Available online at www.ispacs.com/jnaa

Volume 2012, Year 2012 Article ID jnaa-00137, 11 Pages

doi:10.5899/2012/jnaa-00137

Research Article

Common Fixed Point Theorems of C-distance on

Cone Metric Spaces

A. Kaewkhao1,3, W. Sintunavarat 2, P. Kumam 2,3∗

(1) Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200,

Thailand.

(2) Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi

(KMUTT), Bangkok 10140, Thailand.

(3) Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok 10400, Thailand.

Copyright 2012 c⃝ A. Kaewkhao, W. Sintunavarat and P. Kumam. This is an open access article dis-

tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

AbstractWang and Guo (2011) defined the c-distance in a cone metric space and used this conceptto prove a common fixed point theorem. In this paper, we extend and generalize the mainresults of Wang and Guo and also give the applications of our theorem as shown in theexamples.Keywords : Cone metric space; c-distance; Common fixed point.

2000 Mathematics Subject Classification : 47H09, 47H10.

1 Introduction

Since the Banach’s contraction mapping principle very usefulness, it has become a pop-ular tool in solving many problems in mathematical analysis. For some more results ofgeneralization of this principle, refer to [5, 9, 10, 11, 12, 13, 14] and references mentionedtherein. In 2007, Huang and Zhang [3] first introduced the concept of cone metric spaces.Cone metric spaces is a some generalize version of metric spaces, where each pair of pointsis assigned to a member of a real Banach space over the cone. They also established andproved the existence of fixed point theorem which is an extension of the Banach’s contrac-tion mapping principle to cone metric spaces. Later, researchers have studied and extendfixed point problems in cone metric spaces (see [1, 2, 6, 7, 8, 15, 17]). Recently, Wang andGuo [16] introduced the concept of the c-distance in a cone metric spaces, which is a cone

∗Corresponding author. Email address: poom.kum@kmutt.ac.th, Tel: +662-4708998

1

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version of w-distance of Kada et al. [4], and established common fixed point theorem incone metric spaces which is more general than the classical Banach’s contraction mappingprinciple. We state Theorem 2.2 of Wang and Guo in [16] as follows:

Theorem 1.1. Let (X, d) be a cone metric space, P be a normal cone with normal constantK, q be a c-distance on X and S, T : X → X such that T (X) ⊆ S(X) and S(X) be acomplete subspace of X. Suppose that there exists a positive real numbers a1, a2, a3, a4 witha1 + 2a2 + a3 + a4 < 1 such that

q(Tx, Ty) ≼ a1q(Sx, Sy) + a2q(Sx, Ty) + a3q(Sx, Tx) + a4q(Sy, Ty)

for all x, y ∈ X. If S and T satisfy

inf{∥q(Tx, y)∥+ ∥q(Sx, y)∥+ ∥q(Sx, Tx)∥ : x ∈ X} > 0

for all y ∈ X with Ty = y or Sy = y, then S and T have a common fixed point in X.

In this paper, we develop and generalize the common fixed point theorem on c-distanceof Wang and Guo [16]. We also apply our results to conclude the existence of commonfixed point theorem in the example. Main theorem extends and improves Theorem 3.2and many works in fixed point theory in the literature.

2 Preliminaries

Let (E, ∥ · ∥) be a real Banach space and θ denote the zero element in E. A cone P is asubset of E such that

1. P is nonempty closed set and P = {θ};

2. if a, b are nonnegative real numbers and x, y ∈ P , then ax+ by ∈ P ;

3. P ∩ (−P ) = {θ}.

For any cone P ⊆ E, the partial ordering ≼ with respect to P defined by

x ≼ y ⇐⇒ y − x ∈ P.

We shall write x ≺ y to indicate that x ≼ y but x = y, while x ≪ y will stand for y − x ∈int(P ), where int(P ) denotes the interior of P . A cone P is called normal if there is anumber K > 0 such that for all x, y ∈ E,

θ ≼ x ≼ y =⇒ ∥x∥ ≤ K∥y∥.

The least positive number satisfying above is called the normal constant of P .

Remark 2.1. [3], Let E be a real Banach space with a cone P . If x ≼ kx, where x ∈ Pand 0 < k < 1, then x = θ.

Lemma 2.1. If E is a real Banach space with a cone P , then if θ ≼ x ≼ y and k is anonnegative real number, then θ ≼ kx ≼ ky.

