Padcharoen et al. Fixed Point Theory and Applications (2016) 2016:39
DOI 10.1186/s13663-016-0525-4
RESEARCH
Open Access
Fixed point and periodic point results for
α-type F-contractions in modular metric
spaces
Anantachai Padcharoen1 , Dhananjay Gopal2 , Parin Chaipunya1 and Poom Kumam1,3*
*
Correspondence:
[email protected]
1
Theoretical and Computational
Science Center (TaCS-Center),
Department of Mathematics,
Faculty of Science, King Mongkut’s
University of Technology Thonburi
(KMUTT), 126 Pracha-Uthit Road,
Bang Mod, Thung Khru, Bangkok,
10140, Thailand
3
Department of Medical Research,
China Medical University Hospital,
China Medical University, Taichung,
40402, Taiwan
Full list of author information is
available at the end of the article
Abstract
Motivated by Gopal et al. (Acta Math. Sci. 36B(3):1-14, 2016). We introduce the notion
of α -type F-contraction in the setting of modular metric spaces which is independent
from one given in (Hussain et al. in Fixed Point Theory Appl. 2015:158, 2015). Further,
we establish some fixed point and periodic point results for such contraction. The
obtained results encompass various generalizations of the Banach contraction
principle and others.
MSC: 47H09; 47H10; 54H25; 37C25
Keywords: fixed point; α -type F-contraction; modular metric space
1 Introduction and preliminaries
The fixed point technique is one of the important tools with respect to studying the existence and uniqueness of the solution of various mathematical methods appearing in the
practical problems. In particular, the Banach contraction principle provides a constructive method of finding a unique solution for models involving various types of differential
and integral equations. This principle is generalized by several authors in various directions; see [–]. Recently, Gopal et al. [] introduced the concept of α-type F-contraction
in metric space by combining the ideas given in [] and obtained some fixed point results.
On the other hand, to deal with the problems of description of superposition operators,
Chistyakov [] introduced the notion of modular metric spaces and gave some fundamental results on this topic, whereas some authors introduced the analog of the Banach
contraction theorem in modular metric spaces and described the important aspects of
applications of fixed point of mappings in modular metric spaces. Some recent results in
this direction can be found in [, –]. In this paper, we introduce the concept of α-type
F-contraction in the setting of modular metric spaces and establish fixed point and periodic point results for such a contraction. Consequently, our results generalize and improve
some known results from the literature.
Consistent with Chistyakov [], we begin with some basic definitions and results which
will be used in the sequel.
Throughout this paper N, R+ , and R will denote the set of natural numbers, positive real
numbers, and real numbers, respectively.
© 2016 Padcharoen et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Padcharoen et al. Fixed Point Theory and Applications (2016) 2016:39
Page 2 of 12
Let X be a nonempty set. Throughout this paper, for a function w : (, ∞) × X × X →
[, ∞), we write
wλ (x, y) = w(λ, x, y)
for all λ >  and x, y ∈ X.
Definition . [] Let X be a nonempty set. A function w : (, ∞) × X × X → [, ∞] is
said to be a metric modular on X if it satisfies, for all x, y, z ∈ X, the following conditions:
(i) wλ (x, y) =  for all λ >  if and only if x = y;
(ii) wλ (x, y) = ωλ (y, x) for all λ > ;
(iii) wλ+μ (x, y) ≤ wλ (x, z) + wμ (z, y) for all λ, μ > .
If instead of (i) we have only the condition (i )
wλ (x, x) = 
for all λ > , x ∈ X,
then w is said to be a pseudomodular (metric) on X. A modular metric w on X is said to
be regular if the following weaker version of (i) is satisfied:
x = y if and only if
wλ (x, y) =  for some λ > .
Definition . [] Let w be a pseudomodular on X. Fix x ∈ X. The set
Xw = Xw (x ) = x ∈ X : wλ (x, x ) →  as λ → ∞
is said to be modular space (around x ).
Definition . Let Xw be a modular metric space.
(i) The sequence (xn )n∈N in Xw is said to be w-convergent to x ∈ Xω if and only if
wλ (xn , x) → , as n → ∞ for some λ > .
(ii) The sequence (xn )n∈N in Xw is said to be w-Cauchy if wλ (xm , xn ) → , as m, n → ∞
for some λ > .
(iii) A subset C of Xw is said to be w-complete if any w-Cauchy sequence in C is a
convergent sequence and its limit is in C.
(iv) A subset C of Xw is said to be w-bounded if for some λ > , we have
δw (C) = sup{wλ (x, y); x, y ∈ C} < ∞.
Next, we denote by F the family of all functions F : R+ → R satisfying the following
conditions:
(F) F is strictly increasing on R+ ,
(F) for every sequence {sn } in R+ , we have limn→∞ sn =  if and only if
limn→∞ F(sn ) = –∞,
(F) there exists a number k ∈ (, ) such that lims→+ sk F(s) = .
Example . The following functions F : R+ → R belongs to F :
(i) F(t) = ln t, with t > ,
(ii) F(t) = ln t + t, with t > .
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Definition . [] A mapping T : X → X is said to be α-admissible if there exists a function α : X × X → R+ such that
x, y ∈ X,
α(x, y) ≥ 
⇒
α(Tx, Ty) ≥ .
Definition . [] Let G denote the set of all functions G : (R+ ) → R+ satisfying
the condition (G) for all t , t , t , t ∈ R+ with t t t t = , there exists τ >  such that
G(t , t , t , t ) = τ .
Example . The following function G : (R+ ) → R belongs to G :
(i) G(t , t , t , t ) = L min(t , t , t , t ) + τ ,
(ii) G(t , t , t , t ) = τ eL min(t ,t ,t ,t ) , where L ∈ R+ .
Definition . [] Let Xω be a modular metric space and T be a self-mapping on Xω . Suppose that α, η : Xω × Xω → [, ∞) are two functions. We say T is an α-η-GF-contraction
if for x, y ∈ Xω with η(x, Tx) ≤ α(x, y), ωλ/l (Tx, Ty) > , and λ, l > , we have
G ωλ/l (x, Tx), ωλ/l(y, Ty), ωλ/l(x, Ty), ωλ/l(y, Tx) + F ωλ/l(Tx, Ty) ≤ F ωλ/l(x, y) ,
where G ∈ G and F ∈ F .
2 Fixed point results for α -type F-contractions
We begin with the following definitions.
Definition . Let (X, w) be a modular metric space. Let C be a nonempty subset of Xw .
A mapping T : C → C is said to be an α-type F-contraction if there exist τ >  and two
functions F ∈ F , α : C × C → (, ∞) such that, for all x, y ∈ C, satisfying w (Tx, Ty) > ,
the following inequality holds:
τ + α(x, y)F w (Tx, Ty) ≤ F w (x, y) .
(.)
Definition . Let (X, w) be a modular metric space. Let C be a nonempty subset of Xw .
A mapping T : C → C is said to be an α-type F-weak contraction if there exist τ >  and
two functions F ∈ F , α : C ×C → (, ∞) such that, for all x, y ∈ C, satisfying w (Tx, Ty) > ,
the following inequality holds:
τ + α(x, y)F w (Tx, Ty)
w (x, Ty) + w (y, Tx)
.
≤ F max w (x, y), w (x, Tx), w (y, Ty),

