# Fixed point results under c-distance in tvs-cone metric spaces

- Momčilo Đorđević
^{1}, - Dragan Đorić
^{2}, - Zoran Kadelburg
^{3}, - Stojan Radenović
^{4}Email author and - Dragoljub Spasić
^{1}

**2011**:29

https://doi.org/10.1186/1687-1812-2011-29

© Đorđević et al; licensee Springer. 2011

**Received: **5 April 2011

**Accepted: **10 August 2011

**Published: **10 August 2011

## Abstract

Fixed point and common fixed point results for mappings in tvs-cone metric spaces (with the underlying cone which is not normal) under contractive conditions expressed in the terms of *c*-distance are obtained. Respective results concerning mappings without periodic points are also deduced. Examples are given to distinguish these results from the known ones.

**Mathematics Subject Classification (2010)**

47H10, 54H25

## Keywords

*c*-DistanceCommon fixed point

## 1 Introduction

Cone metric spaces were considered by Huang and Zhang in [1], who reintroduced the concept which has been known since the middle of 20th century (see, e.g., [2–4]). Topological vector space-valued version of these spaces was treated in [5–13]; see also [14] for a survey of fixed point results in these spaces.

Fixed point theorems in metric spaces with the so-called *w*-distance were obtained for the first time by Kada et al. in [15] where nonconvex minimization problems were treated. Further results were given, e.g., in [16–18]. Cone metric version of this notion (usually called a *c*-distance) was used, e.g., in [19, 20].

In this paper, we consider fixed point and common fixed point results for mappings in tvs-cone metric spaces (with the underlying cone which is not normal) under contractive conditions expressed in the terms of *c*-distance. Respective results concerning mappings without periodic points are also deduced. Examples are given to distinguish these results from the known ones.

## 2 Preliminaries

Let *E* be a real Hausdorff topological vector space (*tvs* for short) with the zero vector *θ*. A proper nonempty and closed subset *P* of *E* is called a *cone* if *P* + *P* ⊂ *P*, *λP* ⊂ *P* for *λ ≥* 0 and *P* ∩ (-*P*) = {*θ*}. We shall always assume that the cone *P* has a nonempty interior int *P* (such cones are called *solid*).

Each cone *P* induces a partial order ≼ on *E* by *x* ≼ *y* ⇔ *y - ×* ∈ *P*. *x* π *y* will stand for (*x* ≼ *y* and *x* ≠ *y*), while *x* ≪ *y* will stand for *y - ×* ∈ int *P*. The pair (*E*, *P*) is an *ordered topological vector space*.

For a pair of elements *x*, *y* in *E* such that *x* ≼ *y*, put [*x*, *y*] = {*z* ∈ *E* : *x* ≼ *z* ≼ *y*}. A subset *A* of *E* is said to be *order-convex* if [*x*, *y*] ⊂ *A*, whenever *x*, *y* ∈ *A* and *x* ≼ *y*.

Ordered topological vector space (*E*, *P*) is *order-convex* if it has a base of neighborhoods of *θ* consisting of order-convex subsets. In this case, the cone *P* is said to be *normal*. If *E* is a normed space, this condition means that the unit ball is order-convex, which is equivalent to the condition that there is a number *k* such that *x*, *y* ∈ *E* and 0 ≼ *x* ≼ *y* implies that ||*x*|| *≤ k*||*y*||. A proof of the following assertion can be found, e.g., in [2].

**Theorem 1** *If the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space*.

Note that completions of cone metric spaces in the case of nonnormal underlying cone were treated in [21].

From [1, 5–7], we give the following

**Definition 1** Let *X* be a nonempty set and (*E*, *P*) an ordered *tvs*. A function *d* : *X* × *X* → *E* is called a *tvs-cone metric* and (*X*, *d*) is called a *tvs-cone metric space* if the following conditions hold:

(c1) *θ* ≼ *d*(*x*, *y*) for all *x*, *y* ∈ *X* and *d*(*x*, *y*) = *θ* if and only if *x* = *y*;

(c2) *d*(*x*, *y*) = *d*(*y*, *x*) for all *x*, *y* ∈ *X*;

(c3) *d*(*x*, *z*) ≼ *d*(*x*, *y*) + *d*(*y*, *z*) for all *x*, *y*, *z* ∈ *X*.

