Revista Notas de Matemática
Vol.4(2), No. 269, 2008, pp. 66-78
http://www.saber.ula.ve/notasdematematica/
Comisión de Publicaciones
Departamento de Matemáticas
Facultad de Ciencias
Universidad de Los Andes
Cone metric spaces and fixed point theorems of T−contrative
mappings
José R. Morales and Edixón Rojas
Resumen
En este trabajo, se estudia la existencia de puntos fijos para las asignaciones definidas en total,
(secuencialmente compacto) cono espacio métrico, (M,d) que satisface una desigualdad de
contracción general de depender de otra función
Palabras claves: Cono espacios métricos, punto fijo, el mapeo de contracción, de forma
secuencial convergente.
Abstract
In this paper, we study the existence of fixed points for mappings defined on complete,
(sequentially compact) cone metric space, (M,d) satisfying a general contractive inequality
depend on another function
key words. Cone metric spaces, fixed point, contractive mapping, sequentially convergent
AMS(MOS) subject classifications. 46J10, 46J15, 47H10.
1 Introduction
The concept of cone metric space was introduced by Huan Long - Guang and Zhang Xian [2],
where the set of real numbers is replaced by an ordered Banach space. They introduced the basic
definitions and discuss some properties of convergence of sequences in cone metric spaces.
They also obtained various fixed point theorems for contractive single - valued maps in such
spaces. Subsequently, some other mathematicians, ([4],[5],[6],. . . ), have generalized the results of
Guang and Zhang [2].
Recently, A. Beiranvand, S. Moradi, M. Omid and H. Pazandeh [1] introduced a new class
of contractive mappings: T−contraction and T−contrative extending the Banach’s contraction
principle and the Edelstein’s fixed point theorem, (see [3]) respectively.
Cone metric spaces and fixed point theorems of T−contrative mappings 67
The purpose of this paper is to analyze the existence of fixed points for a self-map S defined
on a complete, (sequentially compact) cone metric space, (M,d) satisfying the T−contraction
and T−contractive condition.
Our results extend some fixed points theorems of [1] and [2].
2 Preliminary facts
Consistent with Guang and Zhang [2], we recall the definitions of cone metric space, the notion
of convergence and other results that will be needed in the sequel.
Let E be a real Banach space and P a subset of E. P is called a cone if and only if:
P1.- P is nonempty, closed and P 6= {0};
P2.- a, b ∈ R, a, b ≥ 0 and x, y ∈ P ⇒ ax + by ∈ P ;
P3.- x ∈ P and −x ∈ P ⇒ x = 0 ⇔ P ∩ (−P ) = {0}.
For a given cone P ⊆ E, we can define a partial ordering ≤ on E with respect to P by
x ≤ y , if and only if y − x ∈ P.
We shall write x < y to indicate that x ≤ y but x 6= y, while x ≪ y will stands for y − x ∈ IntP,
where intP denotes the interior of P. The cone P ⊂ E is called normal if there is a number
K > 0 such that for all x, y ∈ E,
0 ≤ x ≤ y, implies ‖x‖ ≤ K‖y‖.
The least positive number satisfying inequality above is called the normal constant of P .
The cone P is called regular if every increasing sequence which is bounded from above is
convergent. That is, if (xn) is a sequence such that
x1 ≤ x2 ≤ . . . ≤ xn ≤ . . . ≤ y
for some y ∈ E, then there is x ∈ E such that ‖xn − x‖ −→ 0, (n → ∞).
In the following we always suppose E is a Banach space, P is a cone with intP 6= ∅ and ≤ is
a partial ordering with respect to P .
68 José R. Morales and Edixón Rojas
Definition 2.1 ([2]) Let M be a nonempty set. Suppose the mapping d : M×M −→ E satisfies:
d1.- 0 < d(x, y) for all x, y ∈ M and d(x, y) = 0 if and only if x = y;
d2.- d(x, y) = d(y, x) for all x, y ∈ M ;
d3.- d(x, y) ≤ d(x, z) + d(y, z) for all x, y, z ∈ M.
Then d is called a cone metric on M and (M,d) is called a cone metric space.
It is obvious that cone metric spaces generalize metric spaces.
Example 2.2 1. ([2, Example 1]) Let E = 2R , P = {(x, y) ∈ E : x, y ≥ 0} ⊂ 2R , M = R
and d : M ×M −→ E such that
( )
d(x, y) = |x− y|, α|x− y|
where α ≥ 0 is a constant. Then (M,d) is a cone metric space.
2. Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0} ⊂ E, M = R and d : M ×M −→ E such that
d(x, y) = |x− y|ϕ
where ϕ(t) = et ∈ E. Then (M,d) is a cone metric space.
