- Research Article
- Open access
- Published:
Linking Contractive Self-Mappings and Cyclic Meir-Keeler Contractions with Kannan Self-Mappings
Fixed Point Theory and Applications volume 2010, Article number: 572057 (2010)
Abstract
Some mutual relations between p-cyclic contractive self-mappings, p-cyclic Kannan self-mappings, and Meir-Keeler p-cyclic contractions are stated. On the other hand, related results about the existence of the best proximity points and existence and uniqueness of fixed points are also formulated.
1. Introduction
In the last years, important attention is being devoted to extend the Fixed Point Theory by weakening the conditions on both the maps and the sets where those maps operate [1, 2]. For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space have fixed points if the weak fixed point property holds [1]. Further, increasing research interest relies on the generalization of Fixed Point Theory to more general spaces than the usual metric spaces such as, for instance, ordered or partially ordered spaces (see, e.g., [3–5]). Also, important fields of application of Fixed Point Theory exist nowadays in the investigation of the stability of complex continuous-time and discrete-time dynamic systems. The theory has been focused, in particular, on systems possessing internal lags, those being described by functional differential equations, those being characterized as hybrid dynamic systems and those being described by coupled continuous-time and discrete-time dynamics, [6–10]. On the other hand, Meir-Keeler self-mappings have received important attention in the context of Fixed Point Theory perhaps due to the associated relaxing in the required conditions for the existence of fixed points compared with the usual contractive mappings [11–14]. It also turns out from their definition that such self-mappings are less restrictive than strict contractive self-mappings so that their associated formalism is applicable to a wider class of real-life problems. Another interest of such self-mappings is their usefulness as a formal tool for the study of (
2)-cyclic contractions, even in the eventual case that the involved subsets of the metric space under study do not intersect, [12] so that there is no fixed point. In such a case, the usual role of fixed points is played by the best proximity points between adjacent subsets in the metric space. The underlying idea is that the best proximity points are fixed points if such subsets intersect while they play a close role to fixed points otherwise. On the other hand, there are also close links between contractive self-mappings and Kannan self-mappings [2, 15–17] with constant
(referred to in the following as
Kannan self-mappings). In fact,
-Kannan self-mappings are contractive for values of the contraction constant being less than 1/3 [17].
The objective of this paper is to formulate some connections between -cyclic contractive self-mappings,
-cyclic Meir-Keeler contractions, and
-cyclic
-Kannan self-mappings. In particular, the existence and uniqueness of potential fixed points and also the best proximity points are investigated. The importance of cyclic maps in some problems as, for instance, in the case that a controlled state-solution trajectory of a dynamic system has to be driven from a set to its adjacent one in a certain time due to technical requirements, is well known. Consider a metric space
and a self-mapping
such that
and
where
and
are nonempty subsets of
. Then,
is a 2-cyclic self-mapping what is said to be a 2-cyclic
-contraction self-mapping if it satisfies in addition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ1_HTML.gif)
for some real . The best proximity point is some
such that
. It turns out that if
, then
; that is,
is a fixed point of
since
[11–13]. If
, then
;
and
is a 2-cyclic nonexpansive self-mapping [12]. Nonexpansive mappings, in general, have received important attention in the last years. For instance, two hybrid methods are used in [18] to prove some strong convergence theorems. Those theorems are used to find a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space. The concept of a strongly relatively nonexpansive sequence in a Banach space is given in [19]. The associate properties are investigated and applied approximating a common fixed point of a countable family of relatively nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces. Also, the so-called
-times reasonable expansive and their properties self-mappings are investigated in [20].
1.1. Notation
,
,
, and
are the sets of nonnegative real and integer numbers and those of positive real and integer numbers, respectively,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ2_HTML.gif)
is the set of fixed points of a
-cyclic self-mapping
in a nonempty subset
of a metric space
.
is the set of the best proximity points in a subset
of a
-cyclic self-mapping
on
, namely, the union of a collection of nonempty subsets of a metric space
which do not intersect.
Contractive self-mappings with constant and Kannan self-mappings with constant
are referred to as
-contractions and
-Kannan self-mappings, respectively.
(referred to as "not
") is the negation of a logic proposition
.