Taking into account Theorem 1, proper generalizations when passing from norm-valued cone metric spaces of [1] to tvs-cone metric spaces can be obtained only in the case of nonnormal cones. We shall make use of the following properties:

(p_{1}) If *u*, *v*, *w* ∈ *E*, *u* ≼ *v* and *v* ≪ *w* then *u* ≪ *w*.

(p_{2}) If *u* ∈ *E* and *θ* ≼ *u* ≪ *c* for each *c* ∈ int *P* then *u* = *θ*.

(p_{3}) If *u*_{
n
} , *v*_{
n
} , *u*, *v* ∈ *E*, *θ* ≼ *u*_{
n
} ≼ *v*_{
n
} for each *n* ∈ ℕ, and *u*_{
n
} → *u*, *v*_{
n
} → *v* (*n* → ∞), then *θ* ≼ *u* ≼ *v*.

(p_{4}) If *x*_{
n
} , *x* ∈ *X*, *u*_{
n
} ∈ *E*, *d*(*x*_{
n
} , *x*) ≼ *u*_{
n
} and *u*_{
n
} → *θ* (*n* → ∞), then *x*_{
n
} → *x* (*n* → ∞).

(p_{5}) If *u* ≼ *λu*, where *u* ∈ *P* and 0 ≤ *λ <* 1, then *u* = *θ*.

(p_{6}) If *c* ≫ *θ* and *u*_{
n
} ∈ *E*, *u*_{
n
} → *θ* (*n* → ∞), then there exists *n*_{0} such that *u*_{
n
} ≪ *c* for all *n* ≥ *n*_{0}.

In the sequel, *E* will always denote a topological vector space, with the zero vector *θ* and with order relation ≼, generated by a solid cone *P*. For notions such as convergent and Cauchy sequences, completeness, continuity etc. in (tvs)-cone metric spaces, we refer to [1, 7, 14] and references therein.

Kada et al. [15] introduced the notion of *w*-distance in metric spaces and proved some fixed point results using this notion (see also [16–18]). Cho et al. [19] transferred it to the setting of cone metric spaces (see also [20]).

**Definition 2**[19] Let (*X*, *d*) be a tvs-cone metric space. A function *q* : *X* × *X* → *E* is called a *c-distance* in *X* if:

(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) If a sequence {*y*_{
n
} } in *X* converges to a point *y* ∈ *X*, and for some *x* ∈ *X* and *u* = *u*_{
x
} ∈ *P*, *q*(*x*, *y*_{
n
} ) ≼ *u* holds for each *n* ∈ ℕ, then *q*(*x*, *y*) ≼ *u*;

(q4) For each *c* ∈ *E* with *θ* ≪ *c*, there exists *e* ∈ *E* with *θ* ≼ *e*, such that *q*(*z*, *x*) ≪ *e* and *q*(*z*, *y*) ≪ *e* implies *d*(*x*, *y*) ≪ *c*.

Each *w*-distance *q* in a metric space (*X*, *d*) (in the sense of [15]) is a *c*-distance in the tvs-cone metric space (*X*, *d*) (with *E* = ℝ and *P* = [0, +∞)). Indeed, only property (q3) has to be checked. Let *y*_{
n
}∈ *X*, *y*_{
n
}→ *y* in the cone metric *d* (*n* → ∞), and let *q*(*x*, *y*_{
n
} ) ≤ *u*_{
x
} ∈ [0, +∞). Since *q* is (as a *w*-distance) lower semi-continuous, we have that *q*(*x*, *y*) ≤ lim inf_{n → ∞}*q*(*x*, *y*_{
n
}) ≤ lim inf_{n → ∞}*u*_{
x
}= *u*_{
x
} , i.e., *q*(*x*, *y*) ≤ *u*_{
x
} holds true.

The first two of the following examples are variations of [[19], Examples 2.7, 2.8], adjusted to the case of a tvs-cone metric.

*Example 1* Let (*X*, *d*) be a tvs-cone metric space such that the metric *d*(*·*,*·*) is a continuous function in second variable. Then, *q*(*x*, *y*) = *d*(*x*, *y*) is a *c*-distance. Indeed, only property (q3) is nontrivial and it follows from *q*(*x*, *y*_{
n
} ) = *d*(*x*, *y*_{
n
} ) ≼ *u*, passing to the limit when *n* → ∞ and using continuity of *d*.

