# On the Extensions of Krasnoselskii-Type Theorems top-Cyclic Self-Mappings in Banach Spaces

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

A set ofnp(≥2)-cyclic and either continuous or contractive self-mappings, with at least one of them being contractive, which are defined on a set of subsets of a Banach space, are considered to build a composed self-mapping of interest. The existence and uniqueness of fixed points and the existence of best proximity points, in the case that the subsets do not intersect, of such composed mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii-type theorems.


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## 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 maps and the sets where those maps operate [1, 2]. For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space has fixed points if the weak fixed point property holds [1]. 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 [3–5]. 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 [6–10]. Another interest of such maps is their usefulness as formal tool for the study of p-cyclic contractions even if the involved subsets of the metric space under study do not intersect [6]. 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. It has to be pointed out that there are close links between contractive self-mappings and Kannan self-mappings [2, 10–13].A rich research is being devoted to the existence and uniqueness of best proximity points of cyclic mappings under different assumptions on the vector space to which the subsets involved in the cyclic mapping belong. For instance, in [14], the concept of 2-cyclic self-mappings is extended to p(≥2)-cyclic self-mappings and results about fixed points are derived in the case that the subsets of the considered complete metric space have a nonempty intersection. Some of the ideas in such a manuscript inspired the definition of p-cyclic self-mappings from X to X with X being the union of the subsets involved in the cyclic representation. On the other hand, the existence and uniqueness of best proximity points of p(≥2)-cyclic φ-contractive self-mappings are investigated in [15] borrowing the previous scenario investigated in [16, 17] for 2-cyclic φ-contractive self-mappings. Generally speaking, the so-called cyclic φ-contractive self-mappings, which are associated with some strictly increasing unbounded map, are based on a concept of weak contractiveness contrarily to the standard, and commonly used, (strict) contractive concept being inspired in the well-known Banach contraction principle. In those papers, the Banach space X under consideration is also assumed to be reflexive and strictly convex. These joint assumptions, which are less restrictive than the assumption that the space is uniformly convex (since uniformly convex Banach spaces are reflexive and strictly convex but the converse is not true in general), are proven to keep intact the essential properties of existence and uniqueness of best proximity points for cyclic φ-contractive self-mappings if the involved subsets are nonempty, weakly closed, and convex. The above formalism is revisited in [18] for cyclic φ-contractive self-mappings in the framework of ordered metric spaces. In [19], characterization of best proximity points is studied for non-self-mappings  S,T:A→B, where A and B are nonempty subsets of a metric space. In general, best proximity points do not fulfil in this context the usual condition x=Sx=Tx. However, they jointly globally optimize the mappings from x to the distances d(x,Tx) and d(x,Sx).In this manuscript,(X,d) is a complete metric space and is considered associated to a Banach space X endowed with translation-invariant and homogeneous metric d:X×X→R0+ and SX∶={Ai⊆X:Aj+p≡Aj;∀i∈p̅,∀j∈Z0+} is a set of p subsets of X.T̂:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic self-mapping if it satisfies T̂(Ai)⊆Ai+1; for all i∈p̅. A valid metric is the norm of the Banach space X.  If the p-cyclic self-mapping T̂:⋃i∈p̅Ai→⋃i∈p̅Ai is nonexpansive (resp., contractive—also referred to as a 2-cyclic contraction) then there exists a real constant k∈[0,1] (resp., k∈[0,1))  such that
(1.1)d(T̂x,T̂y)≤kd(x,y)+(1-k)dist⁡(A,B),∀x∈A,∀y∈B.
The self-mapping T:A∪B→A∪B is said to be a 2-cyclic large contraction if A and B are two given intersecting subsets of X if(1.2)d(Tx,Ty)<d(x,y),∀x∈A,y(≠x)∈B,[∀ε∈R+,∀x∈A,y∈B:d(Tx,Ty)≥ε]⟹[∃δ∈[0,1):d(Tx,Ty)≤δd(x,y)].
This concept of 2-cyclic large contraction on intersecting subsets extends that of large contraction [20], and both concepts extend to p-cyclic contractions. A p-cyclic large contraction satisfies also (1.2) by replacing A→Ai, B→Ai+1; for all i∈p̅ provided that ⋂i∈p̅Ai≠∅. In Section 2 of this paper, we consider a mapping T:(A1∪A2)×(A1∪A2)→X defined by T(x,y)=T1x+T2y such that Tj(x,y)=T(Tj-1(x,y))=T1jx+T2jy; for all j(≥2) ∈Z+;  for all x,y∈Ai, for all i∈2̅ and then T(x,y)∈Ai+1 where Ti:A1∪A2→A1∪A2; i=1,2 are 2-cyclic self-mappings. It is not required for most of the obtained results that T:(A1∪A2)×(A1∪A2)→A1∪A2 so that such a map is not required to be 2-cyclic either. For the obtained results related to boundedness of distances between 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 point, the sets A1 and A2 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 sets A1 and  A2. Also, concerning best proximity points in the case that the subsets do not intersect, it is assumed that the space is restricted to be a uniformly convex Banach space [6–9]. It turns out that since uniformly convex Banach spaces are also strictly convex, the results also hold under this more general assumption used in several papers (see, for instance [15–18]). In Section 3, the results of Section 2 are extended to mappings built in a close way via p(≥2)-cyclic self-mappings on a set of p subsets Ai(i∈p̅) of X.


