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

Authors: M. De la Sen
Journal: Journal of Applied Mathematics
 (2012)

---

## 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.


---

## Body

## 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(≥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(i∈p̅) 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(i∈p̅)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(i∈p̅). Consider a metric space (X,d) associated to the Banach space (X,∥∥) and a self-mapping T:A∪B→A∪Bsuch that T(A)⊆B  and T(B)⊆A, where A and B are nonempty subsets of X. Then, T:A∪B→A∪Bis a 2-cyclic self-mapping. It is said to be a 2-cyclic k-contraction self-mapping if it satisfies, furthermore,(1.1)d(Tx,Ty)≤kd(x,y)+(1-k)dist⁡(A,B);∀x∈A,∀y∈B,for some realk∈[0,1). A best proximity point of convex subsets A or B of X is some z∈cl(A∪B)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 A∩B≠∅or 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 A∪B(in particular, neither z nor Tz  is in A∪B if both of them are open). It turns out that if A∩B≠∅ then z∈Fix(T)⊂A∪B; that is, z is a fixed point of  T since  dist(A,B)=0, [9–11]. If k=1 then d(Tx,Ty)≤d(x,y); forallx∈A, for all y∈B and T:A∪B→A∪Bis a 2-cyclic nonexpansive self-mapping, [10].


### 1.1. Notation

(1.2)R0+:=R+∪{0};Z0+:=Z+∪{0};p̅:={1,2,…,p}⊂Z+,superscriptTdenotes 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 A̅ 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 T:A∪B→A∪B which is simplified as dist(Ai,Ai+1)=dAiAi+1=di; foralli∈p̅ for distances between adjacent subsets of p-cyclic self-mappings T on ⋃i=1pAi.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 ⋃i=1pAi, the union of a collection of nonempty subsets of (X,d)which do not intersect.


## 1.1. Notation

(1.2)R0+:=R+∪{0};Z0+:=Z+∪{0};p̅:={1,2,…,p}⊂Z+,superscriptTdenotes 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 A̅ 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 T:A∪B→A∪B which is simplified as dist(Ai,Ai+1)=dAiAi+1=di; foralli∈p̅ for distances between adjacent subsets of p-cyclic self-mappings T on ⋃i=1pAi.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 ⋃i=1pAi, 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”) d:X×X→R0+. 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 T:X→X which is a 2-cyclic self-mapping restricted as T:Dom(T)⊆X∣A∪B→Im(T)⊆X∣A∪B, where A and B are nonempty subsets of X. Such a restricted self-mapping is sometimes simply denoted as T:A∪B→A∪B. 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(A∪B) and Im(T)⊃cl(A∪B) 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 T:Dom(T)⊆X→Im(T)⊆X in order to define successive iterate points through the self-mapping which do not necessarily belong to A∪B. The following definitions are then used to state the main results.Definition 2.1.
T:Dom(T)⊂X→X  is an accretive mapping if
(2.1)d(x,y)≤d(x+λTx,y+λTy);∀x,y∈Dom⁡(T),
for any λ∈R0+.
Note that, sinceX 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.
T:Dom(T)⊂X→Xis a λ*-accretive mapping, some λ*∈R0+  if
(2.2)d(x,y)≤d(x+λTx,y+λTy);∀x,y∈Dom⁡(T);∀λ∈[0,λ*],
for some λ*∈R0+.       A generalization is as follows:T:Dom(T)⊂X→Xis [λ1*,λ2*]-accretive for some λ1*,λ2*(≥λ1*)∈R0+ if
(2.3)d(x,y)≤d(x+λTx,y+λTy);∀x,y∈Dom⁡(T);∀λ∈[λ1*,λ*].Definition 2.3.
T:Dom(T)⊂X→Xis a weighted λ-accretive mapping, for some function λ:X×X→R0+, if
(2.4)d(x,y)≤d(x+λ(x,y)Tx,y+λ(x,y)Ty);∀x,y∈Dom⁡(T).
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.
T:A∪B→A∪B  is a 2-cyclic k-contractive (resp., nonexpansive) self-mapping if
(2.5)d(Tx,Ty)≤kd(x,y)+(1-k)dist⁡(A,B);∀x∈A,y∈B,
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.
T:X→Xis a α-Kannan self-mapping if
(2.6)d(Tx,Ty)≤α(d(x,Tx)+d(y,Ty));∀x,y∈X,
for some real α∈[0,1/2), [12, 13].Definition 2.6.
T:A∪B→A∪Bis an 2-cyclic (α,β)-Kannan self-mapping for some real α∈[0,1/2) if it satisfies, for some β∈R+. 							(2.7)d(Tx,Ty)≤α(d(x,Tx)+d(y,Ty))+β(1-α)dist⁡(A,B);∀x∈A,∀y∈B.
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 fromX≡A≡R to X as Tx=γx+γ0with γ,γ0∈Rendowed with the Euclidean distance d(x,y)=|x-y|; for all x,y∈X. Then,
(2.8)d(x+λTx,y+λTy)=|x-y+λγ(x-y)|=|1+λγ||x-y|=|1+λγ|d(x,y)≥d(x,y),
for all x,y∈R for any λ∈R0+ provided that γ∈R0+. In this case, T:A∪B→X is accretive. It is also k-contractive if since d(Tx,Ty)=|Tx-Ty|=γ,d(z,y)≤kd(x,y); for all x,y∈R. Also,  if   γ∈R-,  then d(x+λTx,y+λTy)≥|λ|γ|-1|d(x,y)≥d(x,y); for all x,y∈Rif λ|γ|≥2, that is, if λ≥λ1*:=2|γ|-1. Then, T:R→R  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 T:R→R defined by Tx=-t|x|psgnex=-t|x|p-1x with t∈R0+, p∈R0+,and   sgnex=sgnx  if x≠0and sgne0=0. If pt=0,then T:R→R is accretive since
(2.9)d(x+λTx,y+λTy)=d(x,y);∀x,y∈X;∀λ∈R0+.
Furthermore, if   t=0, then 0∈R   is the unique fixed point with  Tjx=0; for all j∈Z+.  If p=0  then, Tjx=tj→z=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
(2.10)T2x=T(Tx)=-t(-t|x|psgnex)psgneTx=tp+1|x|2p(sgnex)p+1,d(x+λTx,y+λTy)=d((1+λtxp-1)x,(1+λtyp-1)y)≥min⁡(|1-λt|x|p-1|,|1-λt|y|p-1|)d(x,y),≥d(x,y);∀x,y∈X,∀λ∈[0,λ*],someλ*∈R+,
holds  if  λ*|t||x|p-1≤1 that is, T:R→R is weighted  λ*(x,y)-accretive with λ*(x,y):=t-1min(|x|1-p,|y|1-p). The restricted self-mapping T:[-1,1]⊂X→[-1,1]is λ*(≡t-1)-accretive. Furthermore, if p≥1,thenT:[-1,1]⊂X→[-1,1] is |t|-contractive if |t|<1 and the iteration Tjx→0as   j→∞with  z=0  being the unique fixed point since
(2.11)d(Tx,Ty)≤|t|min⁡(|x|p-1,|y|p-1)d(x,y)≤|t|d(x,y);∀x,y∈[-1,1].
Note from the definition of the self-mapping Tx=-t|x|p-1x 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 ifX is a normed vector space. A direct result from inspection of Definitions 2.1 and 2.2 is the following.Assertions 1.