*Example 2* Let (*X*, *d*) be a tvs-cone metric space, and let *u* ∈ *X* be fixed. Then, *q*(*x*, *y*) = *d*(*u*, *y*) defines a *c*-distance on *X*. Indeed, (q1) and (q3) are clear. (q2) follows from *q*(*x*, *z*) = *d*(*u*, *z*) ≼ *d*(*u*, *y*) + *d*(*u*, *z*) = *q*(*x*, *y*) + *q*(*y*, *z*). Finally, (q4) is obtained by taking *e* = *c/* 2.

*Example 3* Consider the Banach space *E* = *C*[0, 1] of real-valued continuous functions with the max-norm and ordered by the cone *P* = {*f* ∈ *E* : *f*(*t*) ≥ 0 for *t* ∈ [0, 1]}. This cone is normal in the Banach-space topology on *E*. Let *τ** be the strongest locally convex topology on the vector space *E*. Then, the cone *P* is solid, but it is not normal in the topology *τ**. Indeed, if this were the case, Theorem 1 would imply that the topology *τ** is normed, which is impossible since an infinite dimensional space with the strongest locally convex topology cannot be metrizable (see, e.g., [14]).

*X*= [0, + ∞) and

*d*:

*X*×

*X*→ (

*E*,

*τ**) be defined by

*d*(

*x*,

*y*)(

*t*) = |

*x - y*|

*φ*(

*t*) for a fixed element

*φ*∈

*P*. Then, (

*X*,

*d*) is a tvs-cone metric space which is not a cone metric space in the sense of [1]. We can introduce two

*c*-distances on this space:

They are the examples of *c*-distances in tvs-cone metric spaces which are not *c*-distances in cone metric spaces of [19, 20].

*c*-distance

*q*:

- 1.
*q*(*x*,*y*) =*q*(*y*,*x*) does not necessarily hold for all*x*,*y*∈*X*; - 2.
*q*(*x*,*y*) =*θ*is not necessarily equivalent to*x*=*y*.

## 3 Results

### 3.1 Fixed point and common fixed point results under *c*-distance

We will call a sequence {*u*_{
n
} } in *P* a *c-sequence* if for each *c* ≫ *θ* there exists *n*_{0} ∈ N such that *u*_{
n
} ≪ *c* for *n* ≥ *n*_{0}. It is easy to show that if {*u*_{
n
} } and {*v*_{
n
} } are *c*-sequences in *E* and *α*, *β >* 0, then { *αu*_{
n
} + *βv*_{
n
} } is a *c*-sequence.

Note that in the case that the cone *P* is normal, a sequence in *E* is a *c*-sequence iff it is a *θ* -sequence (see property (p_{6})). However, when the cone is not normal, a *c*-sequence need not be a *θ* -sequence (see [7, 14]). Also, from [7], we know that the cone metric *d* need not be a continuous function.

The following lemma is a tvs-cone metric version of lemmas from [15, 19].

**Lemma 1** *Let* (*X*, *d*) *be a tvs-cone metric space and let q be a c-distance on X. Let* {*x*_{
n
} } *and* {*y*_{
n
} } *be sequences in × and x*, *y*, *z* ∈ *X. Suppose that* {*u*_{
n
} } *and* {*v*_{
n
} } *are c-sequences in P. Then the following hold:*

*(1) If q*(*x*_{
n
} , *y*) ≼ *u*_{
n
} *and q*(*x*_{
n
} , *z*) ≼ *v*_{
n
} *for n* ∈ ℕ, *then y* = *z. In particular, if q*(*x*, *y*) = *θ and q*(*x*, *z*) = *θ, then y* = *z*.

*(2) If q*(*x*_{
n
} , *y*_{
n
} ) ≼ *u*_{
n
} *and q*(*x*_{
n
} , *z*) ≼ *v*_{
n
} *for n* ∈ ℕ, *then* {*y*_{
n
} } *converges to z*.

*(3) If q*(*x*_{
n
} , *x*_{
m
} ) ≼ *u*_{
n
} *for m > n > n*_{0}, *then* {*x*_{
n
} } *is a Cauchy sequence in X*.

*(4) If q*(*y*, *x*_{
n
} ) ≼ *u*_{
n
} *for n* ∈ ℕ, *then* {*x*_{
n
} } *is a Cauchy sequence in X*.

*Proof*We will prove assertions (1) and (2). Proofs of the other two are similar.

