International Journal of Analysis and Applications
ISSN 2291-8639
Volume 6, Number 2 (2014), 132-138
http://www.etamaths.com

TRIPLED FIXED POINT RESULTS FOR T-CONTRACTIONS ON

ABSTRACT METRIC SPACES

HAMIDREZA RAHIMI1, CALOGERO VETRO2, MUJAHID ABBAS3, GHASEM
SOLEIMANI RAD4,∗

Abstract. In this paper we introduce the notion of T-contraction for tripled

fixed points in abstract metric spaces and obtain some tripled fixed point
theorems which extend and generalize well-known comparable results in the

literature. To support our results, we present an example and an applications

to integral equations.

1. Introduction and Preliminaries

In 1922, Banach proved his famous fixed point theorem [5]. Afterward, many
authors considered various definitions of contractive mappings and proved several
fixed point theorems, which are extensions and generalizations of Banach’s theorem
(see, for example, [9, 13, 23]).

On the other hand, non-convex analysis has found some applications in optimiza-
tion theory. Fixed point theory in K-metric and K-normed spaces was developed
by Perov et al. [18], Mukhamadijev and Stetsenko [17] and others (we refer to a
survey by Zabrejko [26]). The main idea consists in using an ordered Banach space
instead of the set of real numbers, as the codomain for a metric. In 2007, Huang
and Zhang [12] reintroduced such as spaces and defined cone metric spaces. Then,
several fixed point results on cone metric spaces were obtained in [1, 2, 22] and
references therein.

In 2009, Beiranvand et al. [6] defined T-contractions in metric spaces. After-
ward, some related fixed point theorems were proved in [15]. Successively, Morales
and Rajes [16] introduced T-Kannan and T-Chatterjea contractive mappings in
cone metric spaces and studied the existence of fixed points for these mappings.
Recently, Rahimi et al. [19, 21] proved fixed theorems for T-contractions involving
two mappings on cone metric spaces.

Recently, Bhaskar and Lakshmikantham [8] introduced the concept of coupled
fixed point in partially ordered metric spaces, starting a fruitful direction of research
followed by many authors, also in the setting of ordered metric and ordered cone
metric spaces; see [14, 24] and the references therein. Finally, Berinde and Borcut
[7] introduced the notion of tripled fixed point (see also [3, 4]) and obtained results
on the existence of tripled fixed points.

2010 Mathematics Subject Classification. 47H10, 46J10, 34A34.
Key words and phrases. Abstract metric space; Tripled fixed point; T-contraction; Sequentially

convergent; Subsequentially convergent.

c©2014 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

132



TRIPLED FIXED POINT RESULTS FOR T -CONTRACTIONS 133

In this paper we introduce the notion of T-contraction in tripled fixed point theo-
ry and prove some related results on abstract metric spaces. It is worth mentioning
that our results do not rely on normality condition on cones involved therein. Our
theorems extend, unify and generalize well-known results in the literature.

Following are some definitions and known results needed in the sequel.

Definition 1.1 ([11, 12]). Let E be a real Banach space and P a subset of E.
Then P is called a cone if and only if

(a) P is closed, nonempty and P 6= {θ};
(b) a,b ∈ R, a,b ≥ 0, x,y ∈ P implies ax + by ∈ P ;
(c) if x ∈ P and −x ∈ P , then x = θ.

Given a cone P ⊂ E, a partial ordering � with respect to P is defined by

x � y ⇐⇒ y −x ∈ P.

We shall write x ≺ y to mean x � y and x 6= y. Also, we write x � y if and only
if y −x ∈ intP , where intP is the interior of P . If intP 6= ∅, the cone P is called
solid. A cone P is called normal if there exists a number K > 0 such that, for all
x,y ∈ E, we have

θ � x � y =⇒‖x‖≤ K‖y‖.

The least positive number satisfying the above inequality is called the normal con-
stant of P .

Definition 1.2. Let X be a nonempty set. Suppose that a mapping d : X×X → E
satisfies the following conditions:

(d1) θ � d(x,y) for all x,y ∈ X and d(x,y) = θ if and only if x = y;
(d2) d(x,y) = d(y,x) for all x,y ∈ X;
(d3) d(x,z) � d(x,y) + d(y,z) for all x,y,z ∈ X.
Then d is called a cone metric [12] or K-metric [26] on X and (X,d) is called a

cone metric space [12] or K-metric space [26].

