Research

# Coupled coincidence points for two mappings in metric spaces and cone metric spaces

Wei Long1*, Billy E Rhoades2 and Miloje Rajović3

Author Affiliations

1 College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, People's Republic of China

2 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

3 Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, 36 000 Kraljevo, Serbia

For all author emails, please log on.

Fixed Point Theory and Applications 2012, 2012:66  doi:10.1186/1687-1812-2012-66

 Received: 24 November 2011 Accepted: 23 April 2012 Published: 23 April 2012

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.

### Abstract

This article is concerned with coupled coincidence points and common fixed points for two mappings in metric spaces and cone metric spaces. We first establish a coupled coincidence point theorem for two mappings and a common fixed point theorem for two w-compatible mappings in metric spaces. Then, by using a scalarization method, we extend our main theorems to cone metric spaces. Our results generalize and complement several earlier results in the literature. Especially, our main results complement a very recent result due to Abbas et al.

### 1 Introduction

Throughout this article, unless otherwise specified, we always suppose that ℕ is the set of positive integers and X is a nonempty set. In addition, for convenience, we denote gx = g(x) for each x X and each mapping g : X X.

Recently, Abbas et al. [1] introduced the following concept of w-compatible mappings:

Definition 1.1. The mappings g : X X and F : X × X X are called w-compatible if g(F(x, y)) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x).

Moreover, they established several coupled coincidence point theorems and common fixed point theorems for such mappings. The problem investigated in [1] is interesting. In fact, recently, the existence of coupled fixed points, coupled coincidence points, coupled common fixed points, and common fixed points for nonlinear mappings with two variables has attracted more and more attention. For example, Bhashkar and Lakshmikantham [2] investigated some coupled fixed point theorems in partially ordered sets, and they also discussed an application of their result by investigating the existence and uniqueness of the solution for a periodic boundary value problem; Sabetghadam et al. [3] extended some results in [2] to cone metric spaces; Lakshmikantham and Ćirić [4] proved several coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces; Karapinar [5] extended some results of [4] to cone metric spaces; Zoran and Mitrović [6] considered this topic in normed spaces and established a coupled best approximation theorem; Ding et al. [7] established some coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces under some generalized contraction conditions; etc.

The aim of this article is to make further studies on such problems, and to generalize and complement some known results. Next, let us recall some related definitions:

Definition 1.2. [1]Let g : X X, F : X × X X be two mappings.

(I) (x, y) ∈ X × X is called a coupled coincidence point of F and g if gx = F(x, y) and gy = F(y, x).

(II) (x, y) ∈ X × X is called a coupled fixed point of F if x = F(x, y) and y = F(y, x).

(III) x X is called a common fixed point of F and g if x = gx = F(x, x).

### 2 Metric spaces

Now, let us present one of our main results.

Theorem 2.1. Let (X, d) be a complete metric space. Assume that g : X X and F : X × X X are two mappings satisfying

(H1) there exists a non-decreasing function ϕ : [0,+∞) → [0,+∞) such that lim n ϕ n ( t ) = 0 for each t > 0, and

d ( F ( x , y ) , F ( u , v ) ) ϕ [ M F g ( x , y , u , v ) ]

for all x, y, u, v X, where

M F g ( x , y , u , v ) = max d ( g x , g u ) , d ( g y , g v ) , d ( g x , F ( x , y ) ) , d ( g u , F ( u , v ) ) , d ( g y , F ( y , x ) ) , d ( g v , F ( v , u ) ) , d ( g x , F ( u , v ) ) + d ( g u , F ( x , y ) ) 2 , d ( g y , F ( v , u ) ) + d ( g v , F ( y , x ) ) 2 ;

(H2) F(X × X) ⊆ g(X), and g(X) is a closed subset of X.

Then F and g have a coupled coincidence point in X.

Proof. First, let us present some properties about ϕ which will be used in the sequel. We claim that ϕ(t) <t for each t > 0. In fact, if ϕ(t0) ≥ t0 for some t0 > 0, then, since ϕ is non-decreasing, ϕn(t0) ≥ t0 for all n ∈ ℕ, which contradicts the condition lim n ϕ n ( t 0 ) = 0 .

Moreover, it is easy to see that ϕ(0) = 0, and thus ϕ(t) ≤ t for all t ≥ 0.

