Journal of Applied MathematicsVolume 2012 (2012), Article ID 817193, 29 pagesdoi:10.1155/2012/817193

Research Article

Fixed and Best Proximity Points of Cyclic Jointly Accretive and Contractive Self-Mappings

M. De la Sen

Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia) Apertado 644 Bilbao, 48080 Bilbao, Spain

Received 21 November 2011; Accepted 15 December 2011

Academic Editor: Yonghong Yao

Copyright © 2012 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.

-cyclic and contractive self-mappings on a set of subsets of a metric space which are simultaneously accretive on the whole metric space are investigated. The joint fulfilment of the -cyclic contractiveness and accretive properties is formulated as well as potential relationships with cyclic self-mappings in order to be Kannan self-mappings. The existence and uniqueness of best proximity points and fixed points is also investigated as well as some related properties of composed self-mappings from the union of any two adjacent subsets, belonging to the initial set of subsets, to themselves.

1. Introduction

In the last years, important attention is being devoted to extend the fixed point theory by weakening the conditions on both the mappings and the sets where those mappings 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]. Another increasing research interest field relies on the generalization of fixed point theory to more general spaces than the usual metric spaces, for instance, ordered or partially ordered spaces (see, e.g., [3–5]). It has also to be pointed out the relevance of fixed point theory in the stability of complex continuous-time and discrete-time dynamic systems [6–8]. 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 [9–12]. Another interest of such mappings is their usefulness as formal tool for the study -cyclic contractions even if the involved subsets of the metric space under study of do not intersect [10]. 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 close links between contractive self-mappings and Kannan self-mappings [2, 13–16]. In fact, Kannan self-mappings are contractive for values of the contraction constant being less than 1/3, [15, 16] and can be simultaneously -cyclic Meir-Keeler contractive self-mappings. The objective of this paper is the investigation of relevant properties of contractive -cyclic self-mappings of the union of set of subsets of a Banach space which are simultaneously-accretive on the whole , while investigating the existence and uniqueness of potential fixed points on the subsets of if they intersect and best proximity points. For such a purpose, the concept of -accretive self-mapping is established in terms of distances as a, in general, partial requirement of that of an accretive self-mapping. Roughly speaking, the self-mapping from to under study can be locally increasing on but it is still -cyclic contractive on the relevant subsets of . For the obtained results of boundedness of distances between the sequences of iterates through , it is not required for the set of subsets of to be either closed or convex. For the obtained results concerning fixed points and best proximity points, the sets are required to be convex but they are not necessarily closed if the self-mapping can be defined on the union of the closures of the sets. Consider a metric space associated to the Banach space and a self-mapping such that and , where and are nonempty subsets of . Then, is a 2-cyclic self-mapping. It is said to be a 2-cyclic -contraction self-mapping if it satisfies, furthermore,

for some real . A best proximity point of convex subsets or of is some such that . If and are closed then either (resp., ) or (resp. ) is in (resp., in ). The distance between subsets and of the metric space if either or if either or is open with . In this case, if is a best proximity point either or is not in (in particular, neither nor is in if both of them are open). It turns out that if then ; that is, is a fixed point of since , [9–11]. If then ; , for all and is a 2-cyclic nonexpansive self-mapping, [10].

1.1. Notation

superscript denotes vector or matrix transpose, is the set of fixed points of a self-mapping on some nonempty convex subset of a metric space cl and denote, respectively, the closure and the complement in of a subset of , and denote, respectively, the domain and image of the self-mapping and is the family of subsets of , dist denotes the distance between the sets and for a 2-cyclic self-mapping which is simplified as dist; for distances between adjacent subsets of -cyclic self-mappings on .

which is the set of best proximity points on a subset of a metric space of a -cyclic self-mapping on , the union of a collection of nonempty subsets of which do not intersect.

2. Some Definitions and Basic Results about 2-Cyclic Contractive and Accretive Mappings

Let be a normed vector space and be an associate metric space endowed with a metric (or distance function or simply “distance”) . For instance, the distance function may be induced by the norm on. If the metric is homogeneous and translation-invariant, then it is possible conversely to define the norm from the metric. Consider a self-mapping which is a 2-cyclic self-mapping restricted as where and are nonempty subsets of . Such a restricted self-mapping is sometimes simply denoted as . Self-mappings which can be extended by continuity to the boundary of its initial domain as well as compact self-mappings, for instance, satisfy such an extendibility assumption. In the cases that the sets and are not closed, it is assumed that and in order to obtain a direct extension of existence of fixed points and best proximity points. This allows, together with the convexity of and , to discuss the existence and uniqueness of fixed points or best proximity points reached asymptotically through the sequences of iterates of the self-mapping . In some results concerning the accretive property, it is needed to extend the self-mapping in order to define successive iterate points through the self-mapping which do not necessarily belong to . The following definitions are then used to state the main results.

