Filomat 27:5 (2013), 881–888
DOI 10.2298/FIL1305881R
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Fixed and periodic point theorems for T-contractions
on cone metric spaces
Hamidreza Rahimi
a
, B. E. Rhoades
b
, Stojan Radenovi´c
c
, Ghasem Soleimani Rad
d
a
Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, P. O. Box 13185/768, Tehran, Iran
b
Department of Mathematics, Indiana University Bloomington, IN 47405-7106, United States
c
Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia
d
Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, P. O. Box 13185/768, Tehran, Iran
Abstract. Recently, Filipovi
´
c et al. [M. Filipovi
´
c, L. Paunovi
´
c, S. Radenovi
´
c, M. Rajovi
´
c, Remarks on
“Cone metric spaces and fixed point theorems of T-Kannan and T-Chatterjea contractive mappings”, Math.
Comput. Modelling. 54 (2011) 1467-1472] proved several fixed and periodic point theorems for solid cones
on cone metric spaces. In this paper several fixed and periodic point theorems for T-contraction of two
maps on cone metric spaces with solid cone are proved. The results of this paper extend and generalize
well-known comparable results in the literature.
1. Introduction and preliminaries
In 1922, Banach proved the following famous fixed point theorem [3]. Suppose that (X, d) is a complete
metric space and a self-map T of X satisfies d(Tx, Ty) ≤ λd(x, y) for all x, y ∈ X where λ ∈ [0, 1); that is, T
is a contractive mapping. Then T has a unique fixed point. Afterward, other people considered various
definitions of contractive mappings and proved several fixed point theorems [4, 8, 11, 12, 17]. In 2007,
Huang and Zhang [9] introduced cone metric space and proved some fixed point theorems. Several fixed
and common fixed point results on cone metric spaces were introduced in [1, 15, 16, 18, 19].
Recently, Morales and Rajes [14] introduced T-Kannan and T-Chatterjea contractive mappings in cone
metric spaces and proved some fixed point theorems. Later, Filipovi
´
c et al. [6] defined T-Hardy-Rogers
contraction in cone metric space and proved some fixed and periodic point theorems. In this work we
prove several fixed and periodic point theorems for a T-contraction of two maps on cone metric spaces.
Our results extend various comparable results of Abbas and Rhoades [2], Filipovi
´
c et al. [6] and, Morales
and Rajes [14].
We begin with some important definitions.
Definition 1.1. (See [7, 9]). 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, non-empty and P , {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.
2010 Mathematics Subject Classification. Primary 47H10; Secondary 47H09.
Keywords. Cone metric space; Fixed point; Periodic point; Property P; Property Q.
Received: 13 July 2012; Accepted: 20 November 2012
Communicated by Vladimir Rako
ˇ
cevi
´
c
Corresponding authors: B. E. Rhoades and Ghasem Soleimani Rad.
Email addresses: rahimi@iauctb.ac.ir (Hamidreza Rahimi), rhoades@indiana.edu (B. E. Rhoades), radens@beotel.rs (Stojan
Radenovi
´
c), gha.soleimani.sci@iauctb.ac.ir (Ghasem Soleimani Rad)
Rahimi et al. / Filomat 27:5 (2013), 881–888 882
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 , y. Also, we write x ≪ y if and only if y − x ∈ intP (where
intP is the interior of P). If intP , ∅, 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,
θ ≼ x ≼ y =⇒ ∥x∥ ≤ K∥y∥.
The least positive number satisfying the above inequality is called the normal constant of P.
Example 1.2. (See [16]).
(i) Let E = C
R
[0, 1] with the supremum norm and P = { f ∈ E : f ≥ 0}. Then P is a normal cone with normal constant
K = 1.
(ii) Let E = C
2
R
[0, 1] with the norm ∥ f∥ = ∥ f ∥
∞
+ ∥ f
′
∥
∞
and consider the cone P = { f ∈ E : f ≥ 0} for every K ≥ 1.
Then P is a non-normal cone.
Definition 1.3. (See [9]). Let X be a nonempty set. Suppose that the mapping d : X × X → E satisfies
(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 on X and (X, d) is called a cone metric space.