Using the notation of a cone, we have following definitions of cone metric space.

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Definition 2.1. [3], Let X be a nonempty set and E be a real Banach space equippedwith the partial ordering ≼ with respect to the cone P ⊆ E. Suppose that the mappingd : X ×X → E satisfies the following conditions:

1. θ ≺ d(x, y) for all x, y ∈ X with x = y and d(x, y) = θ if and only if x = y;

2. d(x, y) = d(y, x) for all x, y ∈ X;

3. d(x, y) ≼ d(x, z) + d(z, y) for all x, y, z ∈ X.

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

Definition 2.2. [3], Let (X, d) be a cone metric space. Let {xn} be a sequence in X andx ∈ X.

1. If for every c ∈ E with θ ≪ c, there exists N ∈ N such that for all n > N ,d(xn, x) ≪ c, then {xn} is said to be convergent and {xn} converges to x, and x isthe limit of {xn}. We denote this by limn→∞ xn = x or xn → x as n → ∞.

2. If for every c ∈ E with θ ≪ c, there exists N ∈ N such that for all m,n > N ,d(xn, xm) ≪ c, then {xn} is called a Cauchy sequence in X.

3. If every Cauchy sequence in X is convergent, then X is called a complete cone metricspace.

For other basic properties on cone metric space, the reader can read in [3]. Next, wegive the notion of c-distance on a cone metric space (X, d) of Wang and Guo in [16], whichis a generalization of w-distance of Kada et al. [4] and some properties.

Definition 2.3. [16], Let (X, d) be a cone metric space. Then a function q : X ×X → Eis called a c-distance on X if the following are satisfied:

(q1) θ ≼ q(x, y) for all x, y ∈ X;

(q2) q(x, z) ≼ q(x, y) + q(y, z) for all x, y, z ∈ X;

(q3) for each x ∈ X and n ≥ 1, if q(x, yn) ≼ u for some u = ux ∈ P , then q(x, y) ≼ uwhenever {yn} is a sequence in X converging to a point y ∈ X;

(q4) for all c ∈ E with θ ≪ c, there exists e ∈ E with θ ≪ e such that q(z, x) ≪ e andq(z, y) ≪ e imply d(x, y) ≪ c.

Remark 2.2. The c-distance q is a w-distance on X if we take (X, d) is a metric space,E = R+, P = [0,∞) and (q3) is replaced by this condition: For any x ∈ X, q(x, ·) :X → R+ is lower semi-continuous. Moreover, (q3) holds whenever q(x, ·) is lower semi-continuous. Thus if (X, d) is a metric space, E = R+, P = [0,∞), then every w-distanceis a c-distance. But the converse is not true in general case. Therefore, the c-distance isa generalization of the w-distance.

Example 2.1. [16], Let (X, d) be a cone metric space and P be a normal cone. Define amapping q : X ×X → E by

q(x, y) = d(x, y)

for all x, y ∈ X. Then q is c-distance.

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Example 2.2. [16], Let (X, d) be a cone metric space and P be a normal cone. Define amapping q : X ×X → E by

q(x, y) = d(u, y)

for all x, y ∈ X, where u is a fixed point in X. Then q is c-distance.

Example 2.3. [16], Let E = C1R[0, 1] with ∥x∥ = ∥x∥∞ + ∥x′∥∞ and

P = {x ∈ E : x(t) ≥ 0 on [0, 1]}

(this cone is not normal). Let X = [0,∞) and defined a mapping d : X ×X → E by

d(x, y) = |x− y|φ

for all x, y ∈ X, where φ : [0, 1] → R such that φ(t) = et. Then (X, d) is a cone metricspace. Define a mapping q : X ×X → E by

q(x, y) = (x+ y)φ

for all x, y ∈ X. Then q is c-distance.

Example 2.4. [16], Let E = R and P = {x ∈ E : x ≥ 0}. Let X = [0,∞) and define amapping d : X ×X → E by

d(x, y) = |x− y|

for all x, y ∈ X. Then (X, d) is a cone metric space. Define a mapping q : X ×X → E by

q(x, y) = y

for all x, y ∈ X. Then q is c-distance.

Remark 2.3. On c-distance q(x, y) = q(y, x) does not necessarily hold and q(x, y) = θ isnot necessarily equivalent to x = y for all x, y ∈ X.

Next lemma is useful for the main result in this paper.