Definition 2.1. is an accretive mapping if for any . Note that, since is also a vector space, is in for all in and all real . This fact facilitates also the motivation of the subsequent definitions as well as the presentation and the various proofs of the mathematical results in this paper. A strong convergence theorem for resolvent accretive operators in Banach spaces has been proved in [17].Two more restrictive (and also of more general applicability) definitions than Definition 2.1 to be then used are now introduced as follows:

Definition 2.2. is a -accretive mapping, some if for some . A generalization is as followsis -accretive for some if

Definition 2.3. is a weighted -accretive mapping, for some function , if The above concepts of accretive mapping generalize that of a nondecreasing function. Contractive and nonexpansive 2-cyclic self-mappings are defined as follows on unions of subsets of .

Definition 2.4. is a 2-cyclic -contractive (resp., nonexpansive) self-mapping if for some real (resp. ), [12, 13]. The concepts of Kannan-self mapping and 2-cyclic -Kannan self-mapping which can be also a contractive mapping, and conversely if , [16], are defined below.

Definition 2.5. is a -Kannan self-mapping if for some real , [12, 13].

Definition 2.6. is an 2-cyclic -Kannan self-mapping for some real if it satisfies, for some . The relevant concepts concerning 2-cyclic self-mappings are extended to -cyclic self-mappings in Section 3. Some simple explanation examples follow.

Example 2.7. Consider the scalar linear mapping from to as with endowed with the Euclidean distance ; for all . Then, for all for any provided that . In this case, is accretive. It is also -contractive if since ; for all . Also, if , then ; for all if , that is, if . Then, is -accretive and -contractive if .

Example 2.8. Consider the metric space with the distance being homogeneous and translation-invariant and a self-mapping defined by with , and if and . If then is accretive since Furthermore, if , then is the unique fixed point with ; for all . If then, as if and then is again the unique fixed point of . In the general case, implies holds if that is, is weighted -accretive with . The restricted self-mapping is -accretive. Furthermore, if then is -contractive if and the iteration as with being the unique fixed point since Note from the definition of the self-mapping on that it is also a 2-cyclic self-mapping from to itself with the property and .

All the given definitions can also be established mutatis-mutandis if is a normed vector space. A direct result from inspection of Definitions 2.1 and 2.2 is the following.

Assertions 1. (1) If is an accretive mapping, then it is -accretive, for all . (2) If is -accretive, then it is -accretive; for all . (3) Any nonexpansive self-mapping is -accretive and conversely.

Theorem 2.9. Let be a Banach vector space withbeing the associated complete metric space endowed with a norm-induced translation-invariant and homogeneous metric. Consider a self-mapping which restricted tois a 2-cyclic -contractive self-mapping where and are nonempty subsets of . Then, the following properties hold. (i) Assume that the self-mapping satisfies the constraint: with satisfying the constraint . Then, the restricted self-mapping satisfies irrespective of and being bounded or not. If, furthermore, and are closed and convex and, then there exists a unique fixed point of such that there exists ; for all for all , implying that . If, in addition, so that , then there exists ; for all , for all for some best proximity points , which depend in general on and . Furthermore, if is a uniformly convex Banach space, then and as ; for all, where and are unique best proximity points in and of . (ii)Assume that and are nondisjoint. Then, is also contractive and -accretive for any nonnegative and any . It is also nonexpansive and -accretive for any nonnegative .(iii)If then is weighted -accretive for for any and its restriction is 2-cyclic 0-contractive.(iv) is weighted -accretive for satisfying for some . The restricted self-mapping is also -contractive with if with . Also, is nonexpansive and weighted -accretive for satisfying if which implies, furthermore, that is bounded.

