1. No category

Piri and Kumam Fixed Point Theory and Applications 2014, 2014:210
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RESEARCH
Open Access
Some fixed point theorems concerning
F-contraction in complete metric spaces
Hossein Piri1 and Poom Kumam2*
*
Correspondence:
[email protected];
[email protected]
2
Department of Mathematics,
Faculty of Science, King Mongkutís
University of Technology Thonburi
(KMUTT), 126 Bangmod, Thrung
Khru, Bangkok, 10140, Thailand
Full list of author information is
available at the end of the article
Abstract
In this paper, we extend the result of Wardowski (Fixed Point Theory Appl. 2012:94,
2012) by applying some weaker conditions on the self map of a complete metric
space and on the mapping F, concerning the contractions defined by Wardowski.
With these weaker conditions, we prove a fixed point result for F-Suzuki contractions
which generalizes the result of Wardowski.
MSC: 74H10; 54H25
Keywords: fixed point; metric space; F-contraction
1 Introduction and preliminaries
Throughout this article, we denote by R the set of all real numbers, by R+ the set of all
positive real numbers, and by N the set of all natural numbers.
In , Polish mathematician Banach [] proved a very important result regarding a
contraction mapping, known as the Banach contraction principle. It is one of the fundamental results in fixed point theory. Due to its importance and simplicity, several authors
have obtained many interesting extensions and generalizations of the Banach contraction
principle (see [–] and references therein). Subsequently, in , M Edelstein proved
the following version of the Banach contraction principle.
Theorem . [] Let (X, d) be a compact metric space and let T : X → X be a selfmapping. Assume that d(Tx, Ty) < d(x, y) holds for all x, y ∈ X with x = y. Then T has a
unique fixed point in X.
In , Suzuki [] proved generalized versions of Edelstein’s results in compact metric
space as follows.
Theorem . [] Let (X, d) be a compact metric space and let T : X → X be a self-mapping.
Assume that for all x, y ∈ X with x = y,

d(x, Tx) < d(x, y)

⇒
d(Tx, Ty) < d(x, y).
Then T has a unique fixed point in X.
In , Wardowski [] introduce a new type of contractions called F-contraction and
prove a new fixed point theorem concerning F-contractions. In this way, Wardowski []
©2014 Piri and Kumam; licensee Springer. 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.
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generalized the Banach contraction principle in a different manner from the well-known
results from the literature. Wardowski defined the F-contraction as follows.
Definition . Let (X, d) be a metric space. A mapping T : X → X is said to be an Fcontraction if there exists τ >  such that
∀x, y ∈ X,
d(Tx, Ty) >  ⇒ τ + F d(Tx, Ty) ≤ F d(x, y) ,
()
where F : R+ → R is a mapping satisfying the following conditions:
(F) F is strictly increasing, i.e. for all x, y ∈ R+ such that x < y, F(x) < F(y);
(F) For each sequence {αn }∞
n= of positive numbers, limn→∞ αn =  if and only if
limn→∞ F(αn ) = –∞;
(F) There exists k ∈ (, ) such that limα→+ α k F(α) = .
We denote by F , the set of all functions satisfying the conditions (F)-(F). For examples
of the function F the reader is referred to [] and [].
Remark . From (F) and () it is easy to conclude that every F-contraction is necessarily
continuous.
Wardowski [] stated a modified version of the Banach contraction principle as follows.
Theorem . [] Let (X, d) be a complete metric space and let T : X → X be an Fcontraction. Then T has a unique fixed point x∗ ∈ X and for every x ∈ X the sequence
{T n x}n∈N converges to x∗ .
Very recently, Secelean [] proved the following lemma.
Lemma . [] Let F : R+ → R be an increasing mapping and {αn }∞
n= be a sequence of
positive real numbers. Then the following assertions hold:
(a) if limn→∞ F(αn ) = –∞, then limn→∞ αn = ;
(b) if inf F = –∞, and limn→∞ αn = , then limn→∞ F(αn ) = –∞.
By proving Lemma ., Secelean showed that the condition (F) in Definition . can be
replaced by an equivalent but a more simple condition,
(F
) inf F = –∞
or, also, by
(F
) there exists a sequence {αn }∞
n= of positive real numbers such that limn→∞ F(αn ) =
–∞.
Remark . Define FB : R+ → R by FB (α) = ln α, then FB ∈ F . Note that with F = FB the
F-contraction reduces to a Banach contraction. Therefore, the Banach contractions are
a particular case of F-contractions. Meanwhile there exist F-contractions which are not
Banach contractions (see [, ]).
In this paper, we use the following condition instead of the condition (F) in Definition .:
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(F
) F is continuous on (, ∞).
We denote by F the set of all functions satisfying the conditions (F), (F
), and (F
).
Example . Let F (α) =
F ∈ F.
–
,
α
F (α) =
–
α
+ α, F (α) =