(.)
Remark . Every α-type F-contraction is an α-type F-weak contraction, but the converse is not necessarily true.
Example . Let Xw = C = [,  ], w = |x – y|, and w = |x – y|. Define T : C → C, α :
C × C → (, ∞), and F : R+ → R by
T(x) =
⎧
⎨,
if x ∈ [,  ],
⎩  , otherwise.

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Then, for x =  and y = , by putting F(t) = ln t with t > , we have

τ + α(, )F wλ T(), T() = τ + α(, ) ln

and
F wλ (, ) = ln().
Clearly, we have
eτ
α(,)



for all τ >  and for all α ∈ (, ∞).
However, since
inf
x∈[,  ],y∈(  ,  ]

w (x, Ty) + w (y, Tx)
max w (x, y), w (x, Tx), w (y, Ty),
= ,


T is an α-type F-weak contraction for the choice
α(x, y) =
⎧
⎨,
if x, y ∈ [,  ] or x, y ∈ (  ,  ],
⎩ log –log  , otherwise,
log –log 
and τ >  such that e–τ =  .
Remark . Definition . (respectively, Definition .) reduces to an F-contraction (respectively, an F-weak contraction) for α(x, y) = .
The next two examples demonstrate that α-type F-contractions (defined above) and
α-η-GF-contractions [] are independent.
Example . Let Xw = C = [, ], w = |x – y|, and wλ = λ |x – y|. Define T : C → C, α :
C × C → (, ∞), and F : R+ → R by
⎧
⎨  , if x ∈ [, ),
T(x) = 
⎩, if x = .
So, define F(t) = ln t with t > . Then T is an α-type F-weak contraction with α(x, y) = 
for all x, y ∈ C and τ >  such that e–τ =  . But T is not an α-η-GF-contraction []. To see
this, consider η : C × C → [, ∞) such that
⎧
⎨, if x =  ,