(1) IfT:D(T)⊂X→Xis an accretive mapping, then it is λ*-accretive, for all λ*∈R0+. (2) If T:D(T)⊂X→Xis λ*-accretive, then it is   λ1*-accretive; for all λ1*∈[0,λ*]. (3) Any nonexpansive self-mapping T:D(T)⊂X→Xis 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 metricd:X×X→R0+. Consider a self-mapping T:X→Xwhich restricted toT:A∪B→A∪Bis 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-mappingT:X→X satisfies the constraint:
(2.12)d(Tx,Ty)≤kd(x,y)+(1-k)dAB≤kd(x+λTx,y+λTy)+(1-k)dAB;∀x∈A,∀y∈B
with k,λ∈R0+ satisfying the constraint k(1+kλ)<1. Then, the restricted self-mapping  T:A∪B→A∪Bsatisfies
(2.13)limsup⁡j→∞d(Tjx,Tjy)<∞;∀x∈A,∀y∈B
irrespective of A and B being bounded or not.If, furthermore, A and B are closed and convex andA∩B≠∅,  then there exists a unique fixed point ω∈A∩B of T:A∪B→A∪B such that there exists  limj→∞d(Tjx,Tjy)=0; for all x∈A,for all y∈B, implying that limj→∞Tjx=limj→∞Tjy=ω.  If, in addition, dist(A,B)>0so that A∩B=∅, then there exists limj→∞d(Tjx,Tjy)=d(z,Tz); for all x∈A, for all y∈B for some best proximity points z∈A, Tz∈B which depend in general on x and y. Furthermore, if (X,∥∥) is a uniformly convex Banach space, then T2jx,T2j+1y→z1∈A and T2jy,T2j+1x→Tz1∈B as →∞; for all(x,y)∈A×B, where z1 and  z2 are unique best proximity points in A and B of T:A∪B→A∪B.
(ii)
Assume thatA and B are nondisjoint. Then, T:A∪B→Xis also kc contractive and λ*-accretive for any nonnegative λ*≤k-2(kc-k) and any kc∈[k,1). It is also nonexpansive and λ*-accretive for any nonnegative λ*≤k-2(1-k).(iii)
Ifk=0 then T:A∪B→X is weighted λ-accretive for λ:X×X→R0+ for any λ*∈R+ and its restriction T:A∪B→A∪B is 2-cyclic 0-contractive.(iv)
T:A∪B→X is weighted λ-accretive for λ:X×X→R0+ satisfying λ(x,y)≤k-2(kc(x,y)-k)(d(x,y)-dAB) for some kc:X×X→[k,∞). The restricted self-mapping T:A∪B→A∪B is also k̅c-contractive with k̅c∈[k,k̅c)⊆[k,1) if kc:X×X→[k,k̅c) with  k̅c<1. Also,T:A∪B→X  is nonexpansive and weighted  λ-accretive for λ:X×X→R0+  satisfying  λ(x,y)≤k-2(kc(x,y)-k)(d(x,y)-dAB) if  kc:X×X→[k,1] which implies, furthermore, that λ:X×X→R0+is bounded.Proof.
Let us denotedAB:=dist(A,B).Consider that the two following relations are verified simultaneously:
(2.14)d(x,y)≤d(x+λTx,y+λTy)forsomeλ∈R0+;∀x∈A,∀y∈B,d(x,y)≤kd(x,y)+(1-k)dABforsomek∈[0,1)∈R,λ∈R0+;∀x∈A,∀y∈B.
Since the distance d:X×X→R0+ 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 0∈X:
(2.15)d(Tx,Ty)≤kd(x,y)+(1-k)dAB≤kd(x+λTx,y+λTy)+(1-k)dAB≤kd(x+λTx,y+λTx+λTy-λTx)+(1-k)dAB≤kd(x+λTx,y+λTx)+kd(y+λTx,y+λTx+λTy-λTx)+(1-k)dAB=kd(x,y)+kλd(0,λTy-λTx)+(1-k)dAB≤kd(x,y)+k2λd(0,y-x)+(1-k)dAB≤kd(x,y)+k2λd(x,y)+(1-k)dAB≤k(1+kλ)d(x,y)+(1-k)dAB≤kcd(x,y)+(1-k)dAB;∀x∈A,∀y∈B;∀λ∈[0,λ*],forλ*≤k-2(1-k),
with kc:=k(1+kλ*)≥k. Note from (2.15) that
(2.16)dAB≤d(Tjx,Tjy)≤kcjd(x,y)+(1-k)dAB(∑i=0j-1kci)=kcjd(x,y)+(1-k)dAB(∑i=0∞kci-∑i=j∞kci)≤kcjd(x,y)+(1-k)(1-kcj)1-kcdAB;∀x∈A,∀y∈B,
and, if kc<1, then
(2.17)dAB≤limsup⁡j→∞d(Tjx,Tjy)≤1-k1-kcdAB=1-k1-k(1+kλ)dAB<∞;∀x∈A,∀y∈B.
If dAB=0 then limj→∞d(Tjx,Tjy)=0. It is first proven that the existence of the limit of the distance implies that of the limit limj→∞Tjz; for all z∈A∪B. Let be xj=Tjx, yj=Tjy withxj,yj∈A∪B. Then,
(2.18)lim⁡j→∞⁡d(Tjx,Tjy)=lim⁡j→∞⁡d(xj,yj)=lim⁡j→∞⁡d(Tlxj,Tlyj)=0;∀l∈Z0+⟹(xj(=Tjx)-yj(=Tjy))⟶Tl(xj-yj)⟶0asj⟶∞;∀l∈Z0+
since T:A∪B→A∪B being contractive is globally Lipschitz continuous. Then, limj→∞d(Tjx,Tjy)=d(limj→∞Tjx,limj→∞Tjy)=0since, because the fact that the metric is translation-invariant, one gets
(2.19)lim⁡j→∞⁡d(Tjx,Tjy)=d(lim⁡j→∞Tjx,lim⁡j→∞Tjy)=lim⁡j→∞d(0,Tjy-Tjx),=d(0,lim⁡j→∞(Tjy-Tjx))=0.
As a result, limj→∞d(Tjx,Tjy)=0 if dAB=0 what implies which limj→∞(Tjx-Tjy)=0; for all x∈A,for all y∈B, since T:A∪B→A∪B is globally Lipschitz continuous since it is contractive.
In addition, there existslimj→∞Tjx=limj→∞Tjy=ω∈A∪B; for all x∈A, for all y∈B. Assume not so that there exists x∈A such that ¬∃limj→∞Tjx and there exists a subsequence on nonnegative integers {jk}k∈Z0+  such that Tjk+1x≠Tjkx. If so, one gets by taking y=Tx∈Bthat d(Tjk(Tx),Tjkx)>0 which contradicts limj→∞d(Tj(Tx),Tjx)=0. Then {Tjx}j∈Z0+ is a Cauchy sequence for any x∈A∪B and then converges to a limit. Furthermore, ω∈A∪B since Tj(A∪B)⊆A∪Bfor any j∈Z0+ and as j→∞ since A and B are nonempty and closed. It has been proven that limj→∞Tjx=limj→∞Tjy=ω∈A∪B; for all x∈A,for all y∈B.
It is now proven thatω=Tω∈Fix(T). Assume not, then, from triangle inequality,
(2.20)0<d(Tω,ω)≤d(ω,Tjω)+d(Tω,Tjω);∀j∈Z0+⟹lim⁡⁡inf⁡j→∞d(ω,Tjω)>0,
which contradicts limj→∞Tjω=ω so that ω=Tω∈Fix(T).  It is now proven that ω∈Fix(T)∩(A∩B). Assume not, such that, for instance, Tjx∈A  and Tj+1x∈A̅∩B. If so, since T(A)⊆B;T(B)⊆A, then the existing limit fulfils limj→∞Tjx∈A∩A̅(=∅) which is impossible so that there would be no existing limitlimj→∞Tjx in A∪B, contradicting the former result of its existence. Then, ω∈Fix(T)∩(A∩B) implying that Fix(T)⊂A∩B.
It is now proven by contradiction thatω=limj→∞Tjx; for all x∈A∪B is the unique fixed point of T:A∪B→A∪B. Assume that ∃ω1(≠ω)∈Fix(T), then limj→∞Tjy1=ω1 for some y1(≠y)∈B with no loss in generality and all x∈A. Thus, limj→∞d(Tjx,Tjy1)=d(ω,ω1)=0⇒ω=ω1 which contradicts ω≠ω1 so that Fix(T)={ω}.
Now, assume thatA 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 x∈A, y∈B, one gets(2.21)d(Tjx,Tjy)≤kjd(x,y)+(1-k)dAB(∑i=0∞kj)=kjd(x,y)+dAB;∀j∈Z,limsup⁡j→∞d(Tjx,Tjy)≤dAB.