Proof. Let us denote Consider that the two following relations are verified simultaneously: Since the distance is translation-invariant and homogeneous, then the substitution of (2.14) yields if and are disjoint sets, after using the subadditive property of distances, the following chained relationships since : with . Note from (2.15) that and, if , then If then . It is first proven that the existence of the limit of the distance implies that of the limit ; for all . Let be , with. Then, since being contractive is globally Lipschitz continuous. Then, since, because the fact that the metric is translation-invariant, one gets As a result, if what implies which ; for all for all , since is globally Lipschitz continuous since it is contractive. In addition, there exists ; for all , for all . Assume not so that there exists such that and there exists a subsequence on nonnegative integers such that . If so, one gets by taking that which contradicts . Then is a Cauchy sequence for any and then converges to a limit. Furthermore, since for any and as since and are nonempty and closed. It has been proven that ; for all for all .It is now proven that. Assume not, then, from triangle inequality, which contradicts so that . It is now proven that . Assume not, such that, for instance, and . If so, since , then the existing limit fulfils which is impossible so that there would be no existing limit in , contradicting the former result of its existence. Then, implying that Fix.It is now proven by contradiction that ; for all is the unique fixed point of . Assume that , then for some with no loss in generality and all . Thus, which contradicts so that Fix.Now, assume that and do not intersect so that dist. Then, one gets from the first inequality in (2.15) that for all , , one getsNote that since , and , then and if is even and and if is oddand if is even and and if is odd.Then, and are not both in either or if and are not both in either or for any . As a result, is impossible so that for some best proximity points and or conversely. Then, where Thus, . It turns out that and dist as . Otherwise, it would exist an infinite subsequence of with being an infinite subset of such that for . On the other hand, since is a normed space, then by taking the norm-translation invariant and homogeneous induced metric and since there exists , it follows that there exist and such that for any given ; for all for all with for any even and , for any odd . As a result, by choosing the positive real constant arbitrarily small, one gets that (a best proximity point of ) and (a best proximity point of ), or vice-versa, as for any given and . A best proximity point fulfils . Best proximity points are unique in and as it is now proven by contradiction. Assume not, for instance, and with no loss in generality, assume that there exist two distinct best proximity points and in . Then and contradict so that necessarily . Since is a uniformly convex Banach space, we take the norm-induced metric to consider such a space as the complete metric space to obtain the following contradiction: since is also a strictly convex Banach space and and are nonempty closed and convex sets. Then, is the unique best proximity point of in and is its unique best proximity point in . Then, Property (i) has been fully proven. Since and are not disjoint, then , and is -contractive and -accretive if with . By taking , note that is nonexpansive and -accretive. Property (ii) has been proven. To prove Property (iii), we now discuss if is possible with and . Note that   for some . Define , if   for some , where and . Three cases can occur in (2.26), namely,(a)If then which is untrue if and and it holds for either or ,(b), then (2.26) is equivalent to Take to be a best proximity point with so that which is untrue if and true for ,(c), then (2.16) is equivalent to ; for all for all , but . Thus, the above constraint is guaranteed to hold in the worst case if which is a contradiction.Property (iii) follows from the above three cases (a)–(c).To prove Property (iv), consider again (2.26) by replacing the real constants and with the real functions and . Note that (2.26) holds through direct calculation if ; for all for all for some . Thus, the self-mapping is weighted -accretive for satisfying for some ; and it is also -contractive with if with and nonexpansive if . On the other hand, note that . If and are bounded and , then Property (iv) has been proven.

Remark 2.10. Note that Theorem 2.9 (iii) allows to overcome the weakness of Theorem 2.9 (ii) when and are disjoint by introducing the concept of weighted accretive mapping since for best proximity points , .

Remark 2.11. Note that the assumption that is a uniformly convex Banach space could be replaced by a condition of strictly convex Banach space since uniformly convex Banach spaces are reflexive and strictly convex, [18]. In both cases, the existence and uniqueness of best proximity points of the 2-cyclic in and are obtained provided that both sets are nonempty, convex, and closed.

Remark 2.12. Note that if either or is not closed, then its best proximity point of is in its closure since ,   leads to and for finitely many and for infinitely many iterations through the self-mapping and Theorem 2.9 is still valid under this extension.

Note that the relevance of iterative processes either in contractive, nonexpansive and pseudocontractive mappings is crucial towards proving convergence of distances and also in the iterative calculations of fixed points of a mapping or common fixed points of several mappings. See, for instance, [19–25] and references therein. Some results on recursive multiestimation schemes have been obtained in [26]. On the other hand, some recent results on Krasnoselskii-type theorems and related to the statement of general rational cyclic contractive conditions for cyclic self-maps in metric spaces have been obtained in [27] and [28], respectively. Finally, the relevance of certain convergence properties of iterative schemes for accretive mappings in Banach spaces has been discussed in [29] and references therein. The following result is concerned with norm constraints related to 2-cyclic accretive self-mappings which can eventually be also contractive or nonexpansive.