η(x, Tx) =
⎩, otherwise
and
G(t , t , t , t , t ) = L min{t , t , t , t , t } + τ .
Padcharoen et al. Fixed Point Theory and Applications (2016) 2016:39
Then, for x =
η
 
,
 


Page 5 of 12
and y = , we have
=≤=α

,  = α(x, y),

wλ (Tx, Ty) =
 
· > ,
 λ
and
G wλ (x, Tx), wλ (y, Ty), wλ (x, Ty), wλ (y, Tx)
= L min wλ (x, Tx), wλ (y, Ty), wλ (x, Ty), wλ (y, Tx) + τ = τ .
Consequently, we have
G wλ (x, Tx), wλ (y, Ty), wλ (x, Ty), wλ (y, Tx) + F wλ (Tx, Ty)
 
 
= τ + ln
·
ln
·
= F wλ (x, y) ,
 λ
 λ
and thus, T is not an α-η-GF-contraction.
Example . Let Xw = C = [, ], w = |x – y|, and wλ = λ |x – y|. Define T : C → C, α, η :
C × C → [, ∞), G : (R+ ) → R+ by
⎧
⎨, if x is rational,
T(x) =
⎩, if x is irrational,
, if x and y both are rational or irrational, α(x, y) =
α(x, y) = x + y and η(x, y) = x+y

 and η(x, y) =  if x is irrational and y is rational (and vice versa), G(t , t , t , t ) =
L min{t , t , t , t } + τ (τ > ), and F(t) = ln t. Then T is an α-η-GF-contraction. But it is
not an α-type F-weak contraction. To see this, consider x =  and y is any irrational.
The motivation of the following definition is in the last step of the proof of the Cauchy
sequence in our theorems.
Definition . Let (X, w) be a modular metric space. Then we will say that w satisfies the M -condition if it is the case that limm,n→∞ wλ (xn , xm ) = , for λ = m implies
limm,n→∞ wλ (xn , xm ) =  (m, n ∈ N, m ≥ n), for some λ > .
Now, we are ready to state our first theorem which generalizes the main theorem of
Gopal et al. [] for modular metric spaces.
Theorem . Let (X, w) be a modular metric space. Assume that w is regular and satisfies
the M -condition. Let C be a nonempty subset of Xw . Assume that C is w-complete and
w-bounded, i.e., δw (C) = sup{w (x, y) : x, y ∈ C} < ∞. Let T : C → C be an α-type F-weak
contraction satisfying the following conditions:
(i) T is α-admissible,
(ii) there exists x ∈ C such that α(x , Tx ) ≥ ,
(iii) T is continuous.
Padcharoen et al. Fixed Point Theory and Applications (2016) 2016:39
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Then T has a fixed point x∗ ∈ C and for every x ∈ C the sequence {T n x }n∈N is convergent
to x∗ .
Proof Let x ∈ C such that α(x , Tx ) ≥  and define a sequence {xn } in C by xn+ = Txn for
all n ∈ N.
Obviously, if there exists n ∈ N such that xn + = xn , then Txn = xn and the proof is
finished. Hence, we suppose that xn+ = xn for every n ∈ N. Now from conditions (ii) and
(i), we have
α(x , x ) = α(x , Tx ) ≥ 
⇒
α(Tx , Tx ) = α(x , x ) ≥ .
By induction we have
α(xn , xn+ ) ≥ .
(.)
Since T is an α-type F-weak contraction, for every n ∈ N, we have
F w (xn+ , xn ) = F w (Txn , Txn– )
≤ α(xn , xn– )F w (Txn , Txn– ) ,
(.)
so
τ + F w (xn+ , xn ) ≤ τ + α(xn , xn+ )F w (Txn , Txn– )
≤ F max w (xn , xn– ), w (xn , Txn ), w (xn– , Txn– ),
w (xn , Txn– ) + w (xn– , Txn )