2. About 2-Cyclic
-Contraction, 2-Cyclic
-Kannan Self-Mappings, Contractions of Meir-Keeler Type, and Some Mutual Relationships
The definition of -Kannan self-mappings
is as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ3_HTML.gif)
for some real [14, 15]. Let us extend the above concept in a natural way to 2-cyclic Kannan self-mappings by considering the definition of 2-cyclic
-contractions (1.1) as follows.
Definition 2.1.
Consider a metric space and a self-mapping
such that
and
where
and
are nonempty subsets of
. Then
is a 2-cyclic
-Kannan self-mapping for some real
if it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ4_HTML.gif)
for some . Definition 2.1 is a natural counterpart of (2.1) for
-Kannan self-mappings by taking into account the definition of a 2-cyclic
-contraction in (1.1).
Remark 2.2.
Let be a metric space and let
be a 2-cyclic
-contraction self-mapping with
and
being nonempty nondisjoint subsets of
. It turns out that
is also a contractive mapping with constant
.
Contraction self-mappings can also be 2-cyclic -Kannan self-mappings and vice-versa as addressed in the two following results:
Proposition 2.3.
Assume that is a 2-cyclic
-contraction self-mapping with
. Then,
is also a 2-cyclic
-Kannan self-mapping,
.
Proof.
The following inequalities follow from (1.1) and the triangle inequality of the distance map :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ5_HTML.gif)
for some , what leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ6_HTML.gif)
for ,
,
. Thus,
is a 2-cyclic
-Kannan self-mapping from Definition 2.1 for
if
. The proof is complete.
Proposition 2.4.
Assume that and
are closed disjoint nonempty bounded simply connected sets with
and
for some
,
. Assume also that
is a 2-cyclic
-Kannan self-mapping with
and
subject to the constraints
and
with
. Then,
is also a 2-cyclic
-contraction self-mapping for any real constant
.
Proof.
Since is a 2-cyclic
-Kannan self-mapping then from Definition 2.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ7_HTML.gif)
Also, one has that ,
, since
;
. Choosing
and
such that
, the triangle inequality for the distance map yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ8_HTML.gif)
so that one has since :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ9_HTML.gif)
provided that under the necessary conditions
and
. The above derivation remains valid by interchanging the roles of the sets
and
since
.
The following two results are direct.
Corollary 2.5.
Proposition 2.4 holds "mutatis-mutandis" if either or
is an open set.
Proof.
It follows under the same reasoning by taking in
if
is open and
in
if
is open.
Corollary 2.6.
Proposition 2.4 and Corollary 2.5 cannot be fulfilled for any if
.
Proof.
It is direct since and
and
are infinite.
Definition 2.7 (see [12]).
A 2-cyclic contraction self-mapping is of Meir-Keeler type if for any given
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ10_HTML.gif)
The subsequent result is concerned with 2-cyclic contraction self-mappings of Meir-Keeler type
Proposition 2.8.
Assume that is a 2-cyclic contraction self-mapping of Meir-Keeler type. Then, the following properties hold:
(i)If , then
and
;
, that is, all the best proximity points are also fixed points.
(ii)If , then either
or
;
.
Also, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ11_HTML.gif)
Proof.
Note from (2.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ12_HTML.gif)
for some ;
. Define
;
. Since
is a 2-cyclic contraction self-mapping of Meir-Keeler type, one has proceeding recursively with (2.10)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ13_HTML.gif)
so that (2.10) together with the constraint implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ14_HTML.gif)
If, in addition, , that is,
, then
(otherwise,
;
would be a contradiction), so that
and
such that
which are the best proximity points and also fixed points.
If , then
for some
which is not obviously a fixed point, since
, so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ15_HTML.gif)
for some defined by
. Furthermore,
since
(resp.,
), what yields
(resp.,
);
and
since
.
Definition 2.9.
A 2-cyclic -Kannan self-mapping
defined for some real
and some
(see Definitions 2.1 and 2.7) is of Meir-Keeler type if for any given
,
such that (2.8) holds.
Proposition 2.3, Definitions 2.1 and 2.9, and Proposition 2.8 yield directly the following result.
Proposition 2.10.
Assume that is a 2-cyclic contraction self-mapping of Meir- Keeler type with
. Then,
is also a 2-cyclic
-Kannan self-mapping,
. If
, then
.
From the definition of -contraction self-mappings and Definition 2.7 for Meir-Keeler type contraction self-mappings, the following result holds.
