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Linking Contractive SelfMappings and Cyclic MeirKeeler Contractions with Kannan SelfMappings
Fixed Point Theory and Applications volume 2010, Article number: 572057 (2010)
Abstract
Some mutual relations between pcyclic contractive selfmappings, pcyclic Kannan selfmappings, and MeirKeeler pcyclic 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 selfmappings 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 continuoustime and discretetime 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 continuoustime and discretetime dynamics, [6–10]. On the other hand, MeirKeeler selfmappings 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 selfmappings are less restrictive than strict contractive selfmappings so that their associated formalism is applicable to a wider class of reallife problems. Another interest of such selfmappings 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 selfmappings and Kannan selfmappings [2, 15–17] with constant (referred to in the following as Kannan selfmappings). In fact, Kannan selfmappings 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 selfmappings, cyclic MeirKeeler contractions, and cyclic Kannan selfmappings. 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 statesolution 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 selfmapping such that and where and are nonempty subsets of . Then, is a 2cyclic selfmapping what is said to be a 2cyclic contraction selfmapping if it satisfies in addition
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 2cyclic nonexpansive selfmapping [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 socalled times reasonable expansive and their properties selfmappings 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,
is the set of fixed points of a cyclic selfmapping in a nonempty subset of a metric space .
is the set of the best proximity points in a subset of a cyclic selfmapping on , namely, the union of a collection of nonempty subsets of a metric space which do not intersect.
Contractive selfmappings with constant and Kannan selfmappings with constant are referred to as contractions and Kannan selfmappings, respectively.
(referred to as "not ") is the negation of a logic proposition .
2. About 2Cyclic Contraction, 2Cyclic Kannan SelfMappings, Contractions of MeirKeeler Type, and Some Mutual Relationships
The definition of Kannan selfmappings is as follows:
for some real [14, 15]. Let us extend the above concept in a natural way to 2cyclic Kannan selfmappings by considering the definition of 2cyclic contractions (1.1) as follows.
Definition 2.1.
Consider a metric space and a selfmapping such that and where and are nonempty subsets of . Then is a 2cyclic Kannan selfmapping for some real if it satisfies
for some . Definition 2.1 is a natural counterpart of (2.1) for Kannan selfmappings by taking into account the definition of a 2cyclic contraction in (1.1).
Remark 2.2.
Let be a metric space and let be a 2cyclic contraction selfmapping with and being nonempty nondisjoint subsets of . It turns out that is also a contractive mapping with constant .
Contraction selfmappings can also be 2cyclic Kannan selfmappings and viceversa as addressed in the two following results:
Proposition 2.3.
Assume that is a 2cyclic contraction selfmapping with . Then, is also a 2cyclic Kannan selfmapping, .
Proof.
The following inequalities follow from (1.1) and the triangle inequality of the distance map :
for some , what leads to
for , , . Thus, is a 2cyclic Kannan selfmapping 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 2cyclic Kannan selfmapping with and subject to the constraints and with . Then, is also a 2cyclic contraction selfmapping for any real constant .
Proof.
Since is a 2cyclic Kannan selfmapping then from Definition 2.1, it follows that
Also, one has that , , since ; . Choosing and such that , the triangle inequality for the distance map yields
so that one has since :
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 "mutatismutandis" 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 2cyclic contraction selfmapping is of MeirKeeler type if for any given , such that
The subsequent result is concerned with 2cyclic contraction selfmappings of MeirKeeler type
Proposition 2.8.
Assume that is a 2cyclic contraction selfmapping of MeirKeeler 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
Proof.
Note from (2.8) that
for some ; . Define ; . Since is a 2cyclic contraction selfmapping of MeirKeeler type, one has proceeding recursively with (2.10)
so that (2.10) together with the constraint implies that
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
for some defined by . Furthermore, since (resp., ), what yields (resp., ); and since .
Definition 2.9.
A 2cyclic Kannan selfmapping defined for some real and some (see Definitions 2.1 and 2.7) is of MeirKeeler 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 2cyclic contraction selfmapping of Meir Keeler type with . Then, is also a 2cyclic Kannan selfmapping, . If , then .
From the definition of contraction selfmappings and Definition 2.7 for MeirKeeler type contraction selfmappings, the following result holds.
Proposition 2.11.
If is a contraction selfmapping of MeirKeeler type, then for any given , and such that
Proof.
Since is a contraction selfmapping which is also of MeirKeeler type, the result follows directly by combining (1.1) and (2.8).
3. Cyclic Contraction, Contractions of MeirKeeler Type, Cyclic Kannan SelfMappings, and Some Mutual Relationships
A set of relevant results for cyclic selfmappings for are obtained in [12]. Those selfmappings obey the subsequent definitions.
Definition 3.1 (see [12]).
Let be nonempty subsets of a metric space ; . Then, is a cyclic selfmapping if ; with ; .
Definition 3.2.
Let be nonempty, subsets of a metric space . Then, is a cyclic contraction selfmapping if ; with ; and, furthermore,
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 selfmapping, that is, ; ; , then ; (i.e., the distances between adjacent sets are identical). Some properties concerned with cyclic nonexpansive selfmappings are stated and proven in the next lemma.
Lemma 3.3.
The following properties hold:

(i)
let be a cyclic contraction selfmapping, 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 selfmapping 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 selfmappings ; , 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
so that
so that
The concepts of cyclic nonexpansive selfmapping and cyclic contraction are generalized in the following. Consider the mappings with ; , , and ; which fulfil the constraint
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
The subsequent definitions extend Definitions 3.13.2 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 selfmapping if ; with ; .
