Ume Fixed Point Theory and Applications (2015) 2015:38
DOI 10.1186/s13663-015-0286-5
RESEARCH
Open Access
Fixed point theorems for Kannan-type maps
Jeong Sheok Ume*
*
Correspondence:
[email protected]
Department of Mathematics,
Changwon National University,
Changwon, 641-773, Korea
Abstract
We introduce the new classes of Kannan-type maps with respect to u-distance and
prove some fixed point theorems for these mappings. Then we present several
examples to illustrate the main theorems.
1 Introduction
A mapping T on a metric space (X, d) is called Kannan if there exists α ∈ [,  ) such that
d(Tx, Ty) ≤ αd(x, Tx) + αd(y, Ty)
(.)
for all x, y ∈ X. Kannan [] proved that if X is complete, then a Kannan mapping has a
fixed point. It is interesting that Kannan’s theorem is independent of the Banach contraction principle []. Also, Kannan’s fixed point theorem is very important because Subrahmanyam [] proved that Kannan’s theorem characterizes the metric completeness. That
is, a metric space X is complete if and only if every Kannan mapping on X has a fixed
point.
Using the concept of Hausdorff metric, Nadler [] proved the fixed point theorem for
multi-valued contraction maps, which is a generalization of the Banach contraction principle []. Since then various fixed point results concerning multi-valued contractions have
appeared; for example, see [–] and the references cited there.
Without using the concept of Hausdorff metric, most recently Dehaish and Latif []
generalized fixed point theorems of Latif and Abdou [], Suzuki [], Suzuki and Takahashi
[].
In , Kada et al. [] introduced the notion of w-distance and improved several classical results including Caristi’s fixed point theorem. Suzuki and Takahashi [] introduced
single-valued and multi-valued weakly contractive maps with respect to w-distance and
proved fixed point results for such maps. Generalizing the concept of w-distance, in ,
Suzuki [] introduced the notion of τ -distance on a metric space and improved several
classical results including the corresponding results of Suzuki and Takahashi []. In ,
Ume [] introduced the new concept of a distance called u-distance, which generalizes
w-distance, Tataru’s distance and τ -distance. Then he proved a new minimization theorem and a new fixed point theorem by using u-distance on a complete metric space.
Distances in uniform spaces were given by Vályi []. More general concepts of distances
were given by Wlodarczyk and Plebaniak [–] and Wlodarczyk [].
In this paper, we introduce the new classes of Kannan-type multi-valued maps without
using the concept of Hausdorff metric and Kannan-type single-valued maps with respect
© 2015 Ume; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly credited.
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to u-distance and prove some fixed point theorems for these mappings. Then we present
several examples to illustrate the main theorems.
2 Preliminaries
Throughout this paper we denote by N the set of all positive integers, by R the set of all
real numbers and by R+ the set of all nonnegative real numbers.
Ume [] introduced u-distance as follows: Let X be metric space with metric d. Then
a function p : X × X → R+ is called u-distance on X if there exists a function θ : X × X ×
R+ × R+ → R+ such that the following hold for x, y, z ∈ X:
(u ) p(x, z) ≤ p(x, y) + p(y, z).
(u ) θ (x, y, , ) =  and θ (x, y, s, t) ≥ min{s, t} for all x, y ∈ X and s, t ∈ R+ , for any x ∈ X and
for every ε > , there exists δ >  such that |s – s | < δ, |t – t | < δ, s, s , t, t ∈ R+ and
y ∈ X imply |θ (x, y, s, t) – θ (x, y, s , t )| < ε.
(u ) limn→∞ xn = x and limn→∞ sup{θ (wn , zn , p(wn , xm ), p(zn , xm )) : m ≥ n} =  imply
p(y, x) ≤ limn→∞ inf p(y, xn ) for all y ∈ X.
(u ) limn→∞ sup{p(xn , wm ) : m ≥ n} = , limn→∞ sup{p(yn , zm ) : m ≥ n} = , limn→∞ θ (xn ,
wn , sn , tn ) =  and limn→∞ θ (yn , zn , sn , tn ) =  imply limn→∞ θ (wn , zn , sn , tn ) =  or
limn→∞ sup{p(wm , xn ) : m ≥ n} = , limn→∞ sup{p(zm , yn ) : m ≥ n} = , limn→∞ θ (xn ,
wn , sn , tn ) =  and limn→∞ θ (yn , zn , sn , tn ) =  imply limn→∞ θ (wn , zn , sn , tn ) = .