Take x0, y0 X. Since F(X × X) ⊆ g(X), one can construct two sequences {xn}, {yn} in X such that

g x n = F ( x n - 1 , y n - 1 ) , g y n = F ( y n - 1 , x n - 1 ) , n = 1 , 2 , . . . .

For any fixed n ∈ ℕ, by (H1), we have

d ( g x n + 1 , g x n ) = d ( F ( x n , y n ) , F ( x n - 1 , y n - 1 ) ) ϕ ( M n ) , (2.1)

and

d ( g y n + 1 , g y n ) = d ( F ( y n , x n ) , F ( y n - 1 , x n - 1 ) ) ϕ ( M n ) , (2.2)

where

M n = max d ( g x n , g x n - 1 ) , d ( g y n , g y n - 1 ) , d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g x n - 1 , g x n + 1 ) 2 , d ( g y n - 1 , g y n + 1 ) 2 .

Since

d ( g x n - 1 , g x n + 1 ) 2 d ( g x n - 1 , g x n ) + d ( g x n , g x n + 1 ) 2 max { d ( g x n - 1 , g x n ) , d ( g x n , g x n + 1 ) }

and

d ( g y n - 1 , g y n + 1 ) 2 d ( g y n - 1 , g y n ) + d ( g y n , g y n + 1 ) 2 max { d ( g y n - 1 , g y n ) , d ( g y n , g y n + 1 ) } ,

we have

M n = max { d ( g x n , g x n - 1 ) , d ( g y n , g y n - 1 ) , d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) } .

Now, let us prove that for each n ∈ ℕ,

M n = max { d ( g x n , g x n - 1 ) , d ( g y n , g y n - 1 ) } . (2.3)

We consider the following three cases:

Case I. If Mn = 0 or Mn = max{d(gxn, gxn-1), d (gyn, gyn-1)}, then (2.3) obviously holds.

Case II. Mn = d(gxn, gxn+1) > 0.

Then, by (2.1),

d ( g x n + 1 , g x n ) ϕ ( d ( g x n , g x n + 1 ) ) < d ( g x n , g x n + 1 ) , (2.4)

Case III. Mn = d(gyn, gyn+1) > 0.

Similar to Case II, by (2.2), we get a contradiction.

Thus, in all cases, (2.3) holds for each n ∈ ℕ. In addition, combining (2.1) and (2.2), we get that for all n ∈ ℕ:

M n + 1 = max { d ( g x n + 1 , g x n ) , d ( g y n + 1 , g y n ) } ϕ ( M n ) ϕ n ( M 1 ) . (2.5)

Let ε > 0 be fixed. Since lim n ϕ n ( M 1 ) = 0 , by (2.5), there exists N ∈ ℕ such that for all n >N,

M n + 1 < ε - ϕ ( ε ) . (2.6)

M n p = max { d ( g x n + p , g x n ) , d ( g y n , g y n + p ) }

for each p ∈ ℕ and each n ∈ ℕ.

Let n >N be fixed. Let us show that for all p ∈ ℕ:

M n p ε . (2.7)

By (2.6), we have

M n 1 = M n + 1 < ε - ϕ ( ε ) < ε .

By (2.5) and (2.6), we get

M n 2 = max { d ( g x n + 2 , g x n ) , d ( g y n + 2 , g y n ) } max { d ( g x n + 2 , g x n + 1 ) , d ( g y n + 2 , g y n + 1 ) } + max { d ( g x n + 1 , g x n ) , d ( g y n + 1 , g y n ) } = M n + 2 + M n + 1 ϕ ( M n + 1 ) + M n + 1 ϕ ( ε ) + ε - ϕ ( ε ) = ε .

