Open Access

Fixed point of α-ψ-contractive type mappings in uniform spaces

Fixed Point Theory and Applications20142014:150

https://doi.org/10.1186/1687-1812-2014-150

Received: 1 April 2014

Accepted: 13 June 2014

Published: 22 July 2014

Abstract

In this paper, we shall investigate the existence and uniqueness of a fixed point of α-ψ-contractive mappings in the context of uniform spaces. We shall also prove some common fixed point theorems by introducing the notion of α-admissible pairs. We shall construct some examples to support our novel results.

MSC:46T99, 47H10, 54H25.

Keywords

α-ψ-contractive α-admissible maps

1 Introduction

One of the interesting metric fixed point results was given by Samet et al. [1] by introducing the notions of α-admissible and α-ψ-contractive type mappings. They reported results via these new notions, and they extended and unified most of the related existing metric fixed point results in the literature. In particular, the authors [2] showed that fixed point results via cyclic contractions are consequences of their related results. Naturally, many authors have started to investigate the existence and uniqueness of a fixed point theorem via admissible mappings and variations of the concept of α-ψ-contractive type mappings, for reference see [119]. The notion of cyclic contraction was introduced by Kirk et al. [20]. The main advantage of the cyclic contraction is that the given mapping does not need to be continuous. It has been appreciated by several authors; see e.g. [2126] and related references therein.

In this paper, we shall consider the characterization of the notions of α-ψ-contractive and α-admissible mappings in the context uniform spaces. Further, we shall prove some fixed point theorems by using these concepts. We shall also use α-admissible pairs to investigate the existence and uniqueness of a common fixed point in the setting of uniform spaces. We shall also establish some examples to illustrate the main results.

For the sake of completeness, we shall recollect some basic definitions and fundamental results. Let X be a nonempty set. A nonempty family, ϑ, of subsets of X × X is called a uniform structure of X it satisfies the following properties:
  1. (i)

    if G is in ϑ, then G contains the diagonal { ( x , x ) | x X } ;

     
  2. (ii)

    if G is in ϑ and H is a subset of X × X which contains G, then H is in ϑ;

     
  3. (iii)

    if G and H are in ϑ, then G H is in ϑ;

     
  4. (iv)

    if G is in ϑ, then there exists H in ϑ, such that, whenever ( x , y ) and ( y , z ) are in H, then ( x , z ) is in G;

     
  5. (v)

    if G is in ϑ, then { ( y , x ) | ( x , y ) G } is also in ϑ.

     

The pair ( X , ϑ ) is called a uniform space and the element of ϑ is called entourage or neighborhood or surrounding. The pair ( X , ϑ ) is called a quasiuniform space (see e.g. [27, 28]) if property (v) is omitted.

Let Δ = { ( x , x ) | x X } be the diagonal of a nonempty set X. For V , W X × X , we shall use the following setting in the sequel:
V W = { ( x , y ) | there exist  z X : ( x , z ) W  and  ( z , y ) V }
and
V 1 = { ( x , y ) | ( y , x ) V } .

For a subset V ϑ , a pair of points x and y are said to be V-close if ( x , y ) V and ( y , x ) V . Moreover, a sequence { x n } in X is called a Cauchy sequence for ϑ, if, for any V ϑ , there exists N 1 such that x n and x m are V-close for n , m N . For ( X , ϑ ) , there is a unique topology τ ( ϑ ) on X generated by V ( x ) = { y X | ( x , y ) V } where V ϑ .

A sequence { x n } in X is convergent to x for ϑ, denoted by lim n x n = x , if, for any V ϑ , there exists n 0 N such that x n V ( x ) for every n n 0 . A uniform space ( X , ϑ ) is called Hausdorff if the intersection of all the V ϑ is equal to Δ of X, that is, if ( x , y ) V for all V ϑ implies x = y . If V = V 1 then we shall say that a subset V ϑ is symmetrical. Throughout the paper, we shall assume that each V ϑ is symmetrical. For more details, see e.g. [27, 2932].

Now, we shall recall the notions of A-distance and E-distance.

Definition 1.1 [29, 30]

Let ( X , ϑ ) be a uniform space. A function p : X × X [ 0 , ) is said to be an A-distance if, for any V ϑ , there exists δ > 0 such that if p ( z , x ) δ and p ( z , y ) δ for some z X , then ( x , y ) V .

