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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 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: YonghongΒ Yao

Copyright Β© 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(β‰₯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., [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 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, 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 𝑝-cyclic Meir-Keeler contractive self-mappings. The objective of this paper is the investigation of relevant properties of contractive 𝑝(β‰₯2)-cyclic self-mappings of the union of set of subsets of a Banach space (𝑋,β€–β€–) which are simultaneouslyπœ†βˆ—-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 πœ†βˆ—-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(π‘–βˆˆπ‘) 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(π‘–βˆˆπ‘)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(π‘–βˆˆπ‘). Consider a metric space (𝑋,𝑑) associated to the Banach space (𝑋,β€–β€–) and a self-mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡such that 𝑇(𝐴)βŠ†π΅ and 𝑇(𝐡)βŠ†π΄, where 𝐴 and 𝐡 are nonempty subsets of 𝑋. Then, π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is a 2-cyclic self-mapping. It is said to be a 2-cyclic π‘˜-contraction self-mapping if it satisfies, furthermore,𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)dist(𝐴,𝐡);βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅,(1.1)

for some real π‘˜βˆˆ[0,1). A best proximity point of convex subsets 𝐴 or 𝐡 of 𝑋 is some π‘§βˆˆcl(𝐴βˆͺ𝐡)such that 𝑑(𝑧,𝑇𝑧)=dist(𝐴,𝐡). If 𝐴 and 𝐡 are closed then either 𝑧 (resp., 𝑇𝑧) or 𝑇𝑧 (resp. 𝑧) is in 𝐴 (resp., in 𝐡). The distance between subsets 𝐴 and 𝐡 of the metric space dist(𝐴,𝐡)=0 if either π΄βˆ©π΅β‰ βˆ…or if either 𝐴 or 𝐡 is open with Fr(𝐴)∩Fr(𝐡)β‰ βˆ…. In this case, if 𝑧 is a best proximity point either 𝑧 or 𝑇𝑧 is not in 𝐴βˆͺ𝐡(in particular, neither 𝑧 nor 𝑇𝑧 is in 𝐴βˆͺ𝐡 if both of them are open). It turns out that if π΄βˆ©π΅β‰ βˆ… then π‘§βˆˆFix(𝑇)βŠ‚π΄βˆͺ𝐡; that is, 𝑧 is a fixed point of 𝑇 since dist(𝐴,𝐡)=0, [9–11]. If π‘˜=1 then 𝑑(𝑇π‘₯,𝑇𝑦)≀𝑑(π‘₯,𝑦); forallπ‘₯∈𝐴, for all π‘¦βˆˆπ΅ and π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is 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,𝐡)=𝑑𝐴𝐡 denotes the distance between the sets 𝐴 and 𝐡 for a 2-cyclic self-mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 which is simplified as dist(𝐴𝑖,𝐴𝑖+1)=𝑑𝐴𝑖𝐴𝑖+1=𝑑𝑖; forallπ‘–βˆˆπ‘ for distances between adjacent subsets of 𝑝-cyclic self-mappings 𝑇 on ⋃𝑝𝑖=1𝐴𝑖.

𝐡𝑃𝑖(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,β€–β€–) be a normed vector space and (𝑋,𝑑)be an associate metric space endowed with a metric (or distance function or simply β€œdistance”) π‘‘βˆΆπ‘‹Γ—π‘‹β†’π‘πŸŽ+. For instance, the distance function may be induced by the norm β€–β€–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 π‘‡βˆΆπ‘‹β†’π‘‹ which is a 2-cyclic self-mapping restricted as π‘‡βˆΆDom(𝑇)βŠ†π‘‹βˆ£π΄βˆͺ𝐡→Im(𝑇)βŠ†π‘‹βˆ£π΄βˆͺ𝐡, where 𝐴 and 𝐡 are nonempty subsets of 𝑋. Such a restricted self-mapping is sometimes simply denoted as π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡. 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 𝐡 are not closed, it is assumed that Dom(𝑇)βŠƒcl(𝐴βˆͺ𝐡) and Im(𝑇)βŠƒcl(𝐴βˆͺ𝐡) 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 𝐡, 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 π‘‡βˆΆDom(𝑇)βŠ†π‘‹β†’Im(𝑇)βŠ†π‘‹ in order to define successive iterate points through the self-mapping which do not necessarily belong to 𝐴βˆͺ𝐡. The following definitions are then used to state the main results.

Definition 2.1. π‘‡βˆΆDom(𝑇)βŠ‚π‘‹β†’π‘‹ is an accretive mapping if 𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦);βˆ€π‘₯,π‘¦βˆˆDom(𝑇),(2.1) for any πœ†βˆˆπ‘0+.
Note that, since 𝑋 is also a vector space, π‘₯+πœ†π‘‡π‘₯is in 𝑋 for all π‘₯ in 𝑋 and all real πœ†. 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. π‘‡βˆΆDom(𝑇)βŠ‚π‘‹β†’π‘‹is a πœ†βˆ—-accretive mapping, some πœ†βˆ—βˆˆπ‘0+ if 𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦);βˆ€π‘₯,π‘¦βˆˆDom(𝑇);βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,(2.2) for some πœ†βˆ—βˆˆπ‘0+. A generalization is as followsβˆΆπ‘‡βˆΆDom(𝑇)βŠ‚π‘‹β†’π‘‹is [πœ†βˆ—1,πœ†βˆ—2]-accretive for some πœ†βˆ—1,πœ†βˆ—2(β‰₯πœ†βˆ—1)βˆˆπ‘0+ if 𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦);βˆ€π‘₯,π‘¦βˆˆDom(𝑇);βˆ€πœ†βˆˆξ€Ίπœ†βˆ—1,πœ†βˆ—ξ€».(2.3)

Definition 2.3. π‘‡βˆΆDom(𝑇)βŠ‚π‘‹β†’π‘‹is a weighted πœ†-accretive mapping, for some function πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+, if 𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†(π‘₯,𝑦)𝑇π‘₯,𝑦+πœ†(π‘₯,𝑦)𝑇𝑦);βˆ€π‘₯,π‘¦βˆˆDom(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. π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a 2-cyclic π‘˜-contractive (resp., nonexpansive) self-mapping if 𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)dist(𝐴,𝐡);βˆ€π‘₯∈𝐴,π‘¦βˆˆπ΅,(2.5) for some real π‘˜βˆˆ[0,1) (resp. π‘˜=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 π‘˜<1/3, [16], are defined below.

Definition 2.5. π‘‡βˆΆπ‘‹β†’π‘‹is a 𝛼-Kannan self-mapping if 𝑑(𝑇π‘₯,𝑇𝑦)≀𝛼(𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦));βˆ€π‘₯,π‘¦βˆˆπ‘‹,(2.6) for some real π›Όβˆˆ[0,1/2), [12, 13].

Definition 2.6. π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is an 2-cyclic (𝛼,𝛽)-Kannan self-mapping for some real π›Όβˆˆ[0,1/2) if it satisfies, for some π›½βˆˆπ‘+. 𝑑(𝑇π‘₯,𝑇𝑦)≀𝛼(𝑑(π‘₯,𝑇π‘₯)+𝑑(𝑦,𝑇𝑦))+𝛽(1βˆ’π›Ό)dist(𝐴,𝐡);βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅.(2.7) The relevant concepts concerning 2-cyclic self-mappings are extended to 𝑝(β‰₯2)-cyclic self-mappings in Section 3. Some simple explanation examples follow.