w (xn– , xn+ )
= F max w (xn– , xn ), w (xn , xn+ ),

w (xn– , xn ) + w (xn , xn+ )
≤ F max w (xn– , xn ), w (xn , xn+ ),

= F max w (xn– , xn ), w (xn , xn+ ) .
(.)
If there exists n ∈ N such that max{w (xn– , xn ), w (xn , xn+ )} = w (xn , xn+ ), then, from (.),
we have
F w (xn , xn+ ) ≤ F w (xn , xn+ ) – τ ,
a contradiction. Therefore max{w (xn– , xn ), w (xn , xn+ )} = w (xn– , xn ), for all n ∈ N.
Hence, from (.), we have
F w (xn , xn+ ) ≤ F w (xn– , xn ) – τ ,
This implies that
F w (xn , xn+ ) ≤ F w (x , x ) – nτ
for all n ∈ N.
(.)
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Taking the limit as n → ∞ in (.) and since C is w-bounded, we have
F w (xn , xn+ ) = –∞;
from (F), we obtain
lim w (xn , xn+ ) = .
(.)
n→∞
From (F), there exists k ∈ (, ) such that
lim
n→∞
k w (xn+ , xn ) F w (xn+ , xn ) = .
(.)
From (.), for all n ∈ N, we deduce that
w (xn+ , xn )
k k
F w (xn+ , xn ) – F w (x , x ) ≤ – w (xn+ , xn ) nτ ≤ .
(.)
By using (.), (.), and taking the limit as n → ∞ in (.), we have
k lim n w (xn+ , xn ) = .
n→∞
Then there exists n ∈ N such that n(w (xn+ , xn ))k ≤  for all n ≥ n , that is,
w (xn , xn+ ) ≤

n/k
for all n ≥ n , for all λ > .
For all m > n > n , we have
wm (xn , xm ) ≤ w (xn , xn+ ) + w (xn+ , xn+ ) + · · · + w (xm , xm+ )
Since the series
lim
m,n→∞
≤



+
+ · · · + /k
/k
/k
n
(n + )
m
<
∞

.
/k
i
i=n
∞

i=n i/k
is convergent, this implies
wm (xn , xm ) = .
Since w satisfies the M -condition. Hence, we have
lim w (xn , xm ) = .
m,n→∞
This shows that {xn } is a w-Cauchy sequence. Since C is w-complete, there exists x∗ ∈ C
such that xn → x∗ as n → ∞. By the continuity of T and since w is regular, we have
w x∗ , Tx∗ = lim w (xn , Txn ) = lim w (xn , xn+ ) = .
n→∞
Hence, x∗ is a fixed point of T.
n→∞
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Theorem . Let (X, w) be a modular metric space. Assume that w is regular and satisfies
the M -condition. Let C be a nonempty subset of Xw . Assume that C is w-complete modular
metric space and w-bounded, i.e., δw (C) = sup{w (x, y) : x, y ∈ C} < ∞. Let T : C → C be an
α-type F-weak contraction satisfying the following conditions:
(i) there exists x ∈ C such that α(x , Tx ) ≥ ,
(ii) T is α-admissible,
(iii) if {xn } is a sequence in Xw such that α(xn , xn+ ) ≥  for all n ∈ N and xn → x as
n → ∞, then α(xn , x) ≥  for all n ∈ N,
(iv) F is continuous.
Then T has a fixed point x∗ ∈ C and for every x ∈ C the sequence {T n x }n∈N is convergent
to x∗ .
Proof Let x ∈ C be such that α(x , Tx ) ≥  and let xn = Txn– for all n ∈ N Following the
proof of Theorem ., we see that {xn } is a w-Cauchy sequence in the w-complete modular
metric space. Then there exists x∗ ∈ C such that xn → x∗ as n → ∞. From (.) and the
hypothesis (iii), we have
α xn , x∗ ≥ 
for all n ∈ N.
Case I: Suppose, for every n ∈ N, there exists in ∈ N such that xin + = Tx∗ and in > in– .
Then we have
x∗ = lim xin + = lim Tx∗in = Tx∗ ,
n→∞
(.)
n→∞
that is, x∗ is a fixed point of T.
Case II: Assume there exists n ∈ N such that xn+ = Tx∗ for all n ≥ n , i.e., w (Txn , Tx∗ ) >
 for all n ≥ n . It follows from (.) and (F) that
τ + F w xn+ , Tx∗ = τ + F w Txn , Tx∗ ≤ τ + α xn , x∗ F w Txn , Tx∗
≤ F max w xn , x∗ , w (xn , Txn ), w x∗ , Tx∗ ,
w (xn , Tx∗ ) + w (x∗ , Txn )