Definition 1.2 [29, 30]

Let ( X , ϑ ) be a uniform space. A function p : X × X [ 0 , ) is said to be an E-distance if
  1. (i)

    p is an A-distance,

     
  2. (ii)

    p ( x , y ) p ( x , z ) + p ( z , y ) , x , y , z X .

     

Example 1.3 [29, 30]

Let ( X , ϑ ) be a uniform space and let d be a metric on X. It is evident that ( X , ϑ d ) is a uniform space where ϑ d is a set of all subsets of X × X containing a ‘band’ U ϵ = { ( x , y ) X 2 | d ( x , y ) < ϵ } for some ϵ > 0 . Moreover, if ϑ ϑ d , then d is an E-distance on ( X , ϑ ) .

Lemma 1.4 [29, 30]

Let ( X , ϑ ) be a Hausdorff uniform space and p be an A-distance on X. Let { x n } and { y n } be sequences in X and { α n } , { β n } be sequences in [ 0 , ) converging to 0. Then, for x , y , z X , the following results hold:
  1. (a)

    If p ( x n , y ) α n and p ( x n , z ) β n for all n N , then y = z . In particular, if p ( x , y ) = 0 and p ( x , z ) = 0 , then y = z .

     
  2. (b)

    If p ( x n , y n ) α n and p ( x n , z ) β n for all n N , then { y n } converges to z.

     
  3. (c)

    If p ( x n , x m ) α n for all n , m N with m > n , then { x n } is a Cauchy sequence in ( X , ϑ ) .

     

Let p be an A-distance. A sequence in a uniform space ( X , ϑ ) with an A-distance is said to be a p-Cauchy if, for every ϵ > 0 , there exists n 0 N such that p ( x n , x m ) < ϵ for all n , m n 0 .

Definition 1.5 [29, 30]

Let ( X , ϑ ) be a uniform space and p be an A-distance on X.
  1. (i)

    X is S-complete if, for every p-Cauchy sequence { x n } , there exists x in X with lim n p ( x n , x ) = 0 .

     
  2. (ii)

    X is p-Cauchy complete if, for every p-Cauchy sequence { x n } , there exists x in X with lim n x n = x with respect to τ ( ϑ ) .

     
  3. (iii)

    T : X X is p-continuous if lim n p ( x n , x ) = 0 implies lim n p ( T ( x n ) , T ( x ) ) = 0 .

     

Remark 1.6 Let ( X , ϑ ) be a Hausdorff uniform space which is S-complete. If a sequence { x n } be a p-Cauchy sequence, then we have lim n p ( x n , x ) = 0 . Regarding Lemma 1.4(b), we derive lim n x n = x with respect to the topology τ ( ϑ ) , and hence S-completeness implies p-Cauchy completeness.

Definition 1.7 [20]

Let X be a nonempty set, m a positive integer and T : X X a mapping. X = i = 1 m A i is said to be a cyclic representation of X with respect to T if
  1. (i)

    A i , i = 1 , 2 , , m are nonempty sets;

     
  2. (ii)

    T ( A 1 ) A 2 , , T ( A m 1 ) A m , T ( A m ) A 1 .

     

2 Main results

Let Ψ be the family of functions ψ : [ 0 , ) [ 0 , ) satisfying the following conditions:

( Ψ 1 ) ψ is nondecreasing;

( Ψ 2 ) n = 1 + ψ n ( t ) < for all t > 0 , where ψ n is the n th iterate of ψ.

These functions are known in the literature as (c)-comparison functions. It is easily proved that if ψ is a (c)-comparison function, then ψ ( t ) < t for any t > 0 .

Definition 2.1 [1]

Let T : X X and α : X × X [ 0 , ) . We shall say that T is α-admissible if, for all x , y X , we have
α ( x , y ) 1 α ( T x , T y ) 1 .

We shall characterize the notion of α-ψ-contractive mapping, introduced by Samet et al. [1], in the context of uniform space as follows.