Note that sinceT(A)⊆B, T(B)⊆A and dist(A,B)=dAB>0, thenx∈A⇒Tjx∈A and Tjx∉B if j is even and Tjx∈B and Tjx∉A if j is oddy∈B⇒Tjy∈Band Tjy∉A if j is even and Tjy∈A and Tjy∉B 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 j∈Z0+. As a result, limj→∞supd(Tjx,Tjy)<dAB is impossible so that(2.22)∃lim⁡j→∞d(Tjx,Tjy)=limsup⁡j→∞d(Tjx,Tjy)=dAB=d(z,Tz),
for some best proximity points z∈A and Tz∈B or conversely. Then,
(2.23)lim⁡j→∞d(Tj+1x,Tj+1y)=lim⁡j→∞d(Tzj,T2zj)=dAB≤klim⁡j→∞d(Tjx,Tjy)+(1-k)d(z,Tz)=klim⁡⁡j→∞d(zj,Tzj)+(1-k)d(z,Tz)=klim⁡⁡j→∞d(zj,Tzj)+(1-k)dAB,
where zj=Tjx Thus, limj→∞d(zj,Tzj)=dAB=d(z,Tz). It turns out that dist(zj,Fr(A∪B))→0 and dist(Tzj,Fr(A∪B))→0 as j→∞. Otherwise, it would exist an infinite subsequence  {d(zj,Tzj)}j∈Ẑ0+of {d(zj,Tzj)}j∈Z0+ with Ẑ0+being an infinite subset of  Z0+such that d(zj,Tzj)>dAB for j∈Ẑ0+. 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 limj→∞d(Tj+1x,Tj+1y)=dAB, it follows that there exist j1∈Z0+ and δ=δ(ε,j1)∈R+ such that
(2.24)2dAB+δ<d(Tjx+Tj+1y,0)≤d(Tjx,0)+d(Tj+1y,0),≤2(dAB+δ)⟹d(Tjx,Tj+1y)<ε,
for any given ε∈R+; for all x∈A,for all y∈B with Tjx∈A,Tj+1y∈A for any even j(≥j1)∈Z0+ and Tjx∈B,Tj+1y∈B, for any odd j(≥j1)∈Z0+.  As a result, by choosing the positive real constant arbitrarily small, one gets that T2jx→T2j+1y→z=z(x,y)∈A (a best proximity point of A) and T2j+1x→T2jy→Tz∈B (a best proximity point of B), or vice-versa, as j→∞ for any given x∈A and y∈B. A best proximity point z∈A∪B 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 z1≠z2 so that necessarily z1=T2z1≠z2=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:
(2.25)dAB=d(z1,Tz1)=d(z1,Tz2)=‖Tz2-z12+Tz2-z12‖<2‖Tz2-z12‖=dAB,
since (X,∥∥) is also a strictly convex Banach space and A and B are nonempty closed and convex sets. Then, z=T2z∈A is the unique best proximity point of  T:A∪B→A∪B 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 T:A∪B→A∪Bis kc-contractive and λ*-accretive if  λ*=k-2(kc-k) with kc∈[k,1). By taking kc=1, note that T:A∪B→X is nonexpansive and k-2(1-k)-accretive. Property (ii) has been proven.
To prove Property (iii), we now discuss if(2.26)d(Tx,Ty)≤k(1+kλ)d(x,y)+(1-k)dAB≤kcd(x,y)+(1-kc)dAB;∀x∈A,∀y∈B.
is possible with 1≥kc and dAB>0. Note that  dAB=dist(A,B)=d(z,Tz) for some  z∈A.  Define  d̅:=max(dA,dB)=kDdAB,  if  dAB≠0 for some kDA,kDB,kD∈R+, where dA:=diamA=kDAdAB and dB:=diamB=kDBdAB. Three cases can occur in (2.26), namely,(a)
Ifk=kc then k2λd(x,y)≤0⇔[kλ=0∨d(x,y)≤0] which is untrue if x≠y and kλ>0 and it holds for either k=0 or  λ=0,(b)
kc>k, then (2.26) is equivalent to
(2.27)d(x,y)≥kc-kkc-k(1+kλ)dAB;∀x∈A,∀y∈B.
Take x∈A to be a best proximity point with so that d(x,Tx)=dAB≥(kc-k)/(kc-k(1+kλ))dAB>dAB which is untrue if  kλ>0 and true for kλ=0,(c)
1≥k(1+kλ)≥kc<k, then (2.16) is equivalent to (k-kc)dAB≥[k(1+kλ)-kc]d(x,y); for all x∈A,for all y∈B, but d(x,y)≤2d̅+dAB=(2kD+1)dAB. Thus, the above constraint is guaranteed to hold in the worst case if k-kc≥(k+k2λ-kc)(2kD+1)>k-kc 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 λ:X×X→R0+ and kc:X×X→[k,1). Note that (2.26) holds through direct calculation if  λ(x,y)≤k-2(kc(x,y)-k)(d(x,y)-dAB); for all x∈A,for all y∈B for some kc:X×X→[k,∞). Thus, the self-mapping T:A∪B→X is weighted λ-accretive for λ:X×X→R0+ satisfying λ(x,y)≤k-2(kc(x,y)-k)(d(x,y)-dAB) for some kc:X×X→[k,∞); and it is also k̅c-contractive with k̅c∈[k,k̅c)⊆[k,1) if kc:X×X→[k,k̅c) with k̅c<1 and nonexpansive if kc:X×X→[k,1]. On the other hand, note that d(x,y)-dAB≤kDAdA+kDBdB≤2kDd̅. If A and B are bounded and kc:X×X→[k,1], then
(2.28)λ(x,y)≤k-2(kc(x,y)-k)(d(x,y)-dAB)≤k-2(kc(x,y)-k)(kDAdA+kDBdB)≤2k-2(kc(x,y)-k)kDd̅≤∞;∀x∈A,∀y∈B.
Property (iv) has been proven.Remark 2.10.
Note that Theorem2.9 (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 z∈A∪B, 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 T:A∪B→A∪B in A and B are obtained provided that both sets are nonempty, convex, and closed.Remark 2.12.
Note that if eitherA or B is not closed, then its best proximity point of  T:A∪B→A∪B is in its closure since T(A)⊆B⊆clB, T(B)⊆A⊆cl A leads to T(A∪B)⊆A∪B⊆cl(A∪B) and Tk(A∪B)⊆cl(A∪B) for finitely many and for infinitely many iterations through the self-mapping T:A∪B→A∪B 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 d:X×X→R0+. Consider the λ*-accretive mapping T:A∪B→Xfor some λ*∈R0+ which restricted as T:A∪B→A∪B is 2-cyclic, where A and B are nonempty subsets of X subject to 0∈A∪B. Then,
(2.29)d((I+λT)jx,0)≥1;∀j∈Z0+,∀x(≠0)∈A∪B,∀λ∈[0,λ*].
If, furthermore, T:A∪B→A∪Bis k-contractive, then
(2.30)1≤d((I+λT)jx,0)<k-1;∀j∈Z+∀x(≠0)∈A∪B,∀λ∈[0,λ*].T:A∪B→A∪Bis guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if
(2.31)d((I+λT)jx,0)=1;∀j∈Z+,∀x(≠0)∈A∪B,∀λ∈[0,λ*],
respectively,
(2.32)lim⁡⁡sup⁡j→∞d((I+λT)jx,0)=1;∀x(≠0)∈A∪B,∀λ∈[0,λ*].(ii)
Let(X,∥∥) be a normed vector space. Consider a λ*-accretive mapping  T:A∪B→X for some λ*∈R0+ which restricted to T:A∪B→A∪B is 2-cyclic, where A and B are nonempty subsets of X subject to 0∈A∪B then
(2.33)‖(I+λT)j‖≥1;∀j∈Z0+,∀λ∈[0,λ*].