≤ F max w xn , x∗ , w (xn , xn+ ), w x∗ , Tx∗ ,
w (xn , x∗ ) + w (x∗ , Tx∗ ) + w (x∗ , xn ) + w (xn , xn+ )

If w (x∗ , Tx∗ ) >  and by the fact that
lim w xn , x∗ = lim w xn+ , x∗ = ,
n→∞
n→∞
there exists n ∈ N such that, for all n ≥ n , we have
max w xn , x∗ , w (xn , xn+ ), w x∗ , Tx∗ ,
w (xn , x∗ ) + w (x∗ , Tx∗ ) + w (x∗ , xn ) + w (xn , xn+ )
= w x∗ , Tx∗ .

.
(.)
Padcharoen et al. Fixed Point Theory and Applications (2016) 2016:39
Page 9 of 12
From (.), we obtain
τ + F w xn+ , Tx∗ ≤ F w x∗ , Tx∗
(.)
for all n ≥ max{n , n }. Since F is continuous, taking the limit as n → ∞ in (.), we have
τ + F w x∗ , Tx∗ ≤ F w x∗ , Tx∗ ,
a contradiction. Thus w (x∗ , Tx∗ ) =  and hence x∗ is a fixed point of T.
Indeed, uniqueness of the fixed point, we will consider the following hypothesis.
(H): for all x, y ∈ Fix(T), α(x, y) ≥ .
Theorem . Adding condition (H) to the hypotheses of Theorem . (respectively, Theorem .) the uniqueness of the fixed point is obtained.
Proof Assume that y∗ ∈ C is an another fixed point of T, such that w (x, y) < ∞ and
w (Tx∗ , Ty∗ ) = w (x∗ , y∗ ) > . Then we have
τ + F w x∗ , y∗ ≤ τ + F w Tx∗ , Ty∗
≤ τ + α x∗ , y∗ F w Tx∗ , Ty∗
≤ F w x∗ , y∗ ,
a contradiction. This implies that x∗ = y∗ .
Example . satisfies all the hypotheses of Theorem ., hence T has a unique fixed
point x =  .
The following result improves the main theorem of the F-contraction for a modular
metric space.
Corollary . Let (X, w) be a modular metric space. Assume that w is regular and satisfies the M -condition. Let C be a nonempty subset of Xw . Assume that C is w-complete
and w-bounded, i.e., δw (C) = sup{w (x, y) : x, y ∈ C} < ∞. Let T : C → C be an α-type
F-contraction satisfying the hypotheses of Theorem ., then T has unique fixed point.
From Example .(i) and Corollary ., we obtain the following result.
Theorem . Let (X, w) be a modular metric space. Assume that w is regular. Let C
be a nonempty subset of Xw . Assume that C is w-complete and w-bounded, i.e., δw (C) =
sup{w (x, y); x, y ∈ C} < ∞. Let T : C → C be a contraction. Then T has a unique fixed
point x . Moreover, the orbit {T n (x)} w-converges to x for x ∈ C.
3 Periodic point results
In this section, we prove some periodic point results for self-mappings on a modular metric space. In the sequel, we need the following definition.
Padcharoen et al. Fixed Point Theory and Applications (2016) 2016:39
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Definition . [] A mapping T : C → C is said to have the property (P) if Fix(T n ) =
Fix(T) for every n ∈ N, where Fix(T) := {x ∈ Xw : Tx = x}.
Theorem . Let (X, w) be a modular metric space. Assume that w is regular and satisfies
the M -condition. Let C be a nonempty subset of Xw . Assume that C is w-complete and wbounded, i.e., δw (C) = sup{w (x, y) : x, y ∈ C} < ∞. Let C be a w-complete and w-bounded
subset of X. Let T : C → C be a mapping satisfying the following conditions:
(i) there exists τ >  and two functions F ∈ F and α : C × C → (, ∞) such that
τ + α(x, Tx)F w Tx, T  x ≤ F w (x, Tx)
(ii)
(iii)
(iv)
(v)
holds for all x ∈ C with w (Tx, T  x) > ,
there exists x ∈ C such that α(x , Tx ) ≥ ,
T is α-admissible,
if {xn } is a sequence in C such that α(xn , xn+ ) ≤  for all n ∈ N and w (xn , x) → , as