The concept of K-metric space is more general than that of metric space, in fact
each metric space is a K-metric space where X = R and P = [0, +∞).

Definition 1.3 ([10]). Let (X,d) be a K-metric space, {xn} a sequence in X and
x ∈ X. Then

(i) {xn} converges to x if, for every c ∈ E with θ � c there exists an n0 ∈ N
such that d(xn,x) � c for all n > n0;

(ii) {xn} is called a Cauchy sequence if, for every c ∈ E with θ � c there exists
an n0 ∈ N such that d(xn,xm) � c for all m,n > n0;

(iii) a K-metric space X is said to be complete if every Cauchy sequence in X
is convergent in X.

Definition 1.4 ([10]). Let (X,d) be a K-metric space, P a solid cone. A mapping
T : X → X is said to be:

(i) sequentially convergent if the sequence {xn} in X is convergent, whenever
{Txn} is convergent;

(ii) subsequentially convergent if the sequence {xn} has a convergent subse-
quence, whenever {Txn} is convergent;



134 H. RAHIMI, C. VETRO, M. ABBAS, G. SOLEIMANI RAD

(iii) continuous if for any sequence {xn} in X with lim
n→+∞

xn = x implies that

lim
n→+∞

Txn = Tx.

Theorem 1.5 ([19, 20]). Let (X,d) be a complete K-metric space, P a solid cone
and T : X → X a continuous and one to one mapping. Moreover, f : X → X be a
mapping satisfying

d(Tfx,Tfy) � α1d(Tx,Ty) + α2[d(Tx,Tfx) + d(Ty,Tfy)]
+α3[d(Tx,Tfy) + d(Ty,Tfx)],

for all x,y ∈ X, where α1,α2,α3 ≥ 0 with α1 + 2α2 + 2α3 < 1. Then
(i) for each x0 ∈ X, {Tfnx0} is a Cauchy sequence, (define the iterate sequence
{xn} by xn+1 = fn+1x0);

(ii) there exists a zx0 ∈ X such that limn→+∞Tfnx0 = zx0 ;
(iii) if T is subsequentially convergent, then {fnx0} has a convergent subse-

quence;
(iv) there exists a unique wx0 ∈ X such that fwx0 = wx0 ; that is, f has a

unique fixed point;
(v) if T is sequentially convergent, then, for each x0 ∈ X, the sequence {fnx0}

converges to wx0 .

Definition 1.6 ([25]). An element (x,y,z) ∈ X ×X ×X is called a tripled fixed
point of a mapping F : X × X × X → X if x = F(x,y,z), y = F(y,z,x) and
z = F(z,x,y).

2. Main results

Definition 2.1. Let (X,d) be a K-metric space and T : X → X a mapping. A
mapping F : X×X×X → X is said to be a T-contraction, if there exist α,β,γ ≥ 0,
with α + β + γ < 1, such that for all x,y,z,x∗,y∗,z∗ ∈ X, we get

(1) d(TF(x,y,z),TF(x∗,y∗,z∗)) � αd(Tx,Tx∗) + βd(Ty,Ty∗) + γd(Tz,Tz∗).

Theorem 2.2. Let (X,d) be a complete K-metric space, P a solid cone and F :
X × X × X → X a T-contraction, where T : X → X is a continuous and one to
one mapping. Then

(i) {TFn(x0,y0,z0)}, {TFn(y0,z0,x0)} and {TFn(z0,x0,y0)} are Cauchy se-
quences for all x0,y0,z0 ∈ X;

(ii) there exist ux0,uy0,uz0 ∈ X such that

lim
n→+∞

TFn(x0,y0,z0) = ux0 , lim
n→+∞

TFn(y0,z0,x0) = uy0, and

lim
n→+∞

TFn(z0,x0,y0) = uz0 ;

(iii) if T is subsequentially convergent, then {TFn(x0,y0,z0)}, {TFn(y0,z0,x0)}
and {TFn(z0,x0,y0)} have a convergent subsequence;

(iv) there exist uniques wx0,wy0,wz0 ∈ X such that

F(wx0,wy0,wz0 ) = wx0, F(wy0,wz0,wx0 ) = wy0, F(wz0,wx0,wy0 ) = wz0 ;

that is, F has a unique tripled fixed point;



TRIPLED FIXED POINT RESULTS FOR T -CONTRACTIONS 135

(v) if T is sequentially convergent, then, for all x0,y0,z0 ∈ X, the sequence
{TFn(x0,y0,z0)} converges to wx0 ∈ X, the sequence {TFn(y0,z0,x0)}
converges to wy0 ∈ X and the sequence {TFn(z0,x0,y0)} converges to
wz0 ∈ X.

