Abstract

𝑝(2)-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 𝑝-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., [35]). 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 [68]. 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 [912]. Another interest of such mappings is their usefulness as formal tool for the study 𝑝-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, 1316]. In fact, Kannan self-mappings are contractive for values of the contraction constant being less than 1/3, [15, 16] and can be simultaneously 𝑝-cyclic Meir-Keeler contractive self-mappings. The objective of this paper is the investigation of relevant properties of contractive 𝑝(2)-cyclic self-mappings of the union of set of subsets of a Banach space (𝑋,) which are simultaneously𝜆-accretive on the whole 𝑋, while investigating the existence and uniqueness of potential fixed points on the subsets of 𝑋 if they intersect and best proximity points. For such a purpose, the concept of 𝜆-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 𝑇 from 𝑋 to 𝑋 under study can be locally increasing on 𝑋 but it is still 𝑝-cyclic contractive on the relevant subsets 𝐴𝑖(𝑖𝑝) of 𝑋. For the obtained results of boundedness of distances between the sequences of iterates through 𝑇, it is not required for the set of subsets of 𝑋 to be either closed or convex. For the obtained results concerning fixed points and best proximity points, the sets 𝐴𝑖(𝑖𝑝)are required to be convex but they are not necessarily closed if the self-mapping 𝑇 can be defined on the union of the closures of the sets𝐴𝑖(𝑖𝑝). Consider a metric space (𝑋,𝑑) associated to the Banach space (𝑋,) and a self-mapping 𝑇𝐴𝐵𝐴𝐵such that 𝑇(𝐴)𝐵 and 𝑇(𝐵)𝐴, where 𝐴 and 𝐵 are nonempty subsets of 𝑋. Then, 𝑇𝐴𝐵𝐴𝐵is a 2-cyclic self-mapping. It is said to be a 2-cyclic 𝑘-contraction self-mapping if it satisfies, furthermore,𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)dist(𝐴,𝐵);𝑥𝐴,𝑦𝐵,(1.1)

for some real 𝑘[0,1). A best proximity point of convex subsets 𝐴 or 𝐵 of 𝑋 is some 𝑧cl(𝐴𝐵)such that 𝑑(𝑧,𝑇𝑧)=dist(𝐴,𝐵). If 𝐴 and 𝐵 are closed then either 𝑧 (resp., 𝑇𝑧) or 𝑇𝑧 (resp. 𝑧) is in 𝐴 (resp., in 𝐵). The distance between subsets 𝐴 and 𝐵 of the metric space dist(𝐴,𝐵)=0 if either 𝐴𝐵or if either 𝐴 or 𝐵 is open with Fr(𝐴)Fr(𝐵). In this case, if 𝑧 is a best proximity point either 𝑧 or 𝑇𝑧 is not in 𝐴𝐵(in particular, neither 𝑧 nor 𝑇𝑧 is in 𝐴𝐵 if both of them are open). It turns out that if 𝐴𝐵 then 𝑧Fix(𝑇)𝐴𝐵; that is, 𝑧 is a fixed point of 𝑇 since dist(𝐴,𝐵)=0, [911]. If 𝑘=1 then 𝑑(𝑇𝑥,𝑇𝑦)𝑑(𝑥,𝑦); forall𝑥𝐴, for all 𝑦𝐵 and 𝑇𝐴𝐵𝐴𝐵is a 2-cyclic nonexpansive self-mapping, [10].

1.1. Notation

𝐑0+=𝐑+{0};𝐙0+=𝐙+{0};𝑝={1,2,,𝑝}𝐙+,(1.2)

superscript 𝑇denotes vector or matrix transpose, Fix(𝑇) is the set of fixed points of a self-mapping 𝑇 on some nonempty convex subset 𝐴 of a metric space (𝑋,𝑑)cl𝐴 and 𝐴 denote, respectively, the closure and the complement in 𝑋 of a subset 𝐴 of 𝑋, Dom(𝑇)and Im(𝑇) denote, respectively, the domain and image of the self-mapping 𝑇 and 2𝑋 is the family of subsets of 𝑋, dist(𝐴,𝐵)=𝑑𝐴𝐵 denotes the distance between the sets 𝐴 and 𝐵 for a 2-cyclic self-mapping 𝑇𝐴𝐵𝐴𝐵 which is simplified as dist(𝐴𝑖,𝐴𝑖+1)=𝑑𝐴𝑖𝐴𝑖+1=𝑑𝑖; forall𝑖𝑝 for distances between adjacent subsets of 𝑝-cyclic self-mappings 𝑇 on 𝑝𝑖=1𝐴𝑖.

𝐵𝑃𝑖(𝑇) which is the set of best proximity points on a subset 𝐴𝑖 of a metric space (𝑋,𝑑) of a 𝑝-cyclic self-mapping 𝑇 on 𝑝𝑖=1𝐴𝑖, the union of a collection of nonempty subsets of (𝑋,𝑑)which do not intersect.

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

Let (𝑋,) be a normed vector space and (𝑋,𝑑)be an associate metric space endowed with a metric (or distance function or simply “distance”) 𝑑𝑋×𝑋𝐑𝟎+. For instance, the distance function may be induced by the norm on𝑋. If the metric is homogeneous and translation-invariant, then it is possible conversely to define the norm from the metric. Consider a self-mapping 𝑇𝑋𝑋 which is a 2-cyclic self-mapping restricted as 𝑇Dom(𝑇)𝑋𝐴𝐵Im(𝑇)𝑋𝐴𝐵, where 𝐴 and 𝐵 are nonempty subsets of 𝑋. Such a restricted self-mapping is sometimes simply denoted as 𝑇𝐴𝐵𝐴𝐵. 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 𝐴 and 𝐵 are not closed, it is assumed that Dom(𝑇)cl(𝐴𝐵) and Im(𝑇)cl(𝐴𝐵) in order to obtain a direct extension of existence of fixed points and best proximity points. This allows, together with the convexity of 𝐴 and 𝐵, to discuss the existence and uniqueness of fixed points or best proximity points reached asymptotically through the sequences of iterates of the self-mapping 𝑇. In some results concerning the accretive property, it is needed to extend the self-mapping 𝑇Dom(𝑇)𝑋Im(𝑇)𝑋 in order to define successive iterate points through the self-mapping which do not necessarily belong to 𝐴𝐵. The following definitions are then used to state the main results.

Definition 2.1. 𝑇Dom(𝑇)𝑋𝑋 is an accretive mapping if 𝑑(𝑥,𝑦)𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦);𝑥,𝑦Dom(𝑇),(2.1) for any 𝜆𝐑0+.
Note that, since 𝑋 is also a vector space, 𝑥+𝜆𝑇𝑥is in 𝑋 for all 𝑥 in 𝑋 and all real 𝜆. 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. 𝑇Dom(𝑇)𝑋𝑋is a 𝜆-accretive mapping, some 𝜆𝐑0+ if 𝑑(𝑥,𝑦)𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦);𝑥,𝑦Dom(𝑇);𝜆0,𝜆,(2.2) for some 𝜆𝐑0+. A generalization is as follows𝑇Dom(𝑇)𝑋𝑋is [𝜆1,𝜆2]-accretive for some 𝜆1,𝜆2(𝜆1)𝐑0+ if 𝑑(𝑥,𝑦)𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦);𝑥,𝑦Dom(𝑇);𝜆𝜆1,𝜆.(2.3)

Definition 2.3. 𝑇Dom(𝑇)𝑋𝑋is a weighted 𝜆-accretive mapping, for some function 𝜆𝑋×𝑋𝐑0+, if 𝑑(𝑥,𝑦)𝑑(𝑥+𝜆(𝑥,𝑦)𝑇𝑥,𝑦+𝜆(𝑥,𝑦)𝑇𝑦);𝑥,𝑦Dom(𝑇).(2.4) 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 𝑋.

Definition 2.4. 𝑇𝐴𝐵𝐴𝐵 is a 2-cyclic 𝑘-contractive (resp., nonexpansive) self-mapping if 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)dist(𝐴,𝐵);𝑥𝐴,𝑦𝐵,(2.5) for some real 𝑘[0,1) (resp. 𝑘=1), [12, 13].
The concepts of Kannan-self mapping and 2-cyclic (𝛼,𝛽)-Kannan self-mapping which can be also a contractive mapping, and conversely if 𝑘<1/3, [16], are defined below.

Definition 2.5. 𝑇𝑋𝑋is a 𝛼-Kannan self-mapping if 𝑑(𝑇𝑥,𝑇𝑦)𝛼(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦));𝑥,𝑦𝑋,(2.6) for some real 𝛼[0,1/2), [12, 13].

Definition 2.6. 𝑇𝐴𝐵𝐴𝐵is an 2-cyclic (𝛼,𝛽)-Kannan self-mapping for some real 𝛼[0,1/2) if it satisfies, for some 𝛽𝐑+. 𝑑(𝑇𝑥,𝑇𝑦)𝛼(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦))+𝛽(1𝛼)dist(𝐴,𝐵);𝑥𝐴,𝑦𝐵.(2.7) The relevant concepts concerning 2-cyclic self-mappings are extended to 𝑝(2)-cyclic self-mappings in Section 3. Some simple explanation examples follow.

Example 2.7. Consider the scalar linear mapping from 𝑋𝐴𝐑 to 𝑋 as 𝑇𝑥=𝛾𝑥+𝛾0with 𝛾,𝛾0𝐑endowed with the Euclidean distance 𝑑(𝑥,𝑦)=|𝑥𝑦|; for all 𝑥,𝑦𝑋. Then, 𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)=||𝑥𝑦+𝜆𝛾(𝑥𝑦)||=||1+𝜆𝛾||||𝑥𝑦||=||1+𝜆𝛾||𝑑(𝑥,𝑦)𝑑(𝑥,𝑦),(2.8) for all 𝑥,𝑦𝐑 for any 𝜆𝐑0+ provided that 𝛾𝐑0+. In this case, 𝑇𝐴𝐵𝑋 is accretive. It is also 𝑘-contractive if since 𝑑(𝑇𝑥,𝑇𝑦)=|𝑇𝑥𝑇𝑦|=𝛾,𝑑(𝑧,𝑦)𝑘𝑑(𝑥,𝑦); for all 𝑥,𝑦𝐑. Also, if 𝛾𝐑, then 𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)|𝜆|𝛾|1|𝑑(𝑥,𝑦)𝑑(𝑥,𝑦); for all 𝑥,𝑦𝐑if 𝜆|𝛾|2, that is, if 𝜆𝜆1=2|𝛾|1. Then, 𝑇𝐑𝐑 is [𝜆1,)-accretive and 𝑘-contractive if |𝛾|𝑘<1.

Example 2.8. Consider the metric space (𝐑,𝑑) with the distance being homogeneous and translation-invariant and a self-mapping 𝑇𝐑𝐑 defined by 𝑇𝑥=𝑡|𝑥|𝑝sgn𝑒𝑥=𝑡|𝑥|𝑝1𝑥 with 𝑡𝐑0+, 𝑝𝐑0+,and sgn𝑒𝑥=sgn𝑥 if 𝑥0and sgn𝑒0=0. If 𝑝𝑡=0,then 𝑇𝐑𝐑 is accretive since 𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)=𝑑(𝑥,𝑦);𝑥,𝑦𝑋;𝜆𝐑0+.(2.9) Furthermore, if 𝑡=0, then 0𝐑 is the unique fixed point with 𝑇𝑗𝑥=0; for all 𝑗𝐙+. If 𝑝=0 then, 𝑇𝑗𝑥=𝑡𝑗𝑧=0as 𝑗 if |𝑡|<1 and then 𝑧=0is again the unique fixed point of 𝑇. In the general case, 𝑇𝑥=𝑡|𝑥|𝑝sgn𝑒𝑥 implies 𝑇2𝑥=𝑇(𝑇𝑥)=𝑡𝑡|𝑥|𝑝sgn𝑒𝑥𝑝sgn𝑒𝑇𝑥=𝑡𝑝+1|𝑥|2𝑝sgn𝑒𝑥𝑝+1,𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)=𝑑1+𝜆𝑡𝑥𝑝1𝑥,1+𝜆𝑡𝑦𝑝1𝑦min||1𝜆𝑡|𝑥|𝑝1||,||1𝜆𝑡||𝑦||𝑝1||𝑑(𝑥,𝑦),𝑑(𝑥,𝑦);𝑥,𝑦𝑋,𝜆0,𝜆,some𝜆𝐑+,(2.10) holds if 𝜆|𝑡||𝑥|𝑝11 that is, 𝑇𝐑𝐑 is weighted 𝜆(𝑥,𝑦)-accretive with 𝜆(𝑥,𝑦)=𝑡1min(|𝑥|1𝑝,|𝑦|1𝑝). The restricted self-mapping 𝑇[1,1]𝑋[1,1]is 𝜆(𝑡1)-accretive. Furthermore, if 𝑝1,then𝑇[1,1]𝑋[1,1] is |𝑡|-contractive if |𝑡|<1 and the iteration 𝑇𝑗𝑥0as 𝑗with 𝑧=0 being the unique fixed point since 𝑑(𝑇𝑥,𝑇𝑦)|𝑡|min|𝑥|𝑝1,||𝑦||𝑝1𝑑(𝑥,𝑦)|𝑡|𝑑(𝑥,𝑦);𝑥,𝑦[1,1].(2.11) Note from the definition of the self-mapping 𝑇𝑥=𝑡|𝑥|𝑝1𝑥 on [1,1] that it is also a 2-cyclic self-mapping from [1,0][0,1] to itself with the property 𝑇([1,0])=[0,1]and 𝑇([0,1])=[1,0].