n → ∞, then w (Txn , Tx) →  as n → ∞,
if z ∈ Fix(T n ) and z ∈/ Fix(T), then α(T n– z, T n z) ≥ . Then T has the property (P).
Proof Let x ∈ C be such that α(x , Tx ) ≥ . Now, for x ∈ C, we define the sequence {xn }
by the rule xn = T n xn = Txn– . By (iii), we have α(x , x ) = α(Tx , Tx ) ≥  and by induction
we write
α(xn , xn+ ) ≥  for all n ∈ N.
(.)
If there exists n ∈ N such that xn = xn + = Txn , then xn is a fixed point of T and the
proof is finished. Thus, we assume xn = xn+ or w (Txn– , T  xn– ) >  for all n ∈ N. From
(.) and (i), we have
τ + F w (xn , xn+ ) = τ + F w Txn– , T  xn–
≤ τ + α(xn– , Txn– )F w Txn– , T  xn–
≤ F w (xn– , Txn– )
or equivalently
F w (xn , xn+ ) ≤ F w (xn– , xn ) – τ .
By using a similar reasoning to the proof of Theorem ., we see that the sequence {xn } is
a w-Cauchy sequence and thus, by w-completeness, there exists x∗ ∈ Xw such that xn → x∗
as n → ∞.
By (iv), we have w (xn+ , Tx∗ ) = w (Txn , Tx∗ ) →  as n → ∞, that is, x∗ = Tx∗ . Hence, T
has a fixed point and Fix(T n ) = Fix(T) is true for n = . Let n >  and assume, by contradiction, that z ∈ Fix(T n ) and z ∈/ Fix(T), such that w (z, Tz) > . Now, applying (v) and (i),
we have
F w (z, Tz) ≤ F w T T n– z , T  T n– z
≤ α T n– z, T n z F w T T n– z , T  T n– z
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and
τ + F w (z, Tz) ≤ τ + F w T T n– z , T  T n– z
≤ F w T n– z, T n z .
Consequently, we have
F w (z, Tz) ≤ F w T n– z, T n z – τ
≤ F w T n– z, T n– z – τ
..
.
≤ F w (z, Tz) – nτ .
By taking the limit as n → ∞ in the above inequality, we have F(w (z, Tz)) = –∞, which
is a contradiction until w (z, Tz) =  and by the regularity of w, we set z = Tz. Hence,
Fix(T n ) = Fix(T) for all n ∈ N.
Taking α(x, y) =  for all x, y ∈ C in Theorem ., we get the following result, which is a
generalization of Theorem  of Abbas et al. [] in the setting of a modular metric.
Corollary . Let (X, w) be a complete modular metric space. Assume that w is regular and
satisfies the M -condition. Let C be a nonempty subset of Xw . Assume that C is w-complete
and w-bounded, i.e., δw (C) = sup{w (x, y) : x, y ∈ C} < ∞. Let T : C → C be a continuous
mapping satisfying
τ + F w Tx, T  x ≤ F w (x, Tx)
for some τ >  and for all x ∈ Xw such that w (Tx, T  x) > . Then T has property (P).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1
Theoretical and Computational Science Center (TaCS-Center), Department of Mathematics, Faculty of Science, King
Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140,
Thailand. 2 Department of Applied Mathematics and Humanities, S.V. National Institute of Technology, Surat, Gujarat
395007, India. 3 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung,
40402, Taiwan.
Acknowledgements
The first author thanks for the support of Petchra Pra Jom Klao Doctoral Scholarship Academic. This work was completed
while the second author (Dr. Gopal) was visiting Theoretical and Computational Science Center (TaCS), Science
Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand,
during 15 October-8 November, 2015. He thanks Professor Poom Kumam and the University for their hospitality and
support.
Received: 4 November 2015 Accepted: 8 March 2016
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