### 1.1. Notation

(1.3)R0+∶=R+∪{0},Z0+∶=Z+∪{0},p̅∶={1,2,…,p}⊂Z+.
Superscript ⊤ denotes vector or matrix transpose, Fix(T) is the set of fixed points of a self-mapping T on some nonempty convex closed  subset A of a metric  space (X,d), clA denotes the closure 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 what is simplified as  dist(Ai,Ai+1)=dAiAi+1=di; for all i∈p̅ for distances between adjacent subsets of p-cyclic self-mappings T on ⋃i=1pAi where Ai(i∈p̅) are subsets of X.BPi(T) 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

(1.3)R0+∶=R+∪{0},Z0+∶=Z+∪{0},p̅∶={1,2,…,p}⊂Z+.
Superscript ⊤ denotes vector or matrix transpose, Fix(T) is the set of fixed points of a self-mapping T on some nonempty convex closed  subset A of a metric  space (X,d), clA denotes the closure 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 what is simplified as  dist(Ai,Ai+1)=dAiAi+1=di; for all i∈p̅ for distances between adjacent subsets of p-cyclic self-mappings T on ⋃i=1pAi where Ai(i∈p̅) are subsets of X.BPi(T) 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. Results for Mappings Defined by 2-Cyclic Self-Mappings