Example 2.7. Consider the scalar linear mapping from 𝑋≑𝐴≑𝐑 to 𝑋 as 𝑇π‘₯=𝛾π‘₯+𝛾0with 𝛾,𝛾0βˆˆπ‘endowed with the Euclidean distance 𝑑(π‘₯,𝑦)=|π‘₯βˆ’π‘¦|; for all π‘₯,π‘¦βˆˆπ‘‹. Then, 𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)=||π‘₯βˆ’π‘¦+πœ†π›Ύ(π‘₯βˆ’π‘¦)||=||1+πœ†π›Ύ||||π‘₯βˆ’π‘¦||=||1+πœ†π›Ύ||𝑑(π‘₯,𝑦)β‰₯𝑑(π‘₯,𝑦),(2.8) for all π‘₯,π‘¦βˆˆπ‘ for any πœ†βˆˆπ‘0+ provided that π›Ύβˆˆπ‘0+. In this case, π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is accretive. It is also π‘˜-contractive if since 𝑑(𝑇π‘₯,𝑇𝑦)=|𝑇π‘₯βˆ’π‘‡π‘¦|=𝛾,𝑑(𝑧,𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦); for all π‘₯,π‘¦βˆˆπ‘. Also, if π›Ύβˆˆπ‘βˆ’, then 𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)β‰₯|πœ†|𝛾|βˆ’1|𝑑(π‘₯,𝑦)β‰₯𝑑(π‘₯,𝑦); for all π‘₯,π‘¦βˆˆπ‘if πœ†|𝛾|β‰₯2, that is, if πœ†β‰₯πœ†βˆ—1∢=2|𝛾|βˆ’1. Then, π‘‡βˆΆπ‘β†’π‘ is [πœ†βˆ—1,∞)-accretive and π‘˜-contractive if |𝛾|β‰€π‘˜<1.

Example 2.8. Consider the metric space (𝐑,𝑑) with the distance being homogeneous and translation-invariant and a self-mapping π‘‡βˆΆπ‘β†’π‘ defined by 𝑇π‘₯=βˆ’π‘‘|π‘₯|𝑝sgn𝑒π‘₯=βˆ’π‘‘|π‘₯|π‘βˆ’1π‘₯ with π‘‘βˆˆπ‘0+, π‘βˆˆπ‘0+,and sgn𝑒π‘₯=sgnπ‘₯ if π‘₯β‰ 0and sgn𝑒0=0. If 𝑝𝑑=0,then π‘‡βˆΆπ‘β†’π‘ is accretive since 𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)=𝑑(π‘₯,𝑦);βˆ€π‘₯,π‘¦βˆˆπ‘‹;βˆ€πœ†βˆˆπ‘0+.(2.9) Furthermore, if 𝑑=0, then 0βˆˆπ‘ is the unique fixed point with 𝑇𝑗π‘₯=0; for all π‘—βˆˆπ™+. If 𝑝=0 then, 𝑇𝑗π‘₯=𝑑𝑗→𝑧=0as π‘—β†’βˆž if |𝑑|<1 and then 𝑧=0is again the unique fixed point of 𝑇. In the general case, 𝑇π‘₯=𝑑|π‘₯|𝑝sgn𝑒π‘₯ implies 𝑇2π‘₯=𝑇(𝑇π‘₯)=βˆ’π‘‘ξ€·βˆ’π‘‘|π‘₯|𝑝sgn𝑒π‘₯𝑝sgn𝑒𝑇π‘₯=𝑑𝑝+1|π‘₯|2𝑝sgn𝑒π‘₯𝑝+1,𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)=𝑑1+πœ†π‘‘π‘₯π‘βˆ’1ξ€Έπ‘₯,ξ€·1+πœ†π‘‘π‘¦π‘βˆ’1𝑦β‰₯minξ‚€||1βˆ’πœ†π‘‘|π‘₯|π‘βˆ’1||,||1βˆ’πœ†π‘‘||𝑦||π‘βˆ’1||𝑑(π‘₯,𝑦),β‰₯𝑑(π‘₯,𝑦);βˆ€π‘₯,π‘¦βˆˆπ‘‹,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,someπœ†βˆ—βˆˆπ‘+,(2.10) holds if πœ†βˆ—|𝑑||π‘₯|π‘βˆ’1≀1 that is, π‘‡βˆΆπ‘β†’π‘ is weighted πœ†βˆ—(π‘₯,𝑦)-accretive with πœ†βˆ—(π‘₯,𝑦)∢=π‘‘βˆ’1min(|π‘₯|1βˆ’π‘,|𝑦|1βˆ’π‘). The restricted self-mapping π‘‡βˆΆ[βˆ’1,1]βŠ‚π‘‹β†’[βˆ’1,1]is πœ†βˆ—(β‰‘π‘‘βˆ’1)-accretive. Furthermore, if 𝑝β‰₯1,thenπ‘‡βˆΆ[βˆ’1,1]βŠ‚π‘‹β†’[βˆ’1,1] is |𝑑|-contractive if |𝑑|<1 and the iteration 𝑇𝑗π‘₯β†’0as π‘—β†’βˆžwith 𝑧=0 being the unique fixed point since 𝑑(𝑇π‘₯,𝑇𝑦)≀|𝑑|minξ€·|π‘₯|π‘βˆ’1,||𝑦||π‘βˆ’1𝑑(π‘₯,𝑦)≀|𝑑|𝑑(π‘₯,𝑦);βˆ€π‘₯,π‘¦βˆˆ[βˆ’1,1].(2.11) Note from the definition of the self-mapping 𝑇π‘₯=βˆ’π‘‘|π‘₯|π‘βˆ’1π‘₯ on [βˆ’1,1] that it is also a 2-cyclic self-mapping from [βˆ’1,0]βˆͺ[0,1] to itself with the property 𝑇([βˆ’1,0])=[0,1]and 𝑇([0,1])=[βˆ’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 π‘‡βˆΆπ·(𝑇)βŠ‚π‘‹β†’π‘‹is an accretive mapping, then it is πœ†βˆ—-accretive, for all πœ†βˆ—βˆˆπ‘0+. (2) If π‘‡βˆΆπ·(𝑇)βŠ‚π‘‹β†’π‘‹is πœ†βˆ—-accretive, then it is πœ†βˆ—1-accretive; for all πœ†βˆ—1∈[0,πœ†βˆ—]. (3) Any nonexpansive self-mapping π‘‡βˆΆπ·(𝑇)βŠ‚π‘‹β†’π‘‹is 0βˆ—-accretive and conversely.

Theorem 2.9. Let (𝑋,β€–β€–)be a Banach vector space with(𝑋,𝑑)being the associated complete metric space endowed with a norm-induced translation-invariant and homogeneous metricπ‘‘βˆΆπ‘‹Γ—π‘‹β†’π‘0+. Consider a self-mapping π‘‡βˆΆπ‘‹β†’π‘‹which restricted toπ‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is a 2-cyclic π‘˜-contractive self-mapping where 𝐴 and 𝐡 are nonempty subsets of 𝑋. Then, the following properties hold.(i) Assume that the self-mapping π‘‡βˆΆπ‘‹β†’π‘‹ satisfies the constraint: 𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘‘(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)+(1βˆ’π‘˜)𝑑𝐴𝐡;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅(2.12) with π‘˜,πœ†βˆˆπ‘0+ satisfying the constraint π‘˜(1+π‘˜πœ†)<1. Then, the restricted self-mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡satisfies limsupπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦<∞;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅(2.13) irrespective of 𝐴 and 𝐡 being bounded or not. If, furthermore, 𝐴 and 𝐡 are closed and convex andπ΄βˆ©π΅β‰ βˆ…, then there exists a unique fixed point πœ”βˆˆπ΄βˆ©π΅ of π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 such that there exists limπ‘—β†’βˆžπ‘‘(𝑇𝑗π‘₯,𝑇𝑗𝑦)=0; for all π‘₯∈𝐴,for all π‘¦βˆˆπ΅, implying that limπ‘—β†’βˆžπ‘‡π‘—π‘₯=limπ‘—β†’βˆžπ‘‡π‘—π‘¦=πœ”. If, in addition, dist(𝐴,𝐡)>0so that 𝐴∩𝐡=βˆ…, then there exists limπ‘—β†’βˆžπ‘‘(𝑇𝑗π‘₯,𝑇𝑗𝑦)=𝑑(𝑧,𝑇𝑧); for all π‘₯∈𝐴, for all π‘¦βˆˆπ΅ for some best proximity points π‘§βˆˆπ΄, π‘‡π‘§βˆˆπ΅ which depend in general on π‘₯ and 𝑦. Furthermore, if (𝑋,β€–β€–) is a uniformly convex Banach space, then 𝑇2𝑗π‘₯,𝑇2𝑗+1𝑦→𝑧1∈𝐴 and 𝑇2𝑗𝑦,𝑇2𝑗+1π‘₯→𝑇𝑧1∈𝐡 as β†’βˆž; for all(π‘₯,𝑦)βˆˆπ΄Γ—π΅, where 𝑧1 and 𝑧2 are unique best proximity points in 𝐴 and 𝐡 of π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡.
(ii)Assume that 𝐴 and 𝐡 are nondisjoint. Then, π‘‡βˆΆπ΄βˆͺ𝐡→𝑋is also π‘˜π‘ contractive and πœ†βˆ—-accretive for any nonnegative πœ†βˆ—β‰€π‘˜βˆ’2(π‘˜π‘βˆ’π‘˜) and any π‘˜π‘βˆˆ[π‘˜,1). It is also nonexpansive and πœ†βˆ—-accretive for any nonnegative πœ†βˆ—β‰€π‘˜βˆ’2(1βˆ’π‘˜).(iii)If π‘˜=0 then π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is weighted πœ†-accretive for πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+ for any πœ†βˆ—βˆˆπ‘+ and its restriction π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is 2-cyclic 0-contractive.(iv)π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is weighted πœ†-accretive for πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+ satisfying πœ†(π‘₯,𝑦)β‰€π‘˜βˆ’2(π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜)(𝑑(π‘₯,𝑦)βˆ’π‘‘π΄π΅) for some π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,∞). The restricted self-mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is also π‘˜π‘-contractive with π‘˜π‘βˆˆ[π‘˜,π‘˜π‘)βŠ†[π‘˜,1) if π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,π‘˜π‘) with π‘˜π‘<1. Also,π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is nonexpansive and weighted πœ†-accretive for πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+ satisfying πœ†(π‘₯,𝑦)β‰€π‘˜βˆ’2(π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜)(𝑑(π‘₯,𝑦)βˆ’π‘‘π΄π΅) if π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,1] which implies, furthermore, that πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+is bounded.