Theorem 2.13. The following properties hold. (i)Let be a metric space endowed with a norm-induced translation-invariant and homogeneous metric . Consider the -accretive mapping for some which restricted as is 2-cyclic, where and are nonempty subsets of subject to . Then, If, furthermore, is -contractive, then is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if respectively, (ii)Let be a normed vector space. Consider a -accretive mapping for some which restricted to is 2-cyclic, where and are nonempty subsets of subject to then If, furthermore, is -contractive, then is nonexpansive (resp., asymptotically nonexpansive, [30]) if respectively,

Proof. To prove Property (i), define an induced by the metric norm as follows since the metric is homogeneous and translation-invariant. Define the norm of , that is, the norm of on restricted to as follows: with the above set being closed, nonempty, and bounded from below. Since is 2-cyclic and is -accretive (Definition 2.2), one gets by proceeding recursively since the metric is homogeneous and , and is the identity operator on , where with the above set being closed, nonempty, and bounded from below. If for some , then we get the contradiction ; for all , for all in (2.38). Thus, ; for all , , for all . If now and are replaced with and for any in (2.30), one gets if is a 2-cyclic -contractive for some real and -accretive mapping: for all , for all , for all , for all , for all . Then, ; for all , for all , for all . If ; for all , for all , for all , it turns out that is -accretive and is a 2-cyclic nonexpansive self-mapping. It is asymptotically nonexpansive if ; for all , . Property (i) has been proven. The proof of Property (ii) for being a normed vector space is identical to that of Property (i) without associating the norms to a metric.

Example 2.14. Assume that , , and the 2-cyclic self-mapping defined by the iteration rule with , ; for all , and . Let be the Euclidean metric.(a)If, then so that for any , ; for all as with , but it is not in which is empty. If and (i.e., there are infinitely many values being less than unity), then the conclusion is identical. If and are redefined as , , then .(b)If ; for all the self-mapping is not expansive and there is no fixed point. (c)If for some , then for, so that is also -accretive and -contractive with . (d)Now, define closed sets and for any given so that . The 2-cyclic self-mapping is re-defined by the iterationif and , for , otherwise, wherefor with the real sequence being subject to , ; , for all and . Then, for any and any , there are two best proximity points and fulfilling and .(e)Redefine so that with , ; ,. In the case that then and are open disjoint subsets (resp., , are closed nondisjoint subsets with ).The 2-cyclic self-mapping is re-defined by the iteration rule: otherwise, where with the real sequence being subject to, ; for ; for all and . The same parallel conclusions to the above ones (a)–(c) follow related to the existence of the unique fixed point in the closure of and but not in its empty intersection if either or is open, respectively, in the intersection of and (the vertical real line of zero abscissa) if they are closed. The same conclusion of (d) is valid for the best proximity points if .

The following result which leads to elementary tests is immediate from Theorem 2.13.

Corollary 2.15. The following properties hold. (i) Let be a normed vector space with being the associate metric space endowed with a norm-induced translation-invariant and homogeneous metric and consider the self-mapping so that the restricted is -accretive for some , where and are nonempty subsets of subject to , and the restricted is 2-cyclic.Then, If, furthermore, is -contractive, then is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if(ii)Let be a normed vector space. Then if is a -accretive mapping and is 2-cyclic for some where and are nonempty subsets of subject to , then If, furthermore, is 2-cyclic k -contractive, then

Outline of ProofIt follows since the basic constraint of being -accretive holds if while it fails if

Remark 2.16. Theorem 2.13 and Corollary 2.15 are easily linked to Theorem 2.9 as follows. Assume that is 2-cyclic -contractive and is a-accretive mapping. Assume that there exists such that . Then, ; for all from (2.47). This is guaranteed under sufficiency-type conditions with with for some real constants, . It is direct to see that if .

Example 2.17. Constraint (2.50) linking Theorem 2.13 and Corollary 2.15 to Theorem 2.9 is tested in a simple case as follows. Let . is a vector space endowed with the Euclidean norm induced by the homogeneous and translation-invariant Euclidean metric. is a linear self-mapping from to represented by a nonsingular constant matrix in . Then, is the spectral (or -) norm of the -contractive self-mapping which is the matrix norm induced by the corresponding vector norm (the vector Euclidean norm being identical to the vector norm as it is wellknown) fulfilling with the symmetric matrix being a matrix having all its eigenvalues positive and less than one, since is nonsingular, upper-bounded by a real constant which is less than one. Thus, is also -accretive for any real constant and -contractive for any real . Assume now that for some integer with ; for all . If , then . Also, for any integer (then is singular) but the last -components of any are zeroed at the first iteration via so that if is the th unit vector in with its th component being one, then Now, assume that the matrix is of rank one with its first column being of the form with , ; for all . Then, (2.54) still holds by changing in the first equation to . Finally, assume that with . Then, the self-mapping is nonexpansive also noncontractive and which is a vector subspace of , that is, there exist infinitely many fixed points each one being reached depending on the initial in with the property for any given with , .

