Open Access

Some common fixed point and invariant approximation results for nonexpansive mappings in convex metric space

Fixed Point Theory and Applications20142014:182

https://doi.org/10.1186/1687-1812-2014-182

Received: 11 June 2014

Accepted: 12 August 2014

Published: 2 September 2014

Abstract

In this work, we introduce a new class of self-maps which satisfy the (E.A.) property with respect to some q M , where M is q-starshaped subset of a convex metric space and common fixed point results are established for this new class of self-maps. After that we obtain some invariant approximation results as an application. Our results represent a very strong variant of the several recent results existing in the literature. We also provide some illustrative examples in the support of proved results.

MSC:46T99, 47H10, 54H25.

Keywords

EA-property common fixed point best approximation compatible maps subcompatible maps

1 Introduction

In 1976, Jungck [1] established some common fixed point results for a pair of commuting self-maps in the setting of complete metric space. The first ever attempt to relax the commutativity of mappings was initiated by Sessa [2] who introduced a class of noncommuting maps called ‘namely’ weak commutativity. Further, in order to enlarge the domain of noncommuting mappings, Jungck [3] in 1986 introduced the concept of ‘compatible maps’ as a generalization of weakly commuting maps.

Definition 1 Two self-maps I and T of a metric space ( X , d ) are called compatible if and only if
lim n d ( I T x n , T I x n ) = 0 ,

whenever { x n } is a sequence in X such that lim n I x n = lim n T x n = t for some t X .

In 2002, Aamri and Moutawakil [4] obtained the notion of (E.A.) property which enables us to study the existence of a common fixed points of self-maps satisfying nonexpansive or Lipschitz type condition in the setting of non-complete metric space.

Definition 2 Two self-maps I and T of a metric space ( X , d ) are said to satisfy the (E.A.) property if there exists a sequence { x n } in X such that
lim n I x n = lim n T x n = t for some  t X .

On the other side, in 1970 Takahashi [5] introduced the notion of convexity into the metric space, studied properties of such spaces and proved several fixed point theorems for nonexpansive mappings. Afterward Guay et al. [6], Beg and Azam [7], Fu and Huang [8], Ding [9], Ćirić et al. [10], and many others have studied fixed point theorem in convex metric spaces. In the recent past, fixed point theorems have been extensively applied to best approximation theory. Meinardus [11] was the first who employed the Schauder’s fixed point theorem to prove a result regarding invariant approximation. Later on, Brosowski [12] generalized the result of Meinardus under different settings. Further significant contribution to this area was made by a number of authors (see [1335]). Many of them considered the pair of commuting or noncommuting mappings in the setting of normed or Banach spaces. In 1992, Beg et al. [36] proved some results on the existence of a common fixed point in the setting of a convex metric space and utilized the same to prove the best approximation results. After that, several authors studied common fixed point and invariant approximation results in the setting of convex metric space (see [3640] and references therein).

In this work, we introduce a new class of self-maps which satisfy the (E.A.) property with respect to some q M , where M is q-starshaped subset of a convex metric space and establish some common fixed point results for this class of self-maps. After that we obtain some invariant approximation results as application. Our results represent a very strong variant of the several recent results existing in the literature.

2 Preliminaries

Firstly, we recall some useful definitions and auxiliary results that will be needed in the sequel. Throughout this paper, and denote the set of natural numbers and the set of real numbers, respectively.

Definition 3 [5]

Let ( X , d ) be a metric space. A continuous mapping W : X × X × [ 0 , 1 ] X is called a convex structure on X if, for all x , y X and λ [ 0 , 1 ] , we have
d ( u , W ( x , y , λ ) ) λ d ( u , x ) + ( 1 λ ) d ( u , y )
(2.1)

for all u X .

A metric space ( X , d ) equipped with a convex structure is called a convex metric space.

Definition 4 A subset M of a convex metric space ( X , d ) is called a convex set if W ( x , y , λ ) M for all x , y M and λ [ 0 , 1 ] . The set M is said to be q-starshaped if there exists q M such that W ( x , q , λ ) M for all x M and λ [ 0 , 1 ] . A set M is called starshaped if it is q-starshaped with respect to any q M .

Clearly, each convex set M is starshaped but the converse assertion is not true. Thus, the class of starshaped sets properly contains the class of convex sets.

Definition 5 A convex metric space ( X , d ) is said to satisfy the Property (I), if for all x , y , z X and λ [ 0 , 1 ] ,
d ( W ( x , z , λ ) , W ( y , z , λ ) ) λ d ( x , y ) .
(2.2)

A normed linear space X and each of its convex subset are simple examples of convex metric spaces with W given by W ( x , y , λ ) = λ x + ( 1 λ ) y for all x , y X and 0 λ 1 . Also, Property (I) is always satisfied in a normed linear space. There are many convex metric spaces which are not normed linear spaces (see [5, 6]). For further information on a convex metric space, refer to [510, 3642].

Definition 6 Let ( X , d ) be a convex metric space and M a subset of X. A mapping I : M M is said to be
  1. (1)

    affine, if M is convex and I ( W ( x , y , λ ) ) = W ( I x , I y , λ ) for all x , y M and λ [ 0 , 1 ] ;

     
  2. (2)

    q-affine, if M is q-starshaped and I ( W ( x , q , λ ) ) = W ( I x , q , λ ) for all x M and λ [ 0 , 1 ] .