Lemma 2.2. [16], Let (X, d) be a cone metric space, q be a c-distance on X and {xn} bea sequence in X. If there exists the sequence {un} in P converging to θ and

q(xn, xm) ≼ un

for m > n, then {xn} is a Cauchy sequence in X.

3 Common fixed point theorems on c-distance

The following is the main result of this paper. We prove a common fixed point theorem byusing the idea of c-distance. Our theorem extends the contractive condition of Theorem1.1 from constant real numbers to some control functions.

Theorem 3.1. Let (X, d) be a cone metric space, P be a normal cone with normal constantK, q be a c-distance on X and S, T : X → X such that T (X) ⊆ S(X) and S(X) be acomplete subspace of X. Suppose that there exists mappings α, β, γ, µ : X → [0, 1) suchthat the following assertions hold:

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1. α(Tx) ≤ α(Sx), β(Tx) ≤ β(Sx), γ(Tx) ≤ γ(Sx) and µ(Tx) ≤ µ(Sx) for all x ∈ X;

2. (α+ 2β + γ + µ)(x) < 1 for all x ∈ X;

3. q(Tx, Ty) ≼ α(Sx)q(Sx, Sy) + β(Sx)q(Sx, Ty) + γ(Sx)q(Sx, Tx) + µ(Sx)q(Sy, Ty)for all x, y ∈ X;

4. inf{∥q(Tx, y)∥ + ∥q(Sx, y)∥ + ∥q(Sx, Tx)∥ : x ∈ X} > 0 for all y ∈ X with Ty = yor Sy = y.

Then S and T have a common fixed point in X.

Proof. Let x0 be an arbitrary point in X. Since T (X) ⊆ S(X), there exists a point x1 ∈ Xsuch that Tx0 = Sx1. By induction, we construct the sequence {xn} in X such that

Sxn = Txn−1 (3.1)

for all n ≥ 1. From (1) and (3), we have

q(Sxn, Sxn+1) = q(Txn−1, Txn)

≼ α(Sxn−1)q(Sxn−1, Sxn) + β(Sxn−1)q(Sxn−1, Txn)

+γ(Sxn−1)q(Sxn−1, Txn−1) + µ(Sxn−1)q(Sxn, Txn)

= α(Txn−2)q(Sxn−1, Sxn) + β(Txn−2)q(Sxn−1, Sxn+1)

+γ(Txn−2)q(Sxn−1, Sxn) + µ(Txn−2)q(Sxn, Sxn+1)

≼ α(Sxn−2)q(Sxn−1, Sxn) + β(Sxn−2)q(Sxn−1, Sxn+1)

+γ(Sxn−2)q(Sxn−1, Sxn) + µ(Sxn−2)q(Sxn, Sxn+1)

...

≼ α(Sx0)q(Sxn−1, Sxn) + β(Sx0)q(Sxn−1, Sxn+1)

+γ(Sx0)q(Sxn−1, Sxn) + µ(Sx0)q(Sxn, Sxn+1)

≼ α(Sx0)q(Sxn−1, Sxn) + β(Sx0)[q(Sxn−1, Sxn) + q(Sxn, Sxn+1)]

+γ(Sx0)q(Sxn−1, Sxn) + µ(Sx0)q(Sxn, Sxn+1)

= [α(Sx0) + β(Sx0) + γ(Sx0)]q(Sxn−1, Sxn))

+[β(Sx0) + µ(Sx0)]q(Sxn, Sxn+1),(3.2)

which implies that

q(Sxn, Sxn+1) ≼(α(Sx0) + β(Sx0) + γ(Sx0)

1− β(Sx0)− µ(Sx0)

)q(xn−1, xn) (3.3)

for all n ≥ 1. Now, we let

k :=

(α(Sx0) + β(Sx0) + γ(Sx0)

1− β(Sx0)− µ(Sx0)

)< 1.

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By repeating (3.3), we get

q(Sxn, Sxn+1) ≼ knq(Sx0, Sx1). (3.4)

Now, for positive integer m and n with m > n ≥ 1, it follows from (3.4) that

q(Sxn, Sxm) ≼ q(Sxn, Sxn+1) + q(Sxn+1, Sxn+2) + · · ·+ q(Sxm−1, Sxm)

≼ knq(Sx0, Sx1) + kn+1q(Sx0, Sx1) + · · ·+ km−1q(Sx0, Sx1)

≼(

kn

1−k

)q(Sx0, Sx1).