Definition 3.7.
Let be nonempty subsets of a metric space with , . Then, ; is a composed cyclic contraction selfmapping if satisfies (3.9), subject to , , , and, furthermore,
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 selfmapping (resp., a composed cyclic contraction selfmapping) 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 selfmapping ; is a composed cyclic contraction selfmapping if the following two conditions hold:
or , otherwise.
If and , then ; is a composed cyclic contraction selfmapping.
Proof.
If ; fulfils (3.9), then for any
since ; ; . Then, is a composed cyclic contraction selfmapping from (3.12) and (3.11) (see Definition 3.7) if
since ; . The second inequality of (3.13) is equivalent to
Again since ; and since ; , then , and
Then (3.15) and (3.14), are equivalent to
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 selfmappings ; are composed cyclic contraction selfmappings possessing fixed points. If, furthermore, is a complete metric space, then each of those selfmappings possesses a unique fixed point.
Corollary 3.10.
Assume that are nonempty subsets of a metric space ; and the composed cyclic contraction selfmapping fulfils Proposition 3.9 for some , subject to ; , ; . Then, the following properties hold provided that ; .

(i)
; are all composed cyclic contraction selfmappings 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 selfmapping 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
Then, the tuple , and thus the corresponding set is unique for each and some since is a selfmapping 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 selfmappings on , if the metric space is complete. Also, there is only a guaranteed fixed point of the composed cyclic contraction selfmappings 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,
Now, the selfmapping 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 selfmappings 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 selfmapping , 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 socalled cyclic Kannan selfmappings 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 selfmapping is a composed cyclic Kannan selfmapping if it satisfies the following property for some real and some :
Proposition 3.14.
Consider the selfmappings with for ,; being composed cyclic contractions satisfying . The following properties hold.
(i)The selfmappings ; and are selfmappings with .
(ii)The selfmappings and are, respectively, cyclic Kannan selfmappings for all and composed cyclic Kannan selfmappings for some real constant and any .
Proof.
(i) From (3.12) and the triangle inequality of the distance mapping
for a given and since ; . Since ,
with since so that is a Kannan selfmapping 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 selfmappings and on and on , respectively, are cyclic Kannan selfmappings; and composed cyclic Kannan selfmappings, 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 MeirKeeler type.
Definition 3.16 (see [12]).
Assume that are nonempty subsets of a metric space with . A cyclic selfmapping is a contraction of MeirKeeler type if for any given , such that
Note that the equivalent contrapositive logic proposition to (3.24) is
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
for any ; , , and the result holds with the replacement .
Proposition 3.17.
Let be a cyclic selfmapping on . Thus, if is a cyclic contraction, then it is also a contraction of MeirKeeler type
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
provided that for any given , since . Then, the cyclic selfmapping on is also a contraction of MeirKeeler 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 selfmappings being cyclic contractions of MeirKeeler type.
Theorem 3.18.
Let be a cyclic contraction of MeirKeeler 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
for . As a result, there exists a finite such that
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
and, furthermore, under the constraint which allows the choice ; .
Proof.
()() A recursion in (3.24) leads to the following recursion of implications for :
for for any arbitrary given and some positive real sequences and which depend on according to the respective implicit dependences:
with , which satisfy the constraints
which imply that
Furthermore, from (3.35) into (3.32), it follows that
so that
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,
Also, since
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):
If , then a sequence satisfying ; is valid from Assertion 1. Thus, , and since may be always taken as on being larger than , then
and together with (3.38), it follows that ; since
under the constraints and ; . Property (iii) has been proven.
It is interesting to discuss when the composed selfmappings on for setdepending selfmappings ; as well as the selfmappings on defined by are guaranteed to be cyclic MeirKeeler 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
If the selfmappings are identical; , then (3.43) becomes in particular:
The following result holds directly from (3.43) and (3.44) and Theorem 3.18.
Theorem 3.19.
The composed selfmappings on for set depending selfmappings ; as well as the selfmappings T on defined by are guaranteed to be as follows.
cyclic MeirKeeler 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 selfmappings if .
Expansive cyclic selfmappings if .
More general conditions than the MeirKeeler ones guaranteeing that the composed selfmappings on , are asymptotic contractions are now discussed.
Theorem 3.20.
Assume that there is a real sequence of finite sum which satisfies the conditions
for some given real constant , whose elements are defined in such a way that the composed selfmapping on satisfies
Then, ; .
Proof.
From the properties of the sequence and (3.46), one gets
for some sufficiently large finite .The constraint on the upperbounds in (3.47) guarantees that the strict upperbound for is less than a strict upperbound for for any sufficiently large provided that
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 :
for some given real constant whose elements are defined in such a way that the composed selfmapping satisfies
Then, for the given .
Proof.
From the properties of the sequence and (3.50), one gets.
for some sufficiently large finite and the given . The constraint on the upperbounds in (3.51) guarantees that the strict upperbound for is less than a strict upperbound for for any sufficiently large provided that
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 selfmapping on has a cyclic MeirKeeler asymptotic contraction for a particular , while Theorem 3.20 guarantees that all the selfmappings are asymptotic contractions. In both cases, the selfmappings 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 200907197. He is also grateful to the Basque Government for its support through Grants GIC07143IT26907and SAIOTEK SPE07UN04. 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 SelfMappings and Cyclic MeirKeeler Contractions with Kannan SelfMappings. 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
Keywords
 Nonexpansive Mapping
 Nonempty Subset
 Real Constant
 Real Sequence
 Maximal Monotone Operator