Linking Contractive Self-Mappings and Cyclic Meir-Keeler Contractions with Kannan Self-Mappings
© M. De la Sen. 2010
Received: 1 September 2009
Accepted: 22 February 2010
Published: 28 February 2010
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.
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 . 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,  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 .
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 . Nonexpansive mappings, in general, have received important attention in the last years. For instance, two hybrid methods are used in  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 . 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 .
2. About 2-Cyclic -Contraction, 2-Cyclic -Kannan Self-Mappings, Contractions of Meir-Keeler Type, and Some Mutual Relationships
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 .
The following two results are direct.
Definition 2.7 (see ).
The subsequent result is concerned with 2-cyclic contraction self-mappings of Meir-Keeler type
Proposition 2.3, Definitions 2.1 and 2.9, and Proposition 2.8 yield directly the following result.
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 . Those self-mappings obey the subsequent definitions.
Definition 3.1 (see ).
A point is said to be the best proximity point if , . 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.
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.
The auxiliary properties of Remark 3.4 below have been used in the proof of Lemma 3.3.
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 ; .
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.
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.
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.
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:
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).
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.
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 3.16 (see ).
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.
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.
The following result holds directly from (3.43) and (3.44) and Theorem 3.18.
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.
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 .
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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