(3.5)

Since 0 ≤ k < 1, we get

(kn

1− k

)q(Sx0, Sx1) → θ as n → 0. From Lemma 2.2, we have

{Sxn} is a Cauchy sequence in S(X). Therefore, there exists a point z ∈ S(X) such thatSxn → z as n → ∞ by the completeness of S(X). From (3.5) and (q3), we have

q(Sxn, z) ≼(

kn

1− k

)q(Sx0, Sx1) (3.6)

for all n ≥ 1. By P is a normal cone with constant K and (3.5), we get

∥q(Sxn, Sxm)∥ ≤ K

(kn

1− k

)∥q(Sx0, Sx1)∥ (3.7)

for all m > n ≥ 1. From (3.6), we get

∥q(Sxn, z)∥ ≤ K

(kn

1− k

)∥q(Sx0, Sx1)∥ (3.8)

for all n ≥ 1.Suppose that Tz = z or Sz = z. Then by hypothesis, (3.7) and (3.8), we get

0 < inf{∥q(Tx, z)∥+ ∥q(Sx, z)∥+ ∥q(Sx, Tx)∥ : x ∈ X}

≤ inf{∥q(Txn, z)∥+ ∥q(Sxn, z)∥+ ∥q(Sxn, Txn)∥ : n ≥ 1}

= inf{∥q(Sxn+1, z)∥+ ∥q(Sxn, z)∥+ ∥q(Sxn, Sxn+1)∥ : n ≥ 1}

≤ inf

{K

(kn+1

1−k

)∥q(Sx0, Sx1)∥+K

(kn

1−k

)∥q(Sx0, Sx1)∥

+K

(kn

1−k

)∥q(Sx0, Sx1)∥ : n ≥ 1

}= 0,

(3.9)

which is a contradiction. Therefore, we can conclude that z = Sz = Tz.

Corollary 3.1. Let (X, d) be a complete cone metric space, P be a normal cone withnormal constant K, q be a c-distance on X and T : X → X. Suppose that there existsmappings α, β, γ, µ : X → [0, 1) such that the following assertions hold:

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1. α(Tx) ≤ α(x), β(Tx) ≤ β(x), γ(Tx) ≤ γ(x) and µ(Tx) ≤ µ(x) for all x ∈ X;

2. (α+ 2β + γ + µ)(x) < 1 for all x ∈ X;

3. q(Tx, Ty) ≼ α(x)q(x, y)+β(x)q(x, Ty)+γ(x)q(x, Tx)+µ(x)q(y, Ty) for all x, y ∈ X;

4. inf{∥q(Tx, y)∥+ ∥q(x, y)∥+ ∥q(x, Tx)∥ : x ∈ X} > 0 for all y ∈ X with Ty = y.

Then T has a fixed point in X.

Proof. Take mapping S in Theorem 3.1 as IX where IX is an identity mapping on X.

Corollary 3.2. [16], Let (X, d) be a cone metric space, P be a normal cone with normalconstant K, q be a c-distance on X and S, T : X → X such that T (X) ⊆ S(X) and S(X)be a complete subspace of X. Suppose that there exists a positive real numbers a1, a2, a3, a4with a1 + 2a2 + a3 + a4 < 1 such that

q(Tx, Ty) ≼ a1q(Sx, Sy) + a2q(Sx, Ty) + a3q(Sx, Tx) + a4q(Sy, Ty)

for all x, y ∈ X. If S and T satisfy

inf{∥q(Tx, y)∥+ ∥q(Sx, y)∥+ ∥q(Sx, Tx)∥ : x ∈ X} > 0

for all y ∈ X with Ty = y or Sy = y, then S and T have a common fixed point in X.

Proof. Take mappings α, β, γ, µ : X → [0, 1) by α(x) = a1, β(x) = a2, γ(x) = a3 andµ(x) = a4 in Theorem 3.1 and then it follows from Theorem 3.1.

4 Some Examples

Next, we give some examples to illustrate the main result.

Example 4.1. Let E = R and P = {x ∈ E : x ≥ 0}. Let X = [0, 1) and define a mappingd : X ×X → E by

d(x, y) = |x− y|

for all x, y ∈ X. Then (X, d) is a cone metric space. Define a mapping q : X ×X → E by

q(x, y) = 2d(x, y)

for all x, y ∈ X. Then q is c-distance. In fact, (q1)-(q3) are immediate. Let c ∈ E with0 ≪ c and put e = c/2. If q(z, x) ≪ e and q(z, y) ≪ e, then we have

d(x, y) ≤ 2d(x, y) = 2|x− y| ≤ 2|x− z|+ 2|z − y| = q(z, x) + q(z, y) ≪ e+ e = c.