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

Assertions 1. (1) If 𝑇𝐷(𝑇)𝑋𝑋is an accretive mapping, then it is 𝜆-accretive, for all 𝜆𝐑0+. (2) If 𝑇𝐷(𝑇)𝑋𝑋is 𝜆-accretive, then it is 𝜆1-accretive; for all 𝜆1[0,𝜆]. (3) Any nonexpansive self-mapping 𝑇𝐷(𝑇)𝑋𝑋is 0-accretive and conversely.

Theorem 2.9. Let (𝑋,)be a Banach vector space with(𝑋,𝑑)being the associated complete metric space endowed with a norm-induced translation-invariant and homogeneous metric𝑑𝑋×𝑋𝐑0+. Consider a self-mapping 𝑇𝑋𝑋which restricted to𝑇𝐴𝐵𝐴𝐵is a 2-cyclic 𝑘-contractive self-mapping where 𝐴 and 𝐵 are nonempty subsets of 𝑋. Then, the following properties hold.(i) Assume that the self-mapping 𝑇𝑋𝑋 satisfies the constraint: 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)+(1𝑘)𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵(2.12) with 𝑘,𝜆𝐑0+ satisfying the constraint 𝑘(1+𝑘𝜆)<1. Then, the restricted self-mapping 𝑇𝐴𝐵𝐴𝐵satisfies limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦<;𝑥𝐴,𝑦𝐵(2.13) irrespective of 𝐴 and 𝐵 being bounded or not. If, furthermore, 𝐴 and 𝐵 are closed and convex and𝐴𝐵, then there exists a unique fixed point 𝜔𝐴𝐵 of 𝑇𝐴𝐵𝐴𝐵 such that there exists lim𝑗𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)=0; for all 𝑥𝐴,for all 𝑦𝐵, implying that lim𝑗𝑇𝑗𝑥=lim𝑗𝑇𝑗𝑦=𝜔. If, in addition, dist(𝐴,𝐵)>0so that 𝐴𝐵=, then there exists lim𝑗𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)=𝑑(𝑧,𝑇𝑧); for all 𝑥𝐴, for all 𝑦𝐵 for some best proximity points 𝑧𝐴, 𝑇𝑧𝐵 which depend in general on 𝑥 and 𝑦. Furthermore, if (𝑋,) is a uniformly convex Banach space, then 𝑇2𝑗𝑥,𝑇2𝑗+1𝑦𝑧1𝐴 and 𝑇2𝑗𝑦,𝑇2𝑗+1𝑥𝑇𝑧1𝐵 as ; for all(𝑥,𝑦)𝐴×𝐵, where 𝑧1 and 𝑧2 are unique best proximity points in 𝐴 and 𝐵 of 𝑇𝐴𝐵𝐴𝐵.
(ii)Assume that 𝐴 and 𝐵 are nondisjoint. Then, 𝑇𝐴𝐵𝑋is also 𝑘𝑐 contractive and 𝜆-accretive for any nonnegative 𝜆𝑘2(𝑘𝑐𝑘) and any 𝑘𝑐[𝑘,1). It is also nonexpansive and 𝜆-accretive for any nonnegative 𝜆𝑘2(1𝑘).(iii)If 𝑘=0 then 𝑇𝐴𝐵𝑋 is weighted 𝜆-accretive for 𝜆𝑋×𝑋𝐑0+ for any 𝜆𝐑+ and its restriction 𝑇𝐴𝐵𝐴𝐵 is 2-cyclic 0-contractive.(iv)𝑇𝐴𝐵𝑋 is weighted 𝜆-accretive for 𝜆𝑋×𝑋𝐑0+ satisfying 𝜆(𝑥,𝑦)𝑘2(𝑘𝑐(𝑥,𝑦)𝑘)(𝑑(𝑥,𝑦)𝑑𝐴𝐵) for some 𝑘𝑐𝑋×𝑋[𝑘,). The restricted self-mapping 𝑇𝐴𝐵𝐴𝐵 is also 𝑘𝑐-contractive with 𝑘𝑐[𝑘,𝑘𝑐)[𝑘,1) if 𝑘𝑐𝑋×𝑋[𝑘,𝑘𝑐) with 𝑘𝑐<1. Also,𝑇𝐴𝐵𝑋 is nonexpansive and weighted 𝜆-accretive for 𝜆𝑋×𝑋𝐑0+ satisfying 𝜆(𝑥,𝑦)𝑘2(𝑘𝑐(𝑥,𝑦)𝑘)(𝑑(𝑥,𝑦)𝑑𝐴𝐵) if 𝑘𝑐𝑋×𝑋[𝑘,1] which implies, furthermore, that 𝜆𝑋×𝑋𝐑0+is bounded.