Proof. Let define

D : X3 ×X3 → P ; D((x1,y1,z1), (x2,y2,z2)) = d(x1,x2) + d(y1,y2) + d(z1,z2);
F : X3 → X3 ; F(x,y,z) = (F(x,y,z),F(y,z,x),F(z,x,y));
T : X3 → X3 ; T(x,y,z) = (Tx,Ty,Tz).

Then (X3,D) is a complete K-metric space, and T is continuous and one-to-one.
It is clear that (x,y,z) ∈ X3 is a tripled fixed point of F if, and only if, it is a fixed
point of F. Suppose that Fn(x,y,z) throughout the text (which is not properly
defined) means exactly Fn(x,y,z). Let k = α + β + γ < 1. Therefore,

D(TF(x,y,z),TF(x∗,y∗,z∗))

= D(T(F(x,y,z),F(y,z,x),F(z,x,y)),T(F(x∗,y∗,z∗),F(y∗,z∗,x∗),F(z∗,x∗,y∗)))

= D((TF(x,y,z),TF(y,z,x),TF(z,x,y)), (TF(x∗,y∗,z∗),TF(y∗,z∗,x∗),TF(z∗,x∗,y∗)))

= d(TF(x,y,z),TF(x∗,y∗,z∗)) + d(TF(y,z,x),TF (y∗,z∗,x∗))

+ d(TF(z,x,y),TF(z∗,x∗,y∗))

� [αd(Tx,Tx∗) + βd(Ty,Ty∗) + d(Tz,Tz∗)]
+ [αd(Ty,Ty∗) + βd(Tz,Tz∗) + γd(Tx,Tx∗)]

+ [αd(Tz,Tz∗) + βd(Tx,Tx∗) + d(Ty,Ty∗)]

= (α + β + γ)d(Tx,Tx∗) + (α + β + γ)d(Ty,Ty∗) + (α + β + γ)d(Tz,Tz∗)

= k(d(Tx,Tx∗) + d(Ty,Ty∗) + d(Tz,Tz∗))

= kD((Tx,Ty,Tz), (Tx∗,Ty∗,Tz∗))

= kD(T(x,y,z),T(x∗,y∗,z∗)).

The proof further follows by Theorem 1.5 (taking α1 = k < 1 and α2 = α3 = 0).
This completes the proof. �

Corollary 2.3. Let (X,d) be a complete K-metric space, P a solid cone, and
T : X → X a continuous and one to one mapping. If F : X ×X ×X → X satisfies

(2) d(TF(x,y,z),TF(x∗,y∗,z∗)) �
k

3
[d(Tx,Tx∗) + d(Ty,Ty∗) + d(Tz,Tz∗)],

for all x,y,z,x∗,y∗,z∗ ∈ X, where k ∈ [0, 1), then the conclusions of Theorem 2.2
hold true.

Proof. The thesis follows easily from Theorem 2.2, by putting α = β = γ = k/3 in
(1). �

Corollary 2.4. Let (X,d) be a complete K-metric space and P a solid cone. If
F : X ×X ×X → X satisfies

(3) d(F(x,y,z),F(x∗,y∗,z∗)) � αd(x,x∗) + βd(y,y∗) + γd(z,z∗),

for all x,y,z,x∗,y∗,z∗ ∈ X, where α,β,γ ≥ 0 with α + β + γ < 1, then, F has a
unique tripled fixed point.



136 H. RAHIMI, C. VETRO, M. ABBAS, G. SOLEIMANI RAD

Proof. The thesis follows easily from Theorem 2.2, by putting T = Ix, where Ix is
the identity mapping on X. �

Corollary 2.5. Let (X,d) be a complete K-metric space and P a solid cone. If
F : X ×X ×X → X satisfies

(4) d(F(x,y,z),F(x∗,y∗,z∗)) �
k

3
[d(x,x∗) + d(y,y∗) + d(z,z∗)],

for all x,y,z,x∗,y∗,z∗ ∈ X, where k ∈ [0, 1), then F has a unique tripled fixed
point.