Definition 2.3 ([2]) Let (M,d) be a cone metric space. Let (xn) be a sequence in M. Then:
(i) (xn) converges to x ∈ M if, for every c ∈ E, with 0 ≪ c there is n0 ∈ N such that for all
n ≥ n0,
d(xn, x) ≪ c.
We denote this by lim xn = x or xn −→ x, (n → ∞).
n→∞
(ii) If for any c ∈ E, there is a number n0 ∈ N such that for all m,n ≥ n0
d(xn, xm) ≪ c,
then (xn) is called a Cauchy sequence in M ;
(iii) (M,d) is a complete cone metric space if every Cauchy sequence is convergent in M.
Cone metric spaces and fixed point theorems of T−contrative mappings 69
The following lemma will be useful for us to prove our main results.
Lemma 2.4 ([2]) Let (M,d) be a cone metric space, P a normal cone with normal constant K
and (xn) is a sequence in M.
(i) (xn) converges a x ∈ M if and only if
lim d(xn, x) = 0;
n→∞
(ii) If (xn) is convergent then it is a Cauchy sequence;
(iii) (xn) is a Cauchy sequence if and only if lim d(xn, xm) = 0;
n,m→∞
(iv) If xn −→ x and xn −→ y, (n → ∞) then x = y;
(v) If xn −→ x and (yn) is another sequence in M such that yn −→ y, then d(xn, yn) −→ d(x, y).
Definition 2.5 Let (M,d) be a cone metric space. It for any sequence (xn) in M, there is a
subsequence (xn ) of (xn) such that (xi n ) is convergent in M. Then M is called a sequentiallyi
compact cone metric space.
Next Definition and subsequent Lemma are given in [1] in the scope of metric spaces, here
we will rewrite it in terms of cone metric spaces.
Definition 2.6 Let (M,d) be a cone metric space, P a normal cone with normal constant K and
T : M −→ M. Then
(i) T is said to be continuous if lim xn = x, implies that lim Txn = Tx for every (xn) in M ;
n→∞ n→∞
(ii) T is said to be sequentially convergent if we have, for every sequence (yn), if T (yn) is
convergent, then (yn) also is convergent;
(iii) T is said to be subsequentially convergent if we have, for every sequence (yn), if T (yn) is
convergent, then (yn) also is convergent.
Lemma 2.7 If (M,d) be a sequence compact cone metric space, then every function T : M −→
M is subsequentially convergent and every continuous function T : M −→ M is sequentially
convergent.
70 José R. Morales and Edixón Rojas
3 Main results
In this section, first we introduce the notions of T−contraction, T−contrative and then we extend
the Banach Contraction Principle and Edelstein’s fixed point Theorem given in [1] and [2].
Definition 3.1 ([1]) Let (M,d) be a cone metric space and T, S : M −→ M two functions. A
mapping S is said to be a T−contraction if there is a ∈ [0, 1) constant such that
d(TSx, TSy) ≤ ad(Tx, Ty) (3.1)
for all x, y ∈ M.
Example 3.2 Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0} ⊂ E, M = R and d(x, y) = |x − y|et,
where et ∈ E. Then (M,d) is a cone metric space. We consider the functions T, S : M −→ M
defined by Tx = e−x and Sx = 2x + 1. Then
(i) It is clear that S is not a contraction;
(ii) S is a T−contraction. In fact,
d(TSx, TSy) = |TSx− TSy|et
1
= |e−x − e−y||e−x − e−y|et
e
≤ 2 |e−x − e−y|et 2= d(Tx, Ty).
e e
The next result extend the Theorem 1 of Guang and Zhang [2], and Theorem 2.6 of Beiran-
vand, Moradi, Omid and Pazandeh [1].
Theorem 3.3 Let (M,d) be a complete cone metric space, P be a normal cone with normal
constant K, in addition let T : M −→ M be an one to one and continuous function and S :
M −→ M a T−contraction continuous function. Then
1. For every x0 ∈ M ,
lim d(TSnx0, TS
n+1x0) = 0;
n→∞
2. There is y0 ∈ M such that
lim TSnx0 = y0;
n→∞
Cone metric spaces and fixed point theorems of T−contrative mappings 71
3. If T is subsequentially convergent, then (Snx0) has a convergent subsequence;
4. There is a unique z0 ∈ M such that
Sz0 = z0;
5. If T is a sequentially convergent, then for each x0 ∈ M the iterate sequence (Snx0) converges
to z0.