If, furthermore, T:A∪B→A∪B is k-contractive, then
(2.34)1≤‖(I+λT)j‖<k-1;∀j∈Z+,∀λ∈[0,λ*].T:A∪B→A∪Bis nonexpansive (resp., asymptotically nonexpansive, [30]) if
(2.35)‖(I+λT)j‖=1;∀j∈Z+,∀λ∈[0,λ*],
respectively,
(2.36)limsup⁡j→∞‖(I+λT)j‖=1;∀λ∈[0,λ*].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  T:A∪B→A∪B, that is, the norm of T on X restricted to A∪B as follows:
(2.37)‖T‖:=min⁡{c∈R0+:‖Tx‖≤c‖x‖;∀x∈A∪B}≡min⁡{c∈R0+:d(Tx,0)≤cd(x,0);∀x∈A∪B},
with the above set being closed, nonempty, and bounded from below. Since T:A∪B→A∪B is 2-cyclic and T:A∪B→X is λ*-accretive (Definition 2.2), one gets by proceeding recursively
(2.38)d(x,y)≤d(x+λTx,y+λTy)≤d(x+λTx+λT(x+λTx),y+λTy)=d((I+λT)2x,(I+λT)2y)≤⋯≤d((I+λT)jx,(I+λT)jy)≤‖(I+λT)j‖d(x,y);∀x∈A,∀y∈B,∀j∈Z0+,∀λ∈[0,λ*],
since the metric is homogeneous and 0∈A∪B, and I is the identity operator on X, where
(2.39)‖(I+λT)j‖:=min⁡{c∈R0+:‖(I+λT)jx‖≤c‖x‖;∀x∈A∪B},≡min⁡{c∈R0+:d((I+λT)jx,0)≤cd(x,0);∀x∈A∪B},
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 x∈A, for all y∈B in (2.38). Thus, ∥(I+λT)j∥=d((I+λT)jx,0)≥1; for all j∈Z0+, forallx(≠0)∈A∪B, for all λ∈[0,λ*]. If now x and y are replaced with Tix and Tiy for any i∈Z0+ in (2.30), one gets if  T:A∪B→A∪B is a 2-cyclic k-contractive for some real k∈[0,1) and λ*-accretive mapping:
(2.40)d(Tix,Tiy)≤d((I+λT)jTix,(I+λT)jTiy)≤‖(I+λT)j‖d(Tix,Tiy)≤ki‖(I+λT)j‖d(x,y)+(1-k)dAB<d(x,y)+(1-k)dAB,
for all x∈A, for all y(≠x)∈B, for all j(≥i)∈Z+, for all i∈Z+, for all λ∈[0,λ*]. Then, 1≤∥(I+λT)j∥=d((I+λT)jx,0)<k-1; for all j∈Z+, for all x(≠0)∈A∪B, for all λ∈[0,λ*]. If ∥(I+λT)j∥=d((I+λT)jx,0)=1; for all j∈Z+, for all x(≠0)∈A∪B, for all λ∈[0,λ*], it turns out that T:A∪B→X is λ*-accretive and T:A∪B→A∪Bis a 2-cyclic nonexpansive self-mapping. It is asymptotically nonexpansive if limj→∞supd((I+λT)jx,0)=1; for all x(≠0)∈A∪B, 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 thatX=R, A=R-,B=R+ and the 2-cyclic self-mapping T(≡)t:(A∪B)×Z0+→(A∪B)×Z0+ defined by the iteration rule  A∪B∋xj+1=kjxj∈A∪B with R∋kj(∈[-k,k])≤k≤1, sgnkj+1=-sgnkj=sgnxj; for all j∈Z0+, and x0∈A∪B. Let d:R0+→R0+ be the Euclidean metric.(a)
Ifk<1, then limj→∞∏i=0j[kj]=0 so that for any x0∈A∪B, xj∈A∪B; for all j∈Z0+xj→z=0∉A∪B as j→∞ with 0∈cl(A∪B), Fix(t)={0}⊂cl(A∩B) but it is not in A∩B which is empty. If k=1 and limj→∞∏i=0j[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}⊂A∩B≠∅.(b)
Ifkj=k=1; for all j∈Z0+ the self-mapping  t:(A∪B)×Z0+→(A∪B)×Z0+is not expansive and there is no fixed point.(c)
Ifk=1-σ for some σ(<1)∈R+, then forR0∋λ∈[0,λ*],
(2.41)d(tx,ty)≤Kd(x,y)≤k(1+λ)|x-y|≤k|x-y|;d(x,y)≤d(x+λx,y+λy)≤(1+λ)|x-y|,
so that t:(A∪B)×Z0+→(A∪B)×Z0+is also λ*-accretive and k1∈[k,1)-contractive with λ*=k1k-1-1.(d)
Now, define closed setsRε+:={r(≥ε)∈R+} and Rε-:={r(≤-ε)∈R+} for any given ε∈R0+so that dAB=ε. The 2-cyclic self-mapping T(≡)t:(A∪B)×Z0+→(A∪B)×Z0+is re-defined by the iterationxj+1=x̅j+1if |xj+1|≥εand xj+1=-εsgnxj, for i=1,2, otherwise, wherex̅j+1=kjxjfor i=1,2 with the real sequence {kj}j∈Z0+being subject to kj(∈[-k,k])≤k≤1, sgnkj+1=-sgnkj=sgnxj(i); i=1,2, for all j∈Z0+and x0∈A∪B.Then, for anyε∈R+and any  x0∈A∪B, there are two best proximity points z=-ε∈A and z1=ε∈B fulfilling -ε=tε=-t2ε and  dAB=d(z,z1)=d(z,tz)=d(tz1,z1).(e)
RedefineX=R2so that R2∋x=(x(1),x(2))T with x(1),x(2)∈R; A=Rε-2,B=Rε+2. In the case that ε=0,then A and B are open disjoint subsets (resp., A=R0-2,B=R0+2 are closed nondisjoint subsets with A∩B={(0,x)T:x∈R}).The 2-cyclic self-mappingT(≡)t:(A∪B)×Z0+→(A∪B)×Z0+is re-defined by the iteration rule:
(2.42)xj+1(i)=x̅j+1(i),if|xj+1(i)|≥ε,xj+1(i)=-εsgn⁡xj(i),fori=1,2,
otherwise, where
(2.43)x̅j+1(i)=kjxj(i),fori=1,2
with the real sequence {kj}j∈Z0+ being subject tokj(∈[-k,k])≤k≤1, sgnkj+1=-sgnkj=sgnxj(i); for i=1,2; for all j∈Z0+ and x0∈A∪B.
The same parallel conclusions to the above ones (a)–(c) follow related to the existence of the unique fixed pointz=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 Theorem2.13.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 d:X×X→R0+ and consider the self-mapping T:X→X  so that the restricted T:A∪B→Xis λ*-accretive for some λ*∈R0+, where A and B are nonempty subsets of X subject to 0∈A∪B, and the restricted T:A∪B→A∪Bis 2-cyclic.Then,
(2.44)d((I+λT)x,0)≥1;∀x(≠0)∈A∪B,∀λ∈[0,λ*].
If, furthermore, T:A∪B→A∪Bis k-contractive, then
(2.45)1≤d((I+λT)x,0)<k-1;∀x(≠0)∈A∪B,∀λ∈[0,λ*].T:A∪B→A∪Bis guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if(ii)
Let(X,∥∥) be a normed vector space. Then if T:A∪B→Xis a λ*-accretive mapping and T:A∪B→A∪Bis 2-cyclic for some λ*∈R0+ where A and B are nonempty subsets of X subject to 0∈A∪B, then
(2.46)‖I+λT‖≥1;∀λ∈[0,λ*].
If, furthermore, T:A∪B→A∪Bis 2-cyclic k (∈[0,1))-contractive, then
(2.47)1≤‖I+λT‖<k-1;∀λ∈[0,λ*].Outline of Proof
It follows since the basic constraint ofT:A∪B→Xbeing λ*-accretive holds if
(2.48)‖I+λT‖≥1⟹‖I+λT‖j≥‖(I+λT)j‖≥1;∀j∈Z+,∀λ∈[0,λ*],
while it fails if
(2.49)‖I+λT‖<1⟹‖(I+λT)j‖≤‖I+λT‖j<1;∀j∈Z+,∀λ∈[0,λ*].Remark 2.16.