Definition 2.2 Let ( X , ϑ ) be a uniform space such that p is an E-distance on X and T : X X be a given mapping. We shall say that T is an α-ψ-contractive mapping if there exist two functions α : X × X [ 0 , ) and ψ Ψ such that
α ( x , y ) p ( T x , T y ) ψ ( p ( x , y ) ) , for all  x , y X .
(2.1)
Theorem 2.3 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p be an E-distance on X. Let T : X X be an α-ψ-contractive mapping satisfying the following conditions:
  1. (i)

    T is α-admissible;

     
  2. (ii)

    there exists x 0 X such that α ( x 0 , T x 0 ) 1 and α ( T x 0 , x 0 ) 1 ;

     
  3. (iii)

    T is p-continuous.

     

Then T has a fixed point u X .

Proof By hypothesis (ii) of the theorem we have x 0 X such that α ( x 0 , T x 0 ) 1 . Define the sequence { x n } in X by x n + 1 = T x n for all n N { 0 } . If x n 0 = x n 0 + 1 for some n 0 , then u = x n 0 is a fixed point of T. So, we can assume that x n x n + 1 for all n. Since T is α-admissible, we have
α ( x 0 , x 1 ) = α ( x 0 , T x 0 ) 1 α ( T x 0 , T x 1 ) = α ( x 1 , x 2 ) 1 .
Inductively, we have
α ( x n , x n + 1 ) 1 , for all  n N { 0 } .
(2.2)
From (2.1) and (2.2), it follows that, for all n N , we have
p ( x n + 1 , x n ) = p ( T x n , T x n 1 ) α ( x n , x n 1 ) p ( T x n , T x n 1 ) ψ ( p ( x n , x n 1 ) ) .
(2.3)
Iteratively, we derive
p ( x n , x n + 1 ) ψ n ( p ( x 0 , x 1 ) ) , for all  n N .
Since p is an E-distance, for m > n , we have
p ( x n , x m ) p ( x n , x n + 1 ) + + p ( x m 1 , x m ) ψ n ( p ( x 0 , x 1 ) ) + ψ n + 1 ( p ( x 0 , x 1 ) ) + + ψ m 1 ( p ( x 0 , x 1 ) ) .
(2.4)
To show that { x n } is a p-Cauchy sequence, consider
S n = k = 0 n ψ k ( p ( x 0 , x 1 ) ) .
Thus from (2.4) we have
p ( x n , x m ) S m 1 S n 1 .
(2.5)
Since ψ Ψ , there exists S [ 0 , ) such that lim n S n = S . Thus by (2.5) we have
lim n , m p ( x n , x m ) = 0 .
(2.6)
Since p is not symmetrical, by repeating the same argument we have
lim n , m p ( x m , x n ) = 0 .
(2.7)

Hence the sequence { x n } is a p-Cauchy in the S-complete space X. Thus, there exists u X such that lim n p ( x n , u ) = 0 , which implies lim n x n = u . Since T is p-continuous, we have lim n p ( T x n , T u ) = 0 , which implies that lim n ( x n + 1 , T u ) = 0 . Hence we have lim n p ( x n , u ) = 0 and lim n ( x n , T u ) = 0 . Thus by Lemma 1.4(a) we have u = T u . □

In the following theorem, we omit the p-continuity by replacing a suitable condition on the obtained iterative sequence.

Theorem 2.4 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X. Let T : X X be an α-ψ-contractive mapping satisfying the following conditions:
  1. (i)

    T is α-admissible;

     
  2. (ii)

    there exists x 0 X such that α ( x 0 , T x 0 ) 1 and α ( T x 0 , x 0 ) 1 ;

     
  3. (iii)

    for any sequence { x n } in X with x n x as n and α ( x n , x n + 1 ) 1 for each n N { 0 } , then α ( x n , x ) 1 for each n N { 0 } .

     

Then T has a fixed point u X .

Proof By following the proof of Theorem 2.3, we know that { x n } is a p-Cauchy in the S-complete space X. Thus, there exists u X such that lim n p ( x n , u ) = 0 , which implies lim n x n = u . By using (2.1) and assumption (iii), we get
p ( x n , T u ) p ( x n , x n + 1 ) + p ( x n + 1 , T u ) p ( x n , x n + 1 ) + α ( x n , u ) p ( T x n , T u ) p ( x n , x n + 1 ) + ψ ( p ( x n , u ) ) .