Proof: For every x1, x2 ∈ M,
d(Tx1, Tx2) ≤ d(Tx1, TSx1) + d(TSx1, TSx2) + d(TSx2, Tx2)
≤ d(Tx1, TSx1) + ad(Tx1, Tx2) + d(TSx2, Tx2)
so,
1
d(Tx1, Tx2) ≤ − [d(Tx1, TSx1) + d(TSx2, Tx2)] . (3.2)1 a
Now, choose x0 ∈ M and define the Picard iteration associated to S, (xn) given by xn+1 = Sxn =
Snx0, n = 0, 1, 2, . . .
d(Txn, Txn+1) = d(TS
nx0, TS
n+1x0) ≤ ad(TSn−1x0, TSnx0)
hence,
d(TSnx , TSn+10 x0) ≤ and(Tx0, TSx0). (3.3)
Since P is a normal cone with normal constant K, we get
‖d(TSnx0, TSn+1x n0)‖ ≤ a K‖d(Tx0, TSx0)‖
which implies that
lim d(TSnx , TSn+10 x0) = 0. (3.4)
n→∞
therefore, for m,n ∈ N with m > n, by (3.2) and (3.3) we have
d(Txn, Txm) = d(TS
nx0, TS
mx0)
[ ]
≤ 1− d(TS
nx0, TS
n+1x0) + d(TS
m+1x0, TS
mx0)
1 a
[ ]
≤ 1− a
nd(Tx , TSx ) + am0 0 d(Tx0, TSx0) ,
1 a
72 José R. Morales and Edixón Rojas
hence,
n m
d(TSx , TSm
a + a
0 x0) ≤ − d(Tx0, TSx0). (3.5)1 a
Taking norm to inequality above, we obtain that
n
‖ n m ‖ ≤ a + a
m
d(TS x0, TS x0) − K‖d(Tx0, TSx0)‖.1 a
Consequently,
lim d(TSnx m0, TS x0) = 0. (3.6)
n,m→∞
Which prove 1. On the other hand, (3.6) implies that (TSnx0) is a Cauchy sequence in M. By
the completeness of M , there is y0 ∈ M such that
lim TSnx0 = y0. (3.7)
n→∞
Proving in this way assertion 2. Now, if T is subsequentially convergent, then (Snx0) has a
convergent subsequence. So, there exist z0 ∈ M and (n ∞i)i=1 such that
lim Snix0 = z0, (3.8)
i→∞
since T is continuous we have,
lim TSnix0 = Tz0 (3.9)
i→∞
from equality (3.7) we conclude that
Tz0 = y0. (3.10)
Since S is continuous, (and also by using (3.8)) then
lim Sni+1x0 = Sz0
i→∞
as well as,
lim TSni+1x0 = TSz0. (3.11)
i→∞
Again by (3.7), the following equality holds,
lim TSni+1x0 = y0
i→∞
Cone metric spaces and fixed point theorems of T−contrative mappings 73
hence, TSz0 = y0 = Tz0. Since T is injective,then Sz0 = z0, so S has a fixed point. Therefore
assertion 3. is proved. On the other hand, since T is one to one and S is a T−contraction, S has
a unique fixed point. i.e., conclusion 4.
Finally, if T is sequentially convergent, (Snx0) is convergent to z0, that is,
lim Snx0 = z0,
n→∞
proving in this way conclusion 5. which finishes the proof of the theorem. 
Corollary 3.4 ([2], Theorem 1) Let (M,d) be a complete cone metric space P ⊂ E be a nor-
mal cone with normal constant K. Suppose S : M −→ M is a contraction function then S has a
unique fixed point in M and for any x n0 ∈ M (S x) converges to the fixed point.
Now, if we take E = R+ in Theorem 3.3 we obtain the following
Corollary 3.5 (Theorem 2.6, [1]) Let (M,d) be a complete metric space and T : M −→ M be
an one to one, continuous and subsequentially convergent mapping. Then for every T−contraction
continuous function S : M −→ M has a unique fixed point. Moreover, if T is sequentially
convergent, then for each x0 ∈ M, the sequence (Snx0) converge to the fixed point of S.
If we take E = R and Tx = x in the Theorem 3.3 then we obtain the Banach’s Contraction
Principle
Corollary 3.6 Let (M,d) be a complete metric space and S : M −→ M is a contraction mapping.
Then S has a unique fixed point.
The following result is the localization of the Theorem 3.3.
Theorem 3.7 Let (M,d) be a complete cone metric space, P ⊂ E be a normal cone with normal
constant K and T : M −→ M be an injective, continuous and subsequentially mapping. For
c ∈ E with 0 ≪ c, x0 ∈ M, set
B(Tx0, c) = {y ∈ M : d(Tx0, y) ≤ c}.