Theorem2.13 and Corollary 2.15 are easily linked to Theorem 2.9 as follows. Assume that T:A∪B→A∪B is 2-cyclic k-contractive and T:A∪B→X is aλ*-accretive mapping. Assume that there exists x∈A∪B such that ∥x∥=d(x,0)≤1. Then, 1≤∥I+λT∥<k-1; for all λ∈[0,λ*] from (2.47). This is guaranteed under sufficiency-type conditions with
(2.50)‖T‖=max⁡‖x‖≤1d(Tx,0)=max⁡d(x,0)≤1d(Tx,0)≤kif1≤‖I+λT‖≤1+λ‖T‖≤1+λk<k-1;∀λ∈[0,λ*],⟺(1+λk)k≤kc<1;∀λ∈[0,λ*],
with λ*=k-2(kc-k) for some real constantskc∈[k,1), k∈[0,1). It is direct to see that  Fix(T)={0∈Rn} if 0∈A∩B.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)=B≡Im(T)⊂X≡Rn. Rn is a vector space endowed with the Euclidean norm induced by the homogeneous and translation-invariant Euclidean metricd:X×X→R0+. T is a linear self-mapping from Rn to Rn  represented by a nonsingular constant matrix T in  Rn×n. Then, ∥T∥is the spectral (or ℓ2-) norm of the k-contractive self-mapping T:X→X which is the matrix norm induced by the corresponding vector norm (the vector Euclidean norm being identical to the ℓ2vector norm as it is wellknown) fulfilling
(2.51)‖T‖=max⁡⁡Dom⁡(T)∋‖x‖2≤1‖Tx‖2=max⁡⁡Dom⁡(T)∋‖x‖2=1‖Tx‖2,=d(Tx,0)=λmax⁡⁡1/2(TTT)≤k<1,d(Tjx,Tjy)=‖Tj(x-y)‖2≤[λmax⁡⁡1/2(TTT)]j‖x-y‖2,=[λmax⁡⁡1/2(TTT)]jd(x,y)∀x,y∈Dom⁡(T)⊂X,
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, T:A∪B→X is also λ*-accretive for any real constant λ*<k-2(1-k) and kc-contractive for any real kc∈[k,1). Assume now that
(2.52)T=diag⁡(k1k2⋯kp0⋯0︷n-p)
for some integer 0<p≤n with
(2.53)A=Dom⁡(T)=X=Rn,B=Im⁡(T)={x∈X:x=(x1x2⋯xp0⋯0︷n-p)T}⊂X=Rn,-k≤ki(≠0)≤k<1; for all i∈p̅. If p=n, then Fix(T)={0∈Rn}. Also, Fix(T)={0∈Rn}for any integer 0<p<n(then T is singular) but the last (n-p)-components of any x∈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
(2.54)eiTTjx≠0;∀i∈p̅,∀x(≠0)∈Rn;∀j∈Z0+,eiTTjx=0;∀i(>p)∈n̅,∀x∈Rn;∀j∈Z0+,Tjx⟶0;∀x∈Rnasj⟶∞.
Now, assume that the matrix T is of rank one with its first column being of the form
(2.55)t1=(k1k2⋯kp0⋯0︷n-p)T
with 0<p<n, -k≤ki(≠0)≤k<1; for all i∈p̅. Then, (2.54) still holds by changing x≠0 in the first equation to x1≠0. Finally, assume that
(2.56)T=diag⁡(k1k2⋯kp1⋯1︷n-p)
with 0<p<n. Then, the self-mapping  T:X→X is nonexpansive also noncontractive and Fix(T)={0∈Rp}⊕Rn-p 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 ∃limj→∞Tjx=(0T,yT)∈Fix(T) for any given x=(zT,yT)T∈Rn with x∈Rp, y∈Rn-p.The following result is concerned with the distance boundedness between iterates through the self-mappingT:A∪B→A∪B.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 d:X×X→R0+. Let T:X∣A∪B→X∣A∪Bbe a 2-cyclic k-contractive self-mapping so that T:A∪B→X is λ*-accretive for some λ*∈R0+ where A and B are nonempty subsets of  X. Then,
(2.57)d(Tjx,Tj+1x)≤m1d(x,0)+m2;∀x∈A∪B;∀j∈Z+,
for some finite real constants m1∈R+, and m2∈R0+, which are independent of x and the jth power, and m2is zero if A and  B intersect. Furthermore, limj→∞supd(Tjx,Tj+1x) is finite irrespective of  x∈A∪B.Proof.
One gets forλ∈[0,λ*], some λ*∈R0+ and x∈A∪B that
(2.58)d(x,Tx)≤d(x+λTx,Tx+λT2x)=d(λTx,Tx+λT2x-x)=d(Tx+(λ-1)Tx,T2x+(λ-1)T2x+Tx-x)=d(Tx,T2x+(1-λ)Tx+(λ-1)T2x+Tx-x)≤d(Tx,T2x)+d(T2x,T2x+(1-λ)Tx+(λ-1)T2x+Tx-x)=d(Tx,T2x)+d((λ-1)Tx,(λ-1)T2x+Tx-x)=d(Tx,T2x)+d(λTx,(λ-1)T2x-x)≤d(Tx,T2x)+d(λTx,λT2x)+d(λT2x,(λ-1)T2x-x)=d(Tx,T2x)+d(λTx,λT2x)+d(T2x,-x)=d(Tx,T2x)+d(λTx,λT2x)+d(T2x,0)+d(x,0)≤kd(x,Tx)+(1-k)dAB+kλd(x,Tx)+λ(1-k)dAB+kd(Tx,0)+(1-k)dAB+d(x,0)≤kd(x,Tx)+(1-k)dAB+kλd(x,Tx)+λ(1-k)dAB+k2d(x,0)+k(1-k)dAB+(1-k)dAB+d(x,0)≤k(1+λ)d(x,Tx)+(1-k)(2+λ+k)dAB+(k2+1)d(x,0);∀λ∈[0,λ*],
so that one has for λ*:=1-k-1ε with ε∈[ε0,1) for some real constant ε0∈[0,1) provided that k∈(0,1):
(2.59)d(x,Tx)≤(2+λ+k)(1-k)1-k(1-λ)dAB+k2+11-k(1-λ)d(x,0)≤1ε0((2+λ+k)(1-k)dAB+(k2+1)d(x,0)),
and if k=0 then
(2.60)d(x,Tx)≤(2+λ)dAB+d(x,0);∀λ∈R0+.
Also,
(2.61)d(Tjx,Tj+1x)≤(2+λ+k)(1-k)1-k(1-λ)dAB+k2+11-k(1-λ)d(Tjx,0)≤[2+λ+k+(k2+1)(∑i=0j-1ki)]1-k1-k(1-λ)dAB+(k2+1)kj1-k(1-λ)d(x,0)≤[2+λ+k+(k2+1)(∑i=0j-1ki)]1-kε0dAB+(k2+1)kjε0d(x,0)=[2+λ+k+(k2+1)(1-kj)1-k]1-kε0dAB+(k2+1)kjε0d(x,0)≤1ε0[(3+λ+21-k)dAB+2d(x,0)];∀λ∈[0,λ*],∀j∈Z+(2.62)lim⁡⁡sup⁡j→∞d(Tjx,Tj+1x)≤(3+λ+k+k2)(1-k)ε0dAB≤5+λε0dAB;∀λ∈[0,λ*]
if k∈(0,1), and
(2.63)d(Tjx,Tj+1x)≤3+λε0dAB;∀λ∈R0+,∀j∈Z+(2.64)lim⁡⁡sup⁡j→∞d(Tjx,Tj+1x)≤3+λε0dAB;∀λ∈[0,λ*]
if k=0.The subsequent result has a close technique for proof to that of Theorem2.18.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 d:X×X→R0+ and let T:X→X be a self-mapping on X which is k-contractive with k∈[0,1/3) and 2-cyclic on A∪B, where Aand B are nonnecessarily disjoint nonempty subsets of X. If such sets A and B intersect then T:X→X is also kc-contractive with kc:=k/(1-2k)=k/(1-(2+λ*)k)∈[0,1) and λ*-accretive with  λ*=∞ if  kc=k=0 and with λ*=k-1-kc-1-2 if  k∈(0,1/3). Irrespective of A and B being disjoint or not, T:A∪B→X is still λ*-accretive and the following inequalities hold:
(2.65)d(Tx,Ty)≤kcd(x,y)+mdAB;∀(x,y)∈A×B,∀λ∈[0,λ*],(2.66)d(Tjx,Tjy)≤kcjd(x,y)+mdAB(∑i=0j-1kci)=kcjd(x,y)+1-kcj1-kcmdAB;∀j∈Z+,∀(x,y)∈A×B,∀λ∈[0,λ*],(2.67)limsup⁡j→∞d(Tjx,Tjy)≤mdAB1-kc<∞;∀j∈Z+,∀(x,y)∈A×B,∀λ∈[0,λ*].Proof.