Letting n in above inequality, we shall have lim n p ( x n , T u ) = 0 . Hence we have lim n p ( x n , u ) = 0 and lim n p ( x n , T u ) = 0 . Thus by Lemma 1.4(a) we have u = T u . □

Example 2.5 Let X = { 1 n : n N } { 0 } be endowed with the usual metric d. Define ϑ = { U ϵ | ϵ > 0 } . It is easy to see that ( X , ϑ ) is a uniform space. Define T : X X by
T x = { 0 if  x = 0 , 1 3 n + 1 if  x = 1 n n > 1 , 1 if  x = 1 ,
(2.8)
and α : X × X [ 0 , ) by
α ( x , y ) = { 1 if  x , y X { 1 } , 0 otherwise ,
(2.9)

and ψ ( t ) = t 3 for all t 0 . One can easily see that T is α-ψ-contractive and α-admissible mapping. Also for x 0 = 1 2 we have α ( x 0 , T x 0 ) = α ( T x 0 , x 0 ) = 1 . Moreover, for any sequence { x n } in X with x n x as n and α ( x n 1 , x n ) = 1 for each n N we have α ( x n , x ) = 1 for each n N . Therefore by Theorem 2.4, T has a fixed point.

In the sequel, we shall investigate the uniqueness of a fixed point. For this purpose, we shall introduce the following condition.
  1. (H)

    For all x , y Fix ( T ) , there exists z X such that α ( z , x ) 1 and α ( z , y ) 1 .

     

Here, Fix ( T ) denotes the set of fixed points of T.

The following theorem guarantees the uniqueness of a fixed point.

Theorem 2.6 Adding the condition (H) to the hypothesis of Theorem  2.3 (respectively, Theorem  2.4), we obtain the uniqueness of fixed point of T.

Proof Suppose, on the contrary, that v X is another fixed point of T. From (H), there exists z X such that
α ( z , u ) 1 and α ( z , v ) 1 .
(2.10)
Owing to the fact that T is α-admissible, from (2.10), we have
α ( T n z , u ) 1 and α ( T n z , v ) 1 , for all  n N { 0 } .
(2.11)
We define the sequence { z n } in X by z n + 1 = T z n = T n z 0 for all n N { 0 } and z 0 = z . From (2.11) and (2.1), we have
p ( z n + 1 , u ) = p ( T z n , T u ) α ( z n , u ) p ( T z n , T u ) ψ ( p ( z n , u ) ) ,
(2.12)
for all n N { 0 } . This implies that
p ( z n , u ) ψ n ( p ( z 0 , u ) ) , for all  n N .
Letting n in the above inequality, we obtain
lim n p ( z n , u ) = 0 .
(2.13)
Similarly,
lim n p ( z n , v ) = 0 .
(2.14)

From (2.13) and (2.14) together with Lemma 1.4(a), it follows that u = v . Thus we have proved that u is the unique fixed point of T. □

Definition 2.7 [9]

A pair of two self-mappings T , S : X X is said to be α-admissible, if, for any x , y X with α ( x , y ) 1 , we have α ( T x , S y ) 1 and α ( S x , T y ) 1 .

Definition 2.8 Let ( X , ϑ ) be a uniform space. A pair of two self-mappings T , S : X X is said to be an α-ψ-contractive pair if
α ( x , y ) max { p ( T x , S y ) , p ( S x , T y ) } ψ ( p ( x , y ) ) ,
(2.15)

for any x , y X , where ψ Ψ .

Theorem 2.9 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X. Suppose that the pair of T , S : X X is an α-ψ-contractive pair satisfying the following conditions:
  1. (i)

    ( T , S ) is α-admissible;

     
  2. (ii)

    there exists x 0 X such that α ( x 0 , T x 0 ) 1 and α ( T x 0 , x 0 ) 1 ;

     
  3. (iii)

    for any sequence { x n } in X with x n x as n and α ( x n , x n + 1 ) 1 for each n N { 0 } , then α ( x n , x ) 1 for each n N { 0 } .

     

Then T and S have a common fixed point.