- (1)
In order to prove that

*y*=*z*, according to (p_{2}), it is enough to show that*d*(*y*,*z*) ≪*c*for each*c*≫*θ*. For the given*c*choose*e*≫*θ*such that property (q4) is satisfied. Choose then*n*_{0}∈ ℕ such that*u*_{ n }≪*e*and*v*_{ n }≪*e*for*n*≥*n*_{0}. Then, by property (p_{1}), we get that*q*(*x*_{ n },*y*) ≪*e*and*q*(*x*_{ n },*z*) ≪*e*and (q4) imply that*d*(*y*,*z*) ≪*c*. - (2)
Let again

*c*≪*θ*be arbitrary and choose a corresponding*e*≫*θ*satisfying property (q4). If*n*_{0}∈ N is such that*u*_{ n }≪*e*and*v*_{ n }≪*e*for*n*≥*n*_{0}, then (p_{1}) implies that*q*(*x*_{ n },*y*_{ n }) ≪*e*and*q*(*x*_{ n },*z*) ≪*e*for*n*≥*n*_{0}. Then, by (q4),*d*(*y*_{ n },*z*) ≪*c*and*y*_{ n }→*z*(*n*→ ∞). ■

Our first result is the following theorem of Hardy-Rogers type.

**Theorem 2**

*Let*(

*X*,

*d*)

*be a complete tvs-cone metric space and let q be a c-distance on X. Suppose that a continuous self-map f*:

*X*→

*X satisfies the following two conditions:*

*for all x*, *y* ∈ *X, where A*, *B*, *C*, *D*, *E are nonnegative constants such that A* + *B* + *C* + 2*D* + 2*E <* 1. *Then f has a fixed point in X. If fu* = *u, then q*(*u*, *u*) = *θ*.

*Proof*Let

*x*

_{0}∈

*X*be arbitrary and form the sequence {

*x*

_{ n }} with

*x*

_{ n }=

*f*

^{ n }

*x*

_{0}. In order to prove that it is a Cauchy sequence, put

*x*=

*x*

_{ n }and

*y*=

*x*

_{n - 1}in (3.1) to get

since *A* + *B* + *C* + 2*D* + 2*E <* 1 and, e.g., *A* + *C* + *E >* 0.

*u*

_{ n }≼

*h*

^{ n }

*u*

_{0}and

*q*(

*x*

_{ n },

*x*

_{n+1}) ≼

*u*

_{ n }≼

*h*

^{ n }(

*q*(

*x*

_{1},

*x*

_{0}) +

*q*(

*x*

_{0},

*x*

_{1})). In the usual way, it follows that

for *m > n*, where {*v*_{
n
} } is a *c*-sequence. Lemma 1.(3) implies that {*x*_{
n
} } is a Cauchy sequence in *X* and, since *X* is complete, *x*_{
n
} → *x** ∈ *X* (*n* → ∞). Continuity of *f* implies that *x*_{n+1}= *fx*_{
n
} → *fx**, and since the limit of a sequence in tvs-cone metric space in unique, we get that *fx** = *x**.

which is, by property (p_{5}) and *A* + *B* + *C* + *D* + *E < A* + *B* + *C* +2*D* +2*E <* 1, possible only if *q*(*u*, *u*) = *θ*. ■

respectively.

*Remark 1*If the underlying cone

*P*of the given tvs-cone metric space (

*X*,

*d*) is normal (and, hence, this space is a cone metric space in the sense of [1], see Theorem 2.1), then continuity of

*f*in Theorem 2 can be replaced by the condition

It may be of interest to note that in this case, property (q3) of *c*-distance has to be used in the course of the proof (see, e.g., the respective procedure in ordered cone metric spaces in [19]), while in our case (when *f* is continuous), this property is not needed.

The next is a result including two mappings and the existence of their common fixed point.

**Theorem 3**

*Let*(

*X*,

*d*)

*be a complete tvs-cone metric space and let q be a c-distance on X. Suppose that continuous self-maps f*,

*g*:

*X*→

*X satisfy the following two conditions:*

*for all x*, *y* ∈ *X, where A*, *B*, *D are nonnegative constants, such that A* + 2*B* + 4*D <* 1. *Then f and g have a common fixed point in X. If fu* = *gu* = *u, then q*(*u*, *u*) = *θ*.