Example 1.4. (See [9]). Let E = R
2
, P = {(x, y) ∈ E|x, y ≥ 0} ⊂ R
2
, X = R and d : X × X → E is such that
d(x, y) = (|x − y|, α|x − y|), where α ≥ 0 is a constant. Then (X, d) is a cone metric space.
Definition 1.5. (See [6]). Let (X, d) be a cone metric space, {x
n
} a sequence in X and x ∈ X. Then
(i) {x
n
} converges to x if, for every c ∈ E with θ ≪ c there exists an n
0
∈ N such that d(x
n
, x) ≪ c for all n > n
0
. We
denote this by lim
n→∞
d(x
n
, x) = θ
(ii) {x
n
} is called a Cauchy sequence if, for every c ∈ E with θ ≪ c there exists an n
0
∈ N such that d(x
n
, x
m
) ≪ c for
all m, n > n
0
. We denote this by lim
n,m→∞
d(x
n
, x
m
) = θ.
The notation θ ≪ c for c ∈ intP of a positive cone is used by Krein and Rutman [13]. Also, a cone metric
space X is said to be complete if every Cauchy sequence in X is convergent in X. In the sequel we shall
always suppose that E is a real Banach space, P is a solid cone in E, and ≼ is a partial ordering with respect
to P.
Lemma 1.6. (See [6]). Let (X, d) be a cone metric space over an ordered real Banach space E. Then the following
properties are often used, particularly when dealing with cone metric spaces in which the cone need not be normal.
(P
1
) If x ≼ y and y ≪ z, then x ≪ z.
(P
2
) If θ ≼ x ≪ c for each c ∈ intP, then x = θ.
(P
3
) If x ≼ λx where x ∈ P and 0 ≤ λ < 1, then x = θ.
(P
4
) Let x
n
→ θ in E and θ ≪ c. Then there exists a positive integer n
0
such that x
n
≪ c for each n > n
0
.
Definition 1.7. (See [6]). Let (X, d) be a cone metric space, P a solid cone and S : X → X. Then
(i) S is said to be sequentially convergent if we have, for every sequence {x
n
}, if {Sx
n
} is convergent, then {x
n
} also is
convergent.
(ii) S is said to be subsequentially convergent if, for every sequence {x
n
} that {Sx
n
} is convergent, {x
n
} has a convergent
subsequence.
(iii) S is said to be continuous if lim
n→∞
x
n
= x implies that lim
n→∞
Sx
n
= Sx, for all {x
n
} in X.
Definition 1.8. (See [6]). Let (X, d) be a cone metric space and T, f : X → X be two mappings. A mapping f is said
to be a T-Hardy-Rogers contraction, if there exist α
i
≥ 0, i = 1, · · · , 5 with α
1
+ α
2
+ α
3
+ α
4
+ α
5
< 1 such that for
all x, y ∈ X ,
d(T f x, T f y) ≼ α
1
d(Tx, Ty) + α
2
d(Tx, T f x) + α
3
d(Ty, T f y) + α
4
d(Tx, T f y) + α
5
d(Ty, T fx). (1)
In Definition 1.8 if one assumes that α
1
= α
4
= α
5
= 0 and α
2
= α
3
, 0 (resp. α
1
= α
2
= α
3
= 0 and α
4
= α
5
, 0),
then one obtains a T-Kannan (resp. T-Chatterjea) contraction.(See [14].)
Rahimi et al. / Filomat 27:5 (2013), 881–888 883
2. Fixed point results
The following is the cone metric space version of a contractive condition of
´
Ciri
´
c for an ordinary metric
space.
Definition 2.1. Let (X, d) be a cone metric space. A mapping f : X → X is said to be a λ-generalized contraction if
and only if for every x, y ∈ X , there exist nonnegative functions q(x, y), r(x, y), s(x, y) and t(x, y) such that
sup
x,y∈X
{q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} ≤ λ < 1
and
d( fx, f y) ≼ q(x, y)d( f x, f y) + r(x, y)d(x, f x) + s(x, y)d(y, f y) + 2t(x, y)[d(x, f y) + d(y, f x)]
holds for all x, y ∈ X .