Proof By hypothesis (ii) of the theorem, we have x 0 X such that α ( x 0 , T x 0 ) 1 and α ( T x 0 , x 0 ) 1 . Since ( T , S ) is an α-admissible pair, we can construct a sequence such that
T x 2 n = x 2 n + 1 , S x 2 n + 1 = x 2 n + 2 and α ( x n , x n + 1 ) 1 , α ( x n + 1 , x n ) 1 , for all  n N { 0 } .
From (2.15) for all n N { 0 } , we have
p ( x 2 n + 1 , x 2 n + 2 ) = p ( T x 2 n , S x 2 n + 1 ) α ( x 2 n , x 2 n + 1 ) max { p ( T x 2 n , S x 2 n + 1 ) , p ( S x 2 n , T x 2 n + 1 ) } ψ ( p ( x 2 n , x 2 n + 1 ) ) .
Hence, we conclude that
p ( x 2 n + 1 , x 2 n + 2 ) ψ ( p ( x 2 n , x 2 n + 1 ) ) .
(2.16)
Similarly, we find that
p ( x 2 n + 2 , x 2 n + 3 ) = p ( S x 2 n + 1 , T x 2 n + 2 ) α ( x 2 n + 1 , x 2 n + 2 ) max { p ( T x 2 n + 1 , S x 2 n + 2 ) , p ( S x 2 n + 1 , T x 2 n + 2 ) } ψ ( p ( x 2 n + 1 , x 2 n + 2 ) ) .
Hence, we derive
p ( x 2 n + 2 , x 2 n + 3 ) ψ ( p ( x 2 n + 1 , x 2 n + 2 ) ) .
(2.17)
Thus from (2.16) and (2.17), and by induction, we get
p ( x n , x n + 1 ) ψ n ( p ( x 0 , x 1 ) ) , for all  n N .
(2.18)
We shall show that { x n } is a p-Cauchy sequence, Since p is an E-distance, for m > n , we have
p ( x n , x m ) p ( x n , x n + 1 ) + + p ( x m 1 , x m ) ψ n ( p ( x 0 , x 1 ) ) + ψ n + 1 ( p ( x 0 , x 1 ) ) + + ψ m 1 ( p ( x 0 , x 1 ) ) .
(2.19)
Now, we shall consider
S n = k = 0 n ψ k ( p ( x 0 , x 1 ) ) .
Thus, from (2.19) we have
p ( x n , x m ) S m 1 S n 1 .
(2.20)
Since ψ Ψ , there exists S [ 0 , ) such that lim n S n = S . Thus, by (2.20) we have
lim n , m p ( x n , x m ) = 0 .
(2.21)
Since p is not symmetrical, by repeating the same argument we have
lim n , m p ( x m , x n ) = 0 .
(2.22)
Hence the sequence { x n } is p-Cauchy in the S-complete space X. Thus, there exists u X such that lim n p ( x n , u ) = 0 , which implies lim n T x 2 n = lim n S x 2 n + 1 = u . By using (2.15) and assumption (iii), we get
p ( x n , T u ) p ( x n , x 2 n + 2 ) + p ( x 2 n + 2 , T u ) = p ( x n , x 2 n + 2 ) + p ( S x 2 n + 1 , T u ) p ( x n , x 2 n + 2 ) + α ( x 2 n + 1 , u ) max { p ( T x 2 n + 1 , S u ) , p ( S x 2 n + 1 , T u ) } p ( x n , x 2 n + 2 ) + ψ ( p ( x 2 n + 1 , u ) ) .
(2.23)

Letting n in (2.23), we have p ( x n , T u ) = 0 . Hence we have lim n p ( x n , u ) = 0 and lim n p ( x n , T u ) = 0 . Thus by Lemma 1.4(a) we have u = T u . Analogously, one can derive u = S u . Therefore u = T u = S u . □

Remark 2.10 Note that Theorem 2.9 is valid if one replaces condition (ii) with

(ii)′ there exists x 0 X such that α ( x 0 , S x 0 ) 1 and α ( S x 0 , x 0 ) 1 .

We shall get the following result by letting S = T in Theorem 2.9.

Corollary 2.11 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X. Suppose that a mapping T : X X is satisfying the condition
α ( x , y ) max { p ( T x , y ) , p ( x , T y ) } ψ ( p ( x , y ) ) ,
for any x , y X , where ψ Ψ . Also suppose that the following conditions are satisfied:
  1. (i)

    T is α-admissible;

     
  2. (ii)

    there exists x 0 X such that α ( x 0 , T x 0 ) 1 and α ( T x 0 , x 0 ) 1 ;

     
  3. (iii)

    for any sequence { x n } in X with x n x as n and α ( x n , x n + 1 ) 1 for each n N { 0 } , then α ( x n , x ) 1 for each n N { 0 } .