(u ) limn→∞ θ (wn , zn , p(wn , xn ), p(zn , xn )) =  and limn→∞ θ (wn , zn , p(wn , yn ), p(zn , yn )) =
 imply limn→∞ d(xn , yn ) =  or limn→∞ θ (an , bn , p(xn , an ), p(xn , bn )) =  and
limn→∞ θ (an , bn , p(yn , an ), p(yn , bn )) =  imply limn→∞ d(xn , yn ) = .
We recall remark, examples, definition and lemmas which will be useful in what follows.
Remark . ([]) (a) Suppose that θ from X × X × R+ × R+ into R+ is a mapping satisfying
(u )∼(u ). Then there exists a mapping η from X × X × R+ × R+ into R+ such that η is
nondecreasing in its third and fourth variable, satisfying (u )η ∼(u )η , where (u )η ∼(u )η
stand for substituting η for θ in (u )∼(u ), respectively.
(b) On account of (a), we may assume that θ is nondecreasing in its third and fourth
variables, respectively, for a function θ from X × X × R+ × R+ into R+ satisfying (u )∼(u ).
(c) Each τ -distance p on a metric space (X, d) is also a u-distance on X.
We present some examples of u-distance which are not τ -distance (for details, see []).
Example . Let X = R+ with the usual metric. Define p : X × X → R+ by p(x, y) = (  )x .
Then p is a u-distance on X but not a τ -distance on X.
Example . Let X be a normed space with · , then a function p : X × X → R+ defined
by p(x, y) = x for every x, y ∈ X is a u-distance on X but not a τ -distance.
It follows from the above examples and (c) of Remark . that u-distance is a proper
extension of τ -distance.
Definition . ([]) Let X be a metric space with a metric d and let p be a u-distance
on X. Then a sequence {xn } in X is called p-Cauchy if there exists a function θ from X ×
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X × R+ × R+ into R+ satisfying (u )∼(u ) and a sequence {zn } of X such that
lim sup θ zn , zn , p(zn , xm ), p(zn , xm ) : m ≥ n = 
n→∞
or
lim sup θ zn , zn , p(xm , zn ), p(xm , zn ) : m ≥ n = .
n→∞
Lemma . ([]) Let X be a metric space with a metric d and let p be a u-distance on X.
If {xn } is a p-Cauchy sequence, then {xn } is a Cauchy sequence.
Lemma . ([]) Let X be a metric space with a metric d and let p be a u-distance on X.
() If sequences {xn } and {yn } of X satisfy limn→∞ p(z, xn ) =  and limn→∞ p(z, yn ) =  for
some z ∈ X, then limn→∞ d(xn , yn ) = .
() If p(z, x) =  and p(z, y) = , then x = y.
() Suppose that sequences {xn } and {yn } of X satisfy limn→∞ p(xn , z) =  and
limn→∞ p(yn , z) =  for some z ∈ X, then limn→∞ d(xn , yn ) = .
() If p(x, z) =  and p(y, z) = , then x = y.
Lemma . ([]) Let X be a metric space with a metric d and let p be a u-distance on X.
Suppose that a sequence {xn } of X satisfies
lim sup p(xn , xm ) : m > n =  or
n→∞
lim sup p(xm , xn ) : m > n = .
n→∞
Then {xn } is a p-Cauchy sequence.
3 Main result
The following lemma plays an important role in proving our theorems.
Lemma . Let (X, d) be a metric space with a u-distance p on X and {an } and {bn } be
sequences of X such that
lim sup p(an , am ) : m > n =  and
n→∞
lim sup p(an , bm ) : m > n = .
n→∞
Then there exist a subsequence {akn } of {an } and a subsequence {bkn } of {bn } such that
limn→∞ d(akn , bkn ) = .
Proof Since p is a u-distance on X,
there exists a mapping θ : X × X × R+ × R+ → R+
such that θ is nondecreasing in its third and
(.)
fourth variable respectively, satisfying (u )∼(u ).
For each n ∈ N , let
αn = sup p(an , am ) : m > n
and βn = sup p(an , bm ) : m > n .
(.)
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By hypotheses and (.), we have
lim (αn + βn ) = .
(.)
n→∞
Let k ∈ N be an arbitrary and fixed element. Then, by (u ), for this ak ∈ X and ε = , there
exists δ >  such that
|s| = s < δ ,
|t| = t < δ ,
y∈X
imply θ (ak , y, s, t) < .
(.)
By virtue of (.) and (.), for this δ > , there exists M ∈ N such that
n ≥ M
implies
αn + βn < δ .
(.)
Let k ∈ N be such that
k ≥ max{ + k , M }.
(.)
Due to (.), we have
k < k
and k ≥ M .
(.)
From (.), (.), (.) and (.) we get
θ (ak , ak , αk + βk , αk + βk ) < .
(.)
In terms of (u ) and (.), for this ak ∈ X and ε =  , there exists δ >  such that |s| = s < δ ,
|t| = t < δ , y ∈ X imply