Next, let us show that M n 3 ε . By (H1), we have

M n + 1 2 = max { d ( g x n + 3 , g x n + 1 ) , d ( g y n + 3 , g y n + 1 ) } = max { d ( F ( x n + 2 , y n + 2 ) , F ( x n , y n ) ) , d ( F ( y n + 2 , x n + 2 ) , F ( y n , x n ) ) } ϕ ( a n ) , (2.8)

where

a n = max d ( g x n + 2 , g x n ) , d ( g y n , g y n + 2 ) , d ( g x n + 2 , g x n + 3 ) , d ( g y n + 2 , g y n + 3 ) , d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g x n + 2 , g x n + 1 ) + d ( g x n , g x n + 3 ) 2 , d ( g y n + 2 , g y n + 1 ) + d ( g y n , g y n + 3 ) 2 = max M n 2 , M n + 3 , M n + 1 , d ( g x n + 2 , g x n + 1 ) + d ( g x n , g x n + 3 ) 2 , d ( g y n + 2 , g y n + 1 ) + d ( g y n , g y n + 3 ) 2 max ε , d ( g x n + 2 , g x n + 1 ) + d ( g x n , g x n + 3 ) 2 , d ( g y n + 2 , g y n + 1 ) + d ( g y n , g y n + 3 ) 2 .

If

a n = d ( g x n + 2 , g x n + 1 ) + d ( g x n , g x n + 3 ) 2 ,

then by (2.5) and (2.8),

d ( g x n + 3 , g x n + 1 ) ϕ ( a n ) a n = d ( g x n + 2 , g x n + 1 ) + d ( g x n , g x n + 3 ) 2 M n + 2 + d ( g x n , g x n + 3 ) 2 ϕ ( ε ) + d ( g x n , g x n + 3 ) 2 ,

which yields

d ( g x n , g x n + 3 ) d ( g x n + 3 , g x n + 1 ) + ( g x n + 1 , g x n ) ϕ ( ε ) + d ( g x n , g x n + 3 ) 2 + ε - ϕ ( ε ) = ε - ϕ ( ε ) 2 + d ( g x n , g x n + 3 ) 2 ,

i.e., d ( g x n , g x n + 3 ) 2 ε - ϕ ( ε ) 2 . Thus,

a n = d ( g x n + 2 , g x n + 1 ) + d ( g x n , g x n + 3 ) 2 M n + 2 + d ( g x n , g x n + 3 ) 2 ϕ ( ε ) 2 + d ( g x n , g x n + 3 ) 2 ε .

If a n = d ( g y n + 2 , g y n + 1 ) + d ( g y n , g y n + 3 ) 2 , one can similarly show that an ε. Hence, in all cases, an ε, so that M n + 1 2 ϕ ( ε ) . Then, by (2.6), we get

M n 3 = max { d ( g x n + 3 , g x n ) , d ( g y n + 3 , g y n ) } max { d ( g x n + 3 , g x n + 1 ) , d ( g y n + 3 , g y n + 1 ) } + max { d ( g x n + 1 , g x n ) , d ( g y n + 1 , g y n ) } = M n + 1 2 + M n + 1 ϕ ( ε ) + ε - ϕ ( ε ) = ε .

In general, in order to prove that M n p ε , one can first show that M n + 1 p - 1 ϕ ( ε ) , and then by the inequality M n p M n + 1 p - 1 + M n + 1 , the conclusion follows easily.

Now, we have proved that (2.7) holds for all p ∈ ℕ, which means that {gxn} and {gyn} are Cauchy sequences. Then, by the completeness of g(X), there exist x, y X such that

lim n g x n = g x , lim n g y n = g y . (2.9)

By (H1) we have

d ( F ( x , y ) , g x ) d ( F ( x , y ) , F ( x n , y n ) ) + d ( g x n + 1 , g x ) ϕ ( c n ) + d ( g x n + 1 , g x ) , (2.10)

and

d ( F ( y , x ) , g y ) d ( F ( y , x ) , F ( y n , x n ) ) + d ( g y n + 1 , g y ) ϕ ( c n ) + d ( g y n + 1 , g y ) , (2.11)

where

c n = max d ( g x , g x n ) , d ( g y , g y n ) , d ( g x , F ( x , y ) ) , d ( g y , F ( y , x ) ) , d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g x , g x n + 1 ) + d ( g x n , F ( x , y ) ) 2 , d ( g y , g y n + 1 ) + d ( g y n , F ( y , x ) ) 2 .

Now, we claim that gx = F(x, y) and gy = F(y, x). In fact, if this is not true, then

max { d ( g x , F ( x , y ) ) , d ( g y , F ( y , x ) ) } > 0 ,

which, together with (2.9), yield that cn = max{d(gx, F(x, y)), d(gy, F(y, x))} when n is sufficiently large. Letting n → ∞ in (2.10) and (2.11), it follows that

d ( F ( x , y ) , g x ) ϕ ( c n ) < max { d ( g x , F ( x , y ) ) , d ( g y , F ( y , x ) ) }

and

d ( F ( y , x ) , g y ) ϕ ( c n ) < max { d ( g x , F ( x , y ) ) , d ( g y , F ( y , x ) ) } .