This shows that (q4) holds. Therefore q is c-distance.

Let S, T : X → X defined by S(x) = x and T (x) =x2

16for all x ∈ X. Take mappings

α, β, γ, µ : X → [0, 1) by

α(x) = β(x) =x+ 1

16, γ(x) =

2x+ 3

16and µ(x) =

3x+ 2

16

for all x ∈ X. Observe that

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(1) α(Tx) = β(Tx) =1

16

(x2

16+1

)≤

1

16(x2+1) ≤

x+ 1

16= α(Sx) = β(Sx) for all x ∈ X.

(2) γ(Tx) =1

16

(2x2

16+ 3

)≤

1

16(2x2 + 3) ≤

2x+ 3

16= γ(Sx) for all x ∈ X.

(3) µ(Tx) =1

16

(3x2

16+ 2

)≤

1

16(3x2 + 2) ≤

3x+ 2

16= µ(Sx) for all x ∈ X.

(4) (α + 2β + γ + µ)(x) =x+ 1

16+ 2

(x+ 1

16

)+

2x+ 3

16+

3x+ 2

16=

8x+ 8

16< 1 for all

x ∈ X.

(5) For all x, y ∈ X, we have

q(Tx, Ty) = 2

∣∣∣∣x2

16 − y2

16

∣∣∣∣≤ 2|x+y||x−y|

16

=(x+y

16

)2|x− y|

≤(x+116

)2|x− y|

≤ α(x)q(Sx, Sy) + β(x)q(Sx, Ty) + γ(x)q(Sx, Tx) + µ(x)q(Sy, Ty).

(6) For any y = Ty, i.e., y > 0, we get

inf{∥q(Tx, y)∥+ ∥q(Sx, y)∥+ ∥q(Sx, Tx)∥ : x ∈ X} = 4

∣∣∣∣y − y2

16

∣∣∣∣ > 0.

Therefore, all the conditions of Theorem 3.1 are satisfied. Thus we can conclude that Sand T have a common fixed point in X. This common fixed point is x = 0.

Example 4.2. Let E = R and P = {x ∈ E : x ≥ 0}. Let X = [0, 1) and define a mappingd : X ×X → E by

d(x, y) = |x− y|

for all x, y ∈ X. Then (X, d) is a cone metric space. Define a c-distance q : X ×X → Eas in Example 4.1.

Let S, T : X → X defined by S(x) = x and T (x) =x2

3for all x ∈ X. Take mappings

α, β, γ, µ : X → [0, 1) by

α(x) =2x+ 1

3and β(x) = γ(x) = µ(x) = 0

for all x ∈ X. Then we have the following statement holds:

(1) α(Tx) =1

3

(2x2

3+ 1

)≤

1

3(2x2 + 1) ≤

2x+ 1

3= α(Sx) for all x ∈ X;

(2) β(Tx) = γ(Tx) = µ(Tx) = 0 ≤ 0 = β(Sx) = γ(Sx) = µ(Sx) for all x ∈ X;

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(3) (α+ 2β + γ + µ)(x) =2x+ 1

3< 1 for all x ∈ X;

(4) For each x, y ∈ X, we have

q(Tx, Ty) = 2

∣∣∣∣x2

3 − y2

3

∣∣∣∣≤ 2|x+y||x−y|

3

=(x+y

3

)2|x− y|

≤(2x+y

3

)2|x− y|

≤(2x+13

)2|x− y|

≤ α(x)q(Sx, Sy) + β(x)q(Sx, Ty) + γ(x)q(Sx, Tx) + µ(x)q(Sy, Ty);

(6) For any y = Ty, i.e., y > 0, we get

inf{∥q(Tx, y)∥+ ∥q(Sx, y)∥+ ∥q(Sx, Tx)∥ : x ∈ X} = 4

∣∣∣∣y − y2

3

∣∣∣∣ > 0.

Therefore, the hypothesis of Theorem 3.1 is satisfied and so we can conclude that S andT have a common fixed point in X. This common fixed point is x = 0.