     

Then T has a fixed point.

Example 2.12 Let ( X , d ) is a dislocated metric space where X = { 1 n : n N } { 0 } and d ( x , y ) = max { x , y } . Define ϑ = { U ϵ | ϵ > 0 } , where U ϵ = { ( x , y ) X 2 : d ( x , y ) < d ( x , x ) + ϵ } . It is easy to see that ( X , ϑ ) is a uniform space. Define T : X X by
T x = { 0 if  x = 0 , 1 2 n + 1 if  x = 1 n n > 1 , 1 if  x = 1 ,
(2.24)
and S : X X by
S x = { 0 if  x = 0 , 1 2 n if  x = 1 n n > 1 , 1 if  x = 1 ,
(2.25)
and α : X × X [ 0 , ) by
α ( x , y ) = { 1 if  x , y X { 1 } , 0 otherwise ,
(2.26)

and ψ ( t ) = t 2 for all t 0 . One can easily see that ( T , S ) is an α-ψ-contractive and α-admissible pair. Also for x 0 = 1 2 we have α ( x 0 , T x 0 ) = α ( T x 0 , x 0 ) = 1 . Moreover, for any sequence { x n } in X with x n x as n and α ( x n , x n + 1 ) 1 for each n N { 0 } , we have α ( x n , x ) 1 for each n N . Therefore by Theorem 2.9, T and S have a common fixed point.

To investigate the uniqueness of a common fixed point, we shall introduce the following condition.
  1. (I)

    For each x , y CFix ( T , S ) , we have α ( x , y ) 1 , where CFix ( T , S ) is the set of all common fixed points of T and S.

     

Theorem 2.13 Adding the condition (I) to the hypothesis of Theorem  2.9, we obtain the uniqueness of the common fixed point of T and S.

Proof On the contrary suppose that u , v X are two distinct common fixed points of T and S. From (I) and (2.15) we have
p ( u , v ) α ( u , v ) max { p ( T u , S v ) , p ( S u , T v ) } ψ ( p ( u , v ) ) < p ( u , v ) ,

which is impossible for p ( u , v ) > 0 . Consequently, we have p ( u , v ) = 0 . Analogously, one can show that p ( u , u ) = 0 . Thus we have u = v , which is a contradiction to our assumption. Hence T and S have a unique common fixed point. □

3 Consequences

3.1 Standard contractions on uniform space

Taking in Theorem 2.6, α ( x , y ) = 1 for all x , y X , we shall obtain immediately the following fixed point theorems.

Corollary 3.1 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and T : X X be a given mapping. Suppose that there exists a function ψ Ψ such that
p ( T x , T y ) ψ ( p ( x , y ) ) ,

for all x , y X . Then T has a unique fixed point.

By substituting ψ ( t ) = k t , where k [ 0 , 1 ) , in Corollary 3.1, we shall get the following.

Corollary 3.2 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X. Suppose that T : X X is a given mapping satisfying
p ( T x , T y ) k p ( x , y ) ,

for all x , y X . Then T has a unique fixed point.

Taking in Theorem 2.13 α ( x , y ) = 1 for all x , y X , we shall obtain immediately the following common fixed point theorem.

Corollary 3.3 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and T , S : X X are given mappings. Suppose that there exists a function ψ Ψ such that
max { p ( T x , S y ) , p ( S x , T y ) } ψ ( p ( x , y ) ) ,

for all x , y X . Then T and S have a unique common fixed point.

Corollary 3.4 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and T : X X be a given mapping. Suppose that there exists a function ψ Ψ such that
max { p ( T x , y ) , p ( x , T y ) } ψ ( p ( x , y ) ) ,

for all x , y X . Then T has a unique fixed point.

3.2 Cyclic contraction on uniform space

Corollary 3.5 Let ( X , ϑ ) be a S-complete Hausdorff uniform space such that p is an E-distance on X and A 1 , A 2 are nonempty closed subsets of X with respect to the topological space ( X , τ ( ϑ ) ) . Let T : Y Y be a mapping, where Y = i = 1 2 A i . Suppose that the following conditions hold:
  1. (i)

    T ( A 1 ) A 2 and T ( A 2 ) A 1 ;

     
  2. (ii)
    there exists a function ψ Ψ such that
    p ( T x , T y ) ψ ( p ( x , y ) ) , for all  ( x , y ) A 1 × A 2 .
     