Theorem 2.2. Suppose that (X, d) is a complete cone metric space, P is a solid cone, and T : X → X is a continuous
and one to one mapping. Moreover, let f and 1 be two mappings of X satisfying
d(T f x, T1y) ≼ q(x, y)d(Tx, Ty) + r(x, y)d(Tx, T f x) + s(x, y)d(Ty, T1y)]
+ t(x, y)[d(Tx, T1y) + d(Ty, T f x)], (2)
for all x, y ∈ X, where q, r, s, and t are nonnegative functions satisfying
sup
x,y∈X
{q(x, y) + r(x, y) + s(x, y) + 2t(x, y)} ≤ λ < 1; (3)
that is, f and 1 are T-contractions. Then
(1) There exists a z
x
∈ X such that lim
n→∞
T f x
2n
= lim
n→∞
T1x
2n+1
= z
x
.
(2) If T is subsequentially convergent, then { fx
2n
} and {1x
2n+1
} have a convergent subsequence.
(3) There exists a unique w
x
∈ X such that f w
x
= 1w
x
= w
x
; that is, f and 1 have a unique common fixed point.
(4) If T is sequentially convergent, then the sequences { fx
2n
} and {1x
2n+1
} converge to w
x
.
Proof. Suppose that x
0
is an arbitrary point of X, and define {x
n
} by
x
1
= fx
0
, x
2
= 1x
1
, · · · , x
2n+1
= fx
2n
, x
2n+2
= 1x
2n+1
f or n = 0, 1, 2, ....
First we shall prove that {Tx
n
} is a Cauchy sequence. Applying the triangle inequality we get
d(Tx
2n+1
, Tx
2n+2
) = d(T f x
2n
, T1x
2n+1
)
≼ q(x
2n
, x
2n+1
)d(Tx
2n
, Tx
2n+1
) + r(x
2n
, x
2n+1
)d(Tx
2n
, T f x
2n
)
+ s(x
2n
, x
2n+1
)d(Tx
2n+1
, T1x
2n+1
) + t(x
2n
, x
2n+1
)[d(Tx
2n
, T1x
2n+1
) + d(Tx
2n+1
, T f x
2n
)]
= q(x
2n
, x
2n+1
)d(Tx
2n
, Tx
2n+1
) + r(x
2n
, x
2n+1
)d(Tx
2n
, Tx
2n+1
)
+ s(x
2n
, x
2n+1
)d(Tx
2n+1
, Tx
2n+2
) + t(x
2n
, x
2n+1
)[d(Tx
2n
, Tx
2n+2
) + d(Tx
2n+1
, Tx
2n+1
)]
≼ (q + r + t)(x
2n
, x
2n+1
)d(Tx
2n
, Tx
2n+1
) + (s + t)(x
2n
, x
2n+1
)d(Tx
2n+1
, Tx
2n+2
).
Consequently
d(Tx
2n+1
, Tx
2n+2
) ≼
q(x
2n
, x
2n+1
) + r(x
2n
, x
2n+1
) + t(x
2n
, x
2n+1
)
1 − s(x
2n
, x
2n+1
) − t(x
2n
, x
2n+1
)
d(Tx
2n
, Tx
2n+1
). (4)
Using (3), we have
q(x, y) + r(x, y) + t(x, y)
1 − s(x, y) − t(x, y)
≤ λ
Rahimi et al. / Filomat 27:5 (2013), 881–888 884
for all x, y ∈ X . Thus, from (4), it follows that
d(Tx
2n+1
, Tx
2n+2
) ≼ λd(Tx
2n
, Tx
2n+1
),
which shows that a generalized contraction is a contraction for certain pairs of points. Following arguments
similar to those given above, we obtain
d(Tx
2n+3
, Tx
2n+2
) ≼ λd(Tx
2n+2
, Tx
2n+1
),
where
q(x, y) + s(x, y) + t(x, y)
1 − r(x, y) − t(x, y)
≤ λ
for all x, y ∈ X . Therefore, for all n,
d(Tx
n
, Tx
n+1
) ≼ λd(Tx
n−1
, Tx
n
) ≼ λ
2
d(Tx
n−2
, Tx
n−1
) ≼ · · · ≼ λ
n
d(Tx
0
, Tx
1
). (5)
Now, for any m > n and λ < 1,
d(Tx
n
, Tx
m
) ≼ d(Tx
n
, Tx
n+1
) + d(Tx
n+1
, Tx
n+2
) + · · · + d(Tx
m−1
, Tx
m
)
≼ (λ
n
+ λ
n+1
+ · · · + λ
m−1
)d(Tx
0
, Tx
1
)
≼
λ
n
1 − λ
d(Tx
0
, Tx
1
) → θ as n → ∞.