Proposition 2.11.
If is a
-contraction self-mapping of Meir-Keeler type, then for any given
,
and
  such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ16_HTML.gif)
Proof.
Since is a contraction self-mapping which is also of Meir-Keeler type, the result follows directly by combining (1.1) and (2.8).
3.
-Cyclic
-Contraction, Contractions of Meir-Keeler Type,
-Cyclic
-Kannan Self-Mappings, and Some Mutual Relationships
A set of relevant results for -cyclic self-mappings for
are obtained in [12]. Those self-mappings obey the subsequent definitions.
Definition 3.1 (see [12]).
Let be nonempty subsets of a metric space
;
. Then,
is a
-cyclic self-mapping if
;
with
;
.
Definition 3.2.
Let be nonempty,
subsets of a metric space
. Then,
is a
-cyclic
-contraction self-mapping if
;
with
;
and, furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ17_HTML.gif)
for some real constant .
A point is said to be the best proximity point if
, [12]. In this paper, it is also proven that if
is a
-cyclic nonexpansive self-mapping, that is,
;
;
, then
;
(i.e., the distances between adjacent sets are identical). Some properties concerned with
-cyclic nonexpansive self-mappings are stated and proven in the next lemma.
Lemma 3.3.
The following properties hold:
-
(i)
let
be a
-cyclic
-contraction self-mapping, then,
(3.2)(3.3)where
;
.
Let the mappings
;
,
, be defined by
(3.4)If
;
,
, with
;
, there is
which is unique if
is complete.
-
(ii)
If
is a
-cyclic nonexpansive self-mapping and, in particular, if t is a
-cyclic
-contraction, then
(3.5)
Proof.
-
(i)
Equation (3.2) follows by constructing a recursion directly from (1.1);
;
,
which can be also written equivalently in the form (3.3) by using the index identity
;
,
. If
, then
;
,
from (3.2) since
, so that there exists
;
. The point
is in
since
by construction of the self-mappings
;
,
since
. Also,
is unique if
is complete. Property (i) has been proven.
-
(ii)
It follows from the recursion
;
, obtained from (3.2) for
, since
is nonexpansive.
The auxiliary properties of Remark 3.4 below have been used in the proof of Lemma 3.3.
Remark 3.4.
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ22_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ23_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ24_HTML.gif)
The concepts of -cyclic nonexpansive self-mapping and
-cyclic
-contraction are generalized in the following. Consider the mappings
with
;
,
,
and
;
which fulfil the constraint
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ25_HTML.gif)
for some ;
and associated composed mappings
defined by
;
, subject to
.
Remark 3.5.
Note that is also defined for points
for some nonempty indexing set
, which contains
, by restricting its domain and image as
for some nonempty indexing set
such that
(since
). An important observation is that a set of constraints of type (3.9) have to be satisfied if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ26_HTML.gif)
The subsequent definitions extend Definitions 3.1-3.2 by removing the necessity of the set inclusions ;
and allowing obtaining of contractions from the composed mappings
;
which are not all necessarily contractions provided that
;
.
Definition 3.6.
Let be nonempty subsets of a metric space with
,
. Then,
;
is a composed
-cyclic self-mapping if
;
with
;
.
Definition 3.7.
Let be nonempty subsets of a metric space
with
,
. Then,
;
is a composed
-cyclic
-contraction self-mapping if
satisfies (3.9), subject to
,
,
, and, furthermore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ27_HTML.gif)
for some real constant and, furthermore, if
;
with
;
.
Definition 3.8.
If fulfils Definition 3.7 with (3.11) being true also for
, then it is said to be a composed
-cyclic nonexpansive mapping.
Note that if is a composed
-cyclic nonexpansive self-mapping (resp., a composed
-cyclic
-contraction self-mapping) for some
then it is so for all
. Composed
-cyclic contractions are characterized according to tests stated and proven in the subsequent result.
Proposition 3.9.
Assume that are nonempty subsets of a metric space
;
. Assume also that
;
,
;
and that
;
fulfils (3.9) for some
;
.
Then, the self-mapping ;
is a composed
-cyclic
-contraction self-mapping if the following two conditions hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ29_HTML.gif)
or , otherwise.
If and
, then
;
is a composed
-cyclic
-contraction self-mapping.
Proof.