Proof. Let us denote π‘‘π΄π΅βˆΆ=dist(𝐴,𝐡).Consider that the two following relations are verified simultaneously: 𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)forsomeπœ†βˆˆπ‘0+;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅,𝑑(π‘₯,𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)𝑑𝐴𝐡forsomeπ‘˜βˆˆ[0,1)βˆˆπ‘…,πœ†βˆˆπ‘0+;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅.(2.14) Since the distance π‘‘βˆΆπ‘‹Γ—π‘‹β†’π‘0+ is translation-invariant and homogeneous, then the substitution of (2.14) yields if 𝐴 and 𝐡 are disjoint sets, after using the subadditive property of distances, the following chained relationships since 0βˆˆπ‘‹: 𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘˜π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘‘(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘‘(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘₯+πœ†π‘‡π‘¦βˆ’πœ†π‘‡π‘₯)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘‘(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘₯)+π‘˜π‘‘(𝑦+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘₯+πœ†π‘‡π‘¦βˆ’πœ†π‘‡π‘₯)+(1βˆ’π‘˜)𝑑𝐴𝐡=π‘˜π‘‘(π‘₯,𝑦)+π‘˜πœ†π‘‘(0,πœ†π‘‡π‘¦βˆ’πœ†π‘‡π‘₯)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘‘(π‘₯,𝑦)+π‘˜2πœ†π‘‘(0,π‘¦βˆ’π‘₯)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘‘(π‘₯,𝑦)+π‘˜2πœ†π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜(1+π‘˜πœ†)𝑑(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)𝑑𝐴𝐡;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅;βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,forπœ†βˆ—β‰€π‘˜βˆ’2(1βˆ’π‘˜),(2.15) with π‘˜π‘βˆΆ=π‘˜(1+π‘˜πœ†βˆ—)β‰₯π‘˜. Note from (2.15) that 𝑑𝐴𝐡≀𝑑𝑇𝑗π‘₯,π‘‡π‘—π‘¦ξ€Έβ‰€π‘˜π‘—π‘π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅βŽ›βŽœβŽπ‘—βˆ’1𝑖=0π‘˜π‘–π‘βŽžβŽŸβŽ =π‘˜π‘—π‘π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅βŽ›βŽœβŽβˆžξ“π‘–=0π‘˜π‘–π‘βˆ’βˆžξ“π‘–=π‘—π‘˜π‘–π‘βŽžβŽŸβŽ β‰€π‘˜π‘—π‘π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)ξ‚€1βˆ’π‘˜π‘—π‘ξ‚1βˆ’π‘˜π‘π‘‘π΄π΅;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅,(2.16) and, if π‘˜π‘<1, then 𝑑𝐴𝐡≀limsupπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦≀1βˆ’π‘˜1βˆ’π‘˜π‘π‘‘π΄π΅=1βˆ’π‘˜1βˆ’π‘˜(1+π‘˜πœ†)𝑑𝐴𝐡<∞;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅.(2.17) If 𝑑𝐴𝐡=0 then limπ‘—β†’βˆžπ‘‘(𝑇𝑗π‘₯,𝑇𝑗𝑦)=0. It is first proven that the existence of the limit of the distance implies that of the limit limπ‘—β†’βˆžπ‘‡π‘—π‘§; for all π‘§βˆˆπ΄βˆͺ𝐡. Let be π‘₯𝑗=𝑇𝑗π‘₯, 𝑦𝑗=𝑇𝑗𝑦 withπ‘₯𝑗,π‘¦π‘—βˆˆπ΄βˆͺ𝐡. Then, limπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦=limπ‘—β†’βˆžπ‘‘ξ€·π‘₯𝑗,𝑦𝑗=limπ‘—β†’βˆžπ‘‘ξ€·π‘‡β„“π‘₯𝑗,𝑇ℓ𝑦𝑗=0;βˆ€β„“βˆˆπ™0+βŸΉξ€·π‘₯𝑗=𝑇𝑗π‘₯ξ€Έβˆ’π‘¦π‘—ξ€·=π‘‡π‘—π‘¦ξ€Έξ€ΈβŸΆπ‘‡β„“ξ€·π‘₯π‘—βˆ’π‘¦π‘—ξ€ΈβŸΆ0asπ‘—βŸΆβˆž;βˆ€β„“βˆˆπ™0+(2.18) since π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 being contractive is globally Lipschitz continuous. Then, limπ‘—β†’βˆžπ‘‘(𝑇𝑗π‘₯,𝑇𝑗𝑦)=𝑑(limπ‘—β†’βˆžπ‘‡π‘—π‘₯,limπ‘—β†’βˆžπ‘‡π‘—π‘¦)=0since, because the fact that the metric is translation-invariant, one gets limπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦=𝑑limπ‘—β†’βˆžπ‘‡π‘—π‘₯,limπ‘—β†’βˆžπ‘‡π‘—π‘¦ξ‚=limπ‘—β†’βˆžπ‘‘ξ€·0,π‘‡π‘—π‘¦βˆ’π‘‡π‘—π‘₯ξ€Έ,=𝑑0,limπ‘—β†’βˆžξ€·π‘‡π‘—π‘¦βˆ’π‘‡π‘—π‘₯=0.(2.19) As a result, limπ‘—β†’βˆžπ‘‘(𝑇𝑗π‘₯,𝑇𝑗𝑦)=0 if 𝑑𝐴𝐡=0 what implies which limπ‘—β†’βˆž(𝑇𝑗π‘₯βˆ’π‘‡π‘—π‘¦)=0; for all π‘₯∈𝐴,for all π‘¦βˆˆπ΅, since π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is globally Lipschitz continuous since it is contractive.
In addition, there exists limπ‘—β†’βˆžπ‘‡π‘—π‘₯=limπ‘—β†’βˆžπ‘‡π‘—π‘¦=πœ”βˆˆπ΄βˆͺ𝐡; for all π‘₯∈𝐴, for all π‘¦βˆˆπ΅. Assume not so that there exists π‘₯∈𝐴 such that Β¬βˆƒlimπ‘—β†’βˆžπ‘‡π‘—π‘₯ and there exists a subsequence on nonnegative integers {π‘—π‘˜}π‘˜βˆˆπ™0+ such that π‘‡π‘—π‘˜+1π‘₯β‰ π‘‡π‘—π‘˜π‘₯. If so, one gets by taking 𝑦=𝑇π‘₯∈𝐡that 𝑑(π‘‡π‘—π‘˜(𝑇π‘₯),π‘‡π‘—π‘˜π‘₯)>0 which contradicts limπ‘—β†’βˆžπ‘‘(𝑇𝑗(𝑇π‘₯),𝑇𝑗π‘₯)=0. Then {𝑇𝑗π‘₯}π‘—βˆˆπ™0+ is a Cauchy sequence for any π‘₯∈𝐴βˆͺ𝐡 and then converges to a limit. Furthermore, πœ”βˆˆπ΄βˆͺ𝐡 since 𝑇𝑗(𝐴βˆͺ𝐡)βŠ†π΄βˆͺ𝐡for any π‘—βˆˆπ™0+ and as π‘—β†’βˆž since 𝐴 and 𝐡 are nonempty and closed. It has been proven that limπ‘—β†’βˆžπ‘‡π‘—π‘₯=limπ‘—β†’βˆžπ‘‡π‘—π‘¦=πœ”βˆˆπ΄βˆͺ𝐡; for all π‘₯∈𝐴,for all π‘¦βˆˆπ΅.
It is now proven thatπœ”=π‘‡πœ”βˆˆFix(𝑇). Assume not, then, from triangle inequality, 0<𝑑(π‘‡πœ”,πœ”)β‰€π‘‘ξ€·πœ”,π‘‡π‘—πœ”ξ€Έ+π‘‘ξ€·π‘‡πœ”,π‘‡π‘—πœ”ξ€Έ;βˆ€π‘—βˆˆπ™0+⟹liminfπ‘—β†’βˆžπ‘‘ξ€·πœ”,π‘‡π‘—πœ”ξ€Έ>0,(2.20) which contradicts limπ‘—β†’βˆžπ‘‡π‘—πœ”=πœ” so that πœ”=π‘‡πœ”βˆˆFix(𝑇). It is now proven that πœ”βˆˆFix(𝑇)∩(𝐴∩𝐡). Assume not, such that, for instance, 𝑇𝑗π‘₯∈𝐴 and 𝑇𝑗+1π‘₯∈𝐴∩𝐡. If so, since 𝑇(𝐴)βŠ†π΅;𝑇(𝐡)βŠ†π΄, then the existing limit fulfils limπ‘—β†’βˆžπ‘‡π‘—π‘₯∈𝐴∩𝐴(=βˆ…) which is impossible so that there would be no existing limitlimπ‘—β†’βˆžπ‘‡π‘—π‘₯ in 𝐴βˆͺ𝐡, contradicting the former result of its existence. Then, πœ”βˆˆFix(𝑇)∩(𝐴∩𝐡) implying that Fix(𝑇)βŠ‚π΄βˆ©π΅.
It is now proven by contradiction that πœ”=limπ‘—β†’βˆžπ‘‡π‘—π‘₯; for all π‘₯∈𝐴βˆͺ𝐡 is the unique fixed point of π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡. Assume that βˆƒπœ”1(β‰ πœ”)∈Fix(𝑇), then limπ‘—β†’βˆžπ‘‡π‘—π‘¦1=πœ”1 for some 𝑦1(≠𝑦)∈𝐡 with no loss in generality and all π‘₯∈𝐴. Thus, limπ‘—β†’βˆžπ‘‘(𝑇𝑗π‘₯,𝑇𝑗𝑦1)=𝑑(πœ”,πœ”1)=0β‡’πœ”=πœ”1 which contradicts πœ”β‰ πœ”1 so that Fix(𝑇)={πœ”}.
Now, assume that 𝐴 and 𝐡 do not intersect so that dist(𝐴,𝐡)=𝑑𝐴𝐡>0. Then, one gets from the first inequality in (2.15) that for all π‘₯∈𝐴, π‘¦βˆˆπ΅, one gets𝑑𝑇𝑗π‘₯,π‘‡π‘—π‘¦ξ€Έβ‰€π‘˜π‘—π‘‘(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅βŽ›βŽœβŽβˆžξ“π‘–=0π‘˜π‘—βŽžβŽŸβŽ =π‘˜π‘—π‘‘(π‘₯,𝑦)+𝑑𝐴𝐡;βˆ€π‘—βˆˆπ™,limsupπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦≀𝑑𝐴𝐡.(2.21)
Note that since 𝑇(𝐴)βŠ†π΅, 𝑇(𝐡)βŠ†π΄ and dist(𝐴,𝐡)=𝑑𝐴𝐡>0, thenπ‘₯βˆˆπ΄β‡’π‘‡π‘—π‘₯∈𝐴 and 𝑇𝑗π‘₯βˆ‰π΅ if 𝑗 is even and 𝑇𝑗π‘₯∈𝐡 and 𝑇𝑗π‘₯βˆ‰π΄ if 𝑗 is oddπ‘¦βˆˆπ΅β‡’π‘‡π‘—π‘¦βˆˆπ΅and π‘‡π‘—π‘¦βˆ‰π΄ if 𝑗 is even and π‘‡π‘—π‘¦βˆˆπ΄ and π‘‡π‘—π‘¦βˆ‰π΅ if 𝑗 is odd.