This is a contradiction. Thus, gx = F(x, y) and gy = F(y, x), i.e., (x, y) is a coupled coincidence point of F and g.

Example 2.2. Let X = [2,+∞), d(x, y) = |x-y|, F(x, y) = x + y, g(x) = x2, and ϕ ( t ) = t 2 . It is easy to verify that all the assumptions of Theorem 2.1 are satisfied. So F and g have a coupled coincidence point. In fact, we have F(2, 2) = g(2).

If F and g are w-compatible, we have the following result:

Theorem 2.3. Suppose that all of the assumptions of Theorem 2.1 are satisfied, and F and g are w-compatible. Then F and g have a unique common fixed point.

Proof. We give the proof in 3 steps.

Step 1. We claim that if

g x 1 = F ( x 1 , y 1 ) , g y 1 = F ( y 1 , x 1 ) , g x 2 = F ( x 2 , y 2 ) , g y 2 = F ( y 2 , x 2 ) ,

then gx1 = gx2 = gy1 = gy2. In fact, by (H1), we have

d ( g x 1 , g x 2 ) = d ( F ( x 1 , y 1 ) , F ( x 2 , y 2 ) ) ϕ ( ω )

and

d ( g y 1 , g y 2 ) = d ( F ( y 1 , x 1 ) , F ( y 2 , x 2 ) ) ϕ ( ω ) ,

where ω = M F g ( x 1 , y 1 , x 2 , y 2 ) = M F g ( y 1 , x 1 , y 2 , x 2 ) = max { d ( g x 1 , g x 2 ) , d ( g y 1 , g y 2 ) } . Then, it follows that

ω = max { d ( g x 1 , g x 2 ) , d ( g y 1 , g y 2 ) } ϕ ( ω ) ,

which gives that ω = 0, i.e., gx1 = gx2 and gy1 = gy2.

By a similar argument, in the case of

g x 1 = F ( x 1 , y 1 ) , g y 1 = F ( y 1 , x 1 ) , g x 2 = F ( x 2 , y 2 ) , g y 2 = F ( y 2 , x 2 ) ,

one can also show that gx1 = gy2 and gy1 = gx2. Then, it follows that

g x 1 = g y 1 = g x 2 = g y 2 .

Step 2. By Theorem 2.1, (x, y) is a coupled coincidence point of F and g, i.e., gx = F(x, y) and gy = F(y, x). Then, by Step 1, we have gx = gy. Let u = gx = gy. Since F and g are w-compatible, we have

g u = g ( g x ) = g ( F ( x , y ) ) = F ( g x , g y ) = F ( u , u ) .

Again by Step 1, one obtains gu = gx. Thus u = gx = gu = F(u, u), i.e., u is a common fixed point of F and g.

Step 3. Let v = gv = F(v, v). By Step 1, one can deduce that gv = gu. So u = gu = gv = v, which means that u is the unique common fixed point of F and g.

### 3 Applications to cone metric spaces

In this section, by a scalarization method used in [7], we apply our main results in metric spaces to cone metric spaces, and obtain some new theorems.

In the following, we always suppose that E is a Banach space, P is a convex cone in E with int P , is the partial ordering induced by P, (X, ρ) is a cone metric space with the underlying cone P, e ∈ intP, and ξe : E → ℝ is defined by

ξ e ( y ) = inf { r : y re - P } , y E .

In addition, x y stands for x - y ∈ intP.

First, let us recall some definitions about cone metric space.

Definition 3.1. [8]Let X be a nonempty set and P be a cone in a Banach space E. Suppose that a mapping d : X × X E satisfies:

(d1) θ ρ(x, y) for all x,y X and ρ(x, y) = θ if and only if x = y, where θ is the zero element of P;

(d2) ρ(x, y) = ρ(y, x) for all x, y X;

(d3) ρ(x, y) ≼ ρ(x, z) + ρ(z, y) for all x, y, z X.

Then ρ is called a cone metric on X and (X, ρ) is called a cone metric space.