The following result is concerned with the distance boundedness between iterates through the self-mapping .

Theorem 2.18. Let be a normed vector space with being the associated metric space endowed with a norm-induced translation-invariant and homogeneous metric . Let be a 2-cyclic -contractive self-mapping so that is -accretive for some where and are nonempty subsets of X. Then, for some finite real constants , and , which are independent of and the th power, and is zero if and intersect. Furthermore, is finite irrespective of .

Proof. One gets for , some and that so that one has for with for some real constant provided that : and if then Also, if , and if .

The subsequent result has a close technique for proof to that of Theorem 2.18.

Theorem 2.19. Let be a normed space with an associate metric space endowed with a norm-induced translation-invariant and homogeneous metric and let be a self-mapping on which is -contractive with and 2-cyclic on , where and are nonnecessarily disjoint nonempty subsets of . If such sets and intersect then is also -contractive with and -accretive with if and with if . Irrespective of and being disjoint or not, is still -accretive and the following inequalities hold:

Proof. Direct calculations yield which leads to the inequalities (2.65)–(2.67) with and where with if and if which holds if and only if . The proof is complete.

Remark 2.20. Compared to Theorem 2.9, Theorem 2.19 guarantees the simultaneous maintenance of the -accretive and contractive properties if the subsets of intersect. Otherwise, the contractive property is not guaranteed if to be -accretive for the nontrivial case of since is larger than in general. However, the guaranteed value of is larger than that guaranteed in Theorem 2.9 to make compatible the accretive and contractive properties of the self-mapping. Also, the relevant properties (2.65)–(2.67) hold irrespective of the sets and being bounded or not. Note, in particular, that the uniformly bounded limit superior distance (2.67) is also independent of the boundedness or not of such subsets of .

The following result follows directly from Theorem 2.9 concerning 2-cyclic Kannan self-mappings which are also contractive (see [16]) which are proven to be accretive.

Theorem 2.21. Let be a normed vector space with and being bounded nonempty subsets of and . Consider a 2-cyclic -contractive self-mapping with . Then, is also a -Kannan self-mapping and is -accretive for, for all .

Proof. Since is a 2-cyclic -contractive self-mapping, then one gets for that the following relationships hold from the distance sub-additive property from the proof of Theorem 2.9(i), (2.15): provided that since if so that is -accretive. Note that the function for a contractive self-mapping is the positive solution of , that is, , which is wellposed since for . Thus, is also a 2-cyclic -Kannan self-mapping from Definition 2.6 since implies with

3. Extended Results for -Cyclic Nonexpansive, Contractive, and Accretive Mappings

This section generalizes the main results of Section 2 to -cyclic self-mappings with . Now, it is assumed that there are nonempty subsets of ; for all which can be disjoint or not and a so-called -cyclic self-mapping such that with . Inspired in the considerations of Remark 2.12 claiming that Theorem 2.9 can be directly extended to the case that the subsets and are not necessarily closed, it is not assumed in the sequel that the subsets of ; for all are necessarily closed. A simple notation for distances between adjacent sets is dist. Definition 2.4 is generalized as follows.

Definition 3.1. is a -cyclic weakly -contractive (resp., weakly nonexpansive) self-mapping if for some real constants (resp., ); for all [12, 13] such that (resp., ).

Definition 3.2. is a -cyclic -contractive (resp., nonexpansive) -cyclic self-mapping if for some real constants (resp., ); for all [12, 13].

Assertion 1. A -cyclic weakly nonexpansive self-mapping may be locally expansive for some ; for all which cannot be best proximity points.

Proof. Assume that . Then, the following inequalities can occur for given, :(1) In this case, and since is impossible, one concludes that so that (3.3) can only hold for best proximity points , for which is nonexpansive. If , then the last inequality of (3.4) becomes so that is also a best proximity point if are convex, for all ,(2) and then is nonexpansive for ;(3) and then is expansive for which cannot be best proximity points since .