If ;
fulfils (3.9), then for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ30_HTML.gif)
since ;
;
. Then,
is a composed
-cyclic
-contraction self-mapping from (3.12) and (3.11) (see Definition 3.7) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ31_HTML.gif)
since ;
. The second inequality of (3.13) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ33_HTML.gif)
Again since ;
and since
;
, then
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ34_HTML.gif)
Then (3.15) and (3.14), are equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ35_HTML.gif)
The first part of the result has been proven since (3.11) holds. The second one is a direct conclusion of the first one for the case .
It is now proven that if and
;
, then all the self-mappings
;
are composed
-cyclic
-contraction self-mappings possessing fixed points. If, furthermore,
is a complete metric space, then each of those self-mappings possesses a unique fixed point.
Corollary 3.10.
Assume that are nonempty subsets of a metric space
;
and the composed
-cyclic
-contraction self-mapping
fulfils Proposition 3.9 for some
, subject to
;
,
;
. Then, the following properties hold provided that
;
.
-
(i)
;
are all composed
-cyclic
-contraction self-mappings which satisfy, in addition,
;
(i.e.,
and
;
) and which possess common fixed points in
, that is,
;
.
-
(ii)
There is a unique set
satisfying the constraints
, subject to
, for any given
and for any given
. Furthermore, each of those sets satisfies the limiting property
;
for each
and any given
.
-
(iii)
consists of a unique point z if
is a complete metric space.
-
(iv)
If
is a complete metric space, then there is a unique set
satisfying
;
with
for any
and any given
.
Proof.
-
(i)
If
;
, then
and
. From Proposition 3.9, constraint (3.11) holds with
and
;
so that the limit
exists and is equal to zero;
,
. Then,
;
since
,
;
.
If
is complete, then
since the fixed point is unique. It is now proven by contradiction that
;
. Assume that
so that
since
. Then,
;
from the definition of the composed self-mapping
as
;
.
As a result,
which is a contradiction to the above assumption and proves the result. Now, it is proven that
;
. Proceed by contradiction. Assume that
for some
. Note that
for the given
since
. Thus,
since
which contradicts
. Then,
. Property (i) has been proven.
-
(ii)
Let
be a fixed point of
for any
. A sequence
;
,
of
points exists obeying the iteration
(3.18)
for some for any
;
subject to
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ37_HTML.gif)
Then, the -tuple
, and thus the corresponding set
is unique for each
and some
since
is a self-mapping on
;
. On the other hand, there exists
such that
for each
and each
. Thus, a unique
-tuple
exists for each
and then a unique set
;
. Property (ii) has been proven. Property (iii) follows directly from Property (i) together with Property (ii) since
is complete. Property (iv) follows directly from Properties (ii) and (iii).
Note that only a point in the unique set , referred to in Corollary 3.10(iv), is a fixed point of the composed
-cyclic
-contraction self-mappings
on
,
if the metric space
is complete. Also, there is only a guaranteed fixed point of the composed
-cyclic
-contraction self-mappings
on
;
, referred to in Corollary 3.10(ii), in each of the corresponding unique sets
if
is not complete.
An extra conclusion result can be obtained from Corollary 3.10 as follows in view of Remark 3.5:
Corollary 3.11.
The images of the mappings are in
;
,
. If, furthermore,
and
;
, then the image of
is in
as
;
. Also,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ38_HTML.gif)
Now, the self-mapping is defined as
for each
such that
for some
;
such that
. It turns out that such a mapping is a
-cyclic
-contraction if the composed self-mappings
on
are composed
-cyclic
-contractions. Note that there always exists a unique
for each given
which, in addition, fulfils
since
;
. The following result is obtained directly from Corollaries 3.10(i) and 3.10(iv).
Corollary 3.12.
Consider the self-mapping , subject to
;
,
, and assume that
and
;
so that
is a
-cyclic
-contraction. Then,
which, furthermore, consists of a single point if
is a complete metric space.
The relation between composed -cyclic
-contractions satisfying Corollaries 3.10–3.12 and the so-called
-cyclic
-Kannan self-mappings defined below is now discussed. Let
be nonempty subsets of a metric space
;
. Consider the mappings
satisfying
for
;
for the nonempty subsets
of the metric space
. Note that this implies that
,
, and
. The following definition which generalizes Definition 2.1 is then used to prove further results.
Definition 3.13.