Then T has a unique fixed point that belongs to A 1 A 2 .

Proof Since A 1 and A 2 are closed subsets of X, ( Y , d ) is an S-complete Hausdorff uniform space. Define the mapping α : Y × Y [ 0 , ) by
α ( x , y ) = { 1 if  ( x , y ) ( A 1 × A 2 ) ( A 2 × A 1 ) , 0 otherwise .
From (ii) and the definition of α, we can write
α ( x , y ) p ( T x , T y ) ψ ( p ( x , y ) ) ,

for all x , y Y . Thus T is an α-ψ-contractive mapping.

Let ( x , y ) Y × Y such that α ( x , y ) 1 . If ( x , y ) A 1 × A 2 , from (i), ( T x , T y ) A 2 × A 1 , which implies that α ( T x , T y ) 1 . If ( x , y ) A 2 × A 1 , from (i), ( T x , T y ) A 1 × A 2 , which implies that α ( T x , T y ) 1 . Thus in all cases, we have α ( T x , T y ) 1 . This implies that T is α-admissible.

Also, from (i), for any a A 1 , we have ( a , T a ) A 1 × A 2 , which implies that α ( a , T a ) 1 .

Now, let { x n } be a sequence in X such that α ( x n , x n + 1 ) 1 for all n and x n x X as n . This implies from the definition of α that
( x n , x n + 1 ) ( A 1 × A 2 ) ( A 2 × A 1 ) , for all  n .
Since ( A 1 × A 2 ) ( A 2 × A 1 ) is a closed subset of X with respect to the topological space ( X , τ ( ϑ ) ) , we get
( x , x ) ( A 1 × A 2 ) ( A 2 × A 1 ) ,

which implies that x A 1 A 2 . Thus we can easily get from the definition of α that α ( x n , x ) 1 for all n.

Finally, let x , y Fix ( T ) . From (i), this implies that x , y A 1 A 2 . So, for any z Y , we have α ( z , x ) 1 and α ( z , y ) 1 . Thus condition (H) is satisfied.

Now, all the hypotheses of Theorem 2.6 are satisfied, and we deduce that T has a unique fixed point that belongs to A 1 A 2 (from (i)). □

Declarations

Acknowledgements

The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.

Authors’ Affiliations

(1)
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology
(2)
Department of Mathematics, Quaid-i-Azam University
(3)
Department of Mathematics, Atilim University
(4)
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University