Remark 3.3. Note from Definitions 3.1 and 3.2 that a -cyclic weakly contractive (resp., contractive) self-mapping is also weakly nonexpansive (resp., weakly contractive). Also, a nonexpansive (resp., contractive) self-mapping is also weakly nonexpansive (resp., weakly contractive). Note that if , is -cyclic weakly nonexpansive and ; for all , then where ; for all , for all . Note that if ; that is, are not best proximity points, then if then since. Thus, a weakly nonexpansive self-mapping is not necessarily nonexpansive for each iteration. However, the composed self-mapping defined as ; for all is nonexpansive in the usual sense since if , then , implies

It has been commented in Remark 2.12 for the case of 2-cyclic self-mappings that results about best proximity and fixed points are extendable to the case that some of the subsets are not closed by using their closures. We use this idea to formulate the main results for -cyclic self-mappings with . The following technical result stands related to the fact that nonexpansive -cyclic self-mappings have identical distances between all the adjacent subsets in the set .

Lemma 3.4. Assume that is -cyclic and nonexpansive. Then, ; for all .

Proof. If (i.e., the closures of the subsets intersect), then the proof is direct since ; for all . Now, assume that for some . Let and best proximity points such that since is a -cyclic nonexpansive self-mapping. Thus, any iterates and are also best proximity points of some subset in; for all . If ; for all does not hold, then from (3.9): Then ;for all which contradicts for some what is a contradiction or ; for all , and .

Note that Lemma 3.4 applies even if the subsets are neither bounded or closed. In this way, note that the contradiction to for some established in the second part of the proof does not necessarily imply that which would require for the subsets , for all to be bounded and, in particular, if such subsets are bounded and closed. The following result stands concerning the limit iterates of -cyclic nonexpansive self-mappings:

Lemma 3.5. The following properties hold. (i)If is a p-cyclic weakly nonexpansive self-mapping, then ifsatisfy (ii) If is a -cyclic nonexpansive self-mapping, then (iii) If is a -cyclic weakly contractive self-mapping, then for all , for all if satisfy the feasibility constraints and . If ; for all , then for all , for all , for all if satisfy the feasibility constraints and (iv) If is a -cyclic contractive self-mapping, then (v) If and is a -cyclic weakly contractive self-mapping, then

Proof. Property (i) follows from (3.7) for . Property (iii) follows from Property (i) since implies as . Property (ii) Follows from Property (i) for ; for all since ; for all from Lemma 3.4. Property (iv) follows from Property (ii) for ; for all since ; for all from Lemma 3.4. Property (v) follows from Property (iii) since if all the subsets ; intersect, then it follows necessarily ; for all so that

Remark 3.6. Note that Lemma 3.5(v) also applies to contractive self-mappings since contractive self-mappings are weakly contractive.

The following result is concerned to the identical distance between adjacent subsets for -cyclic contractive self-mappings. A parallel result is discussed in [10] for Meir-Keeler contractions.

Theorem 3.7. Assume that is a -cyclic weakly -contractive self-mapping and the closures of the subsets Ai; of intersect. Then, it exists a unique fixed point in which is also in if all such subsets; for all of , are closed.

Proof. The existence of a fixed point follows from Lemma 3.5(v). Its uniqueness follows by contradiction. Assume that there exist . Then, for some ,, such that and as . Then, by using triangle inequality for distances, which implies by using Lemma 3.5(v) what contradicts . Therefore, consists of a unique point in which is also in if the sets ; are all closed.

Theorem 3.7 also applies to -cyclic contractive self-mappings since they are weakly contractive. The following result follows from Theorem 2.9, Lemma 3.5 and some parallel result provided in [12].

Theorem 3.8. Let be a uniformly convex Banach space endowed with the translation-invariant and homogeneous metric with nonempty convex subsets, for all of pair-wise disjoint closures. Let be a -cyclic weakly -contractive self-mapping so that the composed 2-cyclic self-mappings, for all are defined as ; for all ; . Then, the following properties hold: (i)Any composed 2-cyclic self-mapping , is -contractive provided that the constraint holds. If, furthermore, it is assumed that and are convex, then the 2-cyclic self-mapping self-mapping is extendable to , and that ; for all . Thus, the iterates and ; for all , for all converge as to best proximity points in and which are also in if is closed, respectively, in if is closed.(ii)If for some given , the sets and are convex and closed, if any, then both best proximity points of of Property (i) are unique and belong, respectively, to and .(iii)Assume that the subsets of are convex, for all . If , then the best proximity points of Property (i) become a unique fixed point for all the composed 2-cyclic self-mappings which are -contractive, . Such a fixed point is in (and also in if all the subsets , , are closed).