θ (ak , y, s, t) < .

(.)
In view of (.) and (.), for this δ > , there exists M ∈ N such that
n ≥ M
implies
αn + βn < δ .
(.)
Let k ∈ N be such that
k ≥ max{ + k , M }.
(.)
On account of (.), (.), (.), we obtain
k < k 

and θ (ak , ak , αk + βk , αk + βk ) < .

(.)
Continuing this process, there exist a subsequence {akn } of {an } and a subsequence {bkn }
of {bn } such that for all n ∈ N ,

θ (akn , akn+ , αkn+ + βkn+ , αkn+ + βkn+ ) < .
n
(.)
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Using (.), (.) and (.), we know that
lim sup p(akn , akm+ ) : m ≥ n
n→∞
≤ lim sup p(akn , al ) : l > kn
n→∞
(.)
= lim αkn =  and
n→∞
lim θ (akn , akn+ , αkn+ + βkn+ , αkn+ + βkn+ ) = .
n→∞
Using (.), (.), (.) and putting xn = yn = akn , wm = zm = akm+ and sn = tn = αkn+ + βkn+
in (u ), we deduce
lim θ akn+ , akn+ , p(akn+ , akn+ ), p(akn+ , akn+ ) =  and
n→∞
lim θ akn+ , akn+ , p(akn+ , bkn+ ), p(akn+ , bkn+ ) = .
(.)
n→∞
Using (.) and putting wn = zn = akn+ , xn = akn+ and yn = bkn+ in (u ), we have
lim d(akn+ , bkn+ ) = .
n→∞
(.)
Due to (.) and (.), there exist a subsequence {akn } of {an } and a subsequence {bkn }
of {bn } such that
lim d(akn , bkn ) = .
n→∞
(.)
Definition . Let (X, d) be a metric space, X be a set of all nonempty subsets of X and
Cl(X) be a set of all nonempty closed subsets of X. Let T : X → X . Then an element z ∈ X
is a fixed point of T if z ∈ Tz.
A mapping T : X → X is called Kannan-type multi-valued p-contractive mapping if
there exist a u-distance p on X and r ∈ [,  ) such that
(i) p(y , y ) ≤ r[p(x , y ) + p(x , y )] for any x , x ∈ X, y ∈ Tx and y ∈ Tx ,
(ii) Ty ⊆ Tx for all x, y ∈ X with y ∈ Tx.
In the next example we shall show that if (X, d) is a complete metric space with a
u-distance p and a mapping T : X → Cl(X) is not Kannan-type multi-valued p-contractive,
in general, T may have no fixed point in X.
Example . Let X = [, ] be a closed interval with the usual metric and p : X × X → R+
and T : X → Cl(X) be mappings defined as follows:
p(x, y) =
Tx =
, x = ,
x, x =
,
x = ,
{  },
x
x
[ (+x) , (+x) ], x =
.
(.)
(.)
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Define θ : X × X × R+ × R+ → R+ by
θ (x, y, s, t) = s
(.)
for all x, y ∈ X and s, t ∈ R+ .
From (.) and (.) easily we can obtain that p is a u-distance on X.
In terms of (.) and (.), we have
p(y , y ) ≤

p(x , y ) + p(x , y )

(.)
for all x , x ∈ X, y ∈ Tx and y ∈ Tx .
To show that (.) is satisfied, we need to consider several possible cases.