The following result is concerned with the above-defined mapT:(A1∪A2)×(A1∪A2)→X constructed with nonexpansive or contractive 2-cyclic self-mappings Ti:A1∪A2→A1∪A2 for i=1,2.Theorem 2.1.
The following properties hold.(i)
IfTi:A1∪A2→A1∪A2 are both 2-cyclic nonexpansive self-mappings then
(2.1)d(Tj(x,y),Tj(x′,y′))≤d(T1jx,T1jx′)+d(T2jy,T2jy′)≤d(x,x′)+d(y,y′)≤diam⁡(A1)+diam⁡(A2)+dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+,(2.2)limsup⁡j→∞d(Tj(x,y),Tj(x′,y′))≤diam⁡(A1)+diam⁡(A2)+dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2.(ii)
IfTi:A1∪A2→A1∪A2 fori=1,2 are both 2-cyclic nonexpansive self-mappings and at least one of them is contractive then Property (i) holds and, furthermore,
(2.3)d(Tj(x,y),Tj(x′,y′))≤max⁡(diam⁡(A1),diam⁡(A2))+(1-min⁡(k1,k2))dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+,(2.4)limsup⁡j→∞d(Tj(x,y),Tj(x′,y′))≤max⁡(diam⁡(A1),diam⁡(A2))+(1-min⁡(k1,k2))dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2.(iii)
IfTi:A1∪A2→A1∪A2 for i=1,2 are both 2-cyclic contractive self-mappings then Properties (i), (ii)  hold and, furthermore,(2.5a)d(Tj(x,y),Tj(x′,y′))≤k1jd(x,x′)+k2jd(y,y′)+(2-k1j-k2j)dist⁡(A1,A2)(2.5b)≤k1d(x,x′)+k2d(y,y′)+(2-k1-k2)dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+,(2.6)limsup⁡j→∞⁡d(Tj(x,y),Tj(x′,y′))≤2dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,(2.7)∃lim⁡j→∞d(Tj(x,y),Tj(x′,y′))=0ifdist⁡(A1,A2)=0,∀x,y∈A1,∀x′,y′∈A2,
(i.e., A1 and A2 are closed and intersect or at least one of them is open while their boundaries intersect).Proof.
Takex,y∈A1 and x′,y′∈A2. Direct calculation yields by taking into account that Ti:A1∪A2→A1∪A2;  i=1,2  are 2-cyclic, then both self-mappings satisfy Ti(A1)⊆A2, Ti(A2)⊆A1 for i=1,2, and (1.1) with respective contraction constants k1 and k2,  and that the metric is translation-invariant and homogeneous:
(2.8)d(T(x,y),T(x′,y′))=d(T1x+T2y,T1x′+T2y′)=d(T1x,T1x′+T2y′-T2y)≤d(T1x,T1x′)+d(T1x′,T1x′+T2y′-T2y)≤d(T1x,T1x′)+d(T2y,T2y′)≤k1d(x,x′)+k2d(y,y′)+(1-k1)dist⁡(A1,A2)+(1-k2)dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+.If k1=k2=1 then one gets from (2.8)
(2.9)d(Tj(x,y),Tj(x′,y′))≤d(T1jx,T1jx′)+d(T2jy,T2jy′),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+≤k1jd(x,x′)+k2jd(y,y′)+∑l=1jk1l(1-k1)dist⁡(A1,A2)+k2l(1-k2)dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+.If  k1=k2=1; that is, if both 2-cyclic self-mappings Ti:A1∪A2→A1∪A2(i=1,2) are nonexpansive. Thus, Property (i) follows from (2.9).  If only one of them is nonexpansive and the other is contractive then max(k1,k2)=1 and 0≤min(k1,k2)<1so that (2.3) follow from (2.9) and Property (ii) is proven. For real constants for some real constants ki∈[0,1),Ti:A1∪A2→A1∪A2(i=1,2) are both contractive and one gets from (2.9)
(2.10)d(Tj(x,y),Tj(x′,y′))≤k1jd(x,x′)+k2jd(y,y′)+∑l=0j-1k1l[(1-k1)dist⁡(A1,A2)+k2l(1-k2)dist⁡(A1,A2)],∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+
what leads to (2.5a), (2.5b), and to ((2.6)), (2.7) by taking j→∞. Property (iii) has been proven.Corollary 2.2.
IfTi:A1∪A2→A1∪A2 are both 2-cyclic asymptotically contractive self-mappings, that is, they are nonexpansive with time-varying real sequences {kij}j∈Z+ in [0,1] for i=1,2 which converge asymptotically to two respective real numbers k1 and k2 in [0,1) as j→∞. Assume that the sequences {kij}j∈Z+ for i=1,2 are such that
(2.11)limsup⁡j→∞⁡(∑l=1j[(∏i=l+1j[k1i])(1-k1l)+(∏i=l+1j[k2i])(1-k2l)])≤K<∞