Direct calculations yield(2.68)d(Tx,Ty)≤d(Tx+λT2x,Ty+λT2y)=d(T2x+(λ-1)T2x,T2y+(λ-1)T2y+Ty-Tx)=d(T2x,T2y+(λ-1)T2y+(1-λ)T2x+Ty-Tx)≤d(T2x,T2y)+d(T2y,T2y+(λ-1)T2y+(1-λ)T2x+Ty-Tx)=d(T2x,T2y)+d(λT2x,λT2y-T2y+T2x+Ty-Tx)≤d(T2x,T2y)+λd(T2x,T2y)+d(λT2y,λT2y-T2y+T2x+Ty-Tx)=d(T2x,T2y)+λd(T2x,T2y)+d(T2y-Ty,T2x-Tx)≤d(T2x,T2y)+λd(T2x,T2y)+d(T2y,T2x)+d(Tx,Ty)≤(2+λ)(kd(Tx,Ty)+(1-k)dAB)+kd(x,y)+(1-k)dAB;∀(x,y)∈A×B,∀λ∈[0,λ*],
which leads to the inequalities (2.65)–(2.67) with kc:=k/(1-(2+λ*)k)∈[0,1) and m:=(3+λ)(1-k)/kkc where kc∈[0,1) with λ*:=∞ if k=kc=0 and λ*:=k-1-kc-1-2≥0 if  kc:=k/(1-2k)∈(0,1) which holds if and only if  k∈(0,1/3). The proof is complete.Remark 2.20.
Compared to Theorem2.9, 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 (1-kc)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 Theorem2.9 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 0∈A∪B. Consider a 2-cyclic (k<1/3)-contractive self-mapping T:A∪B→A∪Bwith k∈[0,1/3). Then, T:A∪B→A∪Bis also a (kc/(1-kc),β)-Kannan self-mapping and T:A∪B→X is (1/3-k)k-2-accretive forkc(∈R+)=k(1+kλ*), for all β(∈R+)≥β0:=(2λ*+1-1+4λ*kc)/2λ*(1-α).Proof.
SinceT:A∪B→A∪B 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):
(2.69)d(Tx,Ty)≤kd(x,y)+(1-k)dAB≤kd(x+λTx,y+λTy)+(1-k)dAB≤kcd(x,y)+(1-k)dAB≤kc(d(x,Tx)+d(y,Ty))+kcd(Tx,Ty)+(1-k)dAB;∀x∈A,∀y∈B⟹d(Tx,Ty)≤α(d(x,Tx)+d(y,Ty))+(1-k)dAB≤α(d(x,Tx)+d(y,Ty))+β(1-α)dAB;∀x∈A,∀y∈B,
provided that
(2.70)α:=kc1-kc<12(⟹kc<13),kc:=k(1+kλ*),β≥β0:=2λ*+1-1+4λ*kc2λ*(1-α),
since 1/3>kc:=k(1+kλ*)≥k if λ*:=(1/3-k)k-2  so that   T:A∪B→X is (1/3-k)k-2-accretive. Note that the function k=k(kc) for a contractive self-mapping is the positive solution of   λ*k2+k-kc=0, that is,  k=(1+2λ*-1+4λ*kc)/2λ*, which is wellposed since 0≤k<1 for 0≤kc<1. Thus, T:A∪B→A∪Bis also a 2-cyclic (kc/(1-kc),β)-Kannan self-mapping from Definition 2.6 since 0≤kc<1/3implies α:=kc/(1-kc)<1/2 with
(2.71)β≥β0:=2λ*+1-1+4λ*kc2λ*(1-α),λ*≤K-2(1-K).


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

This section generalizes the main results of Section2 to p-cyclic self-mappings with p≥2. Now, it is assumed that there are p nonempty subsets Aiof X; for all i∈p̅ which can be disjoint or not and a so-called p-cyclic self-mapping T:⋃i∈p̅Ai→⋃i∈p̅Aisuch that T(Ai)⊆T(Ai+1) with Ap+1≡A1. 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 i∈p̅ 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.
T:⋃i∈p̅Ai→⋃i∈p̅Aiis a p-cyclic weakly k-contractive (resp., weakly nonexpansive) self-mapping if
(3.1)d(Tx,Ty)≤kid(x,y)+(1-ki)di;∀x∈Ai,∀y∈Ai+1;∀i∈p̅,
for some real constants  ki∈R0+(resp.,  ki∈R+); for all i∈p̅ [12, 13]  such that  k:=∏i∈p̅[ki]<1 (resp., k=1).Definition 3.2.
T:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic k-contractive (resp., nonexpansive) p-cyclic self-mapping if
(3.2)d(Tx,Ty)≤kid(x,y)+(1-ki)di;∀x∈Ai,∀y∈Ai+1;∀i∈p̅,
for some real constantski∈[0,1) (resp., ki=1); for all i∈p̅ [12, 13].Assertion 1.
Ap-cyclic weakly nonexpansive self-mapping T:⋃i∈p̅Ai→⋃i∈p̅Ai may be locally expansive for some (x,y)∈Ai×Ai+1; for all i∈p̅ which cannot be best proximity points.Proof.
Assume thatki>1. Then, the following inequalities can occur for givenx∈Ai, y∈Ai+1:(1)
(3.3)d(Tx,Ty)≤kid(x,y)+(1-ki)di≤d(x,y)⟹d(x,y)≤di⟹d(Tx,Ty)≤di,
In this case, and since d(x,y)<di is impossible, one concludes that
(3.4)d(Tx,Ty)≤kid(x,y)+(1-ki)di≤d(x,y)⟹d(x,y)=di⟹d(Tx,Ty)≤di,
so that (3.3) can only hold for best proximity points x∈Ai,y=Tx∈Ai+1 for which T:⋃i∈p̅Ai→⋃i∈p̅Ai is nonexpansive. If di=di+1, then the last inequality of (3.4) becomes d(Tx,Ty)=di=di+1 so that Ty∈Ai+2 is also a best proximity point if Aiare convex, for all i∈p̅,(2)
(3.5)d(Tx,Ty)≤d(x,y)≤kid(x,y)+(1-ki)di⟹d(x,y)≥di,
and then T:⋃i∈p̅Ai→⋃i∈p̅Aiis nonexpansive for (x,y)∈Ai×Ai+1;(3)
(3.6)d(x,y)<d(Tx,Ty)≤kid(x,y)+(1-ki)di⟹d(x,y)>di⟹d(Tx,Ty)>d(x,y)>di,
and then T:⋃i∈p̅Ai→⋃i∈p̅Ai,is expansive for (x,y)∈Ai×Ai+1 which cannot be best proximity points since d(x,y)>di.Remark 3.3.
