Skip to content

Advertisement

Open Access

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

  • Nedal Tahat1Email author,
  • Hassen Aydi2,
  • Erdal Karapinar3 and
  • Wasfi Shatanawi1
Fixed Point Theory and Applications20122012:48

https://doi.org/10.1186/1687-1812-2012-48

Received: 18 October 2011

Accepted: 26 March 2012

Published: 26 March 2012

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.

Keywords

multi-valued mappingscommon fixed pointweakly compatible mappingsgeneralized contraction

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 lim n , m + G x , x n , x m = 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 lim n + 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
d G x , y = G x , y , y + G y , x , x , for all x , y X .
(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 A , B , C = max sup x A G x , B , C , sup x B G x , C , A , sup x C G x , A , B ,
where
G x , B , C = d G x , B + d G B , C + d G x , C , d G x , B = inf d G x , y , y B , d G A , B = inf d G a , b , a A , b B .

Recall that G(x, y, C) = inf {G(x, y, z), z C}. A mapping T : X → 2 X 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 → 2 X .

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 → 2 X 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 lim sup r t + α ( r ) < 1 for every t ≥ 0 such that
H G T x , T y , T z α G g x , g y , g z G g x , g y , g z ,
(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
α n 1 G g x 0 , g x 1 , g x 1 1 - α G g x 0 , g x 1 , g x 1 G g x 0 , g x 1 , g x 1 .
By Lemma 2.1 and the fact that Tx1 g(X), there exists gx2 Tx1 such that
G g x 1 , g x 2 , g x 2 H G T x 0 , T x 1 , T x 1 + α n 1 G g x 0 , g x 1 , g x 1 .
Using the two above inequalities and (2), it follows that
G g x 1 , g x 2 , g x 2 H G T x 0 , T x 1 , T x 1 + α n 1 G g x 0 , g x 1 , g x 1 α G g x 0 , g x 1 , g x 1 G g x 0 , g x 1 , g x 1 + 1 - α G g x 0 , g x 1 , g x 1 G g x 0 , g x 1 , g x 1 = G g x 0 , g x 1 , g x 1 .
If gx1 = gx2, we finished. Assume that gx1 ≠ gx2. Now we choose a positive integer n2> n1 such that
α n 2 G g x 1 , g x 2 , g x 2 1 - α G g x 1 , g x 2 , g x 2 G g x 2 , g x 2 , g x 2 .
Since Tx2 CB(X) and the fact that Tx2 g(X), we may select gx3 Tx2 such that from Lemma 2.1
G g x 2 , g x 3 , g x 3 H G T x 1 , T x 2 , T x 2 + α n 2 G g x 1 , g x 2 , g x 2 ,
and then, similarly to the previous case, we have
G g x 2 , g x 3 , g x 3 H G T x 1 , T x 2 , T x 2 + α n 2 G g x 1 , g x 2 , g x 2 α G g x 1 , g x 2 , g x 2 G g x 1 , g x 2 , g x 2 + 1 - α G g x 1 , g x 2 , g x 2 G g x 1 , g x 2 , g x 2 = G g x 1 , g x 2 , g x 2 .
By repeating this process, for each k *, we may choose a positive integer n k such that
α n k G g x k - 1 , g x k , g x k 1 - α G g x k - 1 , g x k , g x k G g x k - 1 , g x k , g x k .
Again, we may select gx k+1 Tx k such that
G g x k , g x k + 1 , g x k + 1 H G T x k - 1 , T x k , T x k + α n k G g x k - 1 , g x k , g x k .
(3)
The last two inequalities together imply that
G ( g x k , g x k + 1 , g x k + 1 ) G ( g x k - 1 , g x k , g x k ) ,
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
lim k + d k = d .