Proof. Let us denote 𝑑𝐴𝐵=dist(𝐴,𝐵).Consider that the two following relations are verified simultaneously: 𝑑(𝑥,𝑦)𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)forsome𝜆𝐑0+;𝑥𝐴,𝑦𝐵,𝑑(𝑥,𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵forsome𝑘[0,1)𝑅,𝜆𝐑0+;𝑥𝐴,𝑦𝐵.(2.14) Since the distance 𝑑𝑋×𝑋𝐑0+ is translation-invariant and homogeneous, then the substitution of (2.14) yields if 𝐴 and 𝐵 are disjoint sets, after using the subadditive property of distances, the following chained relationships since 0𝑋: 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑥+𝜆𝑇𝑦𝜆𝑇𝑥)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑥)+𝑘𝑑(𝑦+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑥+𝜆𝑇𝑦𝜆𝑇𝑥)+(1𝑘)𝑑𝐴𝐵=𝑘𝑑(𝑥,𝑦)+𝑘𝜆𝑑(0,𝜆𝑇𝑦𝜆𝑇𝑥)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥,𝑦)+𝑘2𝜆𝑑(0,𝑦𝑥)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥,𝑦)+𝑘2𝜆𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘(1+𝑘𝜆)𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑐𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵;𝜆0,𝜆,for𝜆𝑘2(1𝑘),(2.15) with 𝑘𝑐=𝑘(1+𝑘𝜆)𝑘. Note from (2.15) that 𝑑𝐴𝐵𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑘𝑗𝑐𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑗1𝑖=0𝑘𝑖𝑐=𝑘𝑗𝑐𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑖=0𝑘𝑖𝑐𝑖=𝑗𝑘𝑖𝑐𝑘𝑗𝑐𝑑(𝑥,𝑦)+(1𝑘)1𝑘𝑗𝑐1𝑘𝑐𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵,(2.16) and, if 𝑘𝑐<1, then 𝑑𝐴𝐵limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦1𝑘1𝑘𝑐𝑑𝐴𝐵=1𝑘1𝑘(1+𝑘𝜆)𝑑𝐴𝐵<;𝑥𝐴,𝑦𝐵.(2.17) If 𝑑𝐴𝐵=0 then lim𝑗𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)=0. It is first proven that the existence of the limit of the distance implies that of the limit lim𝑗𝑇𝑗𝑧; for all 𝑧𝐴𝐵. Let be 𝑥𝑗=𝑇𝑗𝑥, 𝑦𝑗=𝑇𝑗𝑦 with𝑥𝑗,𝑦𝑗𝐴𝐵. Then, lim𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦=lim𝑗𝑑𝑥𝑗,𝑦𝑗=lim𝑗𝑑𝑇𝑥𝑗,𝑇𝑦𝑗=0;𝐙0+𝑥𝑗=𝑇𝑗𝑥𝑦𝑗=𝑇𝑗𝑦𝑇𝑥𝑗𝑦𝑗0as𝑗;𝐙0+(2.18) since 𝑇𝐴𝐵𝐴𝐵 being contractive is globally Lipschitz continuous. Then, lim𝑗𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)=𝑑(lim𝑗𝑇𝑗𝑥,lim𝑗𝑇𝑗𝑦)=0since, because the fact that the metric is translation-invariant, one gets lim𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦=𝑑lim𝑗𝑇𝑗𝑥,lim𝑗𝑇𝑗𝑦=lim𝑗𝑑0,𝑇𝑗𝑦𝑇𝑗𝑥,=𝑑0,lim𝑗𝑇𝑗𝑦𝑇𝑗𝑥=0.(2.19) As a result, lim𝑗𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)=0 if 𝑑𝐴𝐵=0 what implies which lim𝑗(𝑇𝑗𝑥𝑇𝑗𝑦)=0; for all 𝑥𝐴,for all 𝑦𝐵, since 𝑇𝐴𝐵𝐴𝐵 is globally Lipschitz continuous since it is contractive.
In addition, there exists lim𝑗𝑇𝑗𝑥=lim𝑗𝑇𝑗𝑦=𝜔𝐴𝐵; for all 𝑥𝐴, for all 𝑦𝐵. Assume not so that there exists 𝑥𝐴 such that ¬lim𝑗𝑇𝑗𝑥 and there exists a subsequence on nonnegative integers {𝑗𝑘}𝑘𝐙0+ such that 𝑇𝑗𝑘+1𝑥𝑇𝑗𝑘𝑥. If so, one gets by taking 𝑦=𝑇𝑥𝐵that 𝑑(𝑇𝑗𝑘(𝑇𝑥),𝑇𝑗𝑘𝑥)>0 which contradicts lim𝑗𝑑(𝑇𝑗(𝑇𝑥),𝑇𝑗𝑥)=0. Then {𝑇𝑗𝑥}𝑗𝐙0+ is a Cauchy sequence for any 𝑥𝐴𝐵 and then converges to a limit. Furthermore, 𝜔𝐴𝐵 since 𝑇𝑗(𝐴𝐵)𝐴𝐵for any 𝑗𝐙0+ and as 𝑗 since 𝐴 and 𝐵 are nonempty and closed. It has been proven that lim𝑗𝑇𝑗𝑥=lim𝑗𝑇𝑗𝑦=𝜔𝐴𝐵; for all 𝑥𝐴,for all 𝑦𝐵.
It is now proven that𝜔=𝑇𝜔Fix(𝑇). Assume not, then, from triangle inequality, 0<𝑑(𝑇𝜔,𝜔)𝑑𝜔,𝑇𝑗𝜔+𝑑𝑇𝜔,𝑇𝑗𝜔;𝑗𝐙0+liminf𝑗𝑑𝜔,𝑇𝑗𝜔>0,(2.20) which contradicts lim𝑗𝑇𝑗𝜔=𝜔 so that 𝜔=𝑇𝜔Fix(𝑇). It is now proven that 𝜔Fix(𝑇)(𝐴𝐵). Assume not, such that, for instance, 𝑇𝑗𝑥𝐴 and 𝑇𝑗+1𝑥𝐴𝐵. If so, since 𝑇(𝐴)𝐵;𝑇(𝐵)𝐴, then the existing limit fulfils lim𝑗𝑇𝑗𝑥𝐴𝐴(=) which is impossible so that there would be no existing limitlim𝑗𝑇𝑗𝑥 in 𝐴𝐵, contradicting the former result of its existence. Then, 𝜔Fix(𝑇)(𝐴𝐵) implying that Fix(𝑇)𝐴𝐵.
It is now proven by contradiction that 𝜔=lim𝑗𝑇𝑗𝑥; for all 𝑥𝐴𝐵 is the unique fixed point of 𝑇𝐴𝐵𝐴𝐵. Assume that 𝜔1(𝜔)Fix(𝑇), then lim𝑗𝑇𝑗𝑦1=𝜔1 for some 𝑦1(𝑦)𝐵 with no loss in generality and all 𝑥𝐴. Thus, lim𝑗𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦1)=𝑑(𝜔,𝜔1)=0𝜔=𝜔1 which contradicts 𝜔𝜔1 so that Fix(𝑇)={𝜔}.
Now, assume that 𝐴 and 𝐵 do not intersect so that dist(𝐴,𝐵)=𝑑𝐴𝐵>0. Then, one gets from the first inequality in (2.15) that for all 𝑥𝐴, 𝑦𝐵, one gets𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑘𝑗𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑖=0𝑘𝑗=𝑘𝑗𝑑(𝑥,𝑦)+𝑑𝐴𝐵;𝑗𝐙,limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑑𝐴𝐵.(2.21)
Note that since 𝑇(𝐴)𝐵, 𝑇(𝐵)𝐴 and dist(𝐴,𝐵)=𝑑𝐴𝐵>0, then𝑥𝐴𝑇𝑗𝑥𝐴 and 𝑇𝑗𝑥𝐵 if 𝑗 is even and 𝑇𝑗𝑥𝐵 and 𝑇𝑗𝑥𝐴 if 𝑗 is odd𝑦𝐵𝑇𝑗𝑦𝐵and 𝑇𝑗𝑦𝐴 if 𝑗 is even and 𝑇𝑗𝑦𝐴 and 𝑇𝑗𝑦𝐵 if 𝑗 is odd.
Then, 𝑇𝑗𝑥 and 𝑇𝑗𝑦 are not both in either 𝐴 or 𝐵 if 𝑥 and 𝑦 are not both in either 𝐴 or 𝐵 for any 𝑗𝐙0+. As a result, lim𝑗sup𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)<𝑑𝐴𝐵 is impossible so thatlim𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦=limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦=𝑑𝐴𝐵=𝑑(𝑧,𝑇𝑧),(2.22) for some best proximity points 𝑧𝐴 and 𝑇𝑧𝐵 or conversely. Then, lim𝑗𝑑𝑇𝑗+1𝑥,𝑇𝑗+1𝑦=lim𝑗𝑑𝑇𝑧𝑗,𝑇2𝑧𝑗=𝑑𝐴𝐵𝑘lim𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦+(1𝑘)𝑑(𝑧,𝑇𝑧)=𝑘lim𝑗𝑑𝑧𝑗,𝑇𝑧𝑗+(1𝑘)𝑑(𝑧,𝑇𝑧)=𝑘lim𝑗𝑑𝑧𝑗,𝑇𝑧𝑗+(1𝑘)𝑑𝐴𝐵,(2.23) where 𝑧𝑗=𝑇𝑗𝑥 Thus, lim𝑗𝑑(𝑧𝑗,𝑇𝑧𝑗)=𝑑𝐴𝐵=𝑑(𝑧,𝑇𝑧). It turns out that dist(𝑧𝑗,Fr(𝐴𝐵))0 and dist(𝑇𝑧𝑗,Fr(𝐴𝐵))0 as 𝑗. Otherwise, it would exist an infinite subsequence {𝑑(𝑧𝑗,𝑇𝑧𝑗)}𝑗𝑍0+of {𝑑(𝑧𝑗,𝑇𝑧𝑗)}𝑗𝑍0+ with 𝐙0+being an infinite subset of 𝐙0+such that 𝑑(𝑧𝑗,𝑇𝑧𝑗)>𝑑𝐴𝐵 for 𝑗𝐙0+. On the other hand, since (𝑋,) is a normed space, then by taking the norm-translation invariant and homogeneous induced metric and since there exists lim𝑗𝑑(𝑇𝑗+1𝑥,𝑇𝑗+1𝑦)=𝑑𝐴𝐵, it follows that there exist 𝑗1𝐙0+ and 𝛿=𝛿(𝜀,𝑗1)𝐑+ such that 2𝑑𝐴𝐵+𝛿<𝑑𝑇𝑗𝑥+𝑇𝑗+1𝑦,0𝑑𝑇𝑗𝑥,0+𝑑𝑇𝑗+1𝑦,0,2𝑑𝐴𝐵+𝛿𝑑𝑇𝑗𝑥,𝑇𝑗+1𝑦<𝜀,(2.24) for any given 𝜀𝐑+; for all 𝑥𝐴,for all 𝑦𝐵 with 𝑇𝑗𝑥𝐴,𝑇𝑗+1𝑦𝐴 for any even 𝑗(𝑗1)𝐙0+ and 𝑇𝑗𝑥𝐵,𝑇𝑗+1𝑦𝐵, for any odd 𝑗(𝑗1)𝐙0+. As a result, by choosing the positive real constant arbitrarily small, one gets that 𝑇2𝑗𝑥𝑇2𝑗+1𝑦𝑧=𝑧(𝑥,𝑦)𝐴 (a best proximity point of 𝐴) and 𝑇2𝑗+1𝑥𝑇2𝑗𝑦𝑇𝑧𝐵 (a best proximity point of 𝐵), or vice-versa, as 𝑗 for any given 𝑥𝐴 and 𝑦𝐵. A best proximity point 𝑧𝐴𝐵 fulfils 𝑧=𝑇2𝑧. Best proximity points are unique in 𝐴 and 𝐵 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 𝑧1 and 𝑧2 in 𝐴. Then 𝑇2𝑧1=𝑧1 and 𝑇2𝑧1=𝑧2 contradict 𝑧1𝑧2 so that necessarily 𝑧1=𝑇2𝑧1𝑧2=𝑇2𝑧2. Since (𝑋,) is a uniformly convex Banach space, we take the norm-induced metric to consider such a space as the complete metric space (𝑋,𝑑) to obtain the following contradiction: 𝑑𝐴𝐵=𝑑𝑧1,𝑇𝑧1=𝑑𝑧1,𝑇𝑧2=𝑇𝑧2𝑧12+𝑇𝑧2𝑧12<2𝑇𝑧2𝑧12=𝑑𝐴𝐵,(2.25) since (𝑋,) is also a strictly convex Banach space and 𝐴 and 𝐵 are nonempty closed and convex sets. Then, 𝑧=𝑇2𝑧𝐴 is the unique best proximity point of 𝑇𝐴𝐵𝐴𝐵 in 𝐴 and 𝑇𝑧 is its unique best proximity point in 𝐵. Then, Property (i) has been fully proven. Since 𝐴 and 𝐵 are not disjoint, then 𝑑𝐴𝐵=0, and 𝑇𝐴𝐵𝐴𝐵is 𝑘𝑐-contractive and 𝜆-accretive if 𝜆=𝑘2(𝑘𝑐𝑘) with 𝑘𝑐[𝑘,1). By taking 𝑘𝑐=1, note that 𝑇𝐴𝐵𝑋 is nonexpansive and 𝑘2(1𝑘)-accretive. Property (ii) has been proven.
To prove Property (iii), we now discuss if𝑑(𝑇𝑥,𝑇𝑦)𝑘(1+𝑘𝜆)𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑐𝑑(𝑥,𝑦)+1𝑘𝑐𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵.(2.26) is possible with 1𝑘c and 𝑑AB>0. Note that  𝑑𝐴𝐵=dist(𝐴,𝐵)=𝑑(𝑧,𝑇𝑧) for some 𝑧𝐴. Define 𝑑=max(𝑑𝐴,𝑑𝐵)=𝑘𝐷𝑑𝐴𝐵, if  𝑑𝐴𝐵0 for some 𝑘𝐷𝐴,𝑘𝐷𝐵,𝑘𝐷𝐑+, where 𝑑𝐴=diam𝐴=𝑘𝐷𝐴𝑑𝐴𝐵 and 𝑑𝐵=diam𝐵=𝑘𝐷𝐵𝑑𝐴𝐵. Three cases can occur in (2.26), namely,(a)If 𝑘=𝑘𝑐 then 𝑘2𝜆𝑑(𝑥,𝑦)0[𝑘𝜆=0𝑑(𝑥,𝑦)0] which is untrue if 𝑥𝑦 and 𝑘𝜆>0 and it holds for either 𝑘=0 or 𝜆=0,(b)𝑘𝑐>𝑘, then (2.26) is equivalent to 𝑑(𝑥,𝑦)𝑘𝑐𝑘𝑘𝑐𝑘(1+𝑘𝜆)𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵.(2.27) Take 𝑥𝐴 to be a best proximity point with so that 𝑑(𝑥,𝑇𝑥)=𝑑𝐴𝐵(𝑘𝑐𝑘)/(𝑘𝑐𝑘(1+𝑘𝜆))𝑑𝐴𝐵>𝑑𝐴𝐵 which is untrue if 𝑘𝜆>0 and true for 𝑘𝜆=0,(c)1𝑘(1+𝑘𝜆)𝑘𝑐<𝑘, then (2.16) is equivalent to (𝑘𝑘𝑐)𝑑𝐴𝐵[𝑘(1+𝑘𝜆)𝑘𝑐]𝑑(𝑥,𝑦); for all 𝑥𝐴,for all 𝑦𝐵, but 𝑑(𝑥,𝑦)2𝑑+𝑑𝐴𝐵=(2𝑘𝐷+1)𝑑𝐴𝐵. Thus, the above constraint is guaranteed to hold in the worst case if 𝑘𝑘𝑐(𝑘+𝑘2𝜆𝑘𝑐)(2𝑘𝐷+1)>𝑘𝑘𝑐 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 𝜆 and 𝑘𝑐 with the real functions 𝜆𝑋×𝑋𝐑0+ and 𝑘𝑐𝑋×𝑋[𝑘,1). Note that (2.26) holds through direct calculation if 𝜆(𝑥,𝑦)𝑘2(𝑘𝑐(𝑥,𝑦)𝑘)(𝑑(𝑥,𝑦)𝑑𝐴𝐵); for all 𝑥𝐴,for all 𝑦𝐵 for some 𝑘𝑐𝑋×𝑋[𝑘,). Thus, the self-mapping 𝑇𝐴𝐵𝑋 is weighted 𝜆-accretive for 𝜆𝑋×𝑋𝐑0+ satisfying 𝜆(𝑥,𝑦)𝑘2(𝑘𝑐(𝑥,𝑦)𝑘)(𝑑(𝑥,𝑦)𝑑𝐴𝐵) for some 𝑘𝑐𝑋×𝑋[𝑘,); and it is also 𝑘𝑐-contractive with 𝑘𝑐[𝑘,𝑘𝑐)[𝑘,1) if 𝑘𝑐𝑋×𝑋[𝑘,𝑘𝑐) with 𝑘𝑐<1 and nonexpansive if 𝑘𝑐𝑋×𝑋[𝑘,1]. On the other hand, note that 𝑑(𝑥,𝑦)𝑑𝐴𝐵𝑘𝐷𝐴𝑑𝐴+𝑘𝐷𝐵𝑑𝐵2𝑘𝐷𝑑. If 𝐴 and 𝐵 are bounded and 𝑘𝑐𝑋×𝑋[𝑘,1], then 𝜆(𝑥,𝑦)𝑘2𝑘𝑐(𝑥,𝑦)𝑘𝑑(𝑥,𝑦)𝑑𝐴𝐵𝑘2𝑘𝑐(𝑥,𝑦)𝑘𝑘𝐷𝐴𝑑𝐴+𝑘𝐷𝐵𝑑𝐵2𝑘2𝑘𝑐(𝑥,𝑦)𝑘𝑘𝐷𝑑;𝑥𝐴,𝑦𝐵.(2.28) 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 𝐴 and 𝐵 are disjoint by introducing the concept of weighted accretive mapping since for best proximity points 𝑧𝐴𝐵, 𝜆(𝑧,𝑇𝑧)=0.