Definition 3.2. Let (X, ρ) be a cone metric space. Let {xn} be a sequence in X and x X. If c θ, there exists N ∈ ℕ such that for all n >N, ρ(xn, x) ≪ c, then we say that {xn} converges to x, and we denote it by lim n x n = x or xn x, n → ∞. If c θ, there exists N ∈ ℕ such that for all n, m >N, ρ(xn, xm) ≪ c, then {xn} is called a Cauchy sequence in X. In addition, (X, ρ) is called complete cone metric space if every Cauchy sequence is convergent.

Recall that it has been of great interest for many authors to study fixed point theorems in cone metric spaces, and there is a large literature on this topic. We refer the reader to [1,3,5,7,9-28] and the references therein for some recent developments on this topic.

Next, let us recall some notations and basic results about the scalarization function ξe.

Lemma 3.3. [[7], Lemma 1.1] The following statements are true:

(i) ξe(·) is positively homogeneous and continuous on E;

(ii) y, z E with y z implies ξe (y) ≤ ξe (z);

(ii) ξe (y + z) ≤ ξe (y) + ξe (z) for all y, z E.

Combining Theorems 2.1 and 2.2 of [7] and, we have the following results:

Theorem 3.4. Let (X, ρ) be a cone metric space with underlying cone P. Then, ξe ρ is a metric on X. Moreover, if (X, ρ) is complete, then (X, ξe ρ) is a complete metric space.

By using Theorems 2.1 and 2.3, one can deduce many results on cone metric spaces. For example, we have the following theorem:

Theorem 3.5. Let (X, ρ) be a cone metric space with underlying cone P. Assume that g:X X and F:X × X X are two mappings satisfying that F(X × X) ⊆ g(X), g(X) is a complete cone metric space, and there exists a constant λ ∈ (0,1) such that for each x, y, u, v X, there is a z S F g ( x , y , u , v ) with

ρ ( F ( x , y ) , F ( u , v ) ) λ z ,

where

S F g ( x , y , u , v ) = co ρ ( g x , g u ) , ρ ( g y , g v ) , ρ ( g x , F ( x , y ) ) , ρ ( g u , F ( u , v ) ) , ρ ( g y , F ( y , x ) ) , ρ ( g v , F ( v , u ) ) , ρ ( g x , F ( u , v ) ) + ρ ( g u , F ( x , y ) ) 2 , ρ ( g y , F ( v , u ) ) + ρ ( g v , F ( y , x ) ) 2 ,

and co denotes the convex hull. Then F and g have a coupled coincidence point in X. Moreover, if F and g are w-compatible, then F and g have a unique common fixed point.

Proof. Let d = ξe ρ. By Theorem 3.4, d is a metric on X and (g(X), d) is a complete metric space. Then, by Lemma 3.3, we have

d ( F ( x , y ) , F ( u , v ) ) λ ξ e ( z ) λ M F g ( x , y , u , v ) ,

where M F g ( x , y , u , v ) is defined in Theorem 2.1. Now, letting

ϕ ( t ) = λ t ,

it is easy to see that all of the assumptions of Theorem 2.1 are satisfied. Thus F and g have a coupled coincidence point in X. In addition, if F and g are w-compatible, by Theorem 2.3, F and g have a unique common fixed point.

Remark 3.6. Theorem 3.5 is a complement of [[1], Theorem 2.4]. Moreover, Theorem 3.5 extends some existing results. For example, one can deduce [[3], Theorem 2.2] from Theorem 3.5. In addition, note that Theorems 3.4 and 3.5 are true and in the context of tvs-cone metric spaces (for details see [23,28]).

Remark 3.7. It is needed to note that one can also get Theorem 3.5 by using the method of Minkowski functional, which is introduced in [22].

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### Acknowledgements

The authors thank the referees for their valuable comments that helped to improve the text. Wei Long acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province of China (20114BAB211002), and the Foundation of Jiangxi Provincial Education Department (GJJ12205). Third author is thankful to the Ministry of Science and Technological Development of Serbia.

