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.
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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 , if a mapping satisfies the condition
for all , where , then T has at least one fixed point. Letting 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 be a metric space and . A mapping is said to be a convex structure on X if for each and ,
A metric space together with a convex structure W is called a convex metric space, which is denoted by .
Example 2.2 Let be a normed space. The mapping defined by for each , is a convex structure on X.
Definition 2.3 (see [11])
Let be a convex metric space. A nonempty subset C of X is said to be convex if whenever .
Definition 2.4 (see [3])
Let be a convex metric space and C be a convex subset of X. A self-mapping f on C has a property (I) if for each and .
Example 2.5 If is a normed space, then every affine mapping on a convex subset of X has the property (I).
Definition 2.6 Let . A point is called
-
(i)
a fixed point of f if ,
-
(ii)
a coincidence point of a pair if ,
-
(iii)
a common fixed point of a pair if .
, , and denote the set of all fixed points of f, coincidence points of the pair , and common fixed points of the pair , respectively.
The ordered pair of two self-maps of a metric space is called a Banach operator pair if is f-invariant, namely .
Example 2.8 (i) Let be a metric space and . If the self-maps f, g of X satisfy for all , then is a Banach operator pair.
-
(ii)
It is obvious that a commuting pair of self-maps on X (namely for all ) is a Banach operator pair, but the converse is generally not true. For example, let with the usual norm, and let , for all , then . Moreover, implies that is a Banach operator pair, but the pair 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 , and be a mapping which satisfies the condition
for all , where . 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 , and be a mapping which satisfies the condition
for all , where . 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 be a convex metric space, then the following statements hold:
-
(i)
for all .
-
(ii)
for all .
Proof (i) For any , we have
Therefore, holds.
-
(ii)
Let . By the definition of W and using (i), we have
Therefore,
Similarly,
Therefore, . Now, from (i), we obtain
for all , 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 and f be a self-mapping of C. If there exist a, b, c, k such that
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for all , then f has at least one fixed point.
Proof Suppose is arbitrary. We define a sequence in the following way:
As C is convex, for all . By Lemma 3.1(ii) and (3.3), we have
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for all . Now, by substituting x with and y with in (3.2), we get
for all . Therefore, from (3.4) and (3.5), it follows that
for all . Let c be a nonnegative number. Using the triangle inequality, (3.4) and (3.5), we obtain
for all . Similarly, for the case , we have
for all . Therefore, for each case we have
for all . Now, from (3.6) and (3.7), it follows that
for all . This implies
for all . From (3.1), , and hence, is a contraction sequence in C. Therefore, it is a Cauchy sequence. Since C is a closed subset of a complete space, there exists such that . Therefore, the triangle inequality and (3.4) imply . Now, by substituting x with v and y with in (3.2), we obtain
for all . Letting in the above inequality, it follows that
Since is positive from (3.1), it follows that . Therefore, and the proof of the theorem is complete. □
The following corollary improves and extends Theorem 2.4 in [10].
Corollary 3.3 Let 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
for all , then is a nonempty set.
Proof Set in Theorem 3.2. □
Theorem 3.4 Let 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1812-2012-164/MediaObjects/13663_2012_Article_265_Equ8_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1812-2012-164/MediaObjects/13663_2012_Article_265_Equ9_HTML.gif)
for all . If is a Banach operator pair, g has the property (I) and is a nonempty closed subset of C, then is nonempty.
Proof From (3.9), we obtain
for all . is convex because g has the property (I). It follows from Theorem 3.2 that is nonempty. □
Theorem 3.5 Let be a convex complete metric space and C be a nonempty subset of X. Suppose that f, g are self-mappings of C, is a nonempty closed subset of C, and there exist a, b, c, k such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1812-2012-164/MediaObjects/13663_2012_Article_265_Equ11_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1687-1812-2012-164/MediaObjects/13663_2012_Article_265_Equ12_HTML.gif)
for all . If is a Banach operator pair and g has the property (I), then is nonempty.
Proof Since is a Banach operator pair from (3.12), we have
for all . Because g has the property (I) and is closed, Theorem 3.2 guaranties that is nonempty. □
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The author is grateful to Bu-Ali Sina University for supporting this research.
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Moosaei, M. Fixed point theorems in convex metric spaces. Fixed Point Theory Appl 2012, 164 (2012). https://doi.org/10.1186/1687-1812-2012-164
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DOI: https://doi.org/10.1186/1687-1812-2012-164