Remark 2.11. Note that the assumption that (𝑋,) 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 𝑇𝐴𝐵𝐴𝐵 in 𝐴 and 𝐵 are obtained provided that both sets are nonempty, convex, and closed.

Remark 2.12. Note that if either 𝐴 or 𝐵 is not closed, then its best proximity point of 𝑇𝐴𝐵𝐴𝐵 is in its closure since 𝑇(𝐴)𝐵cl𝐵, 𝑇(𝐵)𝐴cl𝐴 leads to 𝑇(𝐴𝐵)𝐴𝐵cl(𝐴𝐵) and 𝑇𝑘(𝐴𝐵)cl(𝐴𝐵) for finitely many and for infinitely many iterations through the self-mapping 𝑇𝐴𝐵𝐴𝐵 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, [1925] 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 (𝑋,𝑑) be a metric space endowed with a norm-induced translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+. Consider the 𝜆-accretive mapping 𝑇𝐴𝐵𝑋for some 𝜆𝐑0+ which restricted as 𝑇𝐴𝐵𝐴𝐵 is 2-cyclic, where 𝐴 and 𝐵 are nonempty subsets of 𝑋 subject to 0𝐴𝐵. Then, 𝑑(𝐼+𝜆𝑇)𝑗𝑥,01;𝑗𝐙0+,𝑥(0)𝐴𝐵,𝜆0,𝜆.(2.29) If, furthermore, 𝑇𝐴𝐵𝐴𝐵is 𝑘-contractive, then 1𝑑(𝐼+𝜆𝑇)𝑗𝑥,0<𝑘1;𝑗𝐙+𝑥(0)𝐴𝐵,𝜆0,𝜆.(2.30)𝑇𝐴𝐵𝐴𝐵is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if 𝑑(𝐼+𝜆𝑇)𝑗𝑥,0=1;𝑗𝐙+,𝑥(0)𝐴𝐵,𝜆0,𝜆,(2.31) respectively, limsup𝑗𝑑(𝐼+𝜆𝑇)𝑗𝑥,0=1;𝑥(0)𝐴𝐵,𝜆0,𝜆.(2.32)(ii)Let (𝑋,) be a normed vector space. Consider a 𝜆-accretive mapping 𝑇𝐴𝐵𝑋 for some 𝜆𝐑0+ which restricted to 𝑇𝐴𝐵𝐴𝐵 is 2-cyclic, where 𝐴 and 𝐵 are nonempty subsets of 𝑋 subject to 0𝐴𝐵 then (𝐼+𝜆𝑇)𝑗1;𝑗𝐙0+,𝜆0,𝜆.(2.33) If, furthermore, 𝑇𝐴𝐵𝐴𝐵 is 𝑘-contractive, then 1(𝐼+𝜆𝑇)𝑗<𝑘1;𝑗𝐙+,𝜆0,𝜆.(2.34)𝑇𝐴𝐵𝐴𝐵is nonexpansive (resp., asymptotically nonexpansive, [30]) if (𝐼+𝜆𝑇)𝑗=1;𝑗𝐙+,𝜆0,𝜆,(2.35) respectively, limsupj(𝐼+𝜆𝑇)𝑗=1;𝜆0,𝜆.(2.36)

Proof. To prove Property (i), define an induced by the metric norm as follows 𝑥=𝑑(𝑥,0)since the metric is homogeneous and translation-invariant. Define the norm of 𝑇𝐴𝐵𝐴𝐵, that is, the norm of 𝑇 on 𝑋 restricted to 𝐴𝐵 as follows: 𝑇=min𝑐𝐑0+𝑇𝑥𝑐𝑥;𝑥𝐴𝐵min𝑐𝐑0+𝑑(𝑇𝑥,0)𝑐𝑑(𝑥,0);𝑥𝐴𝐵,(2.37) with the above set being closed, nonempty, and bounded from below. Since 𝑇𝐴𝐵𝐴𝐵 is 2-cyclic and 𝑇𝐴𝐵𝑋 is 𝜆-accretive (Definition 2.2), one gets by proceeding recursively 𝑑(𝑥,𝑦)𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)𝑑(𝑥+𝜆𝑇𝑥+𝜆𝑇(𝑥+𝜆𝑇𝑥),𝑦+𝜆𝑇𝑦)=𝑑(𝐼+𝜆𝑇)2𝑥,(𝐼+𝜆𝑇)2𝑦𝑑(𝐼+𝜆𝑇)𝑗𝑥,(𝐼+𝜆𝑇)𝑗𝑦(𝐼+𝜆𝑇)𝑗𝑑(𝑥,𝑦);𝑥𝐴,𝑦𝐵,𝑗𝐙0+,𝜆0,𝜆,(2.38) since the metric is homogeneous and 0𝐴𝐵, and 𝐼 is the identity operator on 𝑋, where (𝐼+𝜆𝑇)𝑗=min𝑐𝐑0+(𝐼+𝜆𝑇)𝑗𝑥𝑐𝑥;𝑥𝐴𝐵,min𝑐𝐑0+𝑑(𝐼+𝜆𝑇)𝑗𝑥,0𝑐𝑑(𝑥,0);𝑥𝐴𝐵,(2.39) with the above set being closed, nonempty, and bounded from below. If(𝐼+𝜆𝑇)𝑗<1 for some (𝐼+𝜆𝑇)𝑗<1, then we get the contradiction 𝑑(𝑥,𝑦)<𝑑(𝑥,𝑦); for all 𝑥𝐴, for all 𝑦𝐵 in (2.38). Thus, (𝐼+𝜆𝑇)𝑗=𝑑((𝐼+𝜆𝑇)𝑗𝑥,0)1; for all 𝑗𝐙0+, forall𝑥(0)𝐴𝐵, for all 𝜆[0,𝜆]. If now 𝑥 and 𝑦 are replaced with 𝑇𝑖𝑥 and 𝑇𝑖𝑦 for any 𝑖𝐙0+ in (2.30), one gets if 𝑇𝐴𝐵𝐴𝐵 is a 2-cyclic 𝑘-contractive for some real 𝑘[0,1) and 𝜆-accretive mapping: 𝑑𝑇𝑖𝑥,𝑇𝑖𝑦𝑑(𝐼+𝜆𝑇)𝑗𝑇𝑖𝑥,(𝐼+𝜆𝑇)𝑗𝑇𝑖𝑦(𝐼+𝜆𝑇)𝑗𝑑𝑇𝑖𝑥,𝑇𝑖𝑦𝑘𝑖(𝐼+𝜆𝑇)𝑗𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵<𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵,(2.40) for all 𝑥𝐴, for all 𝑦(𝑥)𝐵, for all 𝑗(𝑖)𝐙+, for all 𝑖𝐙+, for all 𝜆[0,𝜆]. Then, 1(𝐼+𝜆𝑇)𝑗=𝑑((𝐼+𝜆𝑇)𝑗𝑥,0)<𝑘1; for all 𝑗𝐙+, for all 𝑥(0)𝐴𝐵, for all 𝜆[0,𝜆]. If (𝐼+𝜆𝑇)𝑗=𝑑((𝐼+𝜆𝑇)𝑗𝑥,0)=1; for all 𝑗𝐙+, for all 𝑥(0)𝐴𝐵, for all 𝜆[0,𝜆], it turns out that 𝑇𝐴𝐵𝑋 is 𝜆-accretive and 𝑇𝐴𝐵𝐴𝐵is a 2-cyclic nonexpansive self-mapping. It is asymptotically nonexpansive if lim𝑗sup𝑑((𝐼+𝜆𝑇)𝑗𝑥,0)=1; for all 𝑥(0)𝐴𝐵, forall𝜆[0,𝜆]. Property (i) has been proven. The proof of Property (ii) for (𝑋,) being a normed vector space is identical to that of Property (i) without associating the norms to a metric.

Example 2.14. Assume that 𝑋=𝐑, 𝐴=𝐑,𝐵=𝐑+ and the 2-cyclic self-mapping 𝑇()𝑡(𝐴𝐵)×𝐙0+(𝐴𝐵)×𝐙0+ defined by the iteration rule 𝐴𝐵𝑥𝑗+1=𝑘𝑗𝑥𝑗𝐴𝐵 with 𝐑𝑘𝑗([𝑘,𝑘])𝑘1, sgn𝑘𝑗+1=sgn𝑘𝑗=sgn𝑥𝑗; for all 𝑗𝐙0+, and 𝑥0𝐴𝐵. Let 𝑑𝐑0+𝐑0+ be the Euclidean metric.(a)If𝑘<1, then lim𝑗𝑗𝑖=0[𝑘𝑗]=0 so that for any 𝑥0𝐴𝐵, 𝑥𝑗𝐴𝐵; for all 𝑗𝐙0+𝑥𝑗𝑧=0𝐴𝐵 as 𝑗 with 0cl(𝐴𝐵), Fix(𝑡)={0}cl(𝐴𝐵) but it is not in 𝐴𝐵 which is empty. If 𝑘=1 and lim𝑗𝑗𝑖=0[𝑘𝑗]=0 (i.e., there are infinitely many values |𝑘𝑖| being less than unity), then the conclusion is identical. If 𝐴 and 𝐵 are redefined as 𝐴=𝐑0, 𝐵=𝐑0+, then Fix(𝑡)={0}𝐴𝐵.(b)If 𝑘𝑗=𝑘=1; for all 𝑗𝐙0+ the self-mapping 𝑡(𝐴𝐵)×𝐙0+(𝐴𝐵)×𝐙0+is not expansive and there is no fixed point. (c)If 𝑘=1𝜎 for some 𝜎(<1)𝐑+, then for𝐑0𝜆[0,𝜆], 𝑑(𝑡𝑥,𝑡𝑦)𝐾𝑑(𝑥,𝑦)𝑘(1+𝜆)||𝑥𝑦||𝑘||𝑥𝑦||;𝑑(𝑥,𝑦)𝑑(𝑥+𝜆𝑥,𝑦+𝜆𝑦)(1+𝜆)||𝑥𝑦||,(2.41) so that 𝑡(𝐴𝐵)×𝐙0+(𝐴𝐵)×𝐙0+is also 𝜆-accretive and 𝑘1[𝑘,1)-contractive with 𝜆=𝑘1𝑘11. (d)Now, define closed sets𝐑𝜀+={𝑟(𝜀)𝐑+} and 𝐑𝜀={𝑟(𝜀)𝐑+} for any given 𝜀𝐑0+so that 𝑑𝐴𝐵=𝜀. The 2-cyclic self-mapping 𝑇()𝑡(𝐴𝐵)×𝐙0+(𝐴𝐵)×𝐙0+is re-defined by the iteration𝑥𝑗+1=𝑥𝑗+1if |𝑥𝑗+1|𝜀and 𝑥𝑗+1=𝜀sgn𝑥𝑗, for 𝑖=1,2, otherwise, where𝑥𝑗+1=𝑘𝑗𝑥𝑗for 𝑖=1,2 with the real sequence {𝑘𝑗}𝑗𝐙0+being subject to 𝑘𝑗([𝑘,𝑘])𝑘1, sgn𝑘𝑗+1=sgn𝑘𝑗=sgn𝑥(𝑖)𝑗; 𝑖=1,2, for all 𝑗𝐙0+and 𝑥0𝐴𝐵. Then, for any 𝜀𝐑+and any 𝑥0𝐴𝐵, there are two best proximity points 𝑧=𝜀𝐴 and 𝑧1=𝜀𝐵 fulfilling 𝜀=𝑡𝜀=𝑡2𝜀 and 𝑑𝐴𝐵=𝑑(𝑧,𝑧1)=𝑑(𝑧,𝑡𝑧)=𝑑(𝑡𝑧1,𝑧1).(e)Redefine 𝑋=𝐑2so that 𝐑2𝑥=(𝑥(1),𝑥(2))𝑇 with 𝑥(1), 𝑥(2)𝐑; 𝐴=𝐑2𝜀,𝐵=𝐑2𝜀+. In the case that 𝜀=0,then 𝐴 and 𝐵 are open disjoint subsets (resp., 𝐴=𝐑20,𝐵=𝐑20+ are closed nondisjoint subsets with 𝐴𝐵={(0,𝑥)𝑇𝑥𝐑}).The 2-cyclic self-mapping 𝑇()𝑡(𝐴𝐵)×𝐙0+(𝐴𝐵)×𝐙0+is re-defined by the iteration rule: 𝑥(𝑖)𝑗+1=𝑥(𝑖)𝑗+1,if|||𝑥(𝑖)𝑗+1|||𝜀,𝑥(𝑖)𝑗+1=𝜀sgn𝑥(𝑖)𝑗,for𝑖=1,2,(2.42) otherwise, where 𝑥(𝑖)𝑗+1=𝑘𝑗𝑥(𝑖)𝑗,for𝑖=1,2(2.43) with the real sequence {𝑘𝑗}𝑗𝐙0+ being subject to𝑘𝑗([𝑘,𝑘])𝑘1, sgn𝑘𝑗+1=sgn𝑘𝑗=sgn𝑥(𝑖)𝑗; for 𝑖=1,2; for all 𝑗𝐙0+ and 𝑥0𝐴𝐵.
The same parallel conclusions to the above ones (a)–(c) follow related to the existence of the unique fixed point 𝑧=0 in the closure of 𝐴 and 𝐵 but not in its empty intersection if either 𝐴 or 𝐵 is open, respectively, in the intersection of 𝐴 and 𝐵 (the vertical real line of zero abscissa) if they are closed. The same conclusion of (d) is valid for the best proximity points if 𝜀>0.