*Proof* Let *x*_{0} ∈ *X* be arbitrary and form the sequence {*x*_{
n
} } such that *x*_{2n+1}= *fx*_{2n}and *x*_{2n+2}= *gx*_{2n+1}for *n* ≥ 0. Denote *u*_{
n
} = *q*(*x*_{2n}, *x*_{2n+1})+*q*(*x*_{2n+1}, *x*_{2n}) and *v*_{
n
} = *q*(*x*_{2n+1}, *x*_{2n+2}) + *q*(*x*_{2n+2}, *x*_{2n+1}).

where
, since *A* + *B* + *D >* 0 and *A* + 2*B* + 4*D <* 1.

and we get that {*u*_{
n
} } and {*v*_{
n
} } are *c*-sequences. We have that *q*(*x*_{2n}, *x*_{2n+1}) ≼ *u*_{
n
} , *q*(*x*_{2n+1}, *x*_{2n+2}) ≼ *v*_{
n
} and it follows that *q*(*x*_{
n
} , *x*_{n+1}) ≼ *u*_{
n
} + *v*_{
n
} , where *u*_{
n
} + *v*_{
n
} is a *c*-sequence. Using Lemma 1.(3), we obtain that {*x*_{
n
} } is a Cauchy sequence in *X*. Hence, *x*_{
n
} → *x** ∈ *X* (*n* → ∞). Since *f* and *g* are continuous, it easily follows from the definition of {*x*_{
n
} } that *fx** = *gx** = *x**.

*f*and

*g*have a common fixed point. Suppose that

*u*∈

*X*is any point satisfying

*fu*=

*gu*=

*u*. Then, (3.5) implies that

and, since 0 *< A* + 2*B* + 2*D < A* + 2*B* + 4*D <* 1, property (p_{5}) implies that *q*(*u*, *u*) = *θ*. ■

*f*and

*g*satisfying

where *m*, *n* ∈ ℕ, *A* + 2*B* + 4*D <* 1.

*Remark 2*Similarly as in Remark 1, we note that if the cone

*P*is normal, then continuity of mappings

*f*and

*g*in Theorem 3 can be replaced by conditions

*Example 4* Let *E* = ℝ and *P* = [0, +∞). Let *X* = [0, +∞), *d*(*x*, *y*) = |*x - y*| and define *q*(*x*, *y*) = *x*. It is easy to check that *q* is a *c*-distance on a cone metric space (*X*, *d*).

Take functions *f*, *g* : *X* → *X* defined by
and
. If *x* = 5,
, then
and
. Hence, there is no *A* ∈ (0, 1) (and hence no triple (*A*, *B*, *D*)) such that *d*(*fx*, *gy*) ≤ *Ad*(*x*, *y*) for each *x*, *y* ∈ [0, +∞), i.e., the existence of a common fixed point of *f* and *g* cannot be deduced from the well-known metric version of Theorem 3.

However, conditions of the *c*-distance version (Theorem 3) are satisfied. Indeed, take arbitrary *A*,
and *B* = *D* = 0. Then, for each *x*, *y* ∈ [0, +∞),
and
(see (3.9)). Note that *f* and *g* have a (trivial) common fixed point *u* = 0 and that *q*(*u*, *u*) = *q*(0, 0) = 0.

This example can be easily modified to the tvs-cone metric case. It is enough to define tvs-cone metric on *X* by *d*(*x*, *y*)(*t*) = |*x - y*|*φ*(*t*) with fixed *φ* ∈ *P* = {*f* ∈ *C*[0, 1] : *f*(*t*) ≥ 0 for *t* ∈ [0, 1]} and take *c*-distance *q*_{1}(*x*, *y*)(*t*) = *x · φ* (*t*) (see Example 3).

### 3.2 Mappings without periodic points

The first part of the following result was given with an incorrect proof in [20] (using lim inf which may not be defined in the case of an arbitrary cone metric space).

Recall that a map *f* : *X* → *X* is said to have property (*P*) if it satisfies *F*(*f*) = *F*(*f*^{
n
} ) for each *n* ∈ ℕ, where *F*(*f*) stands for the set of all fixed points of *f*[22].

**Theorem 4**

*Let*(

*X*,

*d*)

*be a tvs-cone metric space and q*:

*X*×

*X*→

*E be a c-distance on X. Suppose that a continuous self-map f*:

*X*→

*X satisfies*

*for some λ* ∈ (0, 1) *and each ×* ∈ *X. Then:*

*1. f has a fixed point and if fu* = *u, then q*(*u*, *u*) = *θ;*

*2. f has property* (*P*).