Case . Let x = x = . Then y ∈ Tx =
, y ∈ Tx =
,



, p(x , y ) =  and p(x , y ) =  and




p(x , y ) + p(x , y ) = [ + ] =  ≥ = p(y , y ). Thus




p(y , y ) ≤ p(x , y ) + p(x , y ) .


,
Case . Let x =  and x = . Then y ∈ Tx =

x

x
y ∈ Tx =
,
, p(y , y ) = y = ,
( + x ) ( + x )

p(y , y ) = y =
(.)
(.)
p(x , y ) =  and p(x , y ) = x . Thus



p(x , y ) + p(x , y ) = [ + x ] ≥ = p(y , y ).



x
x
Case . Let x =  and x = . Then y ∈ Tx =
,
,
( + x ) ( + x )

x
, p(y , y ) = y ≤
, p(x , y ) = x
y ∈ Tx =

( + x )
(.)
and p(x , y ) = . Thus

x

p(x , y ) + p(x , y ) = [x + ] ≥
≥ p(y , y ).


( + x )
x
x
Case . Let x =  and x = . Then y ∈ Tx =
,
,
( + x ) ( + x )
x
x
x
y ∈ Tx =
, p(y , y ) = y ≤
,
( + x ) ( + x )
( + x )
p(x , y ) = x and p(x , y ) = x . Thus

x

x
p(x , y ) + p(x , y ) = [x + x ] ≥ max
,
≥ p(y , y ).


( + x ) ( + x )
(.)
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From (.)∼(.), we have
p(y , y ) ≤

p(x , y ) + p(x , y )

(.)
for all x , x ∈ X, y ∈ Tx and y ∈ Tx .


, 
] {  } =
But there exist x =  ∈ X and y =  ∈ X with y ∈ Tx such that Ty = T  = [ 
T. Therefore T is not Kannan-type multi-valued p-contractive and T does not have a
fixed point.
Using Lemma ., we have the following main theorem.
Theorem . Let (X, d) be a complete metric space and let T : X → Cl(X) be a Kannantype multi-valued p-contractive mapping. Then T has a unique fixed point in X.
Proof Let a ∈ X be arbitrary, a ∈ Ta and a ∈ Ta be chosen. Since T is Kannan-type
p-contractive,
p(a , a ) ≤ r p(a , a ) + p(a , a ) ,
(.)
where r ∈ [,  ).
From (.), we get
p(a , a ) ≤ kp(a , a ),
(.)
r
∈ [, ).
where k = –r
By (.) and (.), we obtain a sequence {an } in X such that
an+ ∈ Tan
and
p(an+ , an+ ) ≤ kp(an , an+ )
(.)
for all n ∈ N .
By repeated application of (.), we have
p(an , an+ ) ≤ k n– p(a , a )
(.)
for all n ∈ N .
Now we shall know that {an } is a Cauchy sequence.
Let n, m ∈ N be such that n < m. Then, by virtue of (.), we deduce
p(an , am ) ≤ p(an , an+ ) + p(an+ , an+ ) + · · · + p(am– , am )
=
m–
p(ai , ai+ ) ≤
i=n
≤
m–
k i– p(a , a )
i=n
n–
k
–k
p(a , a ).
(.)
In view of (.), we get
lim sup p(an , am ) : m > n = .
n→∞
(.)
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On account of Lemma ., Lemma . and (.), {an } is a Cauchy sequence in X. Since X
is complete, there exists b ∈ X such that
lim an = b .
(.)
n→∞
By the same method as that in (.)∼(.), there exists a sequence {bn } in X such that
bn+ ∈ Tbn
and
p(bn , bn+ ) ≤ k n– p(b , b )
(.)
for all n ∈ N .
Combining the hypothesis, (.), (.), (.), (.) and (.), we have
p(an , bm ) ≤ p(an , am ) + p(am , bm )
≤ p(an , am ) + r p(am– , am ) + p(bm– , bm )
n– k
p(a , a ) + r k m– p(a , a ) + k m– p(b , b )
≤
–k
n– k
p(a , a ) + k n– p(a , a ) + k n– p(b , b )
≤
–k