     

In [43] Pant define the concept of reciprocal continuity as follows.

Definition 7 Let ( X , d ) be a metric space and I , T : X X . Then the pair ( I , T ) is said to be reciprocally continuous if and only if
lim n I T x n = I t and lim n T I x n = T t

whenever { x n } is a sequence in X such that lim n I x n = lim n T x n = t for some t X .

It is easy to see that if I and T are continuous, then the pair ( I , T ) is reciprocally continuous but the converse is not true in general (see [[44], Example 2.3]). Moreover, in the setting of common fixed point theorems for compatible pairs of self-mappings satisfying some contractive conditions, continuity of one of the mappings implies their reciprocal continuity.

Definition 8 [45]

A pair ( I , T ) of self-maps of a metric space ( X , d ) is said to be subcompatible if there exists a sequence { x n } such that
lim n I x n = lim n T x n = t for some  t X  and  lim n d ( I T x n , T I x n ) = 0 .

Obviously, compatible maps which satisfy the (E.A.) property are subcompatible but the converse statement does not hold in general (see [40], Example 2.5).

Definition 9 Let ( X , d ) be a metric space, M a nonempty subset of X, and I and T be self-maps of M. A point x M is a coincidence point (common fixed point) of I and T if I x = T x ( I x = T x = x ). The set of coincidence points of I and T is denoted by C ( I , T ) and the set of fixed points of I and T is denoted by F ( I ) and F ( T ) , respectively. The pair { I , T } is called:
  1. (1)

    Commuting if I T x = T I x for all x M .

     
  2. (2)

    Weakly compatible [46] if I T x = T I x for all x C ( I , T ) .

     
  3. (3)

    Banach operator pair [24] if the set F ( I ) is T-invariant, i.e. T ( F ( I ) ) F ( I ) .

     

For more details about these classes, one can refer [27, 47].

Definition 10 [19]

Let M be a q-starshaped subset of convex metric space ( X , d ) such that q F ( I ) and is both I- and T-invariant. Then the self-maps I and T are called R-subweakly commuting on M if for all x M , there exists a real number R > 0 such that d ( I T x , T I x ) R dist ( I x , [ q , T x ] ) , where [ q , x ] = { W ( x , q , λ ) : 0 λ 1 } .

Clearly, R-subweakly commuting maps are compatible but the converse assertion is not necessarily true (see [31], Example 15).

For a nonempty subset M of a metric space ( X , d ) and p X , an element y M is called a best approximation to p if d ( p , y ) = dist ( p , M ) , where dist ( p , M ) = inf { d ( p , z ) : z M } . The set of all best approximations to p is denoted by B M ( p ) .

3 Main results

We start to this section with following definition.

Definition 11 Let M be a q-starshaped subset of a convex metric space ( X , d ) and let I , T : M M with q F ( I ) . The pair ( I , T ) is said to satisfy the (E.A.) property with respect to q if there exists a sequence { x n } in M such that for all λ [ 0 , 1 ]
lim n I x n = lim n T λ x n = t for some  t M ,
(3.1)

where T λ x = W ( T x , q , λ ) .

Obviously, if the pair ( I , T ) satisfies the (E.A.) property with respect to q, then I and T satisfy the (E.A.) property but the converse assertion is not necessarily true. This can be seen by the following simple example.

Example 12 Let X = R 2 be equipped with the metric d : R 2 × R 2 R defined as
d ( ( x 1 , x 2 ) , ( y 1 , y 2 ) ) = | x 1 y 1 | + | x 2 y 2 |
and let M = { ( x , y ) : x 1 , y 1 } . Then ( X , d ) is a convex metric space with W ( x , y , λ ) = λ x + ( 1 λ ) y , 0 λ 1 , and M is q-starshaped with q = ( 2 , 3 ) . Define I , T : M M by
I ( x , y ) = ( x , y ) and T ( x , y ) = ( 2 x 1 , 3 y 2 ) .
Firstly, we show I and T satisfy the (E.A.) property. Take a sequence { z n } = { ( x n , y n ) } in M such that z n ( 1 , 1 ) , then x n 1 and y n 1 . Thus
lim n I z n = lim n T z n = ( 1 , 1 ) M .
(3.2)
Now we will show that the pair ( I , T ) does not satisfy the (E.A.) property with respect to q = ( 2 , 3 ) . Take { z n } = { ( x n , y n ) } be any sequence in M, then lim n T 0 z n = ( 2 , 3 ) and hence
lim n I z n = lim n T 0 z n = ( 2 , 3 ) if and only if x n 2 , y n 3 .
But, for x n 2 , y n 3 , we get
lim n I z n = ( 2 , 3 ) ( 3 , 7 ) = lim n T 1 z n .
Thus, it is not possible to find a sequence { z n } in M such that for each λ [ 0 , 1 ]
lim n I x n = lim n T λ x n = t M .

So, the pair ( I , T ) does not satisfy the (E.A.) property with respect to q = ( 2 , 3 ) .

Remark 13 If M is convex subset of a convex metric space X and p is common fixed point of the self-maps I and T of M, then the pair ( I , T ) satisfies the (E.A.) property with respect to p but converse is not true in general. This can be seen by the following example.