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

Corollary 2.15. The following properties hold.(i) Let (X,) be a normed vector space with (𝑋,𝑑) being the associate metric space endowed with a norm-induced translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+ and consider the self-mapping 𝑇𝑋𝑋 so that the restricted 𝑇𝐴𝐵𝑋is 𝜆-accretive for some 𝜆𝐑0+, where 𝐴 and 𝐵 are nonempty subsets of 𝑋 subject to 0AB, and the restricted 𝑇𝐴𝐵𝐴𝐵is 2-cyclic.Then, 𝑑((𝐼+𝜆𝑇)𝑥,0)1;𝑥(0)𝐴𝐵,𝜆0,𝜆.(2.44) If, furthermore, 𝑇𝐴𝐵𝐴𝐵is 𝑘-contractive, then 1𝑑((𝐼+𝜆𝑇)𝑥,0)<𝑘1;𝑥(0)𝐴𝐵,𝜆0,𝜆.(2.45)𝑇𝐴𝐵𝐴𝐵is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if(ii)Let (𝑋,) be a normed vector space. Then if 𝑇𝐴𝐵𝑋is a 𝜆-accretive mapping and 𝑇𝐴𝐵𝐴𝐵is 2-cyclic for some 𝜆𝐑0+ where 𝐴 and 𝐵 are nonempty subsets of 𝑋 subject to 0𝐴𝐵, then 𝐼+𝜆T1;𝜆0,𝜆.(2.46) If, furthermore, 𝑇𝐴𝐵𝐴𝐵is 2-cyclic k ([0,1))-contractive, then 1𝐼+𝜆𝑇<𝑘1;𝜆0,𝜆.(2.47)

Outline of Proof
It follows since the basic constraint of 𝑇𝐴𝐵𝑋being 𝜆-accretive holds if 𝐼+𝜆𝑇1𝐼+𝜆𝑇𝑗(𝐼+𝜆𝑇)𝑗1;𝑗𝐙+,𝜆0,𝜆,(2.48) while it fails if 𝐼+𝜆𝑇<1(𝐼+𝜆𝑇)𝑗𝐼+𝜆𝑇𝑗<1;𝑗𝐙+,𝜆0,𝜆.(2.49)

Remark 2.16. Theorem 2.13 and Corollary 2.15 are easily linked to Theorem 2.9 as follows. Assume that 𝑇𝐴𝐵𝐴𝐵 is 2-cyclic 𝑘-contractive and 𝑇𝐴𝐵𝑋 is a𝜆-accretive mapping. Assume that there exists 𝑥𝐴𝐵 such that 𝑥=𝑑(𝑥,0)1. Then, 1𝐼+𝜆𝑇<𝑘1; for all 𝜆[0,𝜆] from (2.47). This is guaranteed under sufficiency-type conditions with 𝑇=max𝑥1𝑑(𝑇𝑥,0)=max𝑑(𝑥,0)1𝑑(𝑇𝑥,0)𝑘if1𝐼+𝜆𝑇1+𝜆𝑇1+𝜆𝑘<𝑘1;𝜆0,𝜆,(1+𝜆𝑘)𝑘𝑘𝑐<1;𝜆0,𝜆,(2.50) with 𝜆=𝑘2(𝑘c𝑘) for some real constants𝑘𝑐[𝑘,1), 𝑘[0,1). It is direct to see that Fix(𝑇)={0𝐑𝑛} if 0𝐴𝐵.

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 𝐴Dom(𝑇)=𝐵Im(𝑇)𝑋𝐑𝑛. 𝐑𝑛 is a vector space endowed with the Euclidean norm induced by the homogeneous and translation-invariant Euclidean metricd𝑋×𝑋𝐑0+. 𝑇 is a linear self-mapping from 𝐑𝑛 to 𝐑𝑛 represented by a nonsingular constant matrix 𝐓 in 𝐑𝑛×𝑛. Then, Tis the spectral (or 2-) norm of the 𝑘-contractive self-mapping 𝑇𝑋𝑋 which is the matrix norm induced by the corresponding vector norm (the vector Euclidean norm being identical to the 2vector norm as it is wellknown) fulfilling 𝑇=maxDom(𝑇)𝑥21𝑇𝑥2=maxDom(𝑇)𝑥2=1𝑇𝑥2,=𝑑(𝑇𝑥,0)=𝜆1/2max𝐓𝑇𝐓𝑘<1,𝑑𝑇𝑗𝑥,𝑇𝑗𝑦=𝑇𝑗(𝑥𝑦)2𝜆1/2max𝑇𝑇𝐓𝑗𝑥𝑦2,=𝜆1/2max𝐓𝑇𝐓𝑗𝑑(𝑥,𝑦)𝑥,𝑦Dom(𝑇)𝑋,(2.51) with the symmetric matrix 𝐓𝑇𝐓 being a matrix having all its eigenvalues positive and less than one, since 𝐓 is nonsingular, upper-bounded by a real constant 𝑘 which is less than one. Thus, 𝑇𝐴𝐵𝑋 is also 𝜆-accretive for any real constant 𝜆<𝑘2(1𝑘) and 𝑘𝑐-contractive for any real 𝑘𝑐[𝑘,1). Assume now that 𝐓=diag𝑘1𝑘2𝑘𝑝𝑛𝑝00(2.52) for some integer 0<𝑝𝑛 with 𝐴=Dom(𝑇)=𝑋=𝐑𝑛,𝐵=Im(𝑇)=𝑥𝑋𝑥=𝑥1𝑥2𝑥𝑝𝑛𝑝00𝑇𝑋=𝐑𝑛,(2.53)𝑘𝑘𝑖(0)𝑘<1; for all 𝑖𝑝. If 𝑝=𝑛, then Fix(𝑇)={0𝐑𝑛}. Also, Fix(𝑇)={0𝐑𝑛}for any integer 0<𝑝<𝑛(then 𝐓 is singular) but the last (𝑛𝑝)-components of any 𝑥𝐴=𝑋=𝐑𝑛 are zeroed at the first iteration via 𝐓so that if 𝑒𝑖is the 𝑖th unit vector in 𝐑𝑛with its 𝑖th component being one, then 𝑒𝑇𝑖𝑇𝑗𝑥0;𝑖𝑝,𝑥(0)𝐑𝑛;𝑗𝐙0+,𝑒𝑇𝑖𝑇𝑗𝑥=0;𝑖(>𝑝)𝑛,𝑥𝐑𝑛;𝑗𝐙0+,𝑇𝑗𝑥0;𝑥𝐑𝑛as𝑗.(2.54) Now, assume that the matrix 𝐓 is of rank one with its first column being of the form 𝑡1=𝑘1𝑘2𝑘𝑝𝑛𝑝00𝑇(2.55) with 0<𝑝<𝑛, 𝑘𝑘𝑖(0)𝑘<1; for all 𝑖𝑝. Then, (2.54) still holds by changing 𝑥0 in the first equation to 𝑥10. Finally, assume that 𝐓=diag𝑘1𝑘2𝑘𝑝𝑛𝑝11(2.56) with 0<𝑝<𝑛. Then, the self-mapping 𝑇𝑋𝑋 is nonexpansive also noncontractive and Fix(𝑇)={0𝐑𝑝}𝐑𝑛𝑝 which is a vector subspace of 𝐑𝑛, that is, there exist infinitely many fixed points each one being reached depending on the initial 𝑥 in 𝑋 with the property lim𝑗𝑇𝑗𝑥=(0𝑇,𝑦𝑇)Fix(𝑇) for any given 𝑥=(𝑧𝑇,𝑦𝑇)𝑇𝐑𝑛 with 𝑥𝐑𝑝, 𝑦𝐑𝑛𝑝.

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

Theorem 2.18. Let (𝑋,) be a normed vector space with (𝑋,𝑑) being the associated metric space endowed with a norm-induced translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+. Let 𝑇𝑋𝐴𝐵𝑋𝐴𝐵be a 2-cyclic 𝑘-contractive self-mapping so that 𝑇𝐴𝐵𝑋 is 𝜆-accretive for some 𝜆𝐑0+ where 𝐴 and 𝐵 are nonempty subsets of X. Then, 𝑑𝑇𝑗𝑥,𝑇𝑗+1𝑥𝑚1𝑑(𝑥,0)+𝑚2;𝑥𝐴𝐵;𝑗𝐙+,(2.57) for some finite real constants 𝑚1𝐑+, and 𝑚2𝐑0+, which are independent of 𝑥 and the 𝑗th power, and 𝑚2is zero if 𝐴 and 𝐵 intersect. Furthermore, lim𝑗sup𝑑(𝑇𝑗𝑥,𝑇𝑗+1𝑥) is finite irrespective of 𝑥𝐴𝐵.