A self-mapping is a composed
-cyclic
-Kannan self-mapping if it satisfies the following property for some real
and some
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ39_HTML.gif)
Proposition 3.14.
Consider the self-mappings with
for
,
;
being composed
-cyclic
-contractions satisfying
. The following properties hold.
(i)The self-mappings ;
and
are
self-mappings with
.
(ii)The self-mappings and
are, respectively,
-cyclic
-Kannan self-mappings for all
and composed
-cyclic
-Kannan self-mappings for some real constant
and any
.
Proof.
(i) From (3.12) and the triangle inequality of the distance mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ40_HTML.gif)
for a given and
since
;
. Since
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ41_HTML.gif)
with since
so that
is a
-Kannan self-mapping
from (2.1) and so it is
by construction. Property (i) has been proven. (ii) The proof follows directly since
;
so that (3.23) implies that (3.21) holds.
Remark 3.15.
It turns out that Proposition 3.14(ii) which is slightly modified still holds if the inclusion conditions ;
are removed. In fact, the self-mappings
and
on
and on
, respectively, are
-cyclic
-Kannan self-mappings;
and composed
-cyclic
-Kannan self-mappings, respectively, for some real constant
and
. The proof follows directly from that of Proposition 3.14(ii) and Definition 3.13 (see, in particular, (3.21)).
Definition 2.7 is generalized as follows for the case , and the subsequent theorem compares
-cyclic
-contractions with those of Meir-Keeler type.
Definition 3.16 (see [12]).
Assume that are nonempty subsets of a metric space
with
. A
-cyclic self-mapping
is a contraction of Meir-Keeler type if for any given
,
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ42_HTML.gif)
Note that the equivalent contrapositive logic proposition to (3.24) is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ43_HTML.gif)
which can be used equivalently to state Definition 3.16. The following technical simple result will be then used in the proof of Theorem 3.18 below.
Assertion 1.
If (3.24), and equivalently (3.25), holds for some , for any given
, then they also hold for some
.
Proof.
If for the given
, the result is proven. If
, then (3.25) leads directly to the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ44_HTML.gif)
for any ;
,
 , and the result holds with the replacement
.
Proposition 3.17.
Let be a
-cyclic self-mapping on
. Thus, if
is a
-cyclic
-contraction, then it is also a contraction of Meir-Keeler type
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ45_HTML.gif)
Proof.
Since is a
-cyclic
-contraction, then it is
-cyclic nonexpansive so that
;
. Take any
for some
such that
. Then, since
is a
-cyclic
-contraction, one gets that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ46_HTML.gif)
provided that for any given ,
since
. Then, the
-cyclic self-mapping
on
is also a contraction of Meir-Keeler type from Definition 3.16.
The subsequent result relies on the limiting property to the best proximity points of the distances between points in adjacent sets in self-mappings being -cyclic contractions of Meir-Keeler type.
Theorem 3.18.
Let  be a
-cyclic contraction of Meir-Keeler type. Then the following properties hold.
(i)If for some real constants
then the inequalities
hold for some bounded positive strictly monotone decreasing real sequences
,
which converge to zero, with
,
, and furthermore,
as
;
,
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ47_HTML.gif)
for . As a result, there exists a finite
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ48_HTML.gif)
for any given and
.
(ii)If , then
;
, and
consists of a unique fixed point if
is complete.
(iii)There exists some real constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ49_HTML.gif)
and, furthermore, under the constraint
which allows the choice
;
.
Proof.
()-(
) A recursion in (3.24) leads to the following recursion of implications for
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ50_HTML.gif)
for for any arbitrary given
and some positive real sequences
and
which depend on
according to the respective implicit dependences:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ51_HTML.gif)
with ,
which satisfy the constraints
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ52_HTML.gif)
which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ53_HTML.gif)
Furthermore, from (3.35) into (3.32), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ54_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ55_HTML.gif)
for some , which does not depend neither on the particular initial pair
,
nor on the given arbitrary
since it is the supremum of all the limit superiors. Assume that
. Since
is arbitrary, it may be chosen as
which contradicts
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ56_HTML.gif)
Also, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ57_HTML.gif)
for . If
, then
;
so that
,
,
, and since
from
;
, then
;
. This conclusion is direct from the following reasoning. Assume that
for some arbitrary
which exists since
is a fixed point. Then,
for any
with
;
. Thus,
. Also,
consists of a unique fixed point if
is complete. Properties (i) and (ii) have been proven.