Example 14 Let X = R be endowed with the usual metric and let M = [ 0 , 1 ] . Define I , T : M M by
I ( x ) = { 1 2 if  0 x 1 2 , 1 x if  1 2 x 1 , and T ( x ) = { 1 2 if  0 x < 1 4 , 1 4 if  1 4 x 1 2 , x 4 + 1 8 if  1 2 x 1 .
Clearly, X is a convex metric space with W ( x , y , λ ) = λ x + ( 1 λ ) y and M is 1 2 -starshaped. Take x n = 1 4 1 2 ( n + 1 ) for each n 1 .Then for each λ [ 0 , 1 ] , we have
lim n I ( x n ) = lim n T λ x n = 1 2 M .

Hence the pair ( I , T ) satisfies the (E.A.) property with respect to q = 1 2 , but I and T do not have a common fixed point.

The following lemma is a particular case of Theorem 4.1 of Chauhan and Pant [48].

Lemma 15 Let I and T be self-maps of a metric space ( X , d ) . If the pair ( I , T ) is subcompatible, reciprocally continuous and satisfy
d ( T x , T y ) λ max { d ( I x , I y ) , d ( I x , T x ) , d ( I y , T y ) , d ( I x , T y ) , d ( I y , T x ) }
(3.3)

for some λ ( 0 , 1 ) and all x , y X . Then I and T have a unique common fixed point in X.

Now we prove our first result.

Theorem 16 Let M be a nonempty q-starshaped subset of a convex metric space ( X , d ) with Property (I) and let I and T be continuous self-maps on M such that the pair ( I , T ) satisfies the (E.A.) property with respect to q. Assume that I is q-affine, cl ( T ( M ) ) is compact. If I and T are compatible and satisfy the inequality
d ( T x , T y ) max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) }
(3.4)

for all x , y M , then M F ( T ) F ( I ) ϕ .

Proof For each n N , we define T n : M M by
T n x = W ( T x , q , λ n ) for all  x M ,
(3.5)

where λ n is a sequence in ( 0 , 1 ) such that λ n 1 .

Now, we have to show that for each n N , the pair ( T n , I ) is subcompatible. Since I and T satisfy the (E.A.)-property with respect to q, there exists a sequence { x m } in M such that for all λ [ 0 , 1 ]
lim m I x m = lim m T λ x m = t M ,
(3.6)

where T λ x m = W ( T x m , q , λ ) .

Since λ n ( 0 , 1 ) , in the light of (3.5) and (3.6), for each n N , we have
lim m T n x m = lim m W ( T x m , q , λ n ) = lim m T λ n x m = t M .
Thus, we have
lim m I x m = lim m T n x m = t M .
(3.7)
Now, using the fact that I is q-affine and Property ( I ) is satisfied, we get
d ( T n I x m , I T n x m ) = d ( W ( T I x m , q , λ n ) , I ( W ( T x m , q , λ n ) ) ) = d ( W ( T I x m , q , λ n ) , W ( I T x m , q , λ n ) ) λ n d ( T I x m , I T x m ) .
(3.8)
Since I and T are compatible, in view of (3.6), we have
lim m d ( I T x m , T I x m ) = 0 .
Now, letting m in (3.8), we get
lim m d ( I T n x m , T n I x m ) = 0 .
(3.9)
Hence, on account of (3.7) and (3.9), it follows that the pair ( T n , I ) is subcompatible for each n N . Since I and T are continuous, for each n N , the pair ( T n , I ) is reciprocally continuous. Also, by (3.4),
d ( T n x , T n y ) = d ( W ( T x , q , λ n ) , W ( T y , q , λ n ) ) λ n d ( T x , T y ) , Property (I) λ n max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) } λ n max { d ( I x , I y ) , d ( I x , T n x ) , d ( I y , T n y ) , d ( I x , T n y ) , d ( I y , T n x ) }
(3.10)

for each x , y M and 0 < λ n < 1 . By Lemma 15, for each n N , there exists x n M such that x n = I x n = T n x n .

Now the compactness of cl ( T ( M ) ) implies that there exists a subsequence { T x m } of { T x n } such that T x m z as m . Further, it follows that
x m = T m x m = W ( T x m , q , λ m ) z as  m .

By the continuity of I and T, we obtain I z = z = T z . Thus, M F ( T ) F ( I ) ϕ . □

Now we present a nontrivial example in support of Theorem 16.

Example 17 Let X = R endowed with the usual metric and let M = [ 0 , 1 ] . Define I , T : M M by
I ( x ) = { 1 2 if  0 x 1 2 , 1 x if  1 2 x 1 , and T ( x ) = { 1 2 if  0 x 1 2 , x 2 + 1 4 if  1 2 x 1 .
(3.11)
Then ( X , d ) is a convex metric space with the convex structure W ( x , y , λ ) = λ x + ( 1 λ ) y . Firstly, we check the following:
  1. (a)

    I is q affine with q = 1 2 .

     
  2. (b)

    The pair ( I , T ) satisfies the (E.A.) property with respect to q = 1 2 .

     
  3. (c)

    I and T are compatible.