*Proof*(1) Let

*x*

_{0}∈

*X*and

*x*

_{n+1}=

*fx*

_{ n },

*n*≥ 0. If for some

*n*

_{0}∈ ℕ

_{0}, then is a fixed point of

*f*. Otherwise, we get from (3.10) that

Using Lemma 1.(3) again, one obtains that {*x*_{
n
} } is a Cauchy sequence in *X*.

*x*

_{ n }→

*x**, and continuity of

*f*implies that

*x*

_{n+1}=

*fx*

_{ n }→

*fx**and

*fx**=

*x**.

- (2)

By property (p_{5}), it follows that *q*(*u*, *fu*) = *θ*.

*k*∈ {1, 2,...,

*n*}, we have that

*q*(

*f*

^{ k }

*u*,

*f*

^{k+1}u) ≼

*λ*

^{ k }

*q*(

*u*,

*fu*)

*θ*and so

*q*(

*f*

^{ k }

*u*,

*f*

^{k+1}

*u*) =

*θ*. It follows that

*q*(

*u*,

*f*

^{2}

*u*) ≼

*q*(

*u*,

*fu*)+

*q*(

*fu*,

*f*

^{2}

*u*) =

*θ*, i.e.,

*q*(

*u*,

*f*

^{2}

*u*) =

*θ*and, similarly,

From *q*(*u*, *fu*) = *θ* = *q*(*u*, *f*^{
n
}*u*) = *q*(*u*, *u*) and Lemma 1.(1), we conclude that *fu* = *u*, i.e., *u* ∈ *F*(*f*). ■

Another way to obtain property (*P*) is the following.

**Theorem 5**

*Let*(

*X*,

*d*)

*be a tvs-cone metric space and q*:

*X*×

*X*→

*E be a c-distance on X. Suppose that a continuous self-map f*:

*X*→

*X satisfies*

*for some λ* ∈ (0, 1) *and each ×* ∈ *X. Then f has property* (*P*).

*Proof* Denote *z*_{1}(*x*) = *q*(*x*, *fx*)+ *q*(*fx*, *x*) and *z*_{2}(*x*) = *q*(*fx*, *f*^{2}*x*)+ *q*(*f*^{2}*x*, *fx*).

*z*

_{2}(

*x*) ≼

*λz*

_{1}(

*x*) for each

*x*∈

*X*. Suppose that

*f*

^{ n }

*u*=

*u*. Then,

Since 0 *< λ*^{
n
} *<* 1, property (p_{5}) implies that *z*_{1}(*u*) = *q*(*u*, *fu*) + *q*(*fu*, *u*) = *θ*. Again, the triangle inequality (q2) implies that *q*(*u*, *u*) = *q*(*fu*, *fu*) = *θ*, and by Lemma 1.(1), we get that *fu* = *u*. ■

**Corollary 1** *Let q be a c-distance on a tvs-cone metric space* (*X*, *d*) *and let f* : *X* → *X be continuous and such that for some nonnegative A*, *B*, *C*, *D*, *E such that A* + *B* + *C* + 2*D* + 2*E <* 1, *inequalities* (3.1) *and* (3.2) *hold for all x*, *y* ∈ *X. Then f has property* (*P*).

Adding up, one obtains inequality (3.11) with
, since *A* + *B* + *C* + 2*D* + 2*E <* 1. ■

Similar results concerning property (*Q*) of two self-mappings *f* and *g* (i.e., property that *F*(*f*) ∩ *F*(*g*) = *F*(*f*^{
n
} ) ∩ *F*(*g*^{
n
} ) for each *n* ∈ ℕ) can be obtained.

## Declarations

### Acknowledgements

The authors thank the referees for their valuable comments that helped to improve the text. The authors are thankful to the Ministry of Science and Technological Development of Serbia.