It turns out from (3.35) that for some finite ,
;
. Then, by construction, it follows that there exist some real constant
and some real constant
(both of them are dependent on
and
) which are the respective ratios of the geometric series
and
, such that the following identities hold for any given sequence
that satisfies (3.35):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ58_HTML.gif)
If , then a sequence satisfying
;
is valid from Assertion 1. Thus,
, and since
may be always taken as on being larger than
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ59_HTML.gif)
and together with (3.38), it follows that ;
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ60_HTML.gif)
under the constraints and
;
. Property (iii) has been proven.
It is interesting to discuss when the composed self-mappings on
for set-depending self-mappings
;
as well as the self-mappings
on
defined by
are guaranteed to be
-cyclic Meir-Keeler contractions without requiring that the property holds for each individual
subject to
. For the related discussion, assume that
;
,
for a set of real constants
;
. A direct calculation on
iterations yields directly
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ61_HTML.gif)
If the self-mappings are identical;
, then (3.43) becomes in particular:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ62_HTML.gif)
The following result holds directly from (3.43) and (3.44) and Theorem 3.18.
Theorem 3.19.
The composed self-mappings on
for set- depending self-mappings
;
as well as the self-mappings T on
defined by
are guaranteed to be as follows.
-cyclic Meir-Keeler contractions if
,
Thus, there is an asymptotic convergence from any initial point to the best proximity point in general and to a fixed point if the sets in have a nonempty intersection. The fixed point is unique if
is complete.
Nonexpansive -cyclic self-mappings if
.
Expansive -cyclic self-mappings if
.
More general conditions than the Meir-Keeler ones guaranteeing that the composed self-mappings on
,
are asymptotic contractions are now discussed.
Theorem 3.20.
Assume that there is a real sequence of finite sum
which satisfies the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ63_HTML.gif)
for some given real constant , whose elements are defined in such a way that the composed self-mapping
on satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ64_HTML.gif)
Then, ;
.
Proof.
From the properties of the sequence and (3.46), one gets
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ65_HTML.gif)
for some sufficiently large finite .The constraint on the upperbounds in (3.47) guarantees that the strict upper-bound
for
is less than a strict upper-bound for
for any sufficiently large
provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ66_HTML.gif)
Furthermore, (3.46) holds if
;
. As a result,
as
since the supremum of all limits superior converge to
(see the proof of Theorem 3.19).
Theorem 3.20 may be particularized to -cyclic asymptotic contractions as follows.
Theorem 3.21.
Assume that there is a real sequence of finite sum
which satisfies the following conditions for some
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ67_HTML.gif)
for some given real constant whose elements are defined in such a way that the composed self-mapping satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ68_HTML.gif)
Then, for the given
.
Proof.
From the properties of the sequence and (3.50), one gets.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ69_HTML.gif)
for some sufficiently large finite and the given
. The constraint on the upper-bounds in (3.51) guarantees that the strict upper-bound
for
is less than a strict upper-bound for
for any sufficiently large
provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F572057/MediaObjects/13663_2009_Article_1304_Equ70_HTML.gif)
Furthermore, (3.50) holds for the given
if
;
for the given
. As a result,
as
since the supremum of all limit superiors converges to
(see the proof of Theorem 3.19).
Note that Theorem 3.21 guarantees that the self-mapping on
has a
-cyclic Meir-Keeler asymptotic contraction for a particular
, while Theorem 3.20 guarantees that all the self-mappings
are asymptotic contractions. In both cases, the self-mappings can be locally expansive in the sense that it can happen that
for some finite
, some
, and some
.
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Acknowledgments
The author is grateful to the Spanish Ministry of Education for its partial support to this work through Grant DPI 2009-07197. He is also grateful to the Basque Government for its support through Grants GIC07143-IT-269-07and SAIOTEK S-PE07UN04. The author thanks the reviewers for their useful comments who helped him to improve the former versions of the manuscript.
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De la Sen, M. Linking Contractive Self-Mappings and Cyclic Meir-Keeler Contractions with Kannan Self-Mappings. Fixed Point Theory Appl 2010, 572057 (2010). https://doi.org/10.1155/2010/572057
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DOI: https://doi.org/10.1155/2010/572057