### References

1. Abbas, M, Ali Khan, M, Radenović, S: Common coupled fixed point theorems in cone metric spaces for w-compatible mappings. Appl Math Comput. 217, 195–202 (2010). Publisher Full Text

2. Gnana Bhaskar, T, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006). Publisher Full Text

3. Sabetghadam, F, Masiha, HP, Sanatpour, AH: Some coupled fixed point theorems in cone metric space. Fixed Point Theory Appl. 2009, 8 (Article ID 125426) (2009)

4. Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70, 4341–4349 (2009). Publisher Full Text

5. Karapinar, E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput Math Appl. 59, 3656–3668 (2010). Publisher Full Text

6. Mitrović, ZD: A coupled best approximations theorem in normed spaces. Nonlinear Anal. 72, 4049–4052 (2010). Publisher Full Text

7. Ding, HS, Li, L, Radojević, S: Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces. Fixed Point Theory Appl. 2012, 96 (2012)

8. Du, WS: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72, 2259–2261 (2010). Publisher Full Text

9. Huang, LG, Zhang, X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl. 332, 1468–1476 (2007). Publisher Full Text

10. Abbas, M, Jungck, G: Common fixed point results of noncommuting mappings without continuity in cone metric spaces. J Math Anal Appl. 341, 418–420 (2008)

11. Altun, I, Durmaz, G: Some fixed point results in cone metric spaces. Rendiconti del Circolo Mathematico di Palermo. 58, 319–325 (2009). Publisher Full Text

12. Altun, I, Damjanović, B, Djorić, D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl Math Lett. 23, 310–316 (2010). Publisher Full Text

13. Ding, HS, Li, L: Coupled fixed point theorems in partially ordered cone metric spaces. Filomat. 25, 137–149 (2011)

14. Ding, HS, Li, L, Long, W: Coupled common fixed point theorems for weakly increasing mappings with two variables. In: J Comput Anal Appl to appear

15. Dorić, D, Kadelburg, Z, Radenović, S: Coupled fixed point for mappings without mixed monotone property. Appl Math Lett doi: 10.1016/j.aml.2012.02.022 (2012)

16. Golubović, Z, Kadelburg, Z, Radenović, S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr Appl Anal. 2012, 18 (Article ID 192581) (2012)

17. Ilić, D, Rakočević, V: Common fixed points for maps on cone metric space. J Math Anal Appl. 341, 876–882 (2008). Publisher Full Text

18. Ilić, D, Rakočević, V: Quasi-contraction on a cone metric space. Appl Math Lett. 22, 728–731 (2009). Publisher Full Text

19. Janković, S, Kadelburg, Z, Radenović, S: Rhoades BE: Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces. Fixed Point Theory Appl. 2009, 16 (Article ID 761086) (2009)

20. Jungck, G, Radenović, S, Radojević, S, Rakočević, V: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory Appl. 2009, 13 (Article ID 643840) (2009)

21. Kadelburg, Z, Radenović, S, Rakočvić, V: Remarks on quasi-contraction on a cone metric space. Appl Math Lett. 22, 1674–1679 (2009). Publisher Full Text

22. Kadelburg, Z, Pavlović, M, Radenović, S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput Math Appl. 59, 3148–3159 (2010). Publisher Full Text

23. Kadelburg, Z, Radenović, S, Rakočević, V: A note on the equivalence of some metric and cone metric fixed point results. Appl Math Lett. 24, 370–374 (2011). Publisher Full Text

24. Kadelburg, Z, Radenović, S: Coupled fixed point results under tvs-cone metric and w-cone-distance. Adv Fixed Point Theory (2012, in press)

25. Nashine, HK, Kadelburg, Z, Radenović, S: Coupled common fixed point theorems for w*-compatible mappings in ordered cone metric spaces. Appl Math Comput. 218, 5422–5432 (2012). Publisher Full Text

26. Radenović, S, Rhoades, BE: Fixed point theorem for two non-self mappings in cone metric spaces. Comput Math Appl. 57, 1701–1707 (2009). Publisher Full Text

27. Rezapour, Sh, Hamlbarani, R: Some note on the paper "Cone metric spaces and fixed point theorems of contractive mappings". J Math Anal Appl. 345, 719–724 (2008). Publisher Full Text

28. Rezapour, Sh, Haghi, RH, Shahzad, N: Some notes on fixed points of quasi-contraction maps. Appl Math Lett. 23, 498–502 (2010). Publisher Full Text

29. Zhang, X: Fixed point theorem of generalized quasicontractive mapping in cone metric spaces. Comput Math Appl doi:10.1016/j.camwa.2011.03.107 (2011)