Note from Definitions3.1 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  T:⋃i∈p̅Ai→⋃i∈p̅Ai, is p-cyclic weakly nonexpansive and di=d1; for all i∈p̅, then
(3.7)d(Tjx,Tjy)≤ki+j-1d(Tj-1x,Tj-1y)+(1-ki+j-1)d1,≤(∏i=1j[ki])d(x,y)+(1-∏i=1j[ki])d1;∀x∈Ai,∀y∈Ai+1,
where  kℓp+i=ki; for all ℓ∈Z0+, for all i∈p-1¯∪{0}. Note that if  d(Tj-1x,Tj-1y)>d1; that is, Tj-1x,Tj-1y  are not best proximity points, then if  ki+j-1>1,then d(Tjx,Tjy)>d(Tj-1x,Tj-1y) sinceki+j-1d(Tj-1x,Tj-1y)+(1-ki+j-1)d1>d(Tj-1x,Tj-1y). Thus, a weakly nonexpansive self-mapping is not necessarily nonexpansive for each iteration. However, the composed self-mapping T:⋃i∈p̅Ai→⋃i∈p̅Ai  defined as Tx=T(Tp-1x)=Tpx; for all x∈⋃i∈p̅Ai is nonexpansive in the usual sense since if j=p, then k=∏i∈p̅[ki]=1, implies
(3.8)d(Tjx,Tjy)≤d(Tj-1x,Tj-1y)≤d(x,y);∀x∈Ai,∀y∈Ai+1.It has been commented in Remark2.12 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 p≥2. 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 {Ai⊂X:i∈p̅}.Lemma 3.4.
Assume thatT:⋃i∈p̅Ai→⋃i∈p̅Ai is p-cyclic and nonexpansive. Then, di=d1; for all i∈p̅.Proof.
If⋂i∈p̅clAi≠∅(i.e., the closures of the subsets intersect), then the proof is direct since di=0; for all i∈p̅. Now, assume that 0≤dj<d1≠0 for some j∈p̅. Let z∈A1and  z1=Tz∈A2 best proximity points such that
(3.9)d(Tj+1z,Tj+1z1)=dj≤d(Ti+1z,Ti+1z1)=di≤d(Tz,Tz1)≤d(z,z1)=d1;∀j,i(<j)∈p̅,
since T:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic nonexpansive self-mapping. Thus, any iterates Tjz and Tjz1 are also best proximity points of some subset in{Ai⊂X:i∈p̅}; for all j∈Z+. If dj=d1; for all j∈p,̅ does not hold, then from (3.9):
(3.10)d(Tj+1z,Tj+1z1)<d(Tz,Tz1);∀j∈p̅⟹∃lim⁡j→∞d(Tz,Tjz1)=0.
Then di=0;for all i∈p̅which contradicts 0≤dj<d1≠0 for some j∈p̅ what is a contradiction or di=0; for all i∈p̅, and ⋂i∈p̅Ai≠∅.Note that Lemma3.4 applies even if the subsets are neither bounded or closed. In this way, note that the contradiction to 0≤dj<d1≠0 for some j∈p̅ established in the second part of the proof does not necessarily imply that ⋂i∈p̅clAi≠∅ which would require for the subsets Ai, for all i∈p̅,to be bounded and, in particular, ⋂i∈p̅Ai≠∅ 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)
IfT:⋃i∈p̅Ai→⋃i∈p̅Ai  is a p-cyclic weakly nonexpansive self-mapping, then
(3.11)limsup⁡j→∞d(Tjp+lx,Tjp+ly)≤(∏μ=ii+l[kμ])limsup⁡j→∞d(Tjpx,Tjpy)+∑μ=ii+l(∏σ=μ+1i+l[kσ])(1-kμ)dμ≤lim⁡j→∞[(∏μ=ii+l[kμ])kj]d(x,y)+∑μ=ii+l(∏σ=μ+1i+l[kσ])(1-kμ)dμ=∑μ=ii+l(∏σ=μ+1i+l[kσ])(1-kμ)dμ;∀x∈Ai,∀y∈Ai+1
ifi,ℓ∈p̅satisfy
(3.12)∑μ=ii+l(∏σ=μ+1i+l[kσ])(1-kμ)dμ≥dilimsup⁡j→∞d(Tjpx,Tjpy)≤d(Tlpx,Tlpy)≤d(x,y);∀x∈Ai,∀y∈Ai+1,∀i∈p̅,∀l∈Z+.(ii)
IfT:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic nonexpansive self-mapping, then(3.13)limsup⁡j→∞d(Tjp+lx,Tjp+ly)≤d(Tlpx,Tlpy)≤d(x,y);∀x∈Ai,∀y∈Ai+1,∀i∈p̅,∀l∈Z+(iii)
IfT:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic weakly contractive self-mapping, then
(3.14)limsup⁡j→∞d(Tjp+lx,Tjp+ly)≤∑μ=ii+l(∏σ=μ+1i+l[kσ](1-kμ))dμ;∃lim⁡j→∞d(Tjpx,Tjpy)=di,
for all x∈Ai, for all y∈Ai+1 if  i,ℓ∈p̅ satisfy the feasibility constraints ∑μ=ii+ℓ(∏σ=μ+1i+ℓ[kσ])(1-kμ)dμ≥di and ∑μ=ii+p(∏σ=μ+1i+p[kσ])(1-kμ)dμ=di. If di=d1; for all i∈p̅, then
(3.15)lim⁡⁡sup⁡j→∞d(Tjp+lx,Tjp+ly)≤d1(∑μ=ii+l(∏σ=μ+1i+l[kσ](1-kμ)));∃lim⁡j→∞d(Tjpx,Tjpy)=d1
for all x∈Ai, for all y∈Ai+1, for all ℓ,i∈p̅ if i,ℓ∈p̅ satisfy the feasibility constraints ∑μ=ii+ℓ(∏σ=μ+1i+ℓ[kσ])(1-kμ)≥1and ∑μ=ii+p-1(∏σ=μ+1i+p-1[kσ])(1-kμ)=1(iv)
IfT:⋃i∈p̅Ai→⋃i∈p̅Aiis a p-cyclic contractive self-mapping, then(3.16)limsup⁡j→∞d(Tjp+lx,Tjp+ly)≤d1⟹∃lim⁡j→∞d(Tjp+lx,Tjp+ly)=d1;∀x∈Ai,∀y∈Ai+1,∀l,i∈p̅.(v)
If⋂i∈p̅Ai≠∅ and T:⋃i∈p̅Ai→⋃i∈p̅Aiis a p-cyclic weakly contractive self-mapping, then(3.17)∃lim⁡j→∞d(Tjp+lx,Tjp+ly)=lim⁡j→∞d(Tjpx,Tjpy)=0;∀x∈Ai,∀y∈Ai+1,∀i,l∈p̅.Proof.
Property (i) follows from (3.7) for k=∏i∈p̅[ki]=1. Property (iii) follows from Property (i) since k<1implies (∏μ=ip+i[kμ])j=kj→0 as j→∞. Property (ii) Follows from Property (i) for ki=1; for all i∈p̅ since di=d1; for all i∈p̅ from Lemma 3.4. Property (iv) follows from Property (ii) for ki<1; for all i∈p̅since di=d1; for all i∈p̅  from Lemma 3.4. Property (v) follows from Property (iii) since if all the subsets Ai; i∈p̅ intersect, then it follows necessarily di=d1=0; for all i∈p̅ so that
(3.18)lim⁡j→∞d(Tjpx,Tjpy)=0;∀x∈Ai,∀y∈Ai+1,∀i∈p̅lim⁡j→∞d(Tjp+lx,Tjp+ly)≤(∏μ=il[kμ])lim⁡j→∞d(Tjpx,Tjpy)=0;∀x∈Ai,∀y∈Ai+1,∀i,l∈p̅.Remark 3.6.
Note that Lemma3.5(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 forp-cyclic contractive self-mappings. A parallel result is discussed in [10] for Meir-Keeler contractions.Theorem 3.7.
Assume thatT:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic weakly k-contractive self-mapping and the closures of the p subsets Ai; i∈p̅ of X intersect. Then, it exists a unique fixed point in ⋂i∈p̅clAi which is also in ⋂i∈p̅Ai if all such subsetsAi; for all i∈p̅ of X, are closed.Proof.