Suppose S : M −→ M is a T−contraction continuous mapping for all x, y ∈ B(Tx0, c) and
d(TSx0, Tx0) ≤ (1 − a)c. Then S has a unique fixed point in B(Tx0, c).
Proof: We only need to prove that B(Tx0, c) is complete and TSx ∈ B(Tx0, c) for all Tx ∈
B(Tx0, c). Suppose that (yn) is also a Cauchy sequence in M. By the completeness of M, there
exist y ∈ M such that yn −→ y, (n → ∞).
74 José R. Morales and Edixón Rojas
Thus, we have
d(Tx0, y) ≤ d(yn, Tx0) + d(yn, y) ≤ c + d(yn, y)
since yn −→ y, (n → ∞), d(yn, y) −→ 0. Hence d(Tx0, y) ≤ c and y ∈ B(Tx0, c). Therefore,
B(Tx0, c) is complete.
On the other hand, for every Tx ∈ B(Tx0, c),
d(Tx0, TSx) ≤ d(TSx0, Tx0) + d(TSx0, TSx)
≤ (1 − a)c + ad(Tx0, Tx) ≤ (1 − a)c + ac = c.
I.e., TSx ∈ B(Tx0, c), and the proof is done. 
Corollary 3.8 Let (M,d) be a complete cone metric space, P ⊂ E be a normal cone with normal
constant K and T : M −→ M be an one to one, continuous and subsequentially convergent
mapping. Let suppose that S : M −→ M is a mapping such that, Sn is a T−contraction for some
n ∈ N and furthermore a continuous function. Then S has a unique fixed point in M.
Proof: From Theorem 3.3, we have that Sn has a unique fixed point z0 ∈ M, that is, Snz0 = z0.
But Sn(Sz) = S(Snz) = Sz, so S(z) is also fixed point of Sn. Hence Sz = z, i.e., z is a fixed
point of S. Since the fixed point of S is also fixed point of Sn, then the fixed point of S is unique.

Example 3.9 Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0} ⊂ E,
M = [1,+∞) and d : M × M −→ E defined by d(x, y) = |x − y|et, where ϕ(t) = et ∈ E.
Then (M,d) is a complete cone metric space. Now we will consider the following functions,
√
TS : M −→ M defined by Tx = 1 + lnx and Sx = 2 x.
It is evident that S is not a contraction mapping, but it is a T−contraction because,
1
d(TSx, TSy) = |TSx− TSy|et = | ln x− ln y|et
2
= |Tx− Ty|et ≤ 1d(Tx, Ty).
2
Also, T is one to one, continuous and subsequentially convergent. Therefore, by Theorem 3.3 T
has a unique fixed point, z0 = y.
The following example shows that we can not omit the subsequentially convergence of the
function T in the Theorem 3.3 (5).
Cone metric spaces and fixed point theorems of T−contrative mappings 75
Example 3.10 Consider the example 3.2. Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0}, M = R
and d : M × M −→ E defined by d(x, y) = |x − y|et where et ∈ E. Then (M,d) is a complete
cone metric space. Let T, S : M −→ M be two functions defined by Tx = e−x and Sx = 2x + 1.
It is clear that S is a T−contraction, but T is not subsequentially convergent, because Tn →
0, (n → ∞) but the sequence (n) has not any convergent subsequence and S has not a fixed point.

Definition 3.11 Let (M,d) be a cone metric space and T, S : M −→ M two functions. A
mapping S is said to be a T−contractive if for each x, y ∈ M such that Tx =6 Ty then
d(TSx, TSy) < d(x, y).
It is clear that every T−contraction function is T−contractive, but the converse is not true.
Example 3.12 1. Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0} ⊂ E, M = [1,+∞) and
d : M×M −→ E defined by d(x, y) = |x−y|et, where et ∈ E. Then (M,d) is a cone metric
space.
√
Let T, S : M −→ M be two functions defined by Tx = x and Sx = x. Then:
i.- S is a T−contractive function;
ii.- S is not a T−contraction mapping.
2. Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0} ⊂ E, M = [0, 1/2] and d : M × M −→ E
defined by d(x, y) = |x − y|et, where et ∈ E. Obviously (M,d) is a cone metric space and
x2
the function S : M −→ M defined by Sx = √ is not contractive. If T : M −→ M is
2
defined by Tx = x2, then S is T−contractive, because:
∣ ∣
x4 y4∣ ∣ 1
d(TSx, TSy) = |TSx− TSy|et = ∣ − ∣ et = |x2 + y2||Tx− Ty|et
∣ 2 2 ∣ 2
< |Tx− Ty|et = d(Tx, Ty).