     
Proof (a) Let x [ 0 , 1 2 ] . Then W ( x , 1 2 , λ ) = λ x + ( 1 λ ) 1 2 [ 0 , 1 2 ] and hence
I ( W ( x , 1 2 , λ ) ) = 1 2 = λ 2 + ( 1 λ ) 1 2 = W ( I x , 1 2 , λ ) .
Again, if x [ 1 2 , 1 ] , then W ( x , 1 2 , λ ) [ 1 2 , 1 ] , therefore we have
I ( W ( x , 1 2 , λ ) ) = 1 W ( x , 1 2 , λ ) = 1 + λ 2 λ x = λ ( 1 x ) + ( 1 λ ) 1 2 = λ I x + ( 1 λ ) 1 2 = W ( I x , 1 2 , λ ) .
(3.12)

So, I ( W ( x , 1 2 , λ ) ) = W ( I x , 1 2 , λ ) for all x M and hence I is q-affine with q = 1 2 . □

Proof (b) Clearly, I ( q ) = q for q = 1 2 . Take x n = 1 2 1 n + 1 , n 1 , then for each n, x n [ 0 , 1 2 ) . So for each λ [ 0 , 1 ] , we have
lim n T λ x n = W ( 1 2 , 1 2 , λ ) = 1 2 = lim n I x n .

Thus, the pair ( I , T ) satisfies the (E.A.) property with respect to q = 1 2 . □

Proof (c) If { x n } be a sequence in M such that lim n I x n = lim n T x n = t for some t M , then t lies in the closure of both I ( M ) = [ 0 , 1 2 ] and T ( M ) = [ 1 2 , 3 4 ] , so t = 1 2 . Using the continuity of I and T we obtain
lim n T I x n = T ( 1 2 ) = 1 2 = I ( 1 2 ) = lim n I T x n .

Hence, I and T are compatible. □

Now we will show that the inequality (3.4) holds for each x , y M . If x = y , then d ( T x , T y ) = 0 and hence (3.4) obviously holds. Let x , y M with x y , then we have the following cases.
  1. (1)

    If x , y [ 0 , 1 2 ] , then d ( T x , T y ) = 0 and so inequality (3.4) trivially holds.

     
  2. (2)
    If x , y [ 1 2 , 1 ] , then
    d ( T x , T y ) = 1 2 | x y | < | x y | = d ( I x , I y ) .
     
Thus, the inequality (3.4) holds.
  1. (3)
    If x [ 0 , 1 2 ] and y [ 1 2 , 1 ] , then T x = 1 2 , T y = y 2 + 1 4 , and I x = 1 2 , I y = 1 y . Therefore
    d ( T x , T y ) = 1 2 | y 1 2 | < | y 1 2 | = d ( I x , I y ) .
     
Hence, the inequality (3.4) holds.
  1. (3)

    If x [ 1 2 , 1 ] and y [ 0 , 1 2 ] , then, due to the symmetric property of metric d, the inequality (3.4) follows from case 3.

     

So, for each x , y M , the maps I and T satisfy the inequality (3.4). Also, cl ( T ( M ) ) = [ 1 2 , 3 4 ] is compact, I and T are continuous. Thus, from the above discussion we conclude that I and T satisfy all the hypotheses of Theorem 16 and consequently, M F ( T ) F ( I ) ϕ . Here 1 2 M is a common fixed point of I and T.

Remark 18 Note that, in Example 17, T ( M ) = [ 1 2 , 3 4 ] [ 0 , 1 2 ] = I ( M ) . Also, most of the common fixed point results in which the pair of maps is taken to be commuting, weakly commuting, R-subweakly commuting, compatible, and weakly compatible guarantee the existence of a common fixed point under the hypothesis T ( M ) I ( M ) (for example see [1722, 25, 2830, 3640]). Thus, all these results are not applicable to finding the common fixed point of the maps I and T defined in Example 17.

Now, we present an example that will show if the condition ‘The pair ( I , T ) satisfy the (E.A.) property with respect to q’ of Theorem 16 fails to hold, then I and T may not have a common fixed point.

Example 19 Let X = [ 0 , ) be endowed with the usual metric and M = [ 0 , 2 ] . Define I , T : M M by
I ( x ) = { 1 3 if  0 x 1 3 , x if  1 3 x 2 , and T ( x ) = { 1 6 if  0 x 1 3 , x 2 if  1 3 x 2 .
(3.13)
Then ( X , d ) is a convex metric space with W ( x , y , λ ) = λ x + ( 1 λ ) y and M is q-starshaped with q = 1 3 . Clearly, I and T are continuous and cl ( T ( M ) ) = [ 1 6 , 1 ] is compact. Using a routine calculation as is done in Example 17, it can easily be shown that I is q-affine with q = 1 3 and also the maps I and T satisfy inequality (3.4) for each x , y M . Now we show that the map I and T do not satisfy the (E.A.) property. On the contrary, assume { x n } is a sequence in M such that
lim n I x n = lim n T x n = t for some  t M .
(3.14)
Then t lies in the closure of both I ( M ) = [ 1 3 , 2 ] and T ( M ) = [ 1 6 , 1 ] , so t [ 1 3 , 1 ] . Further, employing the definition of maps I and T with 3.14, we have
t = lim n T x n = 1 2 lim n I x n = t 2 .

This is not true for any t [ 1 3 , 1 ] and hence our assumption is wrong, so there does not exist any sequence { x n } in M such that 3.14 holds. Thus, the maps I and T do not satisfy the (E.A.) property and consequently the pair ( I , T ) does not satisfy (E.A.)-property with respect to q = 1 3 . Moreover, I and T are vacuously compatible and we observe I and T have no common fixed point.

Thus, if we relax the condition ‘The pair ( I , T ) satisfy the (E.A.) property with respect to q’ of Theorem 16, then I and T may not have a common fixed point.

The following corollaries immediately follow from Theorem 16.

Corollary 20 Let M be a nonempty q-starshaped subset of a convex metric space ( X , d ) with Property (I) and let I and T be continuous self-maps on M such that the pair ( I , T ) satisfies the (E.A.) property with respect to q. Assume that I is q-affine, cl ( T ( M ) ) is compact. If I and T are compatible and satisfy the inequality
d ( T x , T y ) max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , 1 / 2 [ dist ( I x , [ T y , q ] ) + dist ( I y , [ T x , q ] ) ] }
(3.15)

for all x , y M , then M F ( T ) F ( I ) ϕ .

Corollary 21 Let M be a nonempty q-starshaped subset of a convex metric space ( X , d ) with Property (I) and let I and T be continuous self-maps on M such that the pair ( I , T ) satisfies the (E.A.) property with respect to q. Assume that I is q-affine, cl ( T ( M ) ) is compact. If I and T are R-subweakly commuting and satisfy the inequality
d ( T x , T y ) max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) }
(3.16)

for all x , y M , then M F ( T ) F ( I ) ϕ .

4 Invariant approximation

Now, we present some invariant approximation results as an application of Theorem 16.

Theorem 22 Let I and T be self-maps of a convex metric space ( X , d ) with Property ( I ) , p F ( I ) F ( T ) , and M be a subset of X such that T ( δ M M ) M , where δM denotes the boundary of M. Suppose that B M ( p ) is nonempty, q-starshaped with I ( B M ( p ) ) B M ( p ) and I is q-affine and continuous on B M ( p ) . If the maps I and T are compatible, satisfy the (E.A.) property with respect to q on B M ( p ) , and also satisfy for all x , y B M ( p ) { p }
d ( T x , T y ) { d ( I x , I p ) if  y = p , max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) } if  y B M ( p ) ,
(4.1)

then I and T have a common fixed point in B M ( p ) , provided cl ( T ( B M ( p ) ) ) is compact and T is continuous on B M ( p ) .

Proof Let x B M ( p ) . Then for all λ ( 0 , 1 ) , we have
d ( p , W ( x , p , λ ) ) λ d ( p , x ) + ( 1 λ ) d ( p , p ) = λ d ( p , x ) < dist ( p , M ) .
Thus, it follows that { W ( x , p , λ ) : λ ( 0 , 1 ) } M = ϕ and so x δ M M . As T ( δ M M ) M , therefore T x M . Since I x B M ( p ) and p F ( I ) F ( T ) , on account of (4.1), we have
d ( T x , p ) = d ( T x , T p ) d ( I x , I p ) = d ( I x , p ) = dist ( p , M ) ,

which shows that T x B M ( p ) , and in all I and T are self-maps on B M ( p ) . In view of Theorem 16 there exists a z B M ( p ) such that z is a common fixed point of I and T. □

Example 23 Consider X = [ 0 , 1 ] equipped with the usual metric, M = ( 0 , 1 2 ] and define a mapping W : X × X × [ 0 , 1 ] X
W ( x , y , λ ) = λ x + ( 1 λ ) y .
Then ( X , d ) is a convex metric space with Property (I). Define I , T : X X by
I ( x ) = { 1 2 if  x [ 0 , 2 3 ) , 0 if  x ( 2 3 , 1 ] , 2 3 if  x = 2 3 and T ( x ) = { 1 2 if  x [ 0 , 1 2 ] ( 2 3 , 1 ] , x if  x ( 1 2 , 2 3 ] .
(4.2)

Clearly, F ( I ) = { 1 2 , 2 3 } , F ( T ) = [ 1 2 , 2 3 ] and T ( δ M M ) = T ( 1 2 ) = 1 2 ( 0 , 1 2 ] = M . Take p = 2 3 F ( I ) F ( T ) = { 1 2 , 2 3 } , then B M ( p ) = { 1 2 } . Here, we observe that B M ( p ) is nonempty, q = 1 2 -starshaped with I ( B M ( p ) ) = { 1 2 } B M ( p ) , and also I is q-affine and continuous on B M ( p ) .

Further, I and T are commuting on B M ( p ) and hence compatible. Also, cl ( T ( B M ( p ) ) ) = { 1 2 } is compact, T is continuous on B M ( p ) and on account of Remark 13, the pair ( I , T ) satisfies the (E.A.) property with respect to q = 1 2 . Moreover, it can easily be checked that I and T satisfy inequality 4.1 for all x , y B M ( p ) { p } . Thus all the conditions of Theorem 22 are satisfied and consequently I and T have a common fixed point in B M ( p ) . Here x = 1 2 is such point.

Corollary 24 Let I and T be self-maps of a convex metric space ( X , d ) with Property ( I ) , p F ( I ) F ( T ) , and M be a subset of X such that T ( δ M M ) M , where δM denotes the boundary of M. Suppose that B M ( p ) is nonempty, q-starshaped with I ( B M ( p ) ) B M ( p ) and I is q-affine and continuous on B M ( p ) . If the maps I and T are R-subweakly commuting, satisfy the (E.A.) property with respect to q on B M ( p ) and also satisfy for all x , y B M ( p ) { p }
d ( T x , T y ) { d ( I x , I p ) if  y = p , max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) } if  y B M ( p ) ,
(4.3)

then I and T have a common fixed point in B M ( p ) , provided cl ( T ( B M ( p ) ) ) is compact and T is continuous on B M ( p ) .

We define D = B M ( p ) C M I ( p ) , where C M I ( p ) = { x M : I x B M ( p ) } .

Theorem 25 Let I and T be self-maps of a convex metric space ( X , d ) with Property ( I ) , p F ( I ) F ( T ) , and M be a subset of X such that T ( δ M M ) M , where δM denotes the boundary of M. Suppose that D is nonempty, q-starshaped with I ( D ) D and I is q-affine and nonexpansive on D. If the maps I and T are compatible, satisfy the (E.A.) property with respect to q on D, and also satisfy for all x , y D { p }
d ( T x , T y ) { d ( I x , I p ) if  y = p , max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) } if  y D ,
(4.4)

then I and T have a common fixed point in B M ( p ) , provided cl ( T ( D ) ) is compact and T is continuous on D.

Proof Let x D . Then x B M ( p ) , and therefore, proceeding as in the proof of Theorem 22, we have T x B M ( p ) . Since I is nonexpansive and p F ( I ) F ( T ) , it follows from (4.4) that
d ( I T x , p ) = d ( I T x , I p ) d ( T x , p ) = d ( T x , T p ) d ( I x , p ) = dist ( p , M ) .

Thus I T x B M ( p ) and so T x C M I ( p ) , which gives T x D . Hence I and T are self-maps on D. Now, in the light of Theorem 16, there exists z B M ( p ) such that z is a common fixed point of I and T. □

Corollary 26 Let I and T be self-maps of a convex metric space ( X , d ) with Property ( I ) , p F ( I ) F ( T ) , and M be a subset of X such that T ( δ M M ) M , where δM denotes the boundary of M. Suppose that D is nonempty, q-starshaped with I ( D ) D , and I is q-affine and nonexpansive on D. If the maps I and T are R-subweakly commuting, satisfy the (E.A.) property with respect to q on D, and also satisfy for all x , y D { p }
d ( T x , T y ) { d ( I x , I p ) if  y = p , max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) } if  y D ,
(4.5)

then I and T have a common fixed point in B M ( p ) , provided cl ( T ( D ) ) is compact and T is continuous on D.

Let D M R , I ( p ) = B M ( p ) G R , I M ( p ) , where G M R , I ( p ) = { x M : d ( I x , p ) ( 2 R + 1 ) dist ( p , M ) } .

Theorem 27 Let I and T be self-maps of a convex metric space ( X , d ) with Property ( I ) , p F ( I ) F ( T ) , and M be a subset of X such that T ( δ M M ) M , where δM denotes the boundary of M. Suppose that D M R , I ( p ) is nonempty, q-starshaped with I ( D M R , I ( p ) ) D M R , I ( p ) , and I is q-affine and continuous on D M R , I ( p ) . If the maps I and T are R-subweakly commuting, satisfy the (E.A.) property with respect to q on D M R , I ( p ) , and also satisfy for all x , y D M R , I ( p ) { p }
d ( T x , T y ) { d ( I x , I p ) if  y = p , max { d ( I x , I y ) , dist ( I x , [ T x , q ] ) , dist ( I y , [ T y , q ] ) , dist ( I x , [ T y , q ] ) , dist ( I y , [ T x , q ] ) } if  y B M ( p ) ,
(4.6)

then I and T have a common fixed point in B M ( p ) , provided cl ( T ( D M R , I ( p ) ) ) is compact and T is continuous on D M R , I ( p ) .

Proof Let x D M R , I ( p ) , then using an argument similar to that in Theorem 22, we have T x B M ( p ) . Since I and T are R-subweakly commuting and p F ( I ) F ( T ) , on account of (4.6) it follows that
d ( I T x , p ) = d ( I T x , T p ) d ( I T x , T I x ) + d ( T I x , T p ) R dist ( T x , [ q , I x ] ) + d ( I 2 x , I p ) R d ( T x , I x ) + d ( I 2 x , I p ) R [ d ( T x , T p ) + d ( I x , T p ) ] + d ( I 2 x , I p ) R [ dist ( p , M ) + dist ( p , M ) ] + dist ( p , M ) = ( 2 R + 1 ) dist ( p , M ) .

Thus T x G R , I M ( p ) . Hence I and T are self-maps on D M R , I ( p ) . Regarding, Theorem 16, there exists z B M ( p ) such that z is a common fixed point of I and T. □

Remark 28
  1. (1)

    The class of compatible and R-subweakly commuting are different from the class of Banach operator pairs and so our results are different from the results of [41].

     
  2. (2)

    In the existing literature, common fixed point results are proved with the assumption T ( M ) I ( M ) but here we replace it with the assumption of the (E.A.) property with respect to some q M .

     

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Maharshi Dayanand University
(2)
Department of Mathematics, Govt. College Nagina

References

  1. Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83(41):261–263.View ArticleMathSciNetGoogle Scholar
  2. Sessa S: On a weak commutativity condition of mappings in fixed point consideration. Publ. Inst. Math. (Belgr.) 1982, 32: 149–153.MathSciNetGoogle Scholar
  3. Jungck G: Compatible mappings and common fixed points. Int. J. Math. Math. Sci. 1986, 4: 771–779.View ArticleMathSciNetGoogle Scholar
  4. Aamri M, El Moutawakil D: Some new common fixed point theorems under strict contractive conditions. J. Math. Anal. Appl. 2002, 270: 181–188. 10.1016/S0022-247X(02)00059-8View ArticleMathSciNetGoogle Scholar
  5. Takahashi WA: A convexity in metric spaces and nonexpansive mapping. I. Kodai Math. Semin. Rep. 1970, 22: 142–149. 10.2996/kmj/1138846111View ArticleGoogle Scholar
  6. Guay MD, Singh KL, Whitfield JHM: Fixed point theorems for nonexpansive mappings in convex metric spaces. Lecture Notes in Pure and Applied Mathematics 80. In Proceedings of Conference on Nonlinear Analysis. Dekker, New York; 1982:179–189.Google Scholar
  7. Beg I, Azam A: Fixed point on starshaped subset of convex metric spaces. Indian J. Pure Appl. Math. 1987, 18: 594–596.MathSciNetGoogle Scholar
  8. Fu JY, Huang NJ: Common fixed point theorems for weakly commuting mappings in convex metric spaces. J. Jiangxi Univ. 1991, 3: 39–43.MathSciNetGoogle Scholar
  9. Ding DP: Iteration processes for nonlinear mappings in convex metric spaces. J. Math. Anal. Appl. 1998, 132: 114–122.View ArticleGoogle Scholar
  10. Ćirić L, Rakočevic V, Radenović S, Rajović M, Lazović R: Common fixed point theorems for non-self mappings in metric spaces of hyperbolic type. J. Comput. Appl. Math. 2010, 233: 2966–2974. 10.1016/j.cam.2009.11.042View ArticleMathSciNetGoogle Scholar
  11. Meinardus G: Invarianz bei linearen approximationen. Arch. Ration. Mech. Anal. 1963, 14: 301–303.MathSciNetGoogle Scholar
  12. Brosowski B: Fixpunksatze in der approximations theorie. Mathematica 1969, 11: 195–220.MathSciNetGoogle Scholar
  13. Singh SP: An application of a fixed point theorem to approximation theory. J. Approx. Theory 1979, 25: 89–90. 10.1016/0021-9045(79)90036-4View ArticleMathSciNetGoogle Scholar
  14. Subrahmanyan PV: An application of a fixed point theorem to best approximation. J. Approx. Theory 1977, 20: 165–172. 10.1016/0021-9045(77)90070-3View ArticleGoogle Scholar
  15. Habiniak L: Fixed point theorems and invariant approximations. J. Approx. Theory 1989, 56: 241–244. 10.1016/0021-9045(89)90113-5View ArticleMathSciNetGoogle Scholar
  16. Sahab SA, Khan MS, Sessa S: A result in best approximation theory. J. Approx. Theory 1988, 55: 349–351. 10.1016/0021-9045(88)90101-3View ArticleMathSciNetGoogle Scholar
  17. Al-Thagafi MA: Common fixed points and best approximation. J. Approx. Theory 1996, 85: 318–323. 10.1006/jath.1996.0045View ArticleMathSciNetGoogle Scholar
  18. Shahzad N: Noncommuting maps and best approximation. Rad. Mat. 2000/01, 10: 77–83.MathSciNetGoogle Scholar
  19. Shahzad N: Invariant approximations and R -subweakly commuting maps. J. Math. Anal. Appl. 2001, 1: 39–45.View ArticleMathSciNetGoogle Scholar
  20. Pathak HK, Cho YJ, Kang SM: An application of fixed point theorems in best approximation theory. Int. J. Math. Math. Sci. 1998, 21(3):467–470. 10.1155/S0161171298000659View ArticleMathSciNetGoogle Scholar
  21. Jungck G, Hussain N: Compatible maps and invariant approximations. J. Math. Anal. Appl. 2007, 325: 1003–1012. 10.1016/j.jmaa.2006.02.058View ArticleMathSciNetGoogle Scholar
  22. Shahzad N: Invariant approximation, generalized I -contractions and R -subweakly commuting maps. Fixed Point Theory Appl. 2005, 1: 79–86.View ArticleMathSciNetGoogle Scholar
  23. Al-Thagafi MA, Shahzad N: Banach operator pair, common fixed points, invariant approximation, and *-nonexpansive multimaps. Nonlinear Anal. 2008, 69: 2733–2739. 10.1016/j.na.2007.08.047View ArticleMathSciNetGoogle Scholar
  24. Chen J, Li Z: Common fixed points for Banach operators in best approximations. J. Math. Anal. Appl. 2007, 336: 1466–1475. 10.1016/j.jmaa.2007.01.064View ArticleMathSciNetGoogle Scholar
  25. Hussain N, Jungck G:Common fixed point and invariant approximation result for noncommuting generalized ( f , g ) -nonexpansive maps. J. Math. Anal. Appl. 2006, 2: 851–861.View ArticleMathSciNetGoogle Scholar
  26. Rathee S, Kumar A: Some common fixed point and invariant approximation results with generalized almost contractions. Fixed Point Theory Appl. 2014., 2014: Article ID 23Google Scholar
  27. Agarwal RP, Bisht RK, Shahzad N: A comparison of various noncommuting conditions in metric fixed point theory and their applications. Fixed Point Theory Appl. 2014., 2014: Article ID 38Google Scholar
  28. O’Regan D, Shahzad N: Invariant approximation for generalized I -contraction. Numer. Funct. Anal. Optim. 2005, 26: 565–575. 10.1080/NFA-200067306View ArticleMathSciNetGoogle Scholar
  29. Shahzad N: On R -subweakly commuting maps and invariant approximations in Banach spaces. Georgian Math. J. 2005, 12: 157–162.MathSciNetGoogle Scholar
  30. Jungck G, Sessa S: Fixed point theorems in best approximation theory. Math. Jpn. 1995, 42: 249–252.MathSciNetGoogle Scholar
  31. Rathee S, Kumar A: Some common fixed point results for modified subcompatible maps and related invariant approximation results. Abstr. Appl. Anal. 2014., 2014: Article ID 505067Google Scholar
  32. Rathee S, Kumar A, Tas K: Invariant approximation results via common fixed point theorems for generalized weak contractions maps. Abstr. Appl. Anal. 2014., 2014: Article ID 752107Google Scholar
  33. Al-Thagafi MA, Shahzad N: Generalized I -nonexpansive selfmaps and invariant approximations. Acta Math. Sin. Engl. Ser. 2008, 24(5):867–876. 10.1007/s10114-007-5598-xView ArticleMathSciNetGoogle Scholar
  34. Shahzad N, Bassindowa G: Fixed point theorems for Suzuki-generalized nonexpansive mappings with application. J. Nonlinear Convex Anal. 2012, 13(4):657–666.MathSciNetGoogle Scholar
  35. Alghamdi MA, Dhompongsa S, Shahzad N: Common fixed points for Suzuki-generalized nonexpansive maps. J. Appl. Math. 2013., 2013: Article ID 206759Google Scholar
  36. Beg I, Shahzad N, Iqbal M: Fixed point theorems and best approximation in convex metric space. Approx. Theory Appl 1992, 8(4):97–105.MathSciNetGoogle Scholar
  37. Nashine HK, Imdad M: Common fixed point and invariant approximations for subcompatible mappings in convex metric space. Math. Commun. 2011, 16: 1–12.MathSciNetGoogle Scholar
  38. Beg I, Abbas M: Common fixed points and invariant approximations in convex metric space. Soochow J. Math. 2007, 33(4):1–10.MathSciNetGoogle Scholar
  39. Narang TD, Chandok S: Common fixed points and invariant approximations of R -subweakly commuting maps in convex metric spaces. Ukr. Math. J. 2011, 62(10):1585–1596. 10.1007/s11253-011-0451-xView ArticleMathSciNetGoogle Scholar
  40. Rouzkard F, Imdad M, Nashine HK: New common fixed point theorems and invariant approximation in convex metric space. Bull. Belg. Math. Soc. Simon Stevin 2012, 19: 311–328.MathSciNetGoogle Scholar
  41. Kutbi MA: Common fixed point and invariant approximation result. Fixed Point Theory Appl. 2013., 2013: Article ID 135Google Scholar
  42. Abbas M, Jovanović M, Radenović S, Sretenović A, Simić S: Abstract metric spaces and approximating fixed points of a pair of contractive type mappings. J. Comput. Anal. Appl. 2011, 13(2):243–253.MathSciNetGoogle Scholar
  43. Pant RP: Common fixed points for four mappings. Bull. Calcutta Math. Soc. 1998, 90: 281–286.MathSciNetGoogle Scholar
  44. Imdad M, Ali J, Tanveer M: Remarks on some recent metrical common fixed point theorems. Appl. Math. Lett. 2011, 24: 1165–1169. 10.1016/j.aml.2011.01.045View ArticleMathSciNetGoogle Scholar
  45. arXiv: http://arxiv.org/abs/0906.3159v1
  46. Jungck G: Common fixed points for noncontinuous nonself-maps on nonmetric spaces. Far East J. Math. Sci. 1996, 4(2):199–215.MathSciNetGoogle Scholar
  47. Kadelburg Z, Radenović S, Shahzad N: A note on various classes of compatible-type pairs of mappings and common fixed point theorems. Abstr. Appl. Anal. 2013., 2013: Article ID 697151Google Scholar
  48. Chauhan S, Pant BD: Fixed point theorems for compatible and subsequentially continuous mappings in Menger spaces. J. Nonlinear Sci. Appl. 2014, 7: 78–89.MathSciNetGoogle Scholar

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© Kumar and Rathee; 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.