Then, 𝑇𝑗π‘₯ and 𝑇𝑗𝑦 are not both in either 𝐴 or 𝐡 if π‘₯ and 𝑦 are not both in either 𝐴 or 𝐡 for any π‘—βˆˆπ™0+. As a result, limπ‘—β†’βˆžsup𝑑(𝑇𝑗π‘₯,𝑇𝑗𝑦)<𝑑𝐴𝐡 is impossible so thatβˆƒlimπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦=limsupπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦=𝑑𝐴𝐡=𝑑(𝑧,𝑇𝑧),(2.22) for some best proximity points π‘§βˆˆπ΄ and π‘‡π‘§βˆˆπ΅ or conversely. Then, limπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—+1π‘₯,𝑇𝑗+1𝑦=limπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘§π‘—,𝑇2𝑧𝑗=π‘‘π΄π΅β‰€π‘˜limπ‘—β†’βˆžπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗𝑦+(1βˆ’π‘˜)𝑑(𝑧,𝑇𝑧)=π‘˜limπ‘—β†’βˆžπ‘‘ξ€·π‘§π‘—,𝑇𝑧𝑗+(1βˆ’π‘˜)𝑑(𝑧,𝑇𝑧)=π‘˜limπ‘—β†’βˆžπ‘‘ξ€·π‘§π‘—,𝑇𝑧𝑗+(1βˆ’π‘˜)𝑑𝐴𝐡,(2.23) where 𝑧𝑗=𝑇𝑗π‘₯ Thus, limπ‘—β†’βˆžπ‘‘(𝑧𝑗,𝑇𝑧𝑗)=𝑑𝐴𝐡=𝑑(𝑧,𝑇𝑧). It turns out that dist(𝑧𝑗,Fr(𝐴βˆͺ𝐡))β†’0 and dist(𝑇𝑧𝑗,Fr(𝐴βˆͺ𝐡))β†’0 as π‘—β†’βˆž. Otherwise, it would exist an infinite subsequence {𝑑(𝑧𝑗,𝑇𝑧𝑗)}π‘—βˆˆξπ‘0+of {𝑑(𝑧𝑗,𝑇𝑧𝑗)}π‘—βˆˆπ‘0+ with 𝐙0+being an infinite subset of 𝐙0+such that 𝑑(𝑧𝑗,𝑇𝑧𝑗)>𝑑𝐴𝐡 for π‘—βˆˆξπ™0+. On the other hand, since (𝑋,β€–β€–) is a normed space, then by taking the norm-translation invariant and homogeneous induced metric and since there exists limπ‘—β†’βˆžπ‘‘(𝑇𝑗+1π‘₯,𝑇𝑗+1𝑦)=𝑑𝐴𝐡, it follows that there exist 𝑗1βˆˆπ™0+ and 𝛿=𝛿(πœ€,𝑗1)βˆˆπ‘+ such that 2𝑑𝐴𝐡+𝛿<𝑑𝑇𝑗π‘₯+𝑇𝑗+1𝑦,0≀𝑑𝑇𝑗π‘₯,0ξ€Έ+𝑑𝑇𝑗+1𝑦,0ξ€Έ,≀2𝑑𝐴𝐡+π›Ώξ€ΈβŸΉπ‘‘ξ€·π‘‡π‘—π‘₯,𝑇𝑗+1𝑦<πœ€,(2.24) for any given πœ€βˆˆπ‘+; for all π‘₯∈𝐴,for all π‘¦βˆˆπ΅ with 𝑇𝑗π‘₯∈𝐴,𝑇𝑗+1π‘¦βˆˆπ΄ for any even 𝑗(β‰₯𝑗1)βˆˆπ™0+ and 𝑇𝑗π‘₯∈𝐡,𝑇𝑗+1π‘¦βˆˆπ΅, for any odd 𝑗(β‰₯𝑗1)βˆˆπ™0+. As a result, by choosing the positive real constant arbitrarily small, one gets that 𝑇2𝑗π‘₯→𝑇2𝑗+1𝑦→𝑧=𝑧(π‘₯,𝑦)∈𝐴 (a best proximity point of 𝐴) and 𝑇2𝑗+1π‘₯→𝑇2π‘—π‘¦β†’π‘‡π‘§βˆˆπ΅ (a best proximity point of 𝐡), or vice-versa, as π‘—β†’βˆž for any given π‘₯∈𝐴 and π‘¦βˆˆπ΅. A best proximity point π‘§βˆˆπ΄βˆͺ𝐡 fulfils 𝑧=𝑇2𝑧. Best proximity points are unique in 𝐴 and 𝐡 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 𝑧1≠𝑧2 so that necessarily 𝑧1=𝑇2𝑧1≠𝑧2=𝑇2𝑧2. Since (𝑋,β€–β€–) 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: 𝑑𝐴𝐡=𝑑𝑧1,𝑇𝑧1ξ€Έ=𝑑𝑧1,𝑇𝑧2ξ€Έ=‖‖‖𝑇𝑧2βˆ’π‘§12+𝑇𝑧2βˆ’π‘§12β€–β€–β€–<2‖‖‖𝑇𝑧2βˆ’π‘§12β€–β€–β€–=𝑑𝐴𝐡,(2.25) since (𝑋,β€–β€–) is also a strictly convex Banach space and 𝐴 and 𝐡 are nonempty closed and convex sets. Then, 𝑧=𝑇2π‘§βˆˆπ΄ is the unique best proximity point of π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 in 𝐴 and 𝑇𝑧 is its unique best proximity point in 𝐡. Then, Property (i) has been fully proven. Since 𝐴 and 𝐡 are not disjoint, then 𝑑𝐴𝐡=0, and π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is π‘˜π‘-contractive and πœ†βˆ—-accretive if πœ†βˆ—=π‘˜βˆ’2(π‘˜π‘βˆ’π‘˜) with π‘˜π‘βˆˆ[π‘˜,1). By taking π‘˜π‘=1, note that π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is nonexpansive and π‘˜βˆ’2(1βˆ’π‘˜)-accretive. Property (ii) has been proven.
To prove Property (iii), we now discuss if𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘˜(1+π‘˜πœ†)𝑑(π‘₯,𝑦)+(1βˆ’π‘˜)π‘‘π΄π΅β‰€π‘˜π‘π‘‘(π‘₯,𝑦)+ξ€·1βˆ’π‘˜π‘ξ€Έπ‘‘π΄π΅;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅.(2.26) is possible with 1β‰₯π‘˜c and 𝑑AB>0. Note that  𝑑𝐴𝐡=dist(𝐴,𝐡)=𝑑(𝑧,𝑇𝑧) for some π‘§βˆˆπ΄. Define π‘‘βˆΆ=max(𝑑𝐴,𝑑𝐡)=π‘˜π·π‘‘π΄π΅, if  𝑑𝐴𝐡≠0 for some π‘˜π·π΄,π‘˜π·π΅,π‘˜π·βˆˆπ‘+, where π‘‘π΄βˆΆ=diam𝐴=π‘˜π·π΄π‘‘π΄π΅ and π‘‘π΅βˆΆ=diam𝐡=π‘˜π·π΅π‘‘π΄π΅. Three cases can occur in (2.26), namely,(a)If π‘˜=π‘˜π‘ then π‘˜2πœ†π‘‘(π‘₯,𝑦)≀0⇔[π‘˜πœ†=0βˆ¨π‘‘(π‘₯,𝑦)≀0] which is untrue if π‘₯≠𝑦 and π‘˜πœ†>0 and it holds for either π‘˜=0 or πœ†=0,(b)π‘˜π‘>π‘˜, then (2.26) is equivalent to 𝑑(π‘₯,𝑦)β‰₯π‘˜π‘βˆ’π‘˜π‘˜π‘βˆ’π‘˜(1+π‘˜πœ†)𝑑𝐴𝐡;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅.(2.27) Take π‘₯∈𝐴 to be a best proximity point with so that 𝑑(π‘₯,𝑇π‘₯)=𝑑𝐴𝐡β‰₯(π‘˜π‘βˆ’π‘˜)/(π‘˜π‘βˆ’π‘˜(1+π‘˜πœ†))𝑑𝐴𝐡>𝑑𝐴𝐡 which is untrue if π‘˜πœ†>0 and true for π‘˜πœ†=0,(c)1β‰₯π‘˜(1+π‘˜πœ†)β‰₯π‘˜π‘<π‘˜, then (2.16) is equivalent to (π‘˜βˆ’π‘˜π‘)𝑑𝐴𝐡β‰₯[π‘˜(1+π‘˜πœ†)βˆ’π‘˜π‘]𝑑(π‘₯,𝑦); for all π‘₯∈𝐴,for all π‘¦βˆˆπ΅, but 𝑑(π‘₯,𝑦)≀2𝑑+𝑑𝐴𝐡=(2π‘˜π·+1)𝑑𝐴𝐡. Thus, the above constraint is guaranteed to hold in the worst case if π‘˜βˆ’π‘˜π‘β‰₯(π‘˜+π‘˜2πœ†βˆ’π‘˜π‘)(2π‘˜π·+1)>π‘˜βˆ’π‘˜π‘ 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 πœ† and π‘˜π‘ with the real functions πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+ and π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,1). Note that (2.26) holds through direct calculation if πœ†(π‘₯,𝑦)β‰€π‘˜βˆ’2(π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜)(𝑑(π‘₯,𝑦)βˆ’π‘‘π΄π΅); for all π‘₯∈𝐴,for all π‘¦βˆˆπ΅ for some π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,∞). Thus, the self-mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is weighted πœ†-accretive for πœ†βˆΆπ‘‹Γ—π‘‹β†’π‘0+ satisfying πœ†(π‘₯,𝑦)β‰€π‘˜βˆ’2(π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜)(𝑑(π‘₯,𝑦)βˆ’π‘‘π΄π΅) for some π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,∞); and it is also π‘˜π‘-contractive with π‘˜π‘βˆˆ[π‘˜,π‘˜π‘)βŠ†[π‘˜,1) if π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,π‘˜π‘) with π‘˜π‘<1 and nonexpansive if π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,1]. On the other hand, note that 𝑑(π‘₯,𝑦)βˆ’π‘‘π΄π΅β‰€π‘˜π·π΄π‘‘π΄+π‘˜π·π΅π‘‘π΅β‰€2π‘˜π·π‘‘. If 𝐴 and 𝐡 are bounded and π‘˜π‘βˆΆπ‘‹Γ—π‘‹β†’[π‘˜,1], then πœ†(π‘₯,𝑦)β‰€π‘˜βˆ’2ξ€·π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜ξ€Έξ€·π‘‘(π‘₯,𝑦)βˆ’π‘‘π΄π΅ξ€Έβ‰€π‘˜βˆ’2ξ€·π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜ξ€Έξ€·π‘˜π·π΄π‘‘π΄+π‘˜π·π΅π‘‘π΅ξ€Έβ‰€2π‘˜βˆ’2ξ€·π‘˜π‘(π‘₯,𝑦)βˆ’π‘˜ξ€Έπ‘˜π·π‘‘β‰€βˆž;βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅.(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 𝐡 are disjoint by introducing the concept of weighted accretive mapping since for best proximity points π‘§βˆˆπ΄βˆͺ𝐡, πœ†(𝑧,𝑇𝑧)=0.