Proof. Since; for all , then for any , and if p is even and and if is odd. Since is -cyclic weakly -contractive then , then is 2-cyclic contractive provided that ; . One has from Lemma 3.5(iv) that ; for all , for the given, where if is closed), (( if is closed)) are best proximity points. Using Theorem 2.9(i) for 2-cyclic self-mappings in uniformly convex Banach spaces endowed with translation-invariant and homogeneous metric, one getsand as ; for all . Property (i) has been proven. Property (ii) was proven in Theorem  3.10, [12] for 2-cyclic -contractive self- mappings in uniformly convex Banach spaces since they can be directly endowed with a norm-induced metric. The proof is valid here for a norm- induced distance in a uniformly convex Banach space since such distances are translation-invariant and homogeneous. It is also valid if the subsets are not closed with the fixed point then being in the nonempty intersection of their closures. Property (iii) follows directly from Lemma 3.5(v), which implies that is -contractive for all , and the fact that all distances between the closures of all pairs of adjacent subsets are zero since is a complete metric space since is a Banach space.

Theorem 3.8 also applies to the composed 2-cyclic self-mappings of -contractive -cyclic self-mappings. However, we have the following extension containing stronger results for such a case:

Theorem 3.9. Let be a uniformly convex Banach space endowed with the norm-induced translation-invariant and homogeneous metric with nonempty subsets , for all of pair-wise disjoint closures. Let be a -cyclic -contractive self-mapping so that the composed 2-cyclic self-mappings , for all , are defined as ; for all ; for all . Assume also that is convex and ; for all . Then, the following properties hold. (i)As , the iterates and ; for all , for all converge to best proximity points in and which are also in if is closed, respectively, in if is closed for any. Also, for any given such that the sets and are convex and closed, if any, then both best proximity points of of Property (i) are unique and belong, respectively, to and . If, furthermore, , then the best proximity points of Property (i) become a unique fixed point for the -cyclic -contractive self-mapping . Such a fixed point is in (and also in if all the subsets , are closed).(ii)All the composed 2-cyclic self-mappings , for all are -contractive. Thus, the iterates and ; for all , for all converge as to best proximity points in and which are also in if is closed, respectively in if is closed. For any given such that the sets and are closed and convex, if any, then both best proximity points of of Property (i) are unique and belong, respectively, to and . If, furthermore, , then the best proximity points of Property (i) become a unique fixed point for all the composed 2-cyclic self-mappings which are -contractive; . Such a fixed point is in (and also in if all the subsets , are closed).

Outline of Proof Property (ii) is the direct version of Theorem 3.8 applicable to the composed 2-cyclic self-mappings which are all -contractive since is a -cyclic -contractive self-mapping. Since -cyclic contractive self-mappings are nonexpansive, all the distances between adjacent subsets are identical (Lemma 3.4) so that there is no mutual constraint on distances contrarily to Theorem 3.8(i). Property (i) is close to Property (ii) by taking into account that is also -contractive.

Definition 2.5 is extended to -cyclic self-mappings as follows.

Definition 3.10. is a 2-cyclic -Kannan self-mapping for some real if it satisfies for some : Now, Theorem 2.9 and Theorems 2.18–2.21 for 2-cyclic accretive and Kannan self-mappings extend directly with direct replacements of their relevant parts as follows:

Theorem 3.11. Let be a Banach space so that is its associate complete metric space endowed with a norm-induced translation-invariant and homogeneous metric . Consider a self-mapping which is also a -cyclic -contractive self-mapping if restricted t, where are nonempty convex subsets of X; for all . Then, Theorem 2.9 holds “mutatis-mutandis” by replacing the subsets and for pairs of adjacent subsets and , , , and . In the same way, Theorems 2.18, 2.19, and 2.21 still hold.

The above result extends directly to each composed 2-cyclic self-mappings ; for all defined from the -cyclic weak -contractive self-mapping since ; are -contractive.

Acknowledgments

The author is grateful to the Spanish Ministry of Education for its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government for its support through Grants IT378-10 and SAIOTEK S-09UN12.

References

T. D. Benavides and S. Phothi, “The fixed point property under renorming in some classes of Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1409–1416, 2010.

T. Suzuki, “A new type of fixed point theorem in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5313–5317, 2009.

Dž. Burgić, S. Kalabušić, and M. R. S. Kulenović, “Global attractivity results for mixed-monotone mappings in partially ordered complete metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 762478, 17 pages, 2009.

J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007.

M. O. Olatinwo, “Some common fixed point theorems for selfmappings satisfying two contractive conditions of integral type in a uniform space,” Central European Journal of Mathematics, vol. 6, no. 2, pp. 335–341, 2008.