Proof. One gets for 𝜆[0,𝜆], some 𝜆𝐑0+ and 𝑥𝐴𝐵 that 𝑑(𝑥,𝑇𝑥)𝑑𝑥+𝜆𝑇𝑥,𝑇𝑥+𝜆𝑇2𝑥=𝑑𝜆𝑇𝑥,𝑇𝑥+𝜆𝑇2𝑥𝑥=𝑑𝑇𝑥+(𝜆1)𝑇𝑥,𝑇2𝑥+(𝜆1)𝑇2𝑥+𝑇𝑥𝑥=𝑑𝑇𝑥,𝑇2𝑥+(1𝜆)𝑇𝑥+(𝜆1)𝑇2𝑥+𝑇𝑥𝑥𝑑𝑇𝑥,𝑇2𝑥+𝑑𝑇2𝑥,𝑇2𝑥+(1𝜆)𝑇𝑥+(𝜆1)𝑇2𝑥+𝑇𝑥𝑥=𝑑𝑇𝑥,𝑇2𝑥+𝑑(𝜆1)𝑇𝑥,(𝜆1)𝑇2𝑥+𝑇𝑥𝑥=𝑑𝑇𝑥,𝑇2𝑥+𝑑𝜆𝑇𝑥,(𝜆1)𝑇2𝑥𝑥𝑑𝑇𝑥,𝑇2𝑥+𝑑𝜆𝑇𝑥,𝜆𝑇2𝑥+𝑑𝜆𝑇2𝑥,(𝜆1)𝑇2𝑥𝑥=𝑑𝑇𝑥,𝑇2𝑥+𝑑𝜆𝑇𝑥,𝜆𝑇2𝑥+𝑑𝑇2𝑥,𝑥=𝑑𝑇𝑥,𝑇2𝑥+𝑑𝜆𝑇𝑥,𝜆𝑇2𝑥+𝑑𝑇2𝑥,0+𝑑(𝑥,0)𝑘𝑑(𝑥,𝑇𝑥)+(1𝑘)𝑑𝐴𝐵+𝑘𝜆𝑑(𝑥,𝑇𝑥)+𝜆(1𝑘)𝑑𝐴𝐵+𝑘𝑑(𝑇𝑥,0)+(1𝑘)𝑑𝐴𝐵+𝑑(𝑥,0)𝑘𝑑(𝑥,𝑇𝑥)+(1𝑘)𝑑𝐴𝐵+𝑘𝜆𝑑(𝑥,𝑇𝑥)+𝜆(1𝑘)𝑑𝐴𝐵+𝑘2𝑑(𝑥,0)+𝑘(1𝑘)𝑑𝐴𝐵+(1𝑘)𝑑𝐴𝐵+𝑑(𝑥,0)𝑘(1+𝜆)𝑑(𝑥,𝑇𝑥)+(1𝑘)(2+𝜆+𝑘)𝑑𝐴𝐵+𝑘2+1𝑑(𝑥,0);𝜆0,𝜆,(2.58) so that one has for 𝜆=1𝑘1𝜀 with 𝜀[𝜀0,1) for some real constant 𝜀0[0,1) provided that 𝑘(0,1): 𝑑(𝑥,𝑇𝑥)(2+𝜆+𝑘)(1𝑘)1𝑘(1𝜆)𝑑𝐴𝐵+𝑘2+11𝑘(1𝜆)𝑑(𝑥,0)1𝜀0(2+𝜆+𝑘)(1𝑘)𝑑𝐴𝐵+𝑘2+1𝑑(𝑥,0),(2.59) and if 𝑘=0 then 𝑑(𝑥,𝑇𝑥)(2+𝜆)𝑑𝐴𝐵+𝑑(𝑥,0);𝜆𝐑0+.(2.60) Also, 𝑑𝑇𝑗𝑥,𝑇𝑗+1𝑥(2+𝜆+𝑘)(1𝑘)1𝑘(1𝜆)𝑑𝐴𝐵+𝑘2+11𝑘(1𝜆)𝑑𝑇𝑗𝑥,02+𝜆+𝑘+𝑘2+1𝑗1𝑖=0𝑘𝑖1𝑘1𝑘(1𝜆)𝑑𝐴𝐵+𝑘2+1𝑘𝑗1𝑘(1𝜆)𝑑(𝑥,0)2+𝜆+𝑘+𝑘2+1𝑗1𝑖=0𝑘𝑖1𝑘𝜀0𝑑𝐴𝐵+𝑘2+1𝑘𝑗𝜀0𝑑(𝑥,0)=2+𝜆+𝑘+𝑘2+11𝑘𝑗1𝑘1𝑘𝜀0𝑑𝐴𝐵+𝑘2+1𝑘𝑗𝜀0𝑑(𝑥,0)1𝜀03+𝜆+21𝑘𝑑𝐴𝐵+2𝑑(𝑥,0);𝜆0,𝜆,𝑗𝐙+(2.61)limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗+1𝑥3+𝜆+𝑘+𝑘2(1𝑘)𝜀0𝑑𝐴𝐵5+𝜆𝜀0𝑑𝐴𝐵;𝜆0,𝜆(2.62) if 𝑘(0,1), and 𝑑𝑇𝑗𝑥,𝑇𝑗+1𝑥3+𝜆𝜀0𝑑𝐴𝐵;𝜆𝐑0+,𝑗𝐙+(2.63)limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗+1𝑥3+𝜆𝜀0𝑑𝐴𝐵;𝜆0,𝜆(2.64) if 𝑘=0.

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

Theorem 2.19. Let (𝑋,) be a normed space with an associate metric space (𝑋,𝑑) endowed with a norm-induced translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+ and let 𝑇𝑋𝑋 be a self-mapping on 𝑋 which is 𝑘-contractive with 𝑘[0,1/3) and 2-cyclic on 𝐴𝐵, where 𝐴and 𝐵 are nonnecessarily disjoint nonempty subsets of 𝑋. If such sets 𝐴 and 𝐵 intersect then 𝑇𝑋𝑋 is also 𝑘𝑐-contractive with 𝑘𝑐=𝑘/(12𝑘)=𝑘/(1(2+𝜆)𝑘)[0,1) and 𝜆-accretive with 𝜆= if 𝑘𝑐=𝑘=0 and with 𝜆=𝑘1𝑘1𝑐2 if 𝑘(0,1/3). Irrespective of 𝐴 and 𝐵 being disjoint or not, 𝑇𝐴𝐵𝑋 is still 𝜆-accretive and the following inequalities hold: 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑐𝑑(𝑥,𝑦)+𝑚𝑑𝐴𝐵;(𝑥,𝑦)𝐴×𝐵,𝜆0,𝜆,(2.65)𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑘𝑗𝑐𝑑(𝑥,𝑦)+𝑚𝑑𝐴𝐵𝑗1𝑖=0𝑘𝑖𝑐=𝑘𝑗𝑐𝑑(𝑥,𝑦)+1𝑘𝑗𝑐1𝑘𝑐𝑚𝑑𝐴𝐵;𝑗𝐙+,(𝑥,𝑦)𝐴×𝐵,𝜆0,𝜆,(2.66)limsup𝑗𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑚𝑑𝐴𝐵1𝑘𝑐<;𝑗𝐙+,(𝑥,𝑦)𝐴×𝐵,𝜆0,𝜆.(2.67)

Proof. Direct calculations yield 𝑑(𝑇𝑥,𝑇𝑦)𝑑𝑇𝑥+𝜆𝑇2𝑥,𝑇𝑦+𝜆𝑇2𝑦=𝑑𝑇2𝑥+(𝜆1)𝑇2𝑥,𝑇2𝑦+(𝜆1)𝑇2𝑦+𝑇𝑦𝑇𝑥=𝑑𝑇2𝑥,𝑇2𝑦+(𝜆1)𝑇2𝑦+(1𝜆)𝑇2𝑥+𝑇𝑦𝑇𝑥𝑑𝑇2𝑥,𝑇2𝑦+𝑑𝑇2𝑦,𝑇2𝑦+(𝜆1)𝑇2𝑦+(1𝜆)𝑇2𝑥+𝑇𝑦𝑇𝑥=𝑑𝑇2𝑥,𝑇2𝑦+𝑑𝜆𝑇2𝑥,𝜆𝑇2𝑦𝑇2𝑦+𝑇2𝑥+𝑇𝑦𝑇𝑥𝑑𝑇2𝑥,𝑇2𝑦+𝜆𝑑𝑇2𝑥,𝑇2𝑦+𝑑𝜆𝑇2𝑦,𝜆𝑇2𝑦𝑇2𝑦+𝑇2𝑥+𝑇𝑦𝑇𝑥=𝑑𝑇2𝑥,𝑇2𝑦+𝜆𝑑𝑇2𝑥,𝑇2𝑦+𝑑𝑇2𝑦𝑇𝑦,𝑇2𝑥𝑇𝑥𝑑𝑇2𝑥,𝑇2𝑦+𝜆𝑑𝑇2𝑥,𝑇2𝑦+𝑑𝑇2𝑦,𝑇2𝑥+𝑑(𝑇𝑥,𝑇𝑦)(2+𝜆)𝑘𝑑(𝑇𝑥,𝑇𝑦)+(1𝑘)𝑑𝐴𝐵+𝑘𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵;(𝑥,𝑦)𝐴×𝐵,𝜆0,𝜆,(2.68) which leads to the inequalities (2.65)–(2.67) with 𝑘𝑐=𝑘/(1(2+𝜆)𝑘)[0,1) and 𝑚=(3+𝜆)(1𝑘)/𝑘𝑘𝑐 where 𝑘𝑐[0,1) with 𝜆= if 𝑘=𝑘𝑐=0 and 𝜆=𝑘1𝑘1𝑐20 if 𝑘𝑐=𝑘/(12𝑘)(0,1) which holds if and only if 𝑘(0,1/3). The proof is complete.

Remark 2.20. Compared to Theorem 2.9, Theorem 2.19 guarantees the simultaneous maintenance of the 𝜆-accretive and contractive properties if the subsets of 𝑋 intersect. Otherwise, the contractive property is not guaranteed if 𝑘>0 to be 𝜆-accretive for the nontrivial case of 𝜆>0 since 𝑚𝑑𝐴𝐵 is larger than (1𝑘𝑐)𝑑𝐴𝐵 in general. However, the guaranteed value of 𝜆 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 𝐴 and 𝐵 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 𝑋.

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 (𝑋,) be a normed vector space with 𝐴 and 𝐵 being bounded nonempty subsets of 𝑋 and 0𝐴𝐵. Consider a 2-cyclic (𝑘<1/3)-contractive self-mapping 𝑇𝐴𝐵𝐴𝐵with 𝑘[0,1/3). Then, 𝑇𝐴𝐵𝐴𝐵is also a (𝑘𝑐/(1𝑘𝑐),𝛽)-Kannan self-mapping and 𝑇𝐴𝐵𝑋 is (1/3𝑘)𝑘2-accretive for𝑘𝑐(𝐑+)=𝑘(1+𝑘𝜆), for all 𝛽(𝐑+)𝛽0=(2𝜆+11+4𝜆𝑘𝑐)/2𝜆(1𝛼).

Proof. Since 𝑇𝐴𝐵𝐴𝐵 is a 2-cyclic 𝑘(<1/3)-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): 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑑(𝑥+𝜆𝑇𝑥,𝑦+𝜆𝑇𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑐𝑑(𝑥,𝑦)+(1𝑘)𝑑𝐴𝐵𝑘𝑐(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦))+𝑘𝑐𝑑(𝑇𝑥,𝑇𝑦)+(1𝑘)𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵𝑑(𝑇𝑥,𝑇𝑦)𝛼(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦))+(1𝑘)𝑑𝐴𝐵𝛼(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦))+𝛽(1𝛼)𝑑𝐴𝐵;𝑥𝐴,𝑦𝐵,(2.69) provided that 𝛼=𝑘𝑐1𝑘𝑐<12𝑘𝑐<13,𝑘𝑐=𝑘1+𝑘𝜆,𝛽𝛽0=2𝜆+11+4𝜆𝑘𝑐2𝜆(1𝛼),(2.70) since 1/3>𝑘𝑐=𝑘(1+𝑘𝜆)𝑘 if 𝜆=(1/3𝑘)𝑘2 so that 𝑇𝐴𝐵𝑋 is (1/3𝑘)𝑘2-accretive. Note that the function 𝑘=𝑘(𝑘𝑐) for a contractive self-mapping is the positive solution of 𝜆𝑘2+𝑘𝑘𝑐=0, that is, 𝑘=(1+2𝜆1+4𝜆𝑘𝑐)/2𝜆, which is wellposed since 0𝑘<1 for 0𝑘𝑐<1. Thus, 𝑇𝐴𝐵𝐴𝐵is also a 2-cyclic (𝑘𝑐/(1𝑘𝑐),𝛽)-Kannan self-mapping from Definition 2.6 since 0𝑘𝑐<1/3implies 𝛼=𝑘𝑐/(1𝑘𝑐)<1/2 with 𝛽𝛽0=2𝜆+11+4𝜆𝑘𝑐2𝜆(1𝛼),𝜆𝐾2(1𝐾).(2.71)

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

This section generalizes the main results of Section 2 to 𝑝-cyclic self-mappings with 𝑝2. Now, it is assumed that there are 𝑝 nonempty subsets 𝐴𝑖of 𝑋; for all 𝑖𝑝 which can be disjoint or not and a so-called 𝑝-cyclic self-mapping 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖such that 𝑇(𝐴𝑖)𝑇(𝐴𝑖+1) with 𝐴𝑝+1𝐴1. Inspired in the considerations of Remark 2.12 claiming that Theorem 2.9 can be directly extended to the case that the subsets 𝐴 and 𝐵 are not necessarily closed, it is not assumed in the sequel that the subsets 𝐴𝑖 of 𝑋; for all 𝑖𝑝 are necessarily closed. A simple notation for distances between adjacent sets is dist(𝐴𝑖,𝐴𝑖+1)=𝑑𝐴𝑖𝐴𝑖+1=𝑑𝑖. Definition 2.4 is generalized as follows.

Definition 3.1. 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖is a 𝑝-cyclic weakly 𝑘-contractive (resp., weakly nonexpansive) self-mapping if 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑖𝑑(𝑥,𝑦)+1𝑘𝑖𝑑𝑖;𝑥𝐴𝑖,𝑦𝐴𝑖+1;𝑖𝑝,(3.1) for some real constants 𝑘𝑖𝐑0+(resp., 𝑘𝑖𝐑+); for all 𝑖𝑝 [12, 13] such that 𝑘=𝑖𝑝[𝑘𝑖]<1 (resp., 𝑘=1).