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

Using the fact that, by hypothesis for t = d, lim sup r d + α ( t ) < 1 , it results that there exists a rank k0 such that for k ≥ k0, we have α(d k ) < h, where
lim sup t d + α ( t ) < h < 1 .
Now, by (3) we deduce that the sequence {d k } satisfies the following recurrence inequality
d k + 1 H G T x k - 1 , T x k , T x k + α n k ( d k ) α ( d k ) d k + α n k ( d k ) , k 1 .
(4)
By induction, from (4), we get
d k + 1 i = 1 k α ( d i ) d 1 + m = 1 k - 1 i = m + 1 k α ( d i ) α n m ( d m ) + α n k ( d k ) , k 1 ,
which, by using the fact that α < 1, can be simplified to
d k + 1 i = 1 k α ( d i ) d 1 + m = 1 k - 1 i = max { k 0 , m + 1 } k α ( d i ) α n m ( d m ) + α n k ( d k ) , k 1 ,
Referring to the proof of Theorem 2.1 in [11] or Lemma 3.2 in [12], we may obtain
i = 1 k α ( d i ) d 1 + m = 1 k - 1 i = max { k 0 , m + 1 } k α ( d i ) α n m ( d m ) + α n k ( d k ) c h k ,
where c is a positive constant. We deduce that
d k + 1 = G ( g x k , g x k + 1 , g x k + 1 ) c h k .
Now for k ≥ k0 and m is a positive arbitrary integer, we have using the property (G4)
G ( g x k , g x k + m , g x k + m ) G ( g x k , g x k + 1 , g x k + 1 ) + G ( g x k + 1 , g x k + 2 , g x k + 2 ) + + G ( g x k + m - 2 , g x k + m - 1 , g x k + m - 1 ) + G ( g x k + m - 1 , g x k + m , g x k + m ) c h k + h k + 1 + + h k + m - 1 c h k 1 - h 0 as k + ,
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
lim n + G ( g x n , g x n , q ) = lim n + G ( g x n , q , q ) = 0 .
(5)
Since q g(X), then there exists p X such that q = gp. From (5), we have
lim n + G ( g x n , g x n , g p ) = lim n + G ( g x n , g p , g p ) = 0 .
(6)
We claim that gp Tp. Indeed, from (2), we have
G ( g x n + 1 , T p , T p ) H G ( T x n , T p , T p ) α ( G ( g x n , g p , g p ) ) G ( g x n , g p , g p ) .
(7)
Letting n → +∞ in (7) and using (6), we get
G ( g p , T p , T p ) = lim n + G ( g x n + 1 , T p , T p ) = 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
G ( g q , g p , g p ) H G ( T q , T p , T p ) α ( G ( g q , g p , g p ) ) G ( g q , g p , g p ) ,
(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 ( T q , T p , T p ) α ( G ( g q , g p , g p ) ) G ( g q , g p , g p ) = 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 ( T x , T y , T z ) α ( G ( x , y , z ) ) G ( x , y , z ) ,
(9)

for all x, y, z X, where α : [0,+∞) → [0,1) satisfies lim sup r t + α ( r ) < 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 ( T x , T y , T z ) k G ( g x , g y , g z ) ,
(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 ( T x , T y , T z ) = G ( T x , T y , T z ) , x , y , z X .
Furthermore, if gp Tp (i.e., gp = Tp) and gq Tq (i.e., gq = Tq), then clearly,
G ( g q , g p , g p ) = G ( T q , T p , T p ) = H G ( T q , T p , T p ) ,

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 ( T x , T y , T z ) α ( G ( x , y , z ) ) G ( x , y , z )
(11)

for all x, y, z X, where α : [0, +∞) → [0, 1) satisfies lim sup r t + α ( r ) < 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 ( T x , T y , T z ) α ( G ( g x , g y , g z ) ) G ( g x , g y , g z )
(12)

for all x, y, z X, where α : [0, +∞) → [0, 1) satisfies lim sup r t + α ( r ) < 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 T x = 0 , 1 16 x and define g : X → X by g x = 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 α ( t ) = 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
d G x , y = G x , y , y + G y , x , x = 2 x - y  for all  x , y X .
To prove (4), let x, y, z X. If x = y = z = 0, then
H G T x , T y , T z = 0 α G g x , g y , g z G g x , g y , g z .
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
H G T x , T y , T z = H G 0 , 1 16 x , 0 , 1 16 y , 0 , 1 16 z = max sup 0 a 1 16 x G a , 0 , 1 16 y , 0 , 1 16 z , sup 0 b 1 16 y G b , 0 , 1 16 z , 0 , 1 16 x , sup 0 c 1 16 z G c , 0 , 1 16 x , 0 , 1 16 y .
Since x ≤ y ≤ z, so 0 , 1 16 x 0 , 1 16 y 0 , 1 16 z This implies that
d G ( [ 0 , 1 16 x ] , [ 0 , 1 16 y ] ) = d G ( [ 0 , 1 16 y ] , [ 0 , 1 16 z ] ) = d G ( [ 0 , 1 16 x ] , [ 0 , 1 16 z ] ) = 0 .
For each 0 a 1 16 x , we have
G a , 0 , 1 16 y , 0 , 1 16 z = d G a , 0 , 1 16 y + d G 0 , 1 16 y , 0 , 1 16 z + d G a , 0 , 1 16 z = 0 .
Also, for each 0 b 1 16 y , we have
G ( b , [ 0 , 1 16 z ] , [ 0 , 1 16 x ] ) = d G ( b , [ 0 , 1 16 z ] ) + d G ( [ 0 , 1 16 z ] , [ 0 , 1 16 x ] ) + d G ( b , [ 0 , 1 16 x ] ) = { 0 if b x 16 2 b x 8 if b x 16 .
This yields that
sup 0 b 1 16 y G b , 0 , 1 16 z , 0 , 1 16 x = y 8 - x 8 .
Moreover, for each 0 c 1 16 z , we have
G c , 0 , 1 16 x , 0 , 1 16 y = d G c , 0 , 1 16 x + d G 0 , 1 16 x , 0 , 1 16 y + d G c , 0 , 1 16 y = 0 if c x 16 2 c - x 8 if x 16 c y 16 4 c - x 8 - y 8 if c y 16 .
This yields that
sup 0 c 1 16 z G c , 1 16 c , 0 , 1 16 y = z 4 - x 8 - y 8 .
We deduce that
H G ( T x , T y , T z ) = z 4 - x 8 - y 8 1 4 ( z - x ) = 1 2 1 2 ( z - x ) 1 2 z - x x + z = 1 2 ( z - x )

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].