p(a , a ) + p(a , a ) + p(b , b )
= k n–
–k
(.)
for all n, m ∈ N with m > n.
By (.), we have
lim sup p(an , bm ) : m > n = .
n→∞
(.)
Due to Lemma ., (.) and (.), there exist a subsequence {akn } of {an } and a subsequence {bkn } of {bn } such that
lim d(akn , bkn ) = .
n→∞
(.)
On account of the hypothesis and (.), we obtain
bn+ ∈ Tb
(.)
for all n ∈ N .
By virtue of the hypothesis, (.), (.) and (.), we have
b ∈ Tb .
(.)
Due to (.), b is a fixed point of T. To prove the unique fixed point of T, let c be another
fixed point of T. Then
c ∈ Tc .
(.)
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Since T is Kannan-type multi-valued p-contractive, by (.) and (.), we have
p(b , b ) ≤ r p(b , b ) + p(b , b ) ,
p(c , c ) ≤ r p(c , c ) + p(c , c ) ,
p(b , c ) ≤ r p(b , b ) + p c , c .
(.)
(.)
(.)
Since r ∈ [,  ), from (.), (.) and (.), we get
p(b , b ) = p(c , c ) = p(b , c ) = .
(.)
By virtue of Lemma . and (.), we have
b = c .
(.)
On account of (.), (.) and (.), T has a unique fixed point.
Now we give an example to support Theorem ..
Example . Let X = [, ] be a closed interval with the usual metric, and p : X × X → R+
and T : X → Cl(X) be mappings defined as follows:
p(x, y) = x,

Tx = , x .

(.)
Let θ : X × X × R+ × R+ → R+ be as in (.). Then, due to (.), we easily can obtain
that p is a u-distance on X.
From (.), we have
p(y , y ) ≤

p(x , y ) + p(x , y )

(.)
for all x , x ∈ X, y ∈ Tx and y ∈ Tx . To show that (.) is satisfied, let x , x ∈ X,
y ∈ Tx and y ∈ Tx . Then p(y , y ) = y ≤  x and  [p(x , y ) + p(x , y )] =  (x + x ) ≥

x . Thus (.) is satisfied. Let x, y ∈ X be such that y ∈ Tx. Then  ≤ y ≤  x and
 
Ty ⊆ [,  x] ⊆ [,  x] = Tx. Thus Ty ⊆ Tx for all x, y ∈ X with y ∈ Tx. Therefore all the
conditions of Theorem . are satisfied and T has a unique fixed point  in X.
Definition . Let (X, d) be a metric space. A mapping T : X → X is called Kannan-type
single-valued p-contractive mapping if there exist a u-distance p on X and r ∈ [,  ) such
that
(iii) p(Tx , Tx ) ≤ r[p(x , Tx ) + p(x , Tx )] for any x , x ∈ X,
(iv) if {xn } is a sequence in X such that xn+ = Txn for each n ∈ N and limn→∞ xn = c ∈ X,
then p(Tc, c) ≤ r[p(Tc, Tc) + p(c, Tc)] and p(c, Tc) ≤ r[p(Tc, Tc) + p(Tc, c)].
In the following example we show that if (X, d) is a complete metric space and a mapping
T : X → X is not Kannan-type single-valued p-contractive, in general, T may have no fixed
point in X.
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Example . Let X = [, ] be a closed interval with the usual metric, and p : X × X → R+
and T : X → X be mappings defined as follows:
p(x, y) =
Tx =
, x = ,
x, x =
,
(.)
x = ,
x = .
(.)