References

  1. Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014View ArticleMathSciNetGoogle Scholar
  2. Karapınar E, Samet B: Generalized α - ψ -contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486Google Scholar
  3. Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212 10.1186/1687-1812-2012-212Google Scholar
  4. Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ciric generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24 10.1186/1687-1812-2013-24Google Scholar
  5. Ali MU, Kamran T:On ( α , ψ ) -contractive multi-valued mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 137 10.1186/1687-1812-2013-137Google Scholar
  6. Amiri P, Rezapour S, Shahzad N: Fixed points of generalized α - ψ -contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2013. 10.1007/s13398-013-0123-9Google Scholar
  7. Minak G, Altun I: Some new generalizations of Mizoguchi-Takahashi type fixed point theorem. J. Inequal. Appl. 2013., 2013: Article ID 493 10.1186/1029-242X-2013-493Google Scholar
  8. Ali MU, Kamran T, Sintunavarat W, Katchang P: Mizoguchi-Takahashi’s fixed point theorem with α , η functions. Abstr. Appl. Anal. 2013., 2013: Article ID 418798Google Scholar
  9. Abdeljawad T: Meir-Keeler α -contractive fixed and common fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 19 10.1186/1687-1812-2013-19Google Scholar
  10. Karapınar E, Kumam P, Salimi P: On α - ψ -Meir-Keeler contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 94 10.1186/1687-1812-2013-94Google Scholar
  11. Chen CM, Karapınar E: Fixed point results for the alpha-Meir-Keeler contractions on partial Hausdorff metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 410 10.1186/1029-242X-2013-410Google Scholar
  12. Salimi P, Latif A, Hussain N: Modified α - ψ -contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151 10.1186/1687-1812-2013-151Google Scholar
  13. Hussain N, Salimi P, Latif A: Fixed point results for single and set-valued α - η - ψ -contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 212 10.1186/1687-1812-2013-212Google Scholar
  14. Mohammadi B, Rezapour S: On modified α - φ -contractions. J. Adv. Math. Stud. 2013, 6: 162–166.MathSciNetGoogle Scholar
  15. Berzig, M, Karapınar, E: Note on ‘Modified α-ψ-contractive mappings with application’. Thai J. Math. 12 (2014)Google Scholar
  16. Ali MU, Kamran T, Karapınar E: ( α , ψ , ξ ) -Contractive multi-valued mappings. Fixed Point Theory Appl. 2014., 2014: Article ID 7 10.1186/1687-1812-2014-7Google Scholar
  17. Ali MU, Kamran T, Karapınar E:A new approach to ( α , ψ ) -contractive nonself multivalued mappings. J. Inequal. Appl. 2014., 2014: Article ID 71 10.1186/1029-242X-2014-71Google Scholar
  18. Ali MU, Kamran T, Kiran Q:Fixed point theorem for ( α , ψ , ϕ ) -contractive mappings on spaces with two metrics. J. Adv. Math. Stud. 2014, 7: 8–11.MathSciNetGoogle Scholar
  19. Ali MU, Kiran Q, Shahzad N: Fixed point theorems for multi-valued mappings involving α -function. Abstr. Appl. Anal. 2014., 2014: Article ID 409467Google Scholar
  20. Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical weak contractive conditions. Fixed Point Theory 2003, 4: 79–89.MathSciNetGoogle Scholar
  21. Karapınar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011, 24: 822–825. 10.1016/j.aml.2010.12.016View ArticleMathSciNetGoogle Scholar
  22. Karapınar E, Sadarangani K:Fixed point theory for cyclic ( ϕ - ψ ) -contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69 10.1186/1687-1812-2011-69Google Scholar
  23. Pacurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002View ArticleMathSciNetGoogle Scholar
  24. Rus IA: Cyclic representations and fixed points. Ann. ‘Tiberiu Popoviciu’ Sem. Funct. Equ. Approx. Convexity 2005, 3: 171–178.Google Scholar
  25. Pacurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181–1187. 10.1016/j.na.2009.08.002View ArticleMathSciNetGoogle Scholar
  26. Hussain N, Karapınar E, Sedghi S, Shobkolaei N, Firouzain S:Cyclic ( ϕ ) -contractions in uniform spaces and related fixed point results. Abstr. Appl. Anal. 2014., 2014: Article ID 976859Google Scholar
  27. Bourbaki N Actualités Scientifiques et Industrielles 1142. In Elements de mathematique. Fasc. II. Livre III: Topologie generale. Chapitre 1: Structures topologiques. Chapitre 2: Structures uniformes. 4th edition. Hermann, Paris; 1965.Google Scholar
  28. Zeidler E 1. In Nonlinear Functional Analysis and Its Applications. Springer, New York; 1986.View ArticleGoogle Scholar
  29. Aamri M, El Moutawakil D: Common fixed point theorems for E -contractive or E -expansive maps in uniform spaces. Acta Math. Acad. Paedagog. Nyházi. 2004, 20(1):83–91.MathSciNetGoogle Scholar
  30. Aamri M, El Moutawakil D: Weak compatibility and common fixed point theorems for A -contractive and E -expansive maps in uniform spaces. Serdica Math. J. 2005, 31: 75–86.MathSciNetGoogle Scholar
  31. Aamri M, Bennani S, El Moutawakil D: Fixed points and variational principle in uniform spaces. Sib. Electron. Math. Rep. 2006, 3: 137–142.MathSciNetGoogle Scholar
  32. Agarwal RP, O’Regan D, Papageorgiou NS: Common fixed point theory for multi-valued contractive maps of Reich type in uniform spaces. Appl. Anal. 2004, 83: 37–47. 10.1080/00036810310001620063View ArticleMathSciNetGoogle Scholar

Copyright

© Ali et al.; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. 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.