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Hashemite University, Zarqa, Jordan
(2)
Université de Sousse, Institut Supérieur d'Informatique et des Technologies de Communication De Hammam Sousse, Hammam Sousse, Tunisie
(3)
Department of Mathematics, Atilim University, İncek, Ankara, Turkey

References

  1. Nadler SB: Multi-valued contraction mappings. Pacific J Math 1969, 30: 475–478.MATHMathSciNetView ArticleGoogle Scholar
  2. Gorniewicz L: Topological fixed point theory of multivalued mappings. Kluwer Academic Publishers, Dordrecht; 1999.MATHView ArticleGoogle Scholar
  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.2006.12.012MATHMathSciNetView ArticleGoogle Scholar
  4. Assad NA, Kirk WA: Fixed point theorems for setvalued mappings of contractive type. Pacific J Math 1972, 43: 553–562.MathSciNetView ArticleGoogle Scholar
  5. Hong SH: Fixed points of multivalued operators in ordered metric spaces with applications. Nonlinear Anal 2010, 72: 3929–3942. 10.1016/j.na.2010.01.013MATHMathSciNetView ArticleGoogle Scholar
  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.2007.03.091MATHMathSciNetView ArticleGoogle Scholar
  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.2008.08.020MATHMathSciNetView ArticleGoogle Scholar
  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)00111-2MATHMathSciNetView ArticleGoogle Scholar
  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.2012.01.015MATHMathSciNetView ArticleGoogle Scholar
  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-XMATHMathSciNetView ArticleGoogle Scholar
  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.2006.03.016MATHMathSciNetView ArticleGoogle Scholar
  12. Kamran T: Multivalued f -weakly Picard mappings. Nonlinear Anal 2007, 67: 2289–2296. 10.1016/j.na.2006.09.010MATHMathSciNetView ArticleGoogle Scholar
  13. Al-Thagafi MA, Shahzad N: Coincidence points, generalized I -nonexpansive multimaps and applications. Nonlinear Anal 2007, 67: 2180–2188. 10.1016/j.na.2006.08.042MATHMathSciNetView ArticleGoogle Scholar
  14. Mustafa Z, Sims B: A new approach to generalized metric spaces. J Nonlinear Convex Anal 2006, 7: 289–297.MATHMathSciNetGoogle Scholar
  15. Mustafa Z: A new structure for generalized metric spaces with applications to fixed point theory. University of Newcastle, Newcastle, UK; 2005. Ph.D. thesisGoogle Scholar
  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 189870MathSciNetView ArticleGoogle Scholar
  17. Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl 2009, 2009: 10. ID 917175MathSciNetView ArticleGoogle Scholar
  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.48.2011.3.1170MATHMathSciNetView ArticleGoogle Scholar
  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 283028MathSciNetView ArticleGoogle Scholar
  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.2009.04.085MATHMathSciNetView ArticleGoogle Scholar
  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.2010.05.009MATHMathSciNetView ArticleGoogle Scholar
  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:10.1155/2010/458086Google Scholar
  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/FIL1102053GMATHGoogle Scholar
  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. 10.1186/1687-1812-2011-41MathSciNetView ArticleGoogle Scholar
  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.2011.01.006Google Scholar
  26. Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G- metric spaces. Filomat 2011, 25: 1–17.MATHMathSciNetView ArticleGoogle Scholar
  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.2011.05.059MATHView ArticleGoogle Scholar
  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.2011.10.007MATHMathSciNetView ArticleGoogle Scholar
  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.2011.11.022MATHMathSciNetView ArticleGoogle Scholar
  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. 10.1186/1687-1812-2012-8MathSciNetView ArticleGoogle Scholar
  31. Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math Comput Model 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036MATHMathSciNetView ArticleGoogle Scholar
  32. Chugh R, Kadian T, Rani A, Rhoades BE: Property P in G -metric spaces. Fixed Point Theory Appl 2010, 2010: 12. (ID 401684)MathSciNetView ArticleGoogle Scholar
  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)MathSciNetView ArticleGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  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)MathSciNetView ArticleGoogle Scholar
  36. Shatanawi W: Some fixed point theorems in ordered G-metric spaces and applications. Abst Appl Anal 2011, 2011: 11. (ID 126205)MathSciNetGoogle Scholar
  37. Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacettepe J Math Stat 2011, 40: 441–447.MATHMathSciNetGoogle Scholar
  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. 10.1186/1687-1812-2011-80MathSciNetView ArticleGoogle Scholar

Copyright

© Tahat et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement