Open Access

Fixed point theorems in convex metric spaces

Fixed Point Theory and Applications20122012:164

https://doi.org/10.1186/1687-1812-2012-164

Received: 28 February 2012

Accepted: 30 August 2012

Published: 25 September 2012

Abstract

In this paper, we study some fixed point theorems for self-mappings satisfying certain contraction principles on a convex complete metric space. In addition, we investigate some common fixed point theorems for a Banach operator pair under certain generalized contractions on a convex complete metric space. Finally, we also improve and extend some recent results.

MSC:47H09, 47H10, 47H19, 54H25.

Keywords

Banach operator pair common fixed point convex metric spaces fixed point

1 Introduction

In 1970, Takahashi [1] introduced the notion of convexity in metric spaces and studied some fixed point theorems for nonexpansive mappings in such spaces. A convex metric space is a generalized space. For example, every normed space and cone Banach space is a convex metric space and convex complete metric space, respectively. Subsequently, Beg [2], Beg and Abbas [3, 4], Chang, Kim and Jin [5], Ciric [6], Shimizu and Takahashi [7], Tian [8], Ding [9], and many others studied fixed point theorems in convex metric spaces.

The purpose of this paper is to study the existence of a fixed point for self-mappings defined on a nonempty closed convex subset of a convex complete metric space that satisfies certain conditions. We also study the existence of a common fixed point for a Banach operator pair defined on a nonempty closed convex subset of a convex complete metric space that satisfies suitable conditions. Our results improve and extend some of Karapinar’s results in [10] from a cone Banach space to a convex complete metric space. For instance, Karapinar proved that for a closed convex subset C of a cone Banach space X with the norm x p = d ( x , 0 ) , if a mapping T : C C satisfies the condition
d ( x , T x ) + d ( y , T y ) q d ( x , y )

for all x , y C , where 2 q < 4 , then T has at least one fixed point. Letting x = y in the above inequality, it is easy to see that T is an identity mapping. In this paper, the above result is improved and extended to a convex complete metric space.

2 Preliminaries

Definition 2.1 (see [11])

Let ( X , d ) be a metric space and I = [ 0 , 1 ] . A mapping W : X × X × I X is said to be a convex structure on X if for each ( x , y , λ ) X × X × I and u X ,
d ( u , W ( x , y , λ ) ) λ d ( u , x ) + ( 1 λ ) d ( u , y ) .

A metric space ( X , d ) together with a convex structure W is called a convex metric space, which is denoted by ( X , d , W ) .

Example 2.2 Let ( X , ) be a normed space. The mapping W : X × X × I X defined by W ( x , y , λ ) = λ x + ( 1 λ ) y for each x , y X , λ I is a convex structure on X.

Definition 2.3 (see [11])

Let ( X , d , W ) be a convex metric space. A nonempty subset C of X is said to be convex if W ( x , y , λ ) C whenever ( x , y , λ ) C × C × I .

Definition 2.4 (see [3])

Let ( X , d , W ) be a convex metric space and C be a convex subset of X. A self-mapping f on C has a property (I) if f ( W ( x , y , λ ) ) = W ( f ( x ) , f ( y ) , λ ) for each x , y C and λ I .

Example 2.5 If ( X , ) is a normed space, then every affine mapping on a convex subset of X has the property (I).

Definition 2.6 Let f , g : X X . A point x X is called
  1. (i)

    a fixed point of f if f ( x ) = x ,

     
  2. (ii)

    a coincidence point of a pair ( f , g ) if f ( x ) = g ( x ) ,

     
  3. (iii)

    a common fixed point of a pair ( f , g ) if f ( x ) = g ( x ) = x .

     

F ( f ) , C ( f , g ) , and F ( f , g ) denote the set of all fixed points of f, coincidence points of the pair ( f , g ) , and common fixed points of the pair ( f , g ) , respectively.

Definition 2.7 (see [12, 13])

The ordered pair ( f , g ) of two self-maps of a metric space ( X , d ) is called a Banach operator pair if F ( g ) is f-invariant, namely f ( F ( g ) ) F ( g ) .

Example 2.8 (i) Let ( X , d ) be a metric space and k 0 . If the self-maps f, g of X satisfy d ( g ( f ( x ) ) , f ( x ) ) k d ( g ( x ) , x ) for all x X , then ( f , g ) is a Banach operator pair.
  1. (ii)

    It is obvious that a commuting pair ( f , g ) of self-maps on X (namely f g ( x ) = g f ( x ) for all x X ) is a Banach operator pair, but the converse is generally not true. For example, let X = R with the usual norm, and let f ( x ) = x 2 2 x , g ( x ) = x 2 x 3 for all x X , then F ( g ) = { 1 , 3 } . Moreover, f ( F ( g ) ) F ( g ) implies that ( f , g ) is a Banach operator pair, but the pair ( f , g ) does not commute.

     

In [10], Karapinar obtained the following theorems.

Theorem 2.9 (see Theorem 2.4 of [10])

Let C be a closed and convex subset of a cone Banach space X with the norm x p = d ( x , 0 ) , and T : C C be a mapping which satisfies the condition
d ( x , T x ) + d ( y , T y ) q d ( x , y )

for all x , y C , where 2 q < 4 . Then, T has at least one fixed point.

Theorem 2.10 (see Theorem 2.6 of [10])

Let C be a closed and convex subset of a cone Banach space X with the norm x p = d ( x , 0 ) , and T : C C be a mapping which satisfies the condition
d ( T x , T y ) + d ( x , T x ) + d ( y , T y ) r d ( x , y )

for all x , y C , where 2 r < 5 . Then, T has at least one fixed point.

3 Main results

To prove the next theorem, we need the following lemma.

Lemma 3.1 Let ( X , d , W ) be a convex metric space, then the following statements hold:
  1. (i)

    d ( x , y ) = d ( x , W ( x , y , λ ) ) + d ( y , W ( x , y , λ ) ) for all ( x , y , λ ) X × X × I .

     
  2. (ii)

    d ( x , W ( x , y , 1 2 ) ) = d ( y , W ( x , y , 1 2 ) ) = 1 2 d ( x , y ) for all x , y X .

     
Proof (i) For any ( x , y , λ ) X × X × I , we have
d ( x , y ) d ( x , W ( x , y , λ ) ) + d ( y , W ( x , y , λ ) ) ( 1 λ ) d ( x , y ) + λ d ( x , y ) = d ( x , y ) .
Therefore, d ( x , y ) = d ( x , W ( x , y , λ ) ) + d ( y , W ( x , y , λ ) ) holds.
  1. (ii)
    Let x , y X . By the definition of W and using (i), we have
    d ( x , W ( x , y , 1 2 ) ) 1 2 d ( x , y ) = 1 2 d ( x , W ( x , y , 1 2 ) ) + 1 2 d ( y , W ( x , y , 1 2 ) ) .
     
Therefore,
1 2 d ( x , W ( x , y , 1 2 ) ) 1 2 d ( y , W ( x , y , 1 2 ) ) .
Similarly,
1 2 d ( y , W ( x , y , 1 2 ) ) 1 2 d ( x , W ( x , y , 1 2 ) ) .
Therefore, d ( x , W ( x , y , 1 2 ) ) = d ( y , W ( x , y , 1 2 ) ) . Now, from (i), we obtain
d ( x , W ( x , y , 1 2 ) ) = d ( y , W ( x , y , 1 2 ) ) = 1 2 d ( x , y )

for all x , y C , and the proof of the lemma is complete. □

The following theorem improves and extends Theorem 2.6 in [10].

Theorem 3.2 Let C be a nonempty closed convex subset of a convex complete metric space ( X , d , W ) and f be a self-mapping of C. If there exist a, b, c, k such that
(3.1)
(3.2)

for all x , y C , then f has at least one fixed point.