From (P
4
) we have (λ
n
/(1 − λ))d(Tx
0
, Tx
1
) ≪ c for all n sufficiently large and θ ≪ c. From (P
1
), we have
d(Tx
n
, Tx
m
) ≪ c. It follows that {Tx
n
} is a Cauchy sequence by Definition 1.5.(ii). Since a cone metric space
X is complete, there exists a z
x
∈ X such that Tx
n
→ z
x
as n → ∞. Thus,
lim
n→∞
T f x
2n
= z
x
, lim
n→∞
T1x
2n+1
= z
x
. (6)
Now, if T is subsequentially convergent, { f x
2n
} (resp. {1x
2n+1
}) has a convergent subsequence. Thus, there
exist w
x
1
∈ X and { f x
2n
i
} (resp. w
x
2
∈ X and {1x
2n
i
+1
}) such that
lim
n→∞
f x
2n
i
= w
x
1
, lim
n→∞
1x
2n
i
+1
= w
x
2
. (7)
Because of the continuity of T, we have
lim
n→∞
T f x
2n
i
= Tw
x
1
, lim
n→∞
T1x
2n
i
+1
= Tw
x
2
. (8)
From (6) and (8) and using the injectivity of T, there exists a w
x
∈ X (set w
x
= w
x
1
= w
x
2
) such that Tw
x
= z
x
.
On the other hand, from (d
3
) and (2) we have
d(Tw
x
, T1w
x
) ≼ d(Tw
x
, T1x
2n
i
+1
) + d(T1x
2n
i
+1
, T f x
2n
i
) + d(T f x
2n
i
, T1w
x
)
≼ d(Tw
x
, Tx
2n
i
+2
) + d(Tx
2n
i
+2
, Tx
2n
i
+1
) + q(x
2n
i
, w
x
)d(Tx
2n
i
, Tw
x
)
+ r(x
2n
i
, w
x
)d(Tx
2n
i
, Tx
2n
i
+1
) + s(x
2n
i
, w
x
)d(Tw
x
, T1w
x
)
+ t(x
2n
i
, w
x
)[d(Tx
2n
i
, T1w
x
) + d(Tw
x
, Tx
2n
i
+1
)]
≼ d(Tw
x
, Tx
2n
i
+2
) + d(Tx
2n
i
+2
, Tx
2n
i
+1
) + (q + t)(x
2n
i
, w
x
)d(Tx
2n
i
, Tw
x
)
+ r(x
2n
i
, w
x
)d(Tx
2n
i
, Tx
2n
i
+1
) + t(x
2n
i
, w
x
)d(Tw
x
, Tx
2n
i
+1
)
+ (s + t)(x
2n
i
, w
x
)d(Tw
x
, T1w
x
). (9)
Now, by (3), (5) and (9) we have
d(Tw
x
, T1w
x
) ≼
1
1 − λ
d(Tw
x
, Tx
2n
i
+2
) +
1
1 − λ
d(Tx
2n
i
+2
, Tx
2n
i
+1
) +
λ
1 − λ
d(Tx
2n
i
, Tw
x
)
+
λ
1 − λ
d(Tx
2n
i
, Tx
2n
i
+1
) +
λ
1 − λ
d(Tw
x
, Tx
2n
i
+1
)
= B
1
d(Tw
x
, Tx
2n
i
+2
) + B
2
λ
2n
i
+1
+ B
3
d(Tx
2n
i
, Tw
x
) + B
4
d(Tw
x
, Tx
2n
i
+1
),
Rahimi et al. / Filomat 27:5 (2013), 881–888 885
where
B
1
=
1
1 − λ
, B
2
=
1
1 − λ
d(Tx
0
, Tx
1
) , B
3
=
λ
1 − λ
, B
4
=
λ
1 − λ
.
