1969, 30: 475–478.

2. Gorniewicz L: Topological fixed point theory of multivalued mappings. Kluwer Academic Publishers, Dordrecht; 1999.

3. Klim D, Wardowski D: Fixed point theorems for set-valued contractions in complete metric spaces. J Math Anal Appl 2007, 334: 132–139. 10.1016/j.jmaa.2006.12.012

4. Assad NA, Kirk WA: Fixed point theorems for setvalued mappings of contractive type. Pacific J Math 1972, 43: 553–562.

5. Hong SH: Fixed points of multivalued operators in ordered metric spaces with applications. Nonlinear Anal 2010, 72: 3929–3942. 10.1016/j.na.2010.01.013

6. Hong SH: Fixed points for mixed monotone multivalued operators in Banach Spaces with applications. J Math Anal Appl 2008, 337: 333–342. 10.1016/j.jmaa.2007.03.091

7. Hong SH, Guan D, Wang L: Hybrid fixed points of multivalued operators in metric spaces with applications. Nonlinear Anal 2009, 70: 4106–4117. 10.1016/j.na.2008.08.020

8. Hong SH: Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications. J Math Anal Appl 2003, 282: 151–162. 10.1016/S0022-247X(03)00111-2

9. Shatanawi W: Some fixed point results for a generalized Ψ -weak contraction mappings in orbitally metric spaces. Chaos Solitons Fract 2012, 45: 520–526. 10.1016/j.chaos.2012.01.015

10. Mizoguchi N, Takahashi W: Fixed point theorems for multi-valued mappings on complete metric spaces. J Math Anal Appl 1989, 141: 177–188. 10.1016/0022-247X(89)90214-X

11. Berinde M, Berinde V: On a general class of multi-valued weakly Picard mappings. J Math Anal Appl 2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016

12. Kamran T: Multivalued f -weakly Picard mappings. Nonlinear Anal 2007, 67: 2289–2296. 10.1016/j.na.2006.09.010

13. Al-Thagafi MA, Shahzad N: Coincidence points, generalized I -nonexpansive multimaps and applications. Nonlinear Anal 2007, 67: 2180–2188. 10.1016/j.na.2006.08.042

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

15. Mustafa Z: A new structure for generalized metric spaces with applications to fixed point theory. University of Newcastle, Newcastle, UK; 2005. Ph.D. thesis

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

17. Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl 2009, 2009: 10. ID 917175

18. Mustafa Z, Khandaqji M, Shatanawi W: Fixed point results on complete G-metric spaces. Studia Scientiarum Mathematicarum Hungarica 2011, 48: 304–319. 10.1556/SScMath.48.2011.3.1170

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

20. 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

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

22. Gajić L, Crvenković ZL: On mappings with contractive iterate at a point in generalized metric spaces. Fixed Point Theory Appl 2010., 2010: (ID 458086), 16 (2010). doi:10.1155/2010/458086

23. Gajić L, Crvenković ZL: A fixed point result for mappings with contractive iterate at a point in G -metric spaces. Filomat 2011, 25: 53–58. doi:10.2298/FIL1102053G

24. Abbas M, Khan SH, Nazir T: Common fixed points of R -weakly commuting maps in generalized metric space. Fixed Point Theory Appl 2011, 2011: 41. 10.1186/1687-1812-2011-41

25. Abbas M, Khan AK, Nazir T: Coupled common fixed point results in two generalized metric spaces. Appl Math Comput 2011. doi:10.1016/j.amc.2011.01.006

26. Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G- metric spaces. Filomat 2011, 25: 1–17.

27. Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math Comput Model 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059

28. Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput Math Appl 2011, 62: 4222–4229. 10.1016/j.camwa.2011.10.007

29. Aydi H, Shatanawi W, Postolache M: Coupled fixed point results for ( Ψ, φ )-weakly contractive mappings in ordered G -metric spaces. Comput Math Appl 2012, 63: 298–309. 10.1016/j.camwa.2011.11.022

30. Cho YJ, Rhoades BE, Saadati R, Samet B, Shatanawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory and Appl 2012, 2012: 8. 10.1186/1687-1812-2012-8

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

32. Chugh R, Kadian T, Rani A, Rhoades BE: Property P in G -metric spaces. Fixed Point Theory Appl 2010, 2010: 12. (ID 401684)

33. Gholizadeh L, Saadati R, Shatanawi W, Vaezpour SM: Contractive Mapping in Generalized, Ordered Metric Spaces with Application in Integral Equations. Math Probl Eng 2011, 2011: 14. (ID 380784)

34. Kaewcharoen A, Kaewkhao A: Common fixed points for single-valued and multi-valued mappings in G -metric spaces. Int J Math Anal 2011, 5: 1775–1790.

35. Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl 2010, 2010: 9. (ID 181650)

36. Shatanawi W: Some fixed point theorems in ordered G-metric spaces and applications. Abst Appl Anal 2011, 2011: 11. (ID 126205)

37. Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacettepe J Math Stat 2011, 40: 441–447.

38. Shatanawi W, Abbas M, Nazir T: Common coupled coincidence and coupled fixed point results in two generalized metric spaces. Fixed point Theory Appl 2011, 2011: 80. 10.1186/1687-1812-2011-80

## Acknowledgements

The authors thank the editor and the referees for their useful comments and suggestions.

## Author information

Authors

### Corresponding author

Correspondence to Nedal Tahat.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

The authors have contributed in obtaining the new results presented in this article. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Tahat, N., Aydi, H., Karapinar, E. et al. Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G-metric spaces. Fixed Point Theory Appl 2012, 48 (2012). https://doi.org/10.1186/1687-1812-2012-48