Remark 2.11. Note that the assumption that (𝑋,β€–β€–) 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 π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 in 𝐴 and 𝐡 are obtained provided that both sets are nonempty, convex, and closed.

Remark 2.12. Note that if either 𝐴 or 𝐡 is not closed, then its best proximity point of π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is in its closure since 𝑇(𝐴)βŠ†π΅βŠ†cl𝐡, 𝑇(𝐡)βŠ†π΄βŠ†cl 𝐴 leads to 𝑇(𝐴βˆͺ𝐡)βŠ†π΄βˆͺπ΅βŠ†cl(𝐴βˆͺ𝐡) and π‘‡π‘˜(𝐴βˆͺ𝐡)βŠ†cl(𝐴βˆͺ𝐡) for finitely many and for infinitely many iterations through the self-mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 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 (𝑋,𝑑) be a metric space endowed with a norm-induced translation-invariant and homogeneous metric π‘‘βˆΆπ‘‹Γ—π‘‹β†’π‘0+. Consider the πœ†βˆ—-accretive mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝑋for some πœ†βˆ—βˆˆπ‘0+ which restricted as π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is 2-cyclic, where 𝐴 and 𝐡 are nonempty subsets of 𝑋 subject to 0∈𝐴βˆͺ𝐡. Then, 𝑑(𝐼+πœ†π‘‡)𝑗π‘₯,0ξ€Έβ‰₯1;βˆ€π‘—βˆˆπ™0+,βˆ€π‘₯(β‰ 0)∈𝐴βˆͺ𝐡,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.29) If, furthermore, π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is π‘˜-contractive, then 1≀𝑑(𝐼+πœ†π‘‡)𝑗π‘₯,0ξ€Έ<π‘˜βˆ’1;βˆ€π‘—βˆˆπ™+βˆ€π‘₯(β‰ 0)∈𝐴βˆͺ𝐡,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.30)π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if 𝑑(𝐼+πœ†π‘‡)𝑗π‘₯,0ξ€Έ=1;βˆ€π‘—βˆˆπ™+,βˆ€π‘₯(β‰ 0)∈𝐴βˆͺ𝐡,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,(2.31) respectively, limsupπ‘—β†’βˆžπ‘‘ξ€·(𝐼+πœ†π‘‡)𝑗π‘₯,0ξ€Έ=1;βˆ€π‘₯(β‰ 0)∈𝐴βˆͺ𝐡,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.32)(ii)Let (𝑋,β€–β€–) be a normed vector space. Consider a πœ†βˆ—-accretive mapping π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 for some πœ†βˆ—βˆˆπ‘0+ which restricted to π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is 2-cyclic, where 𝐴 and 𝐡 are nonempty subsets of 𝑋 subject to 0∈𝐴βˆͺ𝐡 then β€–β€–(𝐼+πœ†π‘‡)𝑗‖‖β‰₯1;βˆ€π‘—βˆˆπ™0+,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.33) If, furthermore, π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is π‘˜-contractive, then 1≀‖‖(𝐼+πœ†π‘‡)𝑗‖‖<π‘˜βˆ’1;βˆ€π‘—βˆˆπ™+,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.34)π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is nonexpansive (resp., asymptotically nonexpansive, [30]) if β€–β€–(𝐼+πœ†π‘‡)𝑗‖‖=1;βˆ€π‘—βˆˆπ™+,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,(2.35) respectively, limsupjβ†’βˆžβ€–β€–(𝐼+πœ†π‘‡)𝑗‖‖=1;βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.36)