## Authors’ Affiliations

## References

- Huang LG, Zhang X:
**Cone metric spaces and fixed point theorems of contractive mappings.***J Math Anal Appl*2007,**332**(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleGoogle Scholar - Vandergraft JS:
**Newton's method for convex operators in partially ordered spaces.***SIAM J Num Anal*1967,**4:**406–432. 10.1137/0704037MathSciNetView ArticleGoogle Scholar - Krasnoseljski MA, Zabrejko PP:
*Geometrical Methods in Nonlinear Analysis.*Springer, Berlin; 1984.View ArticleGoogle Scholar - Zabrejko PP:
**K-metric and K-normed linear spaces, survey.***Collect Math*1997,**48:**825–859.MathSciNetGoogle Scholar - Beg I, Azam A, Arshad M:
**Common fixed points for maps on topological vector space valued cone metric spaces.***Intern J Math Math Sci*2009, 8. (Article ID 560264)Google Scholar - Wei-Shih Du:
**A note on cone metric fixed point theory and its equivalence.***Nonlinear Anal*2010,**72:**2259–2261. 10.1016/j.na.2009.10.026MathSciNetView ArticleGoogle Scholar - Kadelburg Z, Radenović S, Rakočević V:
**Topological vector space valued cone metric spaces and fixed point theorems.***Fixed Point Theory Appl*2010,**2010:**18. (Article ID 170253)View ArticleGoogle Scholar - Abdeljawad Th, Rezapour Sh: On some topological concepts of TVS-cone metric spaces and fixed point theory remarks. arXiv,1102.1419v1[math.GN]Google Scholar
- Abdeljawad Th, Karapinar E:
**A gap in the paper "A note on cone metric fixed point theory and its equivalence.***Nonlinear Anal*2010,**72:**2259–2261. Gazi Univ. J. Sci.**24**(2), 233–234 (2011) 10.1016/j.na.2009.10.026MathSciNetView ArticleGoogle Scholar - Abdeljawad Th, Karapinar E:
**A common fixed point theorem of Gregus type on convex cone metric spaces.***J Comput Anal Appl*2011,**13**(4):609–621.MathSciNetGoogle Scholar - Arandjelović ID, Kečkić DJ:
**On nonlinear qausi-contractions on TVS-valued cone metric spaces.***Appl Math Lett*2011,**24:**1209–1213. 10.1016/j.aml.2011.02.010MathSciNetView ArticleGoogle Scholar - Karapinar E, Yuksel U:
**On common fixed point theorem without commuting conditions in TVS-cone metric spaces.***J Comput Anal Appl*2011,**13:**1123–1131.MathSciNetGoogle Scholar - Simić Su:
**A note on Stone's, Baire's, Ky Fan's and Dugundji's theorem in tvs-cone metric spaces.***Appl Math Lett*2011,**24:**999–1002. 10.1016/j.aml.2011.01.014MathSciNetView ArticleGoogle Scholar - Janković S, Kadelburg Z, Radenović S:
**On cone metric spaces, a survey.***Nonlinear Anal*2011,**74:**2591–2601. 10.1016/j.na.2010.12.014MathSciNetView ArticleGoogle Scholar - Kada O, Suzuki T, Takahashi W:
**Nonconvex minimization theorems and fixed point theorems in complete metric spaces.***Math Japonica*1996,**44:**381–391.MathSciNetGoogle Scholar - Abbas M, Ilić D, Ali Khan M:
**Coupled coincidence point and coupled common fixed point theorems in partially ordered metric spaces with w-distance.***Fixed Point Theory Appl*2010,**2010:**11. (Article ID 134897)Google Scholar - Ilić D, Rakočević V:
**Common fixed point for maps with w-distance.***Appl Math Comput*2008,**199:**599–610. 10.1016/j.amc.2007.10.016MathSciNetView ArticleGoogle Scholar - Razani A, Nezhad ZM, Boujary M:
**A fixed point theorem for w-distance.***Appl Sci*2009,**11:**114–117.MathSciNetGoogle Scholar - Cho YJ, Saadati R, Wang Sh:
**Common fixed point theorems on generalized distance in ordered cone metric spaces.***Comput Math Appl*2011,**61:**1254–1260. 10.1016/j.camwa.2011.01.004MathSciNetView ArticleGoogle Scholar - Lakzian H, Arabyani F:
**Some fixed point theorems in cone metric spaces with w-distance.***Inter J Math Anal*2009,**3**(22):1081–1086.MathSciNetGoogle Scholar - Abdeljawad Th:
**Completions of tvs-cone metric spaces and some fixed point theorems.***Gazi Univ J Sci*2011,**24**(2):235–240.Google Scholar - Jeong GS, Rhoades BE:
**Maps for which**F**(**T**) =**F**(**T^{ n }**).***Fixed Point Theory Appl*2005,**6:**87–131.Google Scholar

## Copyright

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.