V. Azhmyakov, “Convexity of the set of fixed points generated by some control systems,” Journal of Applied Mathematics, vol. 2009, Article ID 291849, 14 pages, 2009.

M. De la Sen, “Total stability properties based on fixed point theory for a class of hybrid dynamic systems,” Fixed Point Theory and Applications, vol. 2009, Article ID 826438, 19 pages, 2009.

M. De la Sen, “About robust stability of dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2008, Article ID 480187, 20 pages, 2008.

A. Meir and E. Keeler, “A theorem on contraction mappings,” Journal of Mathematical Analysis and Applications, vol. 28, pp. 326–329, 1969.

S. Karpagam and S. Agrawal, “Best proximity point theorems for -cyclic Meir-Keeler contractions,” Fixed Point Theory and Applications, vol. 2009, Article ID 197308, 9 pages, 2009.

C.-M. Chen and T.-H. Chang, “Fixed point theorems for a weaker Meir-Keeler type ψ-set contraction in metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 129124, 8 pages, 2009.

A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.

M. Kikkawa and T. Suzuki, “Some similarity between contractions and Kannan mappings,” Fixed Point Theory and Applications, vol. 2009, Article ID 649749, 8 pages, 2008.

Y. Enjouji, M. Nakanishi, and T. Suzuki, “A generalization of Kannan's fixed point theorem,” Fixed Point Theory and Applications, vol. 2009, Article ID 192872, 10 pages, 2009.

M. De la Sen, “Some combined relations between contractive mappings, Kannan mappings, reasonable expansive mappings, and -stability,” Fixed Point Theory and Applications, vol. 2009, Article ID 815637, 25 pages, 2009.

M. De la Sen, “Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 572057, 23 pages, 2010.

W. Takahashi, “Viscosity approximation methods for resolvents of accretive operators in Banach spaces,” Journal of Fixed Point Theory and Applications, vol. 1, no. 1, pp. 135–147, 2007.

A. R. Lovaglia, “Locally uniformly convex Banach spaces,” Transactions of the American Mathematical Society, vol. 78, pp. 225–238, 1955.

Y. Yao, Y.-C. Liou, and C.-P. Chen, “Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings,” Taiwanese Journal of Mathematics, vol. 15, no. 5, pp. 1979–1998, 2011.

S. H. Khan, “A modified iterative process for common fixed points of two finite families of nonexpansive mappings,” Analele Stiintifice Universitatii Ovidius Constanta. Seria Mathematica, vol. 19, no. 1, pp. 161–174, 2011.

S. H. Khan, I. Yildirim, and M. Ozdemir, “Convergence of a generalized iteration process for two finite families of Lipschitzian pseudocontractive mappings,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 707–715, 2011.

M. Abbas, S. H. Khan, A. R. Khan, and R. P. Agarwal, “Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme,” Applied Mathematics Letters, vol. 24, no. 2, pp. 97–102, 2011.

S. H. Khan and J. K. Kim, “Common fixed points of two nonexpansive mappings by a modified faster iteration scheme,” Bulletin of the Korean Mathematical Society, vol. 47, no. 5, pp. 973–985, 2010.

E. Turkmen, S. H. Khan, and M. Ozdemir, “Iterative approximation of common fixed points of two nonself asymptotically nonexpansive mappings,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 487864, 16 pages, 2011.

Y. Yao, Y.-C. Liou, and C.-P. Chen, “A regularized gradient projection method for the minimization problem,” Journal of Applied Mathematics, vol. 2012, no. 259813, 9 pages, 2012.

A. Bilbao-Guillerna, M. De la Sen, A. Ibeas, and S. Alonso-Quesada, “Robustly stable multiestimation scheme for adaptive control and identification with model reduction issues,” Discrete Dynamics in Nature and Society, vol. 2005, no. 1, pp. 31–67, 2005.

M. De La Sen, “On the extensions of Krasnoselskii-type theorems to p-cyclic self-mappings in banach spaces,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 568072, 19 pages, 2011.

M. De la Sen, “On a general contractive condition for cyclic self-mappings,” Journal of Applied Mathematics, vol. 2011, Article ID 542941, 17 pages, 2011.

S. Saejunt, K. Wongchan, and P. Yotkaew, “Another weak convergence theorems for accretive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2011, article 26, 2011.

T.-H. Kim and W. Takahashi, “Strong convergence of modified iteration processes for relatively asymptotically nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 14, no. 6, pp. 2163–2180, 2010.