Proof. To prove Property (i), define an induced by the metric norm as follows β€–π‘₯β€–=𝑑(π‘₯,0)since the metric is homogeneous and translation-invariant. Define the norm of π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡, that is, the norm of 𝑇 on 𝑋 restricted to 𝐴βˆͺ𝐡 as follows: β€–π‘‡β€–βˆΆ=minξ€½π‘βˆˆπ‘0+βˆΆβ€–π‘‡π‘₯‖≀𝑐‖π‘₯β€–;βˆ€π‘₯∈𝐴βˆͺ𝐡≑minξ€½π‘βˆˆπ‘0+βˆΆπ‘‘(𝑇π‘₯,0)≀𝑐𝑑(π‘₯,0);βˆ€π‘₯∈𝐴βˆͺ𝐡,(2.37) with the above set being closed, nonempty, and bounded from below. Since π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is 2-cyclic and π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is πœ†βˆ—-accretive (Definition 2.2), one gets by proceeding recursively 𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†π‘‡π‘₯,𝑦+πœ†π‘‡π‘¦)≀𝑑(π‘₯+πœ†π‘‡π‘₯+πœ†π‘‡(π‘₯+πœ†π‘‡π‘₯),𝑦+πœ†π‘‡π‘¦)=𝑑(𝐼+πœ†π‘‡)2π‘₯,(𝐼+πœ†π‘‡)2𝑦≀⋯≀𝑑(𝐼+πœ†π‘‡)𝑗π‘₯,(𝐼+πœ†π‘‡)𝑗𝑦≀‖‖(𝐼+πœ†π‘‡)𝑗‖‖𝑑(π‘₯,𝑦);βˆ€π‘₯∈𝐴,βˆ€π‘¦βˆˆπ΅,βˆ€π‘—βˆˆπ™0+,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,(2.38) since the metric is homogeneous and 0∈𝐴βˆͺ𝐡, and 𝐼 is the identity operator on 𝑋, where β€–β€–(𝐼+πœ†π‘‡)π‘—β€–β€–βˆΆ=minξ€½π‘βˆˆπ‘0+βˆΆβ€–β€–(𝐼+πœ†π‘‡)𝑗π‘₯‖‖≀𝑐‖π‘₯β€–;βˆ€π‘₯∈𝐴βˆͺ𝐡,≑minξ€½π‘βˆˆπ‘0+βˆΆπ‘‘ξ€·(𝐼+πœ†π‘‡)𝑗π‘₯,0≀𝑐𝑑(π‘₯,0);βˆ€π‘₯∈𝐴βˆͺ𝐡,(2.39) with the above set being closed, nonempty, and bounded from below. Ifβ€–(𝐼+πœ†π‘‡)𝑗‖<1 for some β€–(𝐼+πœ†π‘‡)𝑗‖<1, then we get the contradiction 𝑑(π‘₯,𝑦)<𝑑(π‘₯,𝑦); for all π‘₯∈𝐴, for all π‘¦βˆˆπ΅ in (2.38). Thus, β€–(𝐼+πœ†π‘‡)𝑗‖=𝑑((𝐼+πœ†π‘‡)𝑗π‘₯,0)β‰₯1; for all π‘—βˆˆπ™0+, forallπ‘₯(β‰ 0)∈𝐴βˆͺ𝐡, for all πœ†βˆˆ[0,πœ†βˆ—]. If now π‘₯ and 𝑦 are replaced with 𝑇𝑖π‘₯ and 𝑇𝑖𝑦 for any π‘–βˆˆπ™0+ in (2.30), one gets if π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is a 2-cyclic π‘˜-contractive for some real π‘˜βˆˆ[0,1) and πœ†βˆ—-accretive mapping: 𝑑𝑇𝑖π‘₯,𝑇𝑖𝑦≀𝑑(𝐼+πœ†π‘‡)𝑗𝑇𝑖π‘₯,(𝐼+πœ†π‘‡)𝑗𝑇𝑖𝑦≀‖‖(𝐼+πœ†π‘‡)𝑗‖‖𝑑𝑇𝑖π‘₯,π‘‡π‘–π‘¦ξ€Έβ‰€π‘˜π‘–β€–β€–(𝐼+πœ†π‘‡)𝑗‖‖𝑑(π‘₯,𝑦)+(1βˆ’π‘˜)𝑑𝐴𝐡<𝑑(π‘₯,𝑦)+(1βˆ’π‘˜)𝑑𝐴𝐡,(2.40) for all π‘₯∈𝐴, for all 𝑦(β‰ π‘₯)∈𝐡, for all 𝑗(β‰₯𝑖)βˆˆπ™+, for all π‘–βˆˆπ™+, for all πœ†βˆˆ[0,πœ†βˆ—]. Then, 1≀‖(𝐼+πœ†π‘‡)𝑗‖=𝑑((𝐼+πœ†π‘‡)𝑗π‘₯,0)<π‘˜βˆ’1; for all π‘—βˆˆπ™+, for all π‘₯(β‰ 0)∈𝐴βˆͺ𝐡, for all πœ†βˆˆ[0,πœ†βˆ—]. If β€–(𝐼+πœ†π‘‡)𝑗‖=𝑑((𝐼+πœ†π‘‡)𝑗π‘₯,0)=1; for all π‘—βˆˆπ™+, for all π‘₯(β‰ 0)∈𝐴βˆͺ𝐡, for all πœ†βˆˆ[0,πœ†βˆ—], it turns out that π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is πœ†βˆ—-accretive and π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is a 2-cyclic nonexpansive self-mapping. It is asymptotically nonexpansive if limπ‘—β†’βˆžsup𝑑((𝐼+πœ†π‘‡)𝑗π‘₯,0)=1; for all π‘₯(β‰ 0)∈𝐴βˆͺ𝐡, forallπœ†βˆˆ[0,πœ†βˆ—]. Property (i) has been proven. The proof of Property (ii) for (𝑋,β€–β€–) 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=π‘βˆ’,𝐡=𝐑+ and the 2-cyclic self-mapping 𝑇(≑)π‘‘βˆΆ(𝐴βˆͺ𝐡)×𝐙0+β†’(𝐴βˆͺ𝐡)×𝐙0+ defined by the iteration rule 𝐴βˆͺπ΅βˆ‹π‘₯𝑗+1=π‘˜π‘—π‘₯π‘—βˆˆπ΄βˆͺ𝐡 with π‘βˆ‹π‘˜π‘—(∈[βˆ’π‘˜,π‘˜])β‰€π‘˜β‰€1, sgnπ‘˜π‘—+1=βˆ’sgnπ‘˜π‘—=sgnπ‘₯𝑗; for all π‘—βˆˆπ™0+, and π‘₯0∈𝐴βˆͺ𝐡. Let π‘‘βˆΆπ‘0+→𝐑0+ be the Euclidean metric.(a)Ifπ‘˜<1, then limπ‘—β†’βˆžβˆπ‘—π‘–=0[π‘˜π‘—]=0 so that for any π‘₯0∈𝐴βˆͺ𝐡, π‘₯π‘—βˆˆπ΄βˆͺ𝐡; for all π‘—βˆˆπ™0+π‘₯𝑗→𝑧=0βˆ‰π΄βˆͺ𝐡 as π‘—β†’βˆž with 0∈cl(𝐴βˆͺ𝐡), Fix(𝑑)={0}βŠ‚cl(𝐴∩𝐡) but it is not in 𝐴∩𝐡 which is empty. If π‘˜=1 and limπ‘—β†’βˆžβˆπ‘—π‘–=0[π‘˜π‘—]=0 (i.e., there are infinitely many values |π‘˜π‘–| being less than unity), then the conclusion is identical. If 𝐴 and 𝐡 are redefined as 𝐴=𝐑0βˆ’, 𝐡=𝐑0+, then Fix(𝑑)={0}βŠ‚π΄βˆ©π΅β‰ βˆ….