The existence of a fixed point follows from Lemma3.5(v). Its uniqueness follows by contradiction. Assume that there exist z1,z2(≠z1)∈Fix(T)⊂⋂i∈p̅clAi. Then, for some i∈p̅,∃x∈Ai, y∈Ai+1 such that Tjx→z1 and Tjy→z2as j→∞. Then, by using triangle inequality for distances,
(3.19)d(z1,z2)≤d(z1,Tjx)+d(Tjx,Tjy)+d(z2,Tjy);∀j∈Z0+
which implies by using Lemma 3.5(v)
(3.20)d(z1,z2)≤limsup⁡j→∞(d(z1,Tjx)+d(Tjx,Tjy)+d(z2,Tjy))=0⟹d(z1,z2)⟺z1=z2,
what contradicts z1≠z2. Therefore, Fix(T) consists of a unique point in ⋂i∈p̅clAi which is also in ⋂i∈p̅Ai if the sets Ai; i∈p̅ are all closed.Theorem3.7 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 d:X×X→R0+with nonempty convex subsetsAi⊂X, for all i∈p̅ of pair-wise disjoint closures. Let T:⋃i∈p̅Ai→⋃i∈p̅Ai be a p-cyclic weakly k-contractive self-mapping so that the composed 2-cyclic self-mappings.Ti:Ai∪Ai+1→Ai∪Ai+1, for all i∈p̅ are defined as Tix=T(Tp-1x);  for all x∈Ai∪Ai+1; foralli∈p̅. Then, the following properties hold:(i)
Any composed 2-cyclic self-mappingTi:Ai∪Ai+1→Ai∪Ai+1, i∈p̅ is k-contractive provided that the constraint ∑μ=ii+p(∏σ=μ+1i+p[kσ])(1-kμ)dμ=di holds. If, furthermore, it is assumed that Ai and  Ai+1 are convex, then the 2-cyclic self-mapping Ti:Ai∪Ai+1→Ai∪Ai+1 self-mapping is extendable to Ti:cl(Ai∪Ai+1)→cl(Ai∪Ai+1), and that T(clAi)⊆clAi+1; for all i∈p̅.  Thus, the iterates Tijx=T(Tjp-1x) and Tijy=T(Tjp-1y); for all x∈Ai, for all y∈Ai+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 giveni∈p̅, the sets Ai and Ai+1 are convex and closed, if any, then both best proximity points of Ti:Ai∪Ai+1→Ai∪Ai+1 of Property (i) are unique and belong, respectively, to Ai and Ai+1.(iii)
Assume that the subsetsAi of X are convex, for all i∈p̅. If ⋂i∈p̅clAi≠∅, then the best proximity points of Property (i) become a unique fixed point for all the composed 2-cyclic self-mappings Ti:Ai∪Ai+1→Ai∪Ai+1 which are k-contractive,  foralli∈p̅. Such a fixed point is in ⋂i∈p̅clAi (and also in ⋂i∈p̅Ai if all the subsets Ai, i∈p̅, are closed).Proof.
SinceT(Ai)⊆Ai+1; for all i∈p̅, then for any i∈p̅, x∈Ai⇒Tix∈Ai and  x∈Ai+1⇒Tix∈Ai+1if p is even and x∈Ai⇒Tix∈Ai+1 and x∈Ai+1⇒Tix∈Ai if p is odd. Since T:⋃i∈p̅Ai→⋃i∈p̅Ai is p-cyclic weakly k-contractive then k=∏i=1p[ki]<1, then Ti:Ai∪Ai+1→Ai∪Ai+1is 2-cyclic contractive provided that  ∑μ=ii+p(∏σ=μ+1i+p[kσ])(1-kμ)dμ=di; i∈p̅. One has from Lemma 3.5(iv) that ∃limj→∞d(Ti2jx,Ti2jy)=di=d(z,z1); for all x∈Ai, forally∈Ai+1 for the giveni∈p̅, where z=z(i)∈cl(Ai)(z∈Ai if Ai is closed),  z1=z1(i)∈cl(Ai+1)(z∈cl(Ai)(z∈Ai+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 getsTi2jx→zand Ti2jy→z1as j→∞; for all i∈p̅. 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 Ti:Ai∪Ai+1→Ai∪Ai+1 is k-contractive for all i∈p̅, 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.Theorem3.8 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 d:X×X→R0+ with nonempty subsets Ai⊂X, for all i∈p̅ of pair-wise disjoint closures. Let T:⋃i∈p̅Ai→⋃i∈p̅Ai be a p-cyclic k-contractive self-mapping so that the composed 2-cyclic self-mappings  Ti:Ai∪Ai+1→Ai∪Ai+1, for all i∈p̅,  are defined as  Tix=T(Tp-1x); for all x∈Ai∪Ai+1; for all i∈p̅. Assume also that Ai is convex and T(clAi)⊆clAi+1; for all i∈p̅. Then, the following properties hold.(i)
Asj→∞, the iterates Tjx and Tjy;  for all x∈Ai, for all y∈Ai+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 anyi∈p̅. Also, for any given i∈p̅ such that the sets Ai and Ai+1 are convex and closed, if any, then both best proximity points of   T:Ai∪Ai+1→Ai∪Ai+1of Property (i) are unique and belong, respectively, to Ai and Ai+1. If, furthermore, ⋂i∈p̅clAi≠∅, then the best proximity points of Property (i) become a unique fixed point for the p-cyclic k-contractive self-mapping T:Ai∪Ai+1→Ai∪Ai+1. Such a fixed point is in ⋂i∈p̅clAi (and also in ⋂i∈p̅Ai if all the subsets Ai⊂X, i∈p̅ are closed).(ii)
All the composed 2-cyclic self-mappingsTi:Ai∪Ai+1→Ai∪Ai+1, for all i∈p̅ are k-contractive. Thus, the iterates Tijx=T(Tjp-1x) and Tijy=T(Tjp-1y); for all x∈Ai, for all y∈Ai+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 i∈p̅ such that the sets Ai and Ai+1 are closed and convex, if any, then both best proximity points of Ti:Ai∪Ai+1→Ai∪Ai+1 of Property (i) are unique and belong, respectively, to Ai and Ai+1. If, furthermore, ⋂i∈p̅clAi≠∅,  then the best proximity points of Property (i) become a unique fixed point for all the composed 2-cyclic self-mappings Ti:Ai∪Ai+1→Ai∪Ai+1 which are k-contractive; i∈p̅. Such a fixed point is in ⋂i∈p̅clAi(and also in ⋂i∈p̅Ai if all the subsets Ai, i∈p̅ are closed).Outline of Proof
Property (ii) is the direct version of Theorem3.8 applicable to the composed 2-cyclic self-mappings Ti:Ai∪Ai+1→Ai∪Ai+1 which are all k-contractive since T:⋃i∈p̅Ai→⋃i∈p̅Ai 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 T:⋃i∈p̅Ai→⋃i∈p̅Ai is also k-contractive.Definition2.5 is extended to p-cyclic self-mappings as follows.Definition 3.10.
T:⋃i∈p̅Ai→⋃i∈p̅Ai is a 2-cyclic (α,β)-Kannan self-mapping for some real α∈[0,1/2) if it satisfies for some β∈R+:
(3.21)d(Tx,Ty)≤α(d(x,Tx)+d(y,Ty))+β(1-α)dist⁡(A,B);∀x∈A,∀y∈B.
Now, Theorem 2.9 and Theorems 2.18–2.21 for 2-cyclic accretive and Kannan self-mappings extend directly with direct replacements of their relevant parts as follows:Theorem 3.11.
Let(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 d:X×X→R0+. Consider a self-mapping  T:X→X which is also a p-cyclic k-contractive self-mapping if restricted tT:⋃i∈p̅Ai→⋃i∈p̅Ai, where Ai are nonempty convex subsets of X; for all i∈p̅. Then, Theorem 2.9 holds “mutatis-mutandis” by replacing the subsets A and B for pairs of adjacent subsets Ai and Ai+1, i∈p̅,A∪B→⋃i∈p̅Ai, cl(A∩B)→cl(⋂i∈p̅Ai), A∩B→⋂i∈p̅Ai and k→k:=∏i=1p[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-mappingsTi:Ai∪Ai+1→Ai∪Ai+1; for all i∈p̅ defined from the p-cyclic weak k-contractive self-mapping T:⋃i∈p̅Ai→⋃i∈p̅Ai since T:⋃i∈p̅Ai→⋃i∈p̅Ai; i∈p̅ are k-contractive.



---

*Source: 817193-2012-04-05.xml*