The following result extend the Theorem 2 of [1] and Theorem 2.9 of [2].
Theorem 3.13 Let (M,d) be a compact cone metric space, P be a normal cone with normal con-
stant K and T, S : M −→ M functions such that T is injective, continuous and S is T−contractive
mapping. Then,
76 José R. Morales and Edixón Rojas
i.- S has a unique fixed point;
ii.- For any x0 ∈ M the sequence iterates (Snx0) converges to the fixed point of S.
Proof: In first we are going to show that S is a continuous function. Let lim xn = x, we want
n→∞
to prove that lim Sxn = Sx. Since S is T−contractive, we get
n→∞
d(TSxn, TSx) ≤ d(Txn, Tx)
so,
‖d(TSxn, TSx)‖ ≤ K‖d(Txn, Tx)‖.
Now, since T is continuous, we have
lim ‖d(TSxn, TSx)‖ = 0
n→∞
also that,
lim d(TSxn, TSx) = 0
n→∞
therefore,
lim TSxn = TSx. (3.12)
n→∞
Let (Sxn ) be an arbitrary convergent subsequence of (xn). There is a y ∈ M such thati
lim Sxn = t.i
i→∞
By the continuity of T we infer,
lim TSxn = Ty. (3.13)i
i→∞
By (3.12) and (3.13) we conclude that TSx = Ty. Since T is one to one then, Sx = y. Hence,
every convergence subsequence of (Sxn) converge to Sx. From the fact M a compact cone metric
space, we arrive to the conclusion that S is a continuous function.
Now, because of T and S are continuous functions, then the function ϕ : M −→ P defined
by ϕ(y) = d(TSy, Ty), for all y ∈ M , is continuous on M and from the compactness of M , the
function ϕ attains its minimum, say at x ∈ M.
If Sx 6= x, then
ϕ(Sx) = d(TS2x, TSx) < d(TSx, Tx) = ϕ(x)
Cone metric spaces and fixed point theorems of T−contrative mappings 77
which is a contradiction, So Sx = x proving in this form part i. Choose x0 ∈ M and set
a nn = d(TS x0, Tx). Since
a = d(TSn+1n+1 x0, Tx) = d(TS
n+1x0, TSx) ≤ d(TSnx0, Tx) = an,
then (an) is a non increasing sequence of non negative real numbers and so it has a limit, say a,
that is
a = lim an or lim d(TS
nx0, Tx) = a.
n→∞ n→∞
By compactness, (TSnx0) has a convergent subsequence (TS
nix0) i.e.,
lim TSnix0 = z, (3.14)
i→∞
from the sequentially convergence of T , there exists w ∈ M such that
lim Snix0 = w
i→∞
so,
lim TSnix0 = Tw. (3.15)
i→∞
By (3.14) and (3.15), Tw = z. Then d(Tw, Tx) = a. Now we are going to show that Sw = x. If
Sw 6= x, then
a = lim d(TSnx0, Tx) = lim d(TS
nix0, Tx) = d(TSw, Tx)
n→∞ i→∞
= d(TSw, TSx) < d(Tw, Tx) = a
which is a contradiction. In this way, we get that Sw = x and hence,
a = lim d(TSni+1x0, Tx) = d(TSw, Tx) = 0
i→∞
Therefore, lim TSnx0 = Tx. Finally condition T sequentially convergent implies lim S
nx0 = x,
n→∞ n→∞
which finalize the proof. 
If we take E = R and Tx = x in Theorem 3.13, we obtain the Edelstein’s fixed point theorem
(see, e.g., [3]).
Example 3.14 We must recall example 3.12 (2). Let E = (C[0,1],R), P = {ϕ ∈ E : ϕ ≥ 0} ⊂
E, M = [0, 1] and d(x, y) = |x− y|et, et ∈ E. It is clear that M is a compact cone metric space.
x2
The functions T, S : M −→ M defined by Tx = x2 and Sx = √ satisfy that T is injective and
2
continuous whereas S is T−contractive. So by Theorem 3.13 we have that S has a unique fixed
point, x = 0. 
78 José R. Morales and Edixón Rojas
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JOSÉ R. MORALES
Departamento de Matemáticas, Facultad de Ciencias,
Universidad de Los Andes
Mérida 5101, Venezuela
e-mail: moralesj@ula.ve
EDIXÓN ROJAS
Departamento de Matemáticas, Facultad de Ciencias,
Universidad de Los Andes
Mérida 5101, Venezuela
e-mail: edixonr@ula.ve