,
–eα
F (α) =

.
eα –e–α
Then F , F , F ,
Remark . Note that the conditions (F) and (F
) are independent of each other. Indeed,
for p ≥ , F(α) = α–p satisfies the conditions (F) and (F) but it does not satisfy (F), while it
–
satisfies the condition (F
). Therefore, F F . Again, for a > , t ∈ (, /a), F(α) = (α+[α])
t,
where [α] denotes the integral part of α, satisfies the conditions (F) and (F) but it does
not satisfy (F
), while it satisfies the condition (F) for any k ∈ (/a, ). Therefore, F F.
Also, if we take F(α) = ln α, then F ∈ F and F ∈ F. Therefore, F ∩ F = ∅.
In view of Remark ., it is meaningful to consider the result of Wardowski [] with the
mappings F ∈ F instead F ∈ F . Also, we define the F-Suzuki contraction as follows and
we give a new version of Theorem ..
Definition . Let (X, d) be a metric space. A mapping T : X → X is said to be an FSuzuki contraction if there exists τ >  such that for all x, y ∈ X with Tx = Ty

d(x, Tx) < d(x, y)

⇒
τ + F d(Tx, Ty) ≤ F d(x, y) ,
where F ∈ F.
2 Main results
Theorem . Let T be a self-mapping of a complete metric space X into itself. Suppose
F ∈ F and there exists τ >  such that
∀x, y ∈ X,
d(Tx, Ty) >  ⇒ τ + F d(Tx, Ty) ≤ F d(x, y) .
Then T has a unique fixed point x∗ ∈ X and for every x ∈ X the sequence {T n x }∞
n= con∗
verges to x .
Proof Choose x ∈ X and define a sequence {xn }∞
n= by
x = Tx ,
x = Tx = T  x ,
...,
xn+ = Txn = T n+ x ,
∀n ∈ N.
()
If there exists n ∈ N such that d(xn , Txn ) = , the proof is complete. So, we assume that
 < d(xn , Txn ) = d(Txn– , Txn ),
∀n ∈ N.
For any n ∈ N we have
τ + F d(Txn– , Txn ) ≤ F d(xn– , xn ) ,
i.e.,
F d(Txn– , Txn ) ≤ F d(xn– , xn ) – τ .
()
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Repeating this process, we get
F d(Txn– , Txn ) ≤ F d(xn– , xn ) – τ
= F d(Txn– , Txn– ) – τ
≤ F d(xn– , xn– ) – τ
= F d(Txn– , Txn– ) – τ
≤ F d(xn– , xn– ) – τ
..
.
≤ F d(x , x ) – nτ .
()
From (), we obtain limn→∞ F(d(Txn– , Txn )) = –∞, which together with (F
) and
Lemma . gives limn→∞ d(Txn– , Txn ) = , i.e.,
lim d(xn , Txn ) = .
()
n→∞
Now, we claim that {xn }∞
n= is a Cauchy sequence. Arguing by contradiction, we assume
∞
that there exist >  and sequences {p(n)}∞
n= and {q(n)}n= of natural numbers such that
d(xp(n) , xq(n) ) ≥ ,
p(n) > q(n) > n,
d(xp(n)– , xq(n) ) < ,
∀n ∈ N.
()
So, we have
≤ d(xp(n) , xq(n) ) ≤ d(xp(n) , xp(n)– ) + d(xp(n)– , xq(n) )
≤ d(xp(n) , xp(n)– ) + = d(xp(n)– , Txp(n)– ) + .
It follows from () and the above inequality that
lim d(xp(n) , xq(n) ) = .
()
n→∞
On the other hand, from () there exists N ∈ N, such that
d(xp(n) , Txp(n) ) <

and
d(xq(n) , Txq(n) ) <
,

∀n ≥ N.
()
Next, we claim that
d(Txp(n) , Txq(n) ) = d(xp(n)+ , xq(n)+ ) > ,
∀n ≥ N.
()
Arguing by contradiction, there exists m ≥ N such that
d(xp(m)+ , xq(m)+ ) = .
()
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It follows from (), (), and () that
≤ d(xp(m) , xq(m) ) ≤ d(xp(m) , xp(m)+ ) + d(xp(m)+ , xq(m) )
≤ d(xp(m) , xp(m)+ ) + d(xp(m)+ , xq(m)+ ) + d(xq(m)+ , xq(m) )
= d(xp(m) , Txp(m) ) + d(xp(m)+ , xq(m)+ ) + d(xq(m) , Txq(m) )
<
++ = .