Proof. Result follows from Corollary 2.3, taking T = IX . �

Example 2.6. Let X = [0, 1] and E = C1R[0, 1] endowed with the order induced by
P = {φ ∈ E : φ(t) ≥ 0 for t ∈ [0, 1]}. Define d : X×X → E by d(x,y)(t) = |x−y|2t,
for all x,y ∈ X.
Clearly, (X,d) is a complete K-metric space with a cone having nonempty interior.
Next, define the mappings F : X ×X ×X → X and T : X → X by

Tx =
x

2
, for all x ∈ X and F(x,y,z) =

x + y + z

6
, for all x,y,z ∈ X.

Then F satisfies the contractive condition (2) for k = 1/2; that is,

d(TF(x,y,z),TF(u,v,w)) �
1

6
[d(Tx,Tu) + d(Ty,Tv) + d(Tz,Tw)],

for all x,y,z,u,v,w ∈ X. Consequently, Corollary 2.3 applies to F, which has a
unique tripled fixed point; that is (0, 0, 0).

3. Applications

Let C([0,T],R) be the set of continuous functions defined in [0,T], where T > 0.
Consider the metric given by

d(u,v) = sup
t∈[0,T ]

|u(t) −v(t)|, for all u,v ∈ R.

Note that (C([0,T],R),d) is a complete metric space.

Now, we study the existence and uniqueness of solution to an integral equation, by
using Corollary 2.5. Precisely, we consider the equation

(5) x(t) =

∫ T
0

k(t,s)(f(s,x(s)) + g(s,x(s)) + h(s,x(s))) ds + a(t), t ∈ [0,T].

Now, we state and prove the following theorem.

Theorem 3.1. Assume that the following conditions hold:

(i) k ∈ C([0,T] × [0,T],R) such that sup
s,t∈[0,T ]

|k(t,s)| = M <
1

T
;

(ii) a ∈ C([0,T],R);
(iii) f,g,h ∈ C([0,T] ×R,R);
(iv) for all xi,yi,zi ∈ C([0,T],R), where i = 1, 2, and t ∈ [0,T] we have
|f(t,x1(t)) −f(t,x2(t))| + |g(t,y1(t)) −g(t,y2(t))| + |h(t,z1(t)) −h(t,z2(t))|

≤
1

3
(|x1(t) −x2(t)| + |y1(t) −y2(t)| + |z1(t) −z2(t)|).



TRIPLED FIXED POINT RESULTS FOR T -CONTRACTIONS 137

Then, the integral equation (5) has a unique solution.

Proof. Consider the mapping F : C([0,T],R)×C([0,T],R)×C([0,T],R) → C([0,T],R)
defined by

F(x,y,z)(t) =

∫ T
0

k(t,s)(f(s,x(s)) + g(s,y(s)) + h(s,z(s))) ds + a(t), t ∈ [0,T].

It is easy to show that (x,y,z) is a solution of (5) if and only if (x,y,z) is a tripled
fixed point of F. To establish the existence of such a point, we will use Corollary
2.5. In fact, by condition (iv), we have easily

|F(x1,y1,z1)(t) −F(x2,y2,z2)(t)|

≤
∫ T
0

|k(t,s)|
1

3
(|x1(s) −x2(s)| + |y1(s) −y2(s)| + |z1(s) −z2(s)|) ds

≤
1

3

(∫ T
0

|k(t,s)| ds
)

(d(x1,x2) + d(y1,y2) + d(z1,z2)),

for all xi,yi,zi ∈ C([0,T],R), where i = 1, 2 and t ∈ [0,T]. By (i), it follows that

d(F(x1,y1,z1),F(x2,y2,z2)) ≤
MT

3
(d(x1,x2) + d(y1,y2) + d(z1,z2)),

for all xi,yi,zi ∈ C([0,T],R), where i = 1, 2. Then, condition (4) of Corollary 2.5
is satisfied with k = MT < 1 and hence, applying Corollary 2.5, we obtain the
existence of a unique tripled fixed point of F ; that is, the integral equation (5) has
a unique solution. �

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1Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic

Azad University, P. O. Box 13185/768, Tehran, Iran

2 Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, via
Archirafi 34, 90123 Palermo, Italy

3 Department of Mathematics and Applied Mathematics, University of Pretoria,
Lynnwood road, Pretoria 0002, South Africa

4Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic
Azad University, P. O. Box 13185/768, Tehran, Iran

∗Corresponding author