Proof Suppose x 0 C is arbitrary. We define a sequence { x n } n = 1 in the following way:
x n = W ( x n 1 , f ( x n 1 ) , 1 2 ) , n = 1 , .
(3.3)
As C is convex, x n C for all n N . By Lemma 3.1(ii) and (3.3), we have
(3.4)
(3.5)
for all n N . Now, by substituting x with x n and y with x n 1 in (3.2), we get
a d ( x n , f ( x n ) ) + b d ( x n 1 , f ( x n 1 ) ) + c d ( f ( x n ) , f ( x n 1 ) ) k d ( x n , x n 1 )
for all n N . Therefore, from (3.4) and (3.5), it follows that
2 a d ( x n , x n + 1 ) + 2 b d ( x n , x n 1 ) + c d ( f ( x n ) , f ( x n 1 ) ) k d ( x n , x n 1 )
(3.6)
for all n N . Let c be a nonnegative number. Using the triangle inequality, (3.4) and (3.5), we obtain
2 c d ( x n , x n + 1 ) c d ( x n , x n 1 ) c d ( f ( x n ) , f ( x n 1 ) )
for all n N . Similarly, for the case c < 0 , we have
2 c d ( x n , x n + 1 ) + c d ( x n , x n 1 ) c d ( f ( x n ) , f ( x n 1 ) )
for all n N . Therefore, for each case we have
2 c d ( x n , x n + 1 ) | c | d ( x n , x n 1 ) c d ( f ( x n ) , f ( x n 1 ) )
(3.7)
for all n N . Now, from (3.6) and (3.7), it follows that
2 a d ( x n , x n + 1 ) + 2 b d ( x n , x n 1 ) + 2 c d ( x n , x n + 1 ) | c | d ( x n , x n 1 ) k d ( x n , x n 1 )
for all n N . This implies
d ( x n , x n + 1 ) k 2 b + | c | 2 ( a + c ) d ( x n , x n 1 )
for all n N . From (3.1), k 2 b + | c | 2 ( a + c ) [ 0 , 1 ) , and hence, { x n } n = 1 is a contraction sequence in C. Therefore, it is a Cauchy sequence. Since C is a closed subset of a complete space, there exists v C such that lim n x n = v . Therefore, the triangle inequality and (3.4) imply lim n f ( x n ) = v . Now, by substituting x with v and y with x n in (3.2), we obtain
a d ( v , f ( v ) ) + b d ( x n , f ( x n ) ) + c d ( f ( v ) , f ( x n ) ) k d ( v , x n )
for all n N . Letting n in the above inequality, it follows that
( a + c ) d ( v , f ( v ) ) 0 .

Since a + c is positive from (3.1), it follows that d ( v , f ( v ) ) = 0 . Therefore, f ( v ) = v and the proof of the theorem is complete. □

The following corollary improves and extends Theorem 2.4 in [10].

Corollary 3.3 Let ( X , d , W ) be a convex complete metric space and C be a nonempty closed convex subset of X. Suppose that f is a self-map of C. If there exist a, b, k such that
2 b k < 2 ( a + b ) , a d ( x , f ( x ) ) + b d ( y , f ( y ) ) k d ( x , y )

for all x , y C , then F ( f ) is a nonempty set.

Proof Set c = 0 in Theorem 3.2. □

Theorem 3.4 Let ( X , d , W ) be a convex complete metric space and C be a nonempty subset of X. Suppose that f, g are self-mappings of C, and there exist a, b, c, k such that
(3.8)
(3.9)

for all x , y C . If ( f , g ) is a Banach operator pair, g has the property (I) and F ( g ) is a nonempty closed subset of C, then F ( f , g ) is nonempty.

Proof From (3.9), we obtain
a d ( x , f ( x ) ) + b d ( y , f ( y ) ) + c d ( f ( x ) , f ( y ) ) k d ( x , y )
(3.10)

for all x , y F ( g ) . F ( g ) is convex because g has the property (I). It follows from Theorem 3.2 that F ( f , g ) is nonempty. □

Theorem 3.5 Let ( X , d , W ) be a convex complete metric space and C be a nonempty subset of X. Suppose that f, g are self-mappings of C, F ( g ) is a nonempty closed subset of C, and there exist a, b, c, k such that
(3.11)
(3.12)

for all x , y C . If ( f , g ) is a Banach operator pair and g has the property (I), then F ( f , g ) is nonempty.

Proof Since ( f , g ) is a Banach operator pair from (3.12), we have
a d ( x , f ( x ) ) + b d ( y , f ( y ) ) + c d ( f ( x ) , f ( y ) ) k d ( x , y )

for all x , y F ( g ) . Because g has the property (I) and F ( g ) is closed, Theorem 3.2 guaranties that F ( f , g ) is nonempty. □

Declarations

Acknowledgements

The author is grateful to Bu-Ali Sina University for supporting this research.

Authors’ Affiliations

(1)
Department of Mathematics, Bu-Ali Sina University

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© Moosaei; licensee Springer 2012

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