 
This contradiction establishes the relation (). Therefore, it follows from () and the assumption of the theorem that
τ + F d(Txp(n) , Txq(n) ) ≤ F d(xp(n) , xq(n) ) ,
∀n ≥ N.
()
From (F
), (), and (), we get τ + F() ≤ F(). This contradiction shows that {xn }∞
n= is
∞
a Cauchy sequence. By completeness of (X, d), {xn }n= converges to some point x in X.
Finally, the continuity of T yields
d(Tx, x) = lim d(Txn , xn ) = lim d(xn+ , xn ) = d x∗ , x∗ = .
n→∞
n→∞
Now, let us to show that T has at most one fixed point. Indeed, if x, y ∈ X be two distinct
fixed points of T, that is, Tx = x = y = Ty. Therefore,
d(Tx, Ty) = d(x, y) > ,
then we get
F d(x, y) = F d(Tx, Ty) < τ + F d(Tx, Ty) ≤ F d(x, y) ,
which is a contradiction. Therefore, the fixed point is unique.
Theorem . Let (X, d) be a complete metric space and T : X → X be an F-Suzuki contraction. Then T has a unique fixed point x∗ ∈ X and for every x ∈ X the sequence {T n x }∞
n=
converges to x∗ .
Proof Choose x ∈ X and define a sequence {xn }∞
n= by
x = Tx ,
x = Tx = T  x ,
...,
xn+ = Txn = T n+ x ,
∀n ∈ N.
()
If there exists n ∈ N such that d(xn , Txn ) = , the proof is complete. So, we assume that
 < d(xn , Txn ),
∀n ∈ N.
Therefore,

d(xn , Txn ) < d(xn , Txn ),

∀n ∈ N.
()
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For any n ∈ N we have
τ + F d Txn , T  xn ≤ F d(xn , Txn ) ,
i.e.,
F d(xn+ , Txn+ ) ≤ F d(xn , Txn ) – τ .
Repeating this process, we get
F d(xn , Txn ) ≤ F d(xn– , Txn– ) – τ
≤ F d(xn– , Txn– ) – τ
≤ F d(xn– , Txn– ) – τ
..
.
≤ F d(x , Tx ) – nτ .
()
From (), we obtain limm→∞ F(d(xn , Txn )) = –∞, which together with (F
) and Lemma . gives
lim d(xn , Txn ) = .
()
m→∞
Now, we claim that {xn }∞
n= is a Cauchy sequence. Arguing by contradiction, we assume
∞
that there exist >  and sequences {p(n)}∞
n= and {q(n)}n= of natural numbers such that
p(n) > q(n) > n,
d(xp(n) , xq(n) ) ≥ ,
d(xp(n)– , xq(n) ) < ,
∀n ∈ N.
()
So, we have
≤ d(xp(n) , xq(n) ) ≤ d(xp(n) , xp(n)– ) + d(xp(n)– , xq(n) )
≤ d(xp(n) , xp(n)– ) + = d(xp(n)– , Txp(n)– ) + .
It follows from () and the above inequality that
lim d(xp(n) , xq(n) ) = .
()
n→∞
From () and (), we can choose a positive integer N ∈ N such that


d(xp(n) , Txp(n) ) < < d(xp(n) , xq(n) ),


∀n ≥ N.
So, from the assumption of the theorem, we get
τ + F d(Txp(n) , Txq(n) ) ≤ F d(xp(n) , xq(n) ) ,
∀n ≥ N.
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It follows from () that
τ + F d(xp(n)+ , Txq(n)+ ) ≤ F d(xp(n) , xq(n) ) ,
∀n ≥ N.
()
From (F
), (), and (), we get τ + F() ≤ F(). This contradiction shows that {xn }∞
n= is
∗
converges
to
some
point
x
in
X.
a Cauchy sequence. By completeness of (X, d), {xn }∞
n=
Therefore,
lim d xn , x∗ = .
()
n→∞
Now, we claim that

d(xn , Txn ) < d xn , x∗ or

 d Txn , T  xn < d Txn , x∗ ,

∀n ∈ N.
()
Again, assume that there exists m ∈ N such that

d(xm , Txm ) ≥ d xm , x∗ and

 d Txm , T  xm ≥ d Txm , x∗ .

