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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 817193, 29 pages
Research Article

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

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.


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


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