Definition 3.2. 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a 𝑝-cyclic 𝑘-contractive (resp., nonexpansive) 𝑝-cyclic self-mapping if 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑖𝑑(𝑥,𝑦)+1𝑘𝑖𝑑𝑖;𝑥𝐴𝑖,𝑦𝐴𝑖+1;𝑖𝑝,(3.2) for some real constants𝑘𝑖[0,1) (resp., 𝑘𝑖=1); for all 𝑖𝑝 [12, 13].

Assertion 1. A 𝑝-cyclic weakly nonexpansive self-mapping 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 may be locally expansive for some (𝑥,𝑦)𝐴𝑖×𝐴𝑖+1; for all 𝑖𝑝 which cannot be best proximity points.

Proof. Assume that 𝑘𝑖>1. Then, the following inequalities can occur for given𝑥𝐴𝑖, 𝑦𝐴𝑖+1:(1)𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑖𝑑(𝑥,𝑦)+1𝑘𝑖𝑑𝑖𝑑(𝑥,𝑦)𝑑(𝑥,𝑦)𝑑𝑖𝑑(𝑇𝑥,𝑇𝑦)𝑑𝑖,(3.3) In this case, and since 𝑑(𝑥,𝑦)<𝑑𝑖 is impossible, one concludes that 𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑖𝑑(𝑥,𝑦)+1𝑘𝑖𝑑𝑖𝑑(𝑥,𝑦)𝑑(𝑥,𝑦)=𝑑𝑖𝑑(𝑇𝑥,𝑇𝑦)𝑑𝑖,(3.4) so that (3.3) can only hold for best proximity points 𝑥𝐴𝑖,𝑦=𝑇𝑥𝐴𝑖+1 for which 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is nonexpansive. If 𝑑𝑖=𝑑𝑖+1, then the last inequality of (3.4) becomes 𝑑(𝑇𝑥,𝑇𝑦)=𝑑𝑖=𝑑𝑖+1 so that 𝑇𝑦𝐴𝑖+2 is also a best proximity point if 𝐴𝑖are convex, for all 𝑖𝑝,(2)𝑑(𝑇𝑥,𝑇𝑦)𝑑(𝑥,𝑦)𝑘𝑖𝑑(𝑥,𝑦)+1𝑘𝑖𝑑𝑖𝑑(𝑥,𝑦)𝑑𝑖,(3.5) and then 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖is nonexpansive for (𝑥,𝑦)𝐴𝑖×𝐴𝑖+1;(3)𝑑(𝑥,𝑦)<𝑑(𝑇𝑥,𝑇𝑦)𝑘𝑖𝑑(𝑥,𝑦)+1𝑘𝑖𝑑𝑖𝑑(𝑥,𝑦)>𝑑𝑖𝑑(𝑇𝑥,𝑇𝑦)>𝑑(𝑥,𝑦)>𝑑𝑖,(3.6) and then 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖,is expansive for (𝑥,𝑦)𝐴𝑖×𝐴𝑖+1 which cannot be best proximity points since 𝑑(𝑥,𝑦)>𝑑𝑖.

Remark 3.3. Note from Definitions 3.1 and 3.2 that a 𝑝-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 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖, is 𝑝-cyclic weakly nonexpansive and 𝑑𝑖=𝑑1; for all 𝑖𝑝, then 𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑘𝑖+𝑗1𝑑𝑇𝑗1𝑥,𝑇𝑗1𝑦+1𝑘𝑖+𝑗1𝑑1,𝑗𝑖=1𝑘𝑖𝑑(𝑥,𝑦)+1𝑗𝑖=1𝑘𝑖𝑑1;𝑥𝐴𝑖,𝑦𝐴𝑖+1,(3.7) where 𝑘𝑝+𝑖=𝑘𝑖; for all 𝐙0+, for all 𝑖𝑝1{0}. Note that if 𝑑(𝑇𝑗1𝑥,𝑇𝑗1𝑦)>𝑑1; that is, 𝑇𝑗1𝑥,𝑇𝑗1𝑦 are not best proximity points, then if 𝑘𝑖+𝑗1>1,then 𝑑(𝑇𝑗𝑥,𝑇𝑗𝑦)>𝑑(𝑇𝑗1𝑥,𝑇𝑗1𝑦) since𝑘𝑖+𝑗1𝑑(𝑇𝑗1𝑥,𝑇𝑗1𝑦)+(1𝑘𝑖+𝑗1)𝑑1>𝑑(𝑇𝑗1𝑥,𝑇𝑗1𝑦). Thus, a weakly nonexpansive self-mapping is not necessarily nonexpansive for each iteration. However, the composed self-mapping 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 defined as 𝐓𝑥=𝑇(𝑇𝑝1𝑥)=𝑇𝑝𝑥; for all 𝑥𝑖𝑝𝐴𝑖 is nonexpansive in the usual sense since if 𝑗=𝑝, then 𝑘=𝑖𝑝[𝑘𝑖]=1, implies 𝑑𝑇𝑗𝑥,𝑇𝑗𝑦𝑑𝑇𝑗1𝑥,𝑇𝑗1𝑦𝑑(𝑥,𝑦);𝑥𝐴𝑖,𝑦𝐴𝑖+1.(3.8)

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 𝑝-cyclic self-mappings with 𝑝2. The following technical result stands related to the fact that nonexpansive 𝑝-cyclic self-mappings have identical distances between all the adjacent subsets in the set {𝐴𝑖𝑋𝑖𝑝}.

Lemma 3.4. Assume that 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is 𝑝-cyclic and nonexpansive. Then, 𝑑𝑖=𝑑1; for all 𝑖𝑝.

Proof. If 𝑖𝑝cl𝐴𝑖(i.e., the closures of the subsets intersect), then the proof is direct since 𝑑𝑖=0; for all 𝑖𝑝. Now, assume that 0𝑑𝑗<𝑑10 for some 𝑗𝑝. Let 𝑧𝐴1and 𝑧1=𝑇𝑧𝐴2 best proximity points such that 𝑑𝑇𝑗+1𝑧,𝑇𝑗+1𝑧1=𝑑𝑗𝑑𝑇𝑖+1𝑧,𝑇𝑖+1𝑧1=𝑑𝑖𝑑𝑇𝑧,𝑇𝑧1𝑑𝑧,𝑧1=𝑑1;𝑗,𝑖(<𝑗)𝑝,(3.9) since 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a 𝑝-cyclic nonexpansive self-mapping. Thus, any iterates 𝑇𝑗𝑧 and 𝑇𝑗𝑧1 are also best proximity points of some subset in{𝐴𝑖𝑋𝑖𝑝}; for all 𝑗𝐙+. If 𝑑𝑗=𝑑1; for all 𝑗𝑝, does not hold, then from (3.9): 𝑑𝑇𝑗+1𝑧,𝑇𝑗+1𝑧1<𝑑𝑇𝑧,𝑇𝑧1;𝑗𝑝lim𝑗𝑑𝑇𝑧,𝑇𝑗𝑧1=0.(3.10) Then 𝑑𝑖=0;for all 𝑖𝑝which contradicts 0𝑑𝑗<𝑑10 for some 𝑗𝑝 what is a contradiction or 𝑑𝑖=0; for all 𝑖𝑝, and 𝑖𝑝𝐴𝑖.

Note that Lemma 3.4 applies even if the subsets are neither bounded or closed. In this way, note that the contradiction to 0𝑑𝑗<𝑑10 for some 𝑗𝑝 established in the second part of the proof does not necessarily imply that 𝑖𝑝cl𝐴𝑖 which would require for the subsets 𝐴𝑖, for all 𝑖𝑝,to be bounded and, in particular, 𝑖𝑝𝐴𝑖 if such subsets are bounded and closed. The following result stands concerning the limit iterates of 𝑝-cyclic nonexpansive self-mappings:

Lemma 3.5. The following properties hold.(i)If 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a p-cyclic weakly nonexpansive self-mapping, then limsup𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦𝑖+𝜇=𝑖𝑘𝜇limsup𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦+𝑖+𝜇=𝑖𝑖+𝜎=𝜇+1𝑘𝜎1𝑘𝜇𝑑𝜇lim𝑗𝑖+𝜇=𝑖𝑘𝜇𝑘𝑗𝑑(𝑥,𝑦)+𝑖+𝜇=𝑖𝑖+𝜎=𝜇+1𝑘𝜎1𝑘𝜇𝑑𝜇=𝑖+𝜇=𝑖𝑖+𝜎=𝜇+1𝑘𝜎1𝑘𝜇𝑑𝜇;𝑥𝐴𝑖,𝑦𝐴𝑖+1(3.11) if𝑖,𝑝satisfy 𝑖+𝜇=𝑖𝑖+𝜎=𝜇+1𝑘𝜎1𝑘𝜇𝑑𝜇𝑑𝑖limsup𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦𝑑𝑇𝑝𝑥,𝑇𝑝𝑦𝑑(𝑥,𝑦);𝑥𝐴𝑖,𝑦𝐴𝑖+1,𝑖𝑝,𝐙+.(3.12)(ii) If 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a 𝑝-cyclic nonexpansive self-mapping, thenlimsup𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦𝑑𝑇𝑝𝑥,𝑇𝑝𝑦𝑑(𝑥,𝑦);𝑥𝐴𝑖,𝑦𝐴𝑖+1,𝑖𝑝,𝐙+(3.13)(iii) If 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a 𝑝-cyclic weakly contractive self-mapping, then limsup𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦𝑖+𝜇=𝑖𝑖+𝜎=𝜇+1𝑘𝜎1𝑘𝜇𝑑𝜇;lim𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦=𝑑𝑖,(3.14) for all 𝑥𝐴𝑖, for all 𝑦𝐴𝑖+1 if 𝑖,𝑝 satisfy the feasibility constraints 𝑖+𝜇=𝑖(𝑖+𝜎=𝜇+1[𝑘𝜎])(1𝑘𝜇)𝑑𝜇𝑑𝑖 and 𝑖+𝑝𝜇=𝑖(𝑖+𝑝𝜎=𝜇+1[𝑘𝜎])(1𝑘𝜇)𝑑𝜇=𝑑𝑖. If 𝑑𝑖=𝑑1; for all 𝑖𝑝, then limsup𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦𝑑1𝑖+𝜇=𝑖𝑖+𝜎=𝜇+1𝑘𝜎1𝑘𝜇;lim𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦=𝑑1(3.15) for all 𝑥𝐴𝑖, for all 𝑦𝐴𝑖+1, for all ,ip if i,p satisfy the feasibility constraints 𝑖+𝜇=𝑖(𝑖+𝜎=𝜇+1[𝑘𝜎])(1𝑘𝜇)1and 𝑖+𝑝1𝜇=𝑖(𝑖+𝑝1𝜎=𝜇+1[𝑘𝜎])(1𝑘𝜇)=1(iv) If 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖is a 𝑝-cyclic contractive self-mapping, thenlimsup𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦𝑑1lim𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦=𝑑1;𝑥𝐴𝑖,𝑦𝐴𝑖+1,,𝑖𝑝.(3.16)(v) If 𝑖𝑝𝐴𝑖 and 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖is a 𝑝-cyclic weakly contractive self-mapping, thenlim𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦=lim𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦=0;𝑥𝐴𝑖,𝑦𝐴𝑖+1,𝑖,𝑝.(3.17)

Proof. Property (i) follows from (3.7) for 𝑘=𝑖𝑝[𝑘𝑖]=1. Property (iii) follows from Property (i) since 𝑘<1implies (𝑝+𝑖𝜇=𝑖[𝑘𝜇])𝑗=𝑘𝑗0 as 𝑗. Property (ii) Follows from Property (i) for 𝑘𝑖=1; for all 𝑖𝑝 since 𝑑𝑖=𝑑1; for all 𝑖𝑝 from Lemma 3.4. Property (iv) follows from Property (ii) for 𝑘𝑖<1; for all 𝑖𝑝since 𝑑𝑖=𝑑1; for all 𝑖𝑝 from Lemma 3.4. Property (v) follows from Property (iii) since if all the subsets 𝐴𝑖; 𝑖𝑝 intersect, then it follows necessarily 𝑑𝑖=𝑑1=0; for all 𝑖𝑝 so that lim𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦=0;𝑥𝐴𝑖,𝑦𝐴𝑖+1,𝑖𝑝lim𝑗𝑑𝑇𝑗𝑝+𝑥,𝑇𝑗𝑝+𝑦𝜇=𝑖𝑘𝜇lim𝑗𝑑𝑇𝑗𝑝𝑥,𝑇𝑗𝑝𝑦=0;𝑥𝐴𝑖,𝑦𝐴𝑖+1,𝑖,𝑝.(3.18)

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 𝑝-cyclic contractive self-mappings. A parallel result is discussed in [10] for Meir-Keeler contractions.