()
Therefore,
d xm , x∗ ≤ d(xm , Txm ) ≤ d xm , x∗ + d x∗ , Txm ,
which implies that
d xm , x∗ ≤ d x∗ , Txm .
()
It follows from () and () that
 d xm , x∗ ≤ d x∗ , Txm ≤ d Txm , T  xm .

()
Since  d(xm , Txm ) < d(xm , Txm ), by the assumption of the theorem, we get
τ + F d Txm , T  xm ≤ F d(xm , Txm ) .
Since τ > , this implies that
F d Txm , T  xm < F d(xm , Txm ) .
So, from (F), we get
d Txm , T  xm < d(xm , Txm ).
It follows from (), (), and () that
d Txm , T  xm < d(xm , Txm )
≤ d xm , x∗ + d x∗ , Txm
()
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  ≤ d Txm , T  xm + d Txm , T  xm


= d Txn , T  xn .
This is a contradiction. Hence, () holds. So, from (), for every n ∈ N, either
τ + F d Txn , Tx∗ ≤ F d xn , x∗ ,
or
τ + F d T  xn , Tx∗ ≤ F d Txn , x∗ = F d xn+ , x∗
holds. In the first case, from (), (F
), and Lemma ., we obtain
lim F d Txn , Tx∗ = –∞.
n→∞
It follows from (F
) and Lemma . that limn→∞ d(Txn , Tx∗ ) = . Therefore,
d x∗ , Tx∗ = lim d xn+ , Tx∗ = lim d Txn , Tx∗ = .
n→∞
n→∞
Also, in the second case, from (), (F
), and Lemma ., we obtain
lim F d T  xn , Tx∗ = –∞.
n→∞
It follows from (F
) and Lemma . that limn→∞ d(Txn , Tx∗ ) = . Therefore,
d x∗ , Tx∗ = lim d xn+ , Tx∗ = lim d T  xn , Tx∗ = .
n→∞
n→∞
Hence, x∗ is a fixed point of T. Now let us show that T has at most one fixed point. Indeed,
if x∗ , y∗ ∈ X are two distinct fixed points of T, that is, Tx∗ = x∗ = y∗ = Ty∗ , then d(x∗ , y∗ ) > .
So, we have  =  d(x∗ , Tx∗ ) < d(x∗ , y∗ ) and from the assumption of the theorem, we obtain
F d x∗ , y∗ = F d Tx∗ , Ty∗ < τ + F d Tx∗ , Ty∗ ≤ F d x∗ , y∗ ,
which is a contradiction. Thus, the fixed point is unique.
Example . Consider the sequence {Sn }n∈N as follows:
S =  × ,
S =  ×  +  × ,
Sn =  ×  +  ×  + · · · + n(n + ) =
...,
n(n + )(n + )
,

....
Let X = {Sn : n ∈ N} and d(x, y) = |x – y|. Then (X, d) is complete metric space. Define the
mapping T : X → X by T(S ) = S and T(Sn ) = Sn– for every n > . Since
lim
n→∞
d(T(Sn ), T(S ))
Sn– –  (n – )n(n + ) – 
= lim
=
= ,
n→∞ Sn – 
d(Sn , S )
n(n + )(n + ) – 
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T is not a Banach contraction and a Suzuki contraction. On the other hand taking F(α) =
–
+ α ∈ F, we obtain the result that T is an F-contraction with τ = . To see this, let us
α
consider the following calculation. First observe that

d(Sn , TSn ) < d(Sn , Sm )

⇔
( = n < m) ∨ ( ≤ m < n) ∨ ( < n < m) .
For  = n < m, we have
T(Sm ) – T(S ) = |Sm– – S | =  ×  +  ×  + · · · + (m – )m,
|Sm – S | =  ×  +  ×  + · · · + m(m + ).
Since m >  and
–
×+×+···+(m–)m
<
–
,
×+×+···+m(m+)
()
we have

+  ×  +  ×  + · · · + (m – )m
 ×  +  ×  + · · · + (m – )m

+  ×  +  ×  + · · · + (m – )m
<–
 ×  +  ×  + · · · + m(m + )

+  ×  +  ×  + · · · + (m – )m + m(m + )
≤–
 +  + ··· + m

=–
+  ×  +  ×  + · · · + (m – )m + m(m + ) .
 +  + ··· + m
–
So, from (), we get
–


+ T(Sm ) – T(S ) < –
+ |Sm – S |.
|T(Sm ) – T(S )|
|Sm – S |
For  ≤ m < n, similar to  = n < m, we have
–


