Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 572057, 23 pages
doi:10.1155/2010/572057

Copyright © 2010 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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(1.1) 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,(1.2) 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:(2.1) 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 (2.2) 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 (2.3) for some , what leads to (2.4) 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 (2.5) Also, one has that , since . Choosing and such that , the triangle inequality for the distance map yields (2.6) so that one has since (2.7) 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 (2.8)

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 (2.9)

Proof. Note from (2.8) that (2.10) for some . Define . Since is a 2-cyclic contraction self-mapping of Meir-Keeler type, one has proceeding recursively with (2.10) (2.11) so that (2.10) together with the constraint implies that (2.12) 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 (2.13) 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 (2.14)

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, (3.1) 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 (3.6) so that (3.7) so that (3.8)

The concepts of -cyclic nonexpansive self-mapping and -cyclic -contraction are generalized in the following. Consider the mappings with and which fulfil the constraint (3.9) 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 (3.10)

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, (3.11) 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:(C1)(C2) or , otherwise.

If and , then is a composed -cyclic -contraction self-mapping.

Proof. If fulfils (3.9), then for any (3.12) since . Then, is a composed -cyclic -contraction self-mapping from (3.12) and (3.11) (see Definition 3.7) if (3.13) since . The second inequality of (3.13) is equivalent to (3.14)(3.15) Again since and since , then , and (3.16) Then (3.15) and (3.14), are equivalent to (3.17) 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 (3.19) 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, (3.20)

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 (3.21)

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 (3.22) for a given and since . Since (3.23) 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 (3.24)

Note that the equivalent contrapositive logic proposition to (3.24) is (3.25) 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 (3.26) 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 (3.27)

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 (3.28) 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 (3.29) for . As a result, there exists a finite such that (3.30) 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 (3.31) and, furthermore, under the constraint which allows the choice .

Proof. ()-() A recursion in (3.24) leads to the following recursion of implications for (3.32) for for any arbitrary given and some positive real sequences and which depend on according to the respective implicit dependences: (3.33) with which satisfy the constraints (3.34) which imply that (3.35) Furthermore, from (3.35) into (3.32), it follows that (3.36) so that (3.37) 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, (3.38) Also, since (3.39) 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): (3.40) If , then a sequence satisfying is valid from Assertion 1. Thus, , and since may be always taken as on being larger than , then (3.41) and together with (3.38), it follows that since (3.42) 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(3.43) If the self-mappings are identical; , then (3.43) becomes in particular:(3.44) 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 (3.45) for some given real constant , whose elements are defined in such a way that the composed self-mapping on satisfies (3.46) Then, .

Proof. From the properties of the sequence and (3.46), one gets (3.47) 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 (3.48) 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 (3.49) for some given real constant whose elements are defined in such a way that the composed self-mapping satisfies (3.50) Then, for the given .

Proof. From the properties of the sequence and (3.50), one gets. (3.51) 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 (3.52) 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 .

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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