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

Proof. The existence of a fixed point follows from Lemma 3.5(v). Its uniqueness follows by contradiction. Assume that there exist 𝑧1,𝑧2(𝑧1)Fix(𝑇)𝑖𝑝cl𝐴𝑖. Then, for some 𝑖𝑝,𝑥𝐴𝑖, 𝑦𝐴𝑖+1 such that 𝑇𝑗𝑥𝑧1 and 𝑇𝑗𝑦𝑧2as 𝑗. Then, by using triangle inequality for distances, 𝑑𝑧1,𝑧2𝑑𝑧1,𝑇𝑗𝑥+𝑑𝑇𝑗𝑥,𝑇𝑗𝑦+𝑑𝑧2,𝑇𝑗𝑦;𝑗𝐙0+(3.19) which implies by using Lemma 3.5(v) 𝑑𝑧1,𝑧2limsup𝑗𝑑𝑧1,𝑇𝑗𝑥+𝑑𝑇𝑗𝑥,𝑇𝑗𝑦+𝑑𝑧2,𝑇𝑗𝑦=0𝑑𝑧1,𝑧2𝑧1=𝑧2,(3.20) what contradicts 𝑧1𝑧2. Therefore, Fix(𝑇) consists of a unique point in 𝑖𝑝cl𝐴𝑖 which is also in 𝑖𝑝𝐴𝑖 if the sets 𝐴𝑖; 𝑖𝑝 are all closed.

Theorem 3.7 also applies to 𝑝-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 (X,) be a uniformly convex Banach space endowed with the translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+with nonempty convex subsets𝐴𝑖𝑋, for all 𝑖𝑝 of pair-wise disjoint closures. Let 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 be a 𝑝-cyclic weakly 𝑘-contractive self-mapping so that the composed 2-cyclic self-mappings.𝑇𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1, for all 𝑖𝑝 are defined as 𝑇𝑖𝑥=𝑇(𝑇𝑝1𝑥); for all 𝑥𝐴𝑖𝐴𝑖+1; forall𝑖𝑝. Then, the following properties hold:(i)Any composed 2-cyclic self-mapping 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1, 𝑖𝑝 is 𝑘-contractive provided that the constraint 𝑖+𝑝𝜇=𝑖(𝑖+𝑝𝜎=𝜇+1[𝑘𝜎])(1𝑘𝜇)𝑑𝜇=𝑑𝑖 holds. If, furthermore, it is assumed that 𝐴𝑖 and 𝐴𝑖+1 are convex, then the 2-cyclic self-mapping 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 self-mapping is extendable to 𝐓icl(𝐴𝑖𝐴𝑖+1)cl(𝐴𝑖𝐴𝑖+1), and that T(cl𝐴𝑖)cl𝐴𝑖+1; for all 𝑖𝑝. Thus, the iterates 𝐓𝑖𝑗𝑥=𝑇(𝑇𝑗𝑝1𝑥) and 𝐓𝑖𝑗𝑦=𝑇(𝑇𝑗𝑝1𝑦); for all 𝑥𝐴𝑖, for all 𝑦𝐴𝑖+1 converge as 𝑗 to best proximity points in cl(𝐴𝑖) and cl(𝐴𝑖+1) which are also in 𝐴𝑖 if 𝐴𝑖 is closed, respectively, in 𝐴𝑖+1 if 𝐴𝑖+1 is closed.(ii)If for some given 𝑖𝑝, the sets 𝐴𝑖 and 𝐴𝑖+1 are convex and closed, if any, then both best proximity points of 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 of Property (i) are unique and belong, respectively, to 𝐴𝑖 and 𝐴𝑖+1.(iii)Assume that the subsets 𝐴𝑖 of 𝑋 are convex, for all 𝑖𝑝. If 𝑖𝑝cl𝐴𝑖, then the best proximity points of Property (i) become a unique fixed point for all the composed 2-cyclic self-mappings 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 which are 𝑘-contractive, forall𝑖𝑝. Such a fixed point is in 𝑖𝑝cl𝐴𝑖 (and also in 𝑖𝑝𝐴𝑖 if all the subsets 𝐴𝑖, 𝑖𝑝, are closed).

Proof. Since𝑇(𝐴𝑖)𝐴𝑖+1; for all 𝑖𝑝, then for any 𝑖𝑝, 𝑥𝐴𝑖𝐓𝑖𝑥𝐴𝑖 and 𝑥𝐴𝑖+1𝑇𝑖𝑥𝐴𝑖+1if p is even and 𝑥𝐴𝑖𝑇𝑖𝑥𝐴𝑖+1 and 𝑥𝐴𝑖+1𝑇𝑖𝑥𝐴𝑖 if 𝑝 is odd. Since 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is 𝑝-cyclic weakly 𝑘-contractive then 𝑘=𝑝𝑖=1[𝑘𝑖]<1, then 𝑇𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1is 2-cyclic contractive provided that 𝑖+𝑝𝜇=𝑖(𝑖+𝑝𝜎=𝜇+1[𝑘𝜎])(1𝑘𝜇)𝑑𝜇=𝑑𝑖; 𝑖𝑝. One has from Lemma 3.5(iv) that lim𝑗𝑑(𝑇2𝑗𝑖𝑥,𝑇2𝑗𝑖𝑦)=𝑑𝑖=𝑑(𝑧,𝑧1); for all 𝑥𝐴𝑖, forall𝑦𝐴𝑖+1 for the given𝑖𝑝, where 𝑧=𝑧(𝑖)cl(𝐴𝑖)(𝑧𝐴𝑖 if 𝐴𝑖 is closed), 𝑧1=𝑧1(𝑖)cl(𝐴𝑖+1)(𝑧cl(𝐴𝑖)(𝑧𝐴𝑖+1 if 𝐴𝑖+1 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 gets𝐓2𝑗𝑖𝑥zand 𝐓2𝑗𝑖𝑦𝑧1as 𝑗; for all 𝑖𝑝. Property (i) has been proven. Property (ii) was proven in Theorem  3.10, [12] for 2-cyclic 𝑘-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 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 is 𝑘-contractive for all 𝑖𝑝, and the fact that all distances between the closures of all pairs of adjacent subsets are zero since (𝑋,𝑑) is a complete metric space since 𝑋 is a Banach space.

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

Theorem 3.9. Let (X,) be a uniformly convex Banach space endowed with the norm-induced translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+ with nonempty subsets 𝐴𝑖𝑋, for all 𝑖𝑝 of pair-wise disjoint closures. Let 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 be a 𝑝-cyclic 𝑘-contractive self-mapping so that the composed 2-cyclic self-mappings 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1, for all 𝑖𝑝, are defined as 𝐓𝑖𝑥=𝑇(𝑇𝑝1𝑥); for all 𝑥𝐴𝑖𝐴𝑖+1; for all 𝑖𝑝. Assume also that 𝐴𝑖 is convex and 𝑇(cl𝐴𝑖)cl𝐴𝑖+1; for all 𝑖𝑝. Then, the following properties hold.(i)As 𝑗, the iterates 𝑇𝑗𝑥 and 𝑇𝑗𝑦; for all 𝑥𝐴𝑖, for all 𝑦𝐴𝑖+1converge to best proximity points in cl(𝐴𝑖) and cl(𝐴𝑖+1) which are also in 𝐴𝑖 if 𝐴𝑖 is closed, respectively, in 𝐴𝑖+1 if 𝐴𝑖+1 is closed for any𝑖𝑝. Also, for any given 𝑖𝑝 such that the sets 𝐴𝑖 and 𝐴𝑖+1 are convex and closed, if any, then both best proximity points of 𝑇𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1of Property (i) are unique and belong, respectively, to 𝐴𝑖 and 𝐴𝑖+1. If, furthermore, 𝑖𝑝cl𝐴𝑖, then the best proximity points of Property (i) become a unique fixed point for the 𝑝-cyclic 𝑘-contractive self-mapping 𝑇𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1. Such a fixed point is in 𝑖𝑝cl𝐴𝑖 (and also in 𝑖𝑝𝐴𝑖 if all the subsets 𝐴𝑖𝑋, 𝑖𝑝 are closed).(ii)All the composed 2-cyclic self-mappings 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1, for all 𝑖𝑝 are 𝑘-contractive. Thus, the iterates 𝐓𝑖𝑗𝑥=𝑇(𝑇𝑗𝑝1𝑥) and 𝐓𝑖𝑗𝑦=𝑇(𝑇𝑗𝑝1𝑦); for all 𝑥𝐴𝑖, for all 𝑦𝐴𝑖+1converge as 𝑗 to best proximity points in cl(𝐴𝑖)and cl(𝐴𝑖+1) which are also in 𝐴𝑖 if 𝐴𝑖 is closed, respectively in 𝐴𝑖+1 if 𝐴𝑖+1 is closed. For any given 𝑖𝑝 such that the sets 𝐴𝑖 and 𝐴𝑖+1 are closed and convex, if any, then both best proximity points of 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 of Property (i) are unique and belong, respectively, to 𝐴𝑖 and 𝐴𝑖+1. If, furthermore, 𝑖𝑝cl𝐴𝑖, then the best proximity points of Property (i) become a unique fixed point for all the composed 2-cyclic self-mappings 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 which are 𝑘-contractive; 𝑖𝑝. Such a fixed point is in 𝑖𝑝cl𝐴𝑖(and also in 𝑖𝑝𝐴𝑖 if all the subsets 𝐴𝑖, 𝑖𝑝 are closed).

Outline of Proof
Property (ii) is the direct version of Theorem 3.8 applicable to the composed 2-cyclic self-mappings 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1 which are all 𝑘-contractive since 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a 𝑝-cyclic 𝑘-contractive self-mapping. Since 𝑝-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 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is also 𝑘-contractive.

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

Definition 3.10. 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 is a 2-cyclic (𝛼,𝛽)-Kannan self-mapping for some real 𝛼[0,1/2) if it satisfies for some 𝛽𝐑+: 𝑑(𝑇𝑥,𝑇𝑦)𝛼(𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦))+𝛽(1𝛼)dist(𝐴,𝐵);𝑥𝐴,𝑦𝐵.(3.21) Now, Theorem 2.9 and Theorems 2.182.21 for 2-cyclic accretive and Kannan self-mappings extend directly with direct replacements of their relevant parts as follows:

Theorem 3.11. Let (X,) be a Banach space so that (𝑋,𝑑)is its associate complete metric space endowed with a norm-induced translation-invariant and homogeneous metric 𝑑𝑋×𝑋𝐑0+. Consider a self-mapping 𝑇𝑋𝑋 which is also a 𝑝-cyclic 𝑘-contractive self-mapping if restricted t𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖, where 𝐴𝑖 are nonempty convex subsets of X; for all 𝑖𝑝. Then, Theorem 2.9 holds “mutatis-mutandis” by replacing the subsets 𝐴 and 𝐵 for pairs of adjacent subsets 𝐴𝑖 and 𝐴𝑖+1, 𝑖𝑝,𝐴𝐵𝑖𝑝𝐴𝑖, cl(𝐴𝐵)cl(𝑖𝑝𝐴𝑖), 𝐴𝐵𝑖𝑝𝐴𝑖 and 𝑘𝑘=𝑝𝑖=1[𝑘𝑖]. 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 𝐓𝑖𝐴𝑖𝐴𝑖+1𝐴𝑖𝐴𝑖+1; for all 𝑖𝑝 defined from the 𝑝-cyclic weak 𝑘-contractive self-mapping 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖 since 𝑇𝑖𝑝𝐴𝑖𝑖𝑝𝐴𝑖; 𝑖𝑝 are 𝑘-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.