(b)If π‘˜π‘—=π‘˜=1; for all π‘—βˆˆπ™0+ the self-mapping π‘‘βˆΆ(𝐴βˆͺ𝐡)×𝐙0+β†’(𝐴βˆͺ𝐡)×𝐙0+is not expansive and there is no fixed point. (c)If π‘˜=1βˆ’πœŽ for some 𝜎(<1)βˆˆπ‘+, then for𝐑0βˆ‹πœ†βˆˆ[0,πœ†βˆ—], 𝑑(𝑑π‘₯,𝑑𝑦)≀𝐾𝑑(π‘₯,𝑦)β‰€π‘˜(1+πœ†)||π‘₯βˆ’π‘¦||β‰€π‘˜||π‘₯βˆ’π‘¦||;𝑑(π‘₯,𝑦)≀𝑑(π‘₯+πœ†π‘₯,𝑦+πœ†π‘¦)≀(1+πœ†)||π‘₯βˆ’π‘¦||,(2.41) so that π‘‘βˆΆ(𝐴βˆͺ𝐡)×𝐙0+β†’(𝐴βˆͺ𝐡)×𝐙0+is also πœ†βˆ—-accretive and π‘˜1∈[π‘˜,1)-contractive with πœ†βˆ—=π‘˜1π‘˜βˆ’1βˆ’1. (d)Now, define closed setsπ‘πœ€+∢={π‘Ÿ(β‰₯πœ€)βˆˆπ‘+} and π‘πœ€βˆ’βˆΆ={π‘Ÿ(β‰€βˆ’πœ€)βˆˆπ‘+} for any given πœ€βˆˆπ‘0+so that 𝑑𝐴𝐡=πœ€. The 2-cyclic self-mapping 𝑇(≑)π‘‘βˆΆ(𝐴βˆͺ𝐡)×𝐙0+β†’(𝐴βˆͺ𝐡)×𝐙0+is re-defined by the iterationπ‘₯𝑗+1=π‘₯𝑗+1if |π‘₯𝑗+1|β‰₯πœ€and π‘₯𝑗+1=βˆ’πœ€sgnπ‘₯𝑗, for 𝑖=1,2, otherwise, whereπ‘₯𝑗+1=π‘˜π‘—π‘₯𝑗for 𝑖=1,2 with the real sequence {π‘˜π‘—}π‘—βˆˆπ™0+being subject to π‘˜π‘—(∈[βˆ’π‘˜,π‘˜])β‰€π‘˜β‰€1, sgnπ‘˜π‘—+1=βˆ’sgnπ‘˜π‘—=sgnπ‘₯(𝑖)𝑗; 𝑖=1,2, for all π‘—βˆˆπ™0+and π‘₯0∈𝐴βˆͺ𝐡. Then, for any πœ€βˆˆπ‘+and any π‘₯0∈𝐴βˆͺ𝐡, there are two best proximity points 𝑧=βˆ’πœ€βˆˆπ΄ and 𝑧1=πœ€βˆˆπ΅ fulfilling βˆ’πœ€=π‘‘πœ€=βˆ’π‘‘2πœ€ and 𝑑𝐴𝐡=𝑑(𝑧,𝑧1)=𝑑(𝑧,𝑑𝑧)=𝑑(𝑑𝑧1,𝑧1).(e)Redefine 𝑋=𝐑2so that 𝐑2βˆ‹π‘₯=(π‘₯(1),π‘₯(2))𝑇 with π‘₯(1), π‘₯(2)βˆˆπ‘; 𝐴=𝐑2πœ€βˆ’,𝐡=𝐑2πœ€+. In the case that πœ€=0,then 𝐴 and 𝐡 are open disjoint subsets (resp., 𝐴=𝐑20βˆ’,𝐡=𝐑20+ are closed nondisjoint subsets with 𝐴∩𝐡={(0,π‘₯)π‘‡βˆΆπ‘₯βˆˆπ‘}).The 2-cyclic self-mapping 𝑇(≑)π‘‘βˆΆ(𝐴βˆͺ𝐡)×𝐙0+β†’(𝐴βˆͺ𝐡)×𝐙0+is re-defined by the iteration rule: π‘₯(𝑖)𝑗+1=π‘₯(𝑖)𝑗+1,if|||π‘₯(𝑖)𝑗+1|||β‰₯πœ€,π‘₯(𝑖)𝑗+1=βˆ’πœ€sgnπ‘₯(𝑖)𝑗,for𝑖=1,2,(2.42) otherwise, where π‘₯(𝑖)𝑗+1=π‘˜π‘—π‘₯(𝑖)𝑗,for𝑖=1,2(2.43) with the real sequence {π‘˜π‘—}π‘—βˆˆπ™0+ being subject toπ‘˜π‘—(∈[βˆ’π‘˜,π‘˜])β‰€π‘˜β‰€1, sgnπ‘˜π‘—+1=βˆ’sgnπ‘˜π‘—=sgnπ‘₯(𝑖)𝑗; for 𝑖=1,2; for all π‘—βˆˆπ™0+ and π‘₯0∈𝐴βˆͺ𝐡.
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 𝐡 but not in its empty intersection if either 𝐴 or 𝐡 is open, respectively, in the intersection of 𝐴 and 𝐡 (the vertical real line of zero abscissa) if they are closed. The same conclusion of (d) is valid for the best proximity points if πœ€>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,β€–β€–) be a normed vector space with (𝑋,𝑑) being the associate metric space endowed with a norm-induced translation-invariant and homogeneous metric π‘‘βˆΆπ‘‹Γ—π‘‹β†’π‘0+ and consider the self-mapping π‘‡βˆΆπ‘‹β†’π‘‹ so that the restricted π‘‡βˆΆπ΄βˆͺ𝐡→𝑋is πœ†βˆ—-accretive for some πœ†βˆ—βˆˆπ‘0+, where 𝐴 and 𝐡 are nonempty subsets of 𝑋 subject to 0∈AβˆͺB, and the restricted π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is 2-cyclic.Then, 𝑑((𝐼+πœ†π‘‡)π‘₯,0)β‰₯1;βˆ€π‘₯(β‰ 0)∈𝐴βˆͺ𝐡,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.44) If, furthermore, π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is π‘˜-contractive, then 1≀𝑑((𝐼+πœ†π‘‡)π‘₯,0)<π‘˜βˆ’1;βˆ€π‘₯(β‰ 0)∈𝐴βˆͺ𝐡,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.45)π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if(ii)Let (𝑋,β€–β€–) be a normed vector space. Then if π‘‡βˆΆπ΄βˆͺ𝐡→𝑋is a πœ†βˆ—-accretive mapping and π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is 2-cyclic for some πœ†βˆ—βˆˆπ‘0+ where 𝐴 and 𝐡 are nonempty subsets of 𝑋 subject to 0∈𝐴βˆͺ𝐡, then ‖𝐼+πœ†Tβ€–β‰₯1;βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.46) If, furthermore, π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡is 2-cyclic k (∈[0,1))-contractive, then 1≀‖𝐼+πœ†π‘‡β€–<π‘˜βˆ’1;βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.47)