+ T(Sm ) – T(S ) < –
+ |Sm – S |.
|T(Sm ) – T(S )|
|Sm – S |
For  < n < m, we have
T(Sm ) – T(Sn ) = n(n + ) + (n + )(n + ) + · · · + (m – )m,
|Sm – Sn | = (n + )(n + ) + (n + )(n + ) + · · · + m(m + ).
()
Since m > n > , we have
(m + )m ≥ (n + )(n + ) = n(n + ) + (n + ) ≥ n(n + ) + .
We know that
–
–
n(n+)+(n+)(n+)+···+(m–)m
<
–
.
(n+)(n+)+(n+)(n+)+···+m(m+)

n(n + ) + (n + )(n + ) + · · · + (m – )m
+ n(n + ) + (n + )(n + ) + · · · + (m – )m
Therefore
Piri and Kumam Fixed Point Theory and Applications 2014, 2014:210
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Page 10 of 11

(n + )(n + ) + (n + )(n + ) + · · · + m(m + )
+ n(n + ) + (n + )(n + ) + · · · + (m – )m
<–

(n + )(n + ) + (n + )(n + ) · · · + m(m + )
+  + n(n + ) + (n + )(n + ) + · · · + (m – )m
=–

(n + )(n + ) + (n + )(n + ) + · · · + m(m + )
+ m(m + ) + (n + )(n + ) + · · · + (m – )m
≤–

(n + )(n + ) + (n + )(n + ) + · · · + m(m + )
+ (n + )(n + ) + · · · + (m – )m .
=–
So from (), we get
–


+ T(Sm ) – T(Sn ) < –
+ |Sm – Sn |.
|T(Sm ) – T(Sn )|
|Sm – Sn |
Therefore τ + F(d(T(Sm ), T(Sn ))) ≤ d(Sm , Sn ) for all m, n ∈ N. Hence T is an F-contraction
and T(S ) = S .
–
+ α, and F (α) = √α+[α]
+ α in the above
For F (α) = ln(α), F (α) = ln(α) + α, F (α) = –
α
example, we compare the rate of convergence of the Banach contraction (F -contraction)
and F-contractions for F ∈ F ∩ F, F ∈ (F – F ), and F ∈ (F – F) in Table .
Table 1 The generated iterations start from a point x0 = S30 . CF denotes
F(d(S1 , Sn )) – F(d(T(S1 ), T(Sn )))
n
xn
CF1
CF2
CF3
CF4
3
4
5
.
..
27
28
29
30
31
32
33
.
..
314
315
316
317
318
.
..
3 × 103
7308
6552
5850
.
..
20
8
2
2
2
2
2
.
..
2
2
2
2
2
.
..
2
1.098612
0.727214
0.581922
.
..
0.109231
0.105388
0.101807
0.09846093
0.0923895
0.08962648
0.08702411
.
..
0.00953902
0.00950879
0.00947875
0.00944889
0.00941922
.
..
0.00099983
13.09861
20.74721
30.58192
.
..
756.1092
812.1054
870.1018
930.0983
1056.092
1122.09
1190.087
.
..
98910.01
99540.01
100172
100806
101442
.
..
9003000
12.111111
20.02924
30.01161
.
..
756.0000
812.0000
870.0000
930
1056
1122
1190
.
..
98910
99540
100172
100806
101442
.
..
9003000
12.12201
20.05196
30.02896
.
..
756.0005
812.0004
870.0004
930.0004
1056
1122
1190
.
..
98910
99540
100172
100806
101442
.
..
9003000
n→∞
T(2) = 2
tends to 0
≥τ =1
≥τ =1
≥τ =1
Piri and Kumam Fixed Point Theory and Applications 2014, 2014:210
http://www.fixedpointtheoryandapplications.com/content/2014/1/210
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1
Department of Mathematics, University of Bonab, Bonab, 5551761167, Iran. 2 Department of Mathematics, Faculty of
Science, King Mongkutís University of Technology Thonburi (KMUTT), 126 Bangmod, Thrung Khru, Bangkok, 10140,
Thailand.
Acknowledgements
The second author was supported by the Higher Education Research Promotion and National Research University Project
of Thailand, Office of the Higher Education Commission (Under NUR Project ‘Theoretical and Computational fixed points
for Optimization problems’ No. 57000621).
Received: 11 April 2014 Accepted: 10 September 2014 Published: 13 Oct 2014
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10.1186/1687-1812-2014-210
Cite this article as: Piri and Kumam: Some fixed point theorems concerning F-contraction in complete metric spaces.
Fixed Point Theory and Applications 2014, 2014:210
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