,

x
,
(+x)
Define θ : X × X × R+ × R+ → R+ by
θ (x, y, s, t) = s.
(.)
By the same methods as in Example ., we know that p is a u-distance on X and T is not
Kannan-type single-valued p-contractive and T has no fixed point in X.
Theorem . Let (X, d) be a metric space with a u-distance p on X.
Let T : X → X be a Kannan-type single-valued p-contractive mapping such that there
exist a sequence {xn } of X and c ∈ X satisfying xn+ = Txn for each n ∈ N and limn→∞ xn =
c ∈ X. Then c is a fixed point of T, i.e., Tc = c.
Proof By hypotheses, we obtain
p(Tc, Tc) ≤ r p(c, Tc) + p(c, Tc) = rp(c, Tc),
p(Tc, c) ≤ r p(Tc, Tc) + p(c, Tc)
≤ r rp(c, Tc) + p(c, Tc)
= r(r + )p(c, Tc),
p(c, Tc) ≤ r p(Tc, Tc) + p(Tc, c)
≤ r rp(c, Tc) + r(r + )p(c, Tc)
= r + r p(c, Tc).
(.)
(.)
(.)
Since r ∈ [,  ), r, r(r + ), (r + r ) ∈ [, ) and thus, by (.), (.) and (.), we
have
p(Tc, Tc) = p(Tc, c) = p(c, Tc).
(.)
In view of Lemma . and (.),
Tc = c.
This means that c is a fixed point of T.
(.)
Theorem . Let (X, d) be a complete metric space and let T : X → X be a Kannan-type
single-valued p-contractive mapping.
Then T has a unique fixed point in X.
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Proof Since T is Kannan type single-valued p-contractive, there exists a sequence {xn } of
X such that
xn+ = Txn
and p(xn+ , xn+ ) ≤ kp(xn , xn+ )
(.)
for all n ∈ N , where k ∈ [, ).
By repeated application of (.), we have
p(xn , xn+ ) ≤ k n– p(x , x )
(.)
for all n ∈ N .
On account of (.), we get
p(xn , xm ) ≤ p(xn , xn+ ) + p(xn+ , xn+ ) + · · · + p(xm– , xm )
≤
m–
p(xi , xi+ ) ≤
i=n
≤
m–
k i– p(x , x )
i=n
n–
k
–k
p(x , x )
(.)
for all n, m ∈ N with n < m.
In view of (.), we deduce that
lim sup p(xn , xm ) : m > n = .
n→∞
(.)
By virtue of Lemma ., Lemma . and (.), we know that {xn } is a Cauchy sequence.
Since X is complete, there exists c ∈ X such that
lim xn = c.
n→∞
(.)
On account of the hypothesis, Theorem ., (.) and (.), we know that c is a fixed
point, i.e.,
Tc = c.
(.)
By the same method as that in (.)∼(.), we can prove that T has a unique fixed
point X.
From Theorem ., we have the following corollary.
Corollary . ([]) Let (X, d) be a complete metric space and let T : X → X be a mapping
such that
d(Tx, Ty) ≤ r d(x, Tx) + d(y, Ty)
for all x, y ∈ X and some r ∈ [,  ).
Then T has a unique fixed point in X.
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Proof By the same methods as in (.)∼(.), we deduce that
lim sup d(xn , xm ) : m > n = ,
n→∞
(.)
where xn+ = Txn for all n ∈ N .
Since X is complete, there exists c ∈ X such that
lim xn = c.
(.)
n→∞
Due to the hypothesis and (.), we get
d(xn+ , Tc) = d(Txn , Tc)
≤ r d(xn , Txn ) + d(c, Tc)
= r d(xn , xn+ ) + d(c, Tc)
(.)
for all n ∈ N and some r ∈ [,  ).
Taking the limit as n → ∞ in (.), we obtain
d(c, Tc) ≤ rd(c, Tc)
(.)
for some r ∈ [,  ).
From (.), we have
d(c, Tc) = .
(.)
Since metric d is a u-distance, by view of (.), (.), (.) and the hypothesis, conditions of Corollary . satisfy all conditions of Theorem ..
Therefore T has a unique fixed point.
Finally we shall present an example to show that all conditions of Theorem . are satisfied, but all conditions of Corollary . are not satisfied.
Example . Let X = [, ] be a closed interval with the usual metric, and p : X × X → R+
and T : X × X be mappings defined as follows:
p(x, y) = x,
Tx =
(.)

x.

Then, due to (.), we easily can obtain that p is a u-distance on X, but not metric and
p(Tx, Ty) ≤

p(x, Tx) + p(y, Ty)

for all x, y ∈ X.
Suppose that {xn } is a sequence of X such that xn+ = Txn for all x ∈ N .
(.)
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Then, by (.), we have
xn+ = Txn =

xn

(.)
for all n ∈ N .
By virtue of (.),
n–

x = .
n→∞ 
lim xn = lim
n→∞
(.)
On account of (.)∼(.), all conditions of Theorem . are satisfied, but all conditions
of Corollary . are not satisfied since p is not metric.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author completed the paper himself. The author read and approved the final manuscript.
Acknowledgements
The author would like to express his gratitude to the referees for giving valuable comments and suggestions. This
research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Science, ICT & Future Planning (2013R1A1A2057665).
Received: 16 December 2014 Accepted: 23 February 2015
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