Remark 4.1. Although, Theorem 1.1 is essential tool in the cone metric d to concludethe existence of fixed points and common fixed points of some mappings. Sometimes theconstant numbers which satisfy Theorem 1.1 is very difficult to find. Therefore, it is themost interest to define such mappings α, β, γ, µ as another auxiliary tool of the cone metricd.

5 Acknowledgements

This research is partially supported by the Centre of Excellence in Mathematics, theCommission on High Education, Thailand. The second author would like to thank theResearch Professional Development Project Under the Science Achievement Scholarshipof Thailand (SAST). Finally, the authors would like to express their thanks to the refereesfor their helpful comments and suggestions.

References

[1] M. Abbas, G. Jungck, Common fixed point results for noncombetating mappings with-out continuity in cone metric space, J. Math. Anal. Appl. 341 (2008) 416-420.http://dx.doi.org/10.1016/j.jmaa.2007.09.070.

[2] M. Abbas, B.E. Rhoades, Fixed and periodic point results in cone metric spaces, Appl.Math. Lett. 22 (2009) 511-515.http://dx.doi.org/10.1016/j.aml.2008.07.001.

Author's Copy

10 Journal of Nonlinear Analysis and Application

[3] L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractivemappings, J. Math. Anal. Appl. 332 (2007) 1468-1476.http://dx.doi.org/10.1016/j.jmaa.2005.03.087.

[4] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed pointtheorems in complete metric spaces, Math. Japon. 44 (1996) 381-391.

[5] C. Mongkolkeha, W. Sintunavarat, P. Kumam, Fixed point theorems for contrac-tion mappings in modular metric spaces, Fixed Point Theory and Applications 2011,2011:93.http://dx.doi.org/10.1186/1687-1812-2011-93.

[6] S. Radenovic, Common fixed points under contractive conditions in cone metric spaces,Comput. Math. Appl. 58 (2009) 1273-1278.http://dx.doi.org/10.1016/j.camwa.2009.07.035.

[7] S. Radenovic, B.E. Rhoades, Fixed point theorem for two non-self mappings in conemetric spaces, Comput. Math. Appl. 57 (2009) 1701-1707.http://dx.doi.org/10.1016/j.camwa.2009.03.058.

[8] W. Sintunavarat, P. Kumam, Common fixed points of f -weak contractions in conemetric spaces, Bulletin of the Iranian Mathematical Society, (in press).

[9] W. Sintunavarat, P. Kumam, Weak condition for generalized multi-valued (f, α, β)-weak contraction mappings, Appl. Math. Lett. 24 (2011) 460-465.http://dx.doi.org/10.1016/j.aml.2010.10.042.

[10] W. Sintunavarat, P. Kumam, Gregus type fixed points for a tangential multi-valuedmappings satisfying contractive conditions of integral type, Journal of Inequalities andApplications 2011, 2011:3.http://dx.doi.org/10.1186/1029-242X-2011-3.

[11] W. Sintunavarat, P. Kumam, Common fixed point theorems for hybrid generalizedmulti-valued contraction mappings, Appl. Math. Lett. 25 (2012) 52-57.http://dx.doi.org/10.1016/j.aml.2011.05.047.

[12] W. Sintunavarat, P. Kumam, Common fixed point theorems for generalized JH-operator classes and invariant approximations, Journal of Inequalities and Applications2011, 2011:67.http://dx.doi.org/10.1186/1029-242X-2011-67.

[13] W. Sintunavarat, P. Kumam, Common fixed point theorem for cyclic generalizedmulti-valued contraction mappings, Appl. Math. Lett. (2012)http://dx.doi.org/10.1016/j.aml.2012.02.045.

[14] T. Suzuki, A generalized Banach contraction principle that characterizes metric com-pleteness, Proc. Amer. Math. Soc. 136 (5) (2008) 1861-1869.http://dx.doi.org/10.1090/S0002-9939-07-09055-7.

[15] P. Vetro, Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 56(2007) 464-468.http://dx.doi.org/10.1007/BF03032097.

Author's Copy

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[16] S. Wang, B. Guo, Distance in cone metric spaces and common fixed point theorems,Appl. Math. Lett. 24 (2011) 1735-1739.http://dx.doi.org/10.1016/j.aml.2011.04.031.

[17] D. Wardowski, Endpoint and fixed points of set-valued contractions in cone metricspaces, Nonlinear Anal. 71 (2009) 512-516.http://dx.doi.org/10.1016/j.na.2008.10.089.

Author's Copy