Let θ ≪ c. Since λ
2n
i
+1
→ θ and Tx
n
i
→ Tw
x
as i → ∞, there exists a natural number n
0
such that, for each
i ≥ n
0
,
(
by Definition 1.5.(i)
)
we have
d(Tw
x
, Tx
2n
i
+2
) ≪
c
4B
1
, λ
2n
i
≪
c
4B
2
, d(Tx
2n
i
, Tw
x
) ≪
c
4B
3
, d(Tw
x
, Tx
2n
i
+1
) ≪
c
4B
4
.
By (P
1
), we obtain
d(Tw
x
, T1w
x
) ≪
c
4
+
c
4
+
c
4
+
c
4
= c.
Thus, d(Tw
x
, T1w
x
) ≪ c for each c ∈ intP. Using (P
2
), we obtain d(Tw
x
, T1w
x
) = θ; that is, Tw
x
= T1w
x
. Since
T is one to one, 1w
x
= w
x
. Now we shall show that fw
x
= w
x
.
d(T f w
x
, Tw
x
) = d(T f w
x
, T1w
x
)
≼ q(w
x
, w
x
)d(Tw
x
, Tw
x
) + r(w
x
, w
x
)d(Tw
x
, T f w
x
) + s(w
x
, w
x
)d(Tw
x
, T1w
x
)
+ t(w
x
, w
x
)[d(Tw
x
, T1w
x
) + d(Tw
x
, T f w
x
)]
= (r + t)(w
x
, w
x
)d(Tw
x
, T f w
x
) ≼ λd(Tw
x
, T f w
x
).
Using (P
3
), it follows that d(T f w
x
, Tw
x
) = θ, which implies the equality T f w
x
= Tw
x
. Since T is one to one,
then fw
x
= w
x
. Thus fw
x
= 1w
x
= w
x
; that is, w
x
is a common fixed point of f and 1. Now we shall show
that w
x
is the unique common fixed point. Suppose that w
′
x
is another common fixed point of f and 1. Then
d(Tw
x
, Tw
′
x
) = d(T f w
x
, T1w
′
x
)
≼ q(w
x
, w
′
x
)d(Tw
x
, Tw
′
x
) + r(w
x
, w
′
x
)d(Tw
x
, T f w
x
) + s(w
x
, w
′
x
)d(Tw
′
x
, T1w
′
x
)
+ t(w
x
, w
′
x
)[d(Tw
x
, T1w
′
x
) + d(Tw
′
x
, T f w
x
)]
= (q + 2t)(w
x
, w
′
x
)d(Tw
x
, Tw
′
x
) ≼ λd(Tw
x
, Tw
′
x
).
Using (P
3
), it follows that d(Tw
x
, Tw
′
x
) = θ, which implies the equality Tw
x
= Tw
′
x
. Since T is one to one,
w
x
= w
′
x
. Thus f and 1 have a unique common fixed point.
Ultimately, if T is sequentially convergent, then we can replace n by n
i
. Thus we have
lim
n→∞
f x
2n
= w
x
, lim
n→∞
1x
2n+1
= w
x
.
Therefore if T is sequentially convergent, then the sequences { fx
2n
} and {1x
2n+1
} converge to w
x
.
The following results is obtained from Theorem 2.2.
Corollary 2.3. Suppose that (X, d) is a complete cone metric space, P is a solid cone, and T : X → X is a continuous
and one to one mapping. Moreover, let f and 1 be two maps of X satisfying
d(T f x, T1y) ≼ αd(Tx, Ty) + β[d(Tx, T f x) + d(Ty, T1y)] + γ[d(Tx, T1y) + d(Ty, T f x)], (10)
for all x, y ∈ X, where
α, β, γ ≥ 0 and α + 2β + 2γ < 1; (11)
that is, f and 1 are T-contractions. Then
(1) There exists a z
x
∈ X such that lim
n→∞
T f x
2n
= lim
n→∞
T1x
2n+1
= z
x
.
(2) If T is subsequentially convergent, then { fx
2n
} and {1x
2n+1
} have a convergent subsequence.
(3) There exists a unique w
x
∈ X such that f w
x
= 1w
x
= w
x
; that is, f and 1 have a unique common fixed point.
(4) If T is sequentially convergent, then the sequences { fx
2n
} and {1x
2n+1
} converge to w
x
.