Outline of Proof
It follows since the basic constraint of π‘‡βˆΆπ΄βˆͺ𝐡→𝑋being πœ†βˆ—-accretive holds if ‖𝐼+πœ†π‘‡β€–β‰₯1βŸΉβ€–πΌ+πœ†π‘‡β€–π‘—β‰₯β€–β€–(𝐼+πœ†π‘‡)𝑗‖‖β‰₯1;βˆ€π‘—βˆˆπ™+,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,(2.48) while it fails if ‖𝐼+πœ†π‘‡β€–<1βŸΉβ€–β€–(𝐼+πœ†π‘‡)𝑗‖‖≀‖𝐼+πœ†π‘‡β€–π‘—<1;βˆ€π‘—βˆˆπ™+,βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€».(2.49)

Remark 2.16. Theorem 2.13 and Corollary 2.15 are easily linked to Theorem 2.9 as follows. Assume that π‘‡βˆΆπ΄βˆͺ𝐡→𝐴βˆͺ𝐡 is 2-cyclic π‘˜-contractive and π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is aπœ†βˆ—-accretive mapping. Assume that there exists π‘₯∈𝐴βˆͺ𝐡 such that β€–π‘₯β€–=𝑑(π‘₯,0)≀1. Then, 1≀‖𝐼+πœ†π‘‡β€–<π‘˜βˆ’1; for all πœ†βˆˆ[0,πœ†βˆ—] from (2.47). This is guaranteed under sufficiency-type conditions with ‖𝑇‖=maxβ€–π‘₯‖≀1𝑑(𝑇π‘₯,0)=max𝑑(π‘₯,0)≀1𝑑(𝑇π‘₯,0)β‰€π‘˜if1≀‖𝐼+πœ†π‘‡β€–β‰€1+πœ†β€–π‘‡β€–β‰€1+πœ†π‘˜<π‘˜βˆ’1;βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,⟺(1+πœ†π‘˜)π‘˜β‰€π‘˜π‘<1;βˆ€πœ†βˆˆξ€Ί0,πœ†βˆ—ξ€»,(2.50) with πœ†βˆ—=π‘˜βˆ’2(π‘˜cβˆ’π‘˜) for some real constantsπ‘˜π‘βˆˆ[π‘˜,1), π‘˜βˆˆ[0,1). It is direct to see that Fix(𝑇)={0βˆˆπ‘π‘›} if 0∈𝐴∩𝐡.

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 𝐴≑Dom(𝑇)=𝐡≑Im(𝑇)βŠ‚π‘‹β‰‘π‘π‘›. 𝐑𝑛 is a vector space endowed with the Euclidean norm induced by the homogeneous and translation-invariant Euclidean metricdβˆΆπ‘‹Γ—π‘‹β†’π‘0+. 𝑇 is a linear self-mapping from 𝐑𝑛 to 𝐑𝑛 represented by a nonsingular constant matrix 𝐓 in 𝐑𝑛×𝑛. Then, β€–Tβ€–is the spectral (or β„“2-) norm of the π‘˜-contractive self-mapping π‘‡βˆΆπ‘‹β†’π‘‹ which is the matrix norm induced by the corresponding vector norm (the vector Euclidean norm being identical to the β„“2vector norm as it is wellknown) fulfilling ‖𝑇‖=maxDom(𝑇)βˆ‹β€–π‘₯β€–2≀1‖𝑇π‘₯β€–2=maxDom(𝑇)βˆ‹β€–π‘₯β€–2=1‖𝑇π‘₯β€–2,=𝑑(𝑇π‘₯,0)=πœ†1/2maxξ€·π“π‘‡π“ξ€Έβ‰€π‘˜<1,𝑑𝑇𝑗π‘₯,𝑇𝑗𝑦=‖‖𝑇𝑗(π‘₯βˆ’π‘¦)β€–β€–2β‰€ξ€Ίπœ†1/2max𝑇𝑇𝐓𝑗‖π‘₯βˆ’π‘¦β€–2,=ξ€Ίπœ†1/2max𝐓𝑇𝐓𝑗𝑑(π‘₯,𝑦)βˆ€π‘₯,π‘¦βˆˆDom(𝑇)βŠ‚π‘‹,(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 π‘˜ which is less than one. Thus, π‘‡βˆΆπ΄βˆͺ𝐡→𝑋 is also πœ†βˆ—-accretive for any real constant πœ†βˆ—<π‘˜βˆ’2(1βˆ’π‘˜) and π‘˜π‘-contractive for any real π‘˜π‘βˆˆ[π‘˜,1). Assume now that 𝐓=diagβŽ›βŽœβŽœβŽπ‘˜1π‘˜2β‹―π‘˜π‘π‘›βˆ’π‘ξ„½ξ…ξ…‚ξ…ξ„Ύ0β‹―0⎞⎟⎟⎠(2.52) for some integer 0<𝑝≀𝑛 with 𝐴=Dom(𝑇)=𝑋=𝐑𝑛,𝐡=Im(𝑇)=⎧βŽͺ⎨βŽͺ⎩π‘₯βˆˆπ‘‹βˆΆπ‘₯=βŽ›βŽœβŽœβŽπ‘₯1π‘₯2β‹―π‘₯π‘π‘›βˆ’π‘ξ„½ξ…‚ξ„Ύ0β‹―0βŽžβŽŸβŽŸβŽ π‘‡βŽ«βŽͺ⎬βŽͺβŽ­βŠ‚π‘‹=𝐑𝑛,(2.53)βˆ’π‘˜β‰€π‘˜π‘–(β‰ 0)β‰€π‘˜<1; for all π‘–βˆˆπ‘. If 𝑝=𝑛, then Fix(𝑇)={0βˆˆπ‘π‘›}. Also, Fix(𝑇)={0βˆˆπ‘π‘›}for any integer 0<𝑝<𝑛(then 𝐓 is singular) but the last (π‘›βˆ’π‘)-components of any π‘₯∈𝐴=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 𝑒𝑇𝑖𝑇𝑗π‘₯β‰ 0;βˆ€π‘–βˆˆπ‘,βˆ€π‘₯(β‰ 0