is satisfied for some real K∈R+. Then, (2.1) holds and, furthermore, the following properties hold:
(i)
(2.12)d(Tj(x,y),Tj(x′,y′))≤2εj+Kdist⁡(A1,A2)≤2ε+Kjdist⁡(A1,A2),∀j(≥j0)∈Z+,∀x,y∈A1,∀x′,y′∈A2for any bounded positive decreasing real sequence εj, with arbitrary prescribed upper-bound ε, which converges asymptotically to zero as j→∞ for some finite j0=j0(ε)∈Z+ and some bounded nondecreasing positive real sequence Kj of upper-bound K.(ii)
(2.13)limsup⁡j→∞d(Tj(x,y),Tj(x′,y′)),∀x,y∈A1,∀x′,y′∈A2,∃lim⁡j→∞d(Tj(x,y),Tj(x′,y′))=0,ifdist⁡(A1,A2)=0,∀x,y∈A1,∀x′,y′∈A2.Proof.
Note that(2.14)d(Tj(x,y),Tj(x′,y′))≤(∏l=1j[k1l])d(x,x′)+(∏l=1j[k2l])d(y,y′)+∑l=1j[(∏i=l+1j[k1i])(1-k1l)+(∏i=l+1j[k2i])(1-k2l)]dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+.
Since the terms in the sum the summation term of the right-hand-side of the above equation are all nonnegative, it follows that:
(2.15)limsup⁡j→∞(∑l=1j[(∏i=l+1j[k1i])(1-k1l)+(∏i=l+1j[k2i])(1-k2l)])≥∑l=1j[(∏i=l+1j[k1i])(1-k1l)+(∏i=l+1j[k2i])(1-k2l)],∀j∈Z+.
Note also that ∏ℓ=1j[kiℓ]≤1 for i=1,2, for all j∈Z+, ∃j0=j0(ε)∈Z+ such that ∏ℓ=1j[kiℓ]≤ε<1; for all j≥j0 for any given real constant ε∈(0,1), an infinite subsequence {∏ℓ=1ji[kiℓ]}ji∈Z+of {∏ℓ=1j[kiℓ]}j∈Z+which is monotone decreasing and limj→∞∏ℓ=1j[kiℓ]=0 for i=1,2. Thus, if the sequences {kij} in [0,1] for i=1,2  are such that there is some finite K∈R+ for which
(2.16)∑l=1j[(∏i=l+1j[k1i])(1-k1l)+(∏i=l+1j[k2i])(1-k2l)]≤limsup⁡j→∞(∑l=1j[(∏i=l+1j[k1i])(1-k1l)+(∏i=l+1j[k2i])(1-k2l)])≤K,∀j∈Z+
according to (2.15) then if (2.11) holds, one gets from (2.11) that (2.12) holds for any given ε∈R+ and some positive decreasing real sequence {εj}j∈Z+and some nondecreasing positive real sequence  {Kj}j∈Z+subject to Kj≤Kj+1≤K, so that limsupj→∞Kj≤K, and εj+1≤εj≤ε which converges then to zero as j→∞ since it has an infinite monotone decreasing subsequence {εji}ji(j)∈Z+ for each j∈Z+, satisfying to 0<ji+1-ji<∞ for any i,j∈Z+, then limji→∞εji=limj→∞εj=0. As a result, (2.13) follows, whose upper-bound is zero if dist(A1,A2)=0.Remark 2.3.
Note that a simple comparison between ((2.6)) in Theorem 2.1 and (2.12) in Corollary 2.2 concludes that K in (2.12) can be taken as small as 2 if the sequences {kij}j∈Z+in Corollary 2.2 are constant and equal to ki∈(0,1) for i=1,2. Therefore, a logic procedure of accomplishing with (2.11) is to check for the existence of valid constants K possessing a lower-bound 2.If the constants(k1+k2) (i=1,2) in Theorem 2.1, or the time-varying sequences {k1j+k2j}j∈Z+for i=1,2 of Corollary 2.2, are less than unity then the following result holds.Corollary 2.4.
The following properties hold.(i)
Ti:A1∪A2→A1∪A2 for i=1,2 are both 2-cyclic contractive self-mappings with time-varying real sequences {kij}j∈Z+ (i=1,2) such that any element of the sum sequence {k1j+k2j}j∈Z+is in [0,1] for i=1,2  with  k≤maxj∈Ẑ+(k1j+k2j)∈[0,1)and k′:=minj∈Z+(k1j+k2j)≥0 where Ẑ+ is some infinite subset of Z+. Then,
(2.17)d(Tj(x,y),Tj(x′,y′))≤kjmax⁡(d(x,x′)+d(y,y′))+2-ρk1-kdist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+,
where ρ∶=k′/k if kk′≠0, ρ=1 if k=k′=0, and ρ=0 if  k≠0 and k′=0(2.18)limsup⁡j→∞d(Tj(x,y),Tj(x′,y′))≤2-ρk1-kdist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2.(ii)
LetTi:A1∪A2→A1∪A2 be both 2-cyclic nonexpansive and asymptotically contractive self-mappings for i=1,2 with  the sequences {kij}j∈Z+in [0,1] for i=1,2 and having limits k̅i for i=1,2 satisfying k̅∶=max(k̅1,k̅2)<1/2,k′̅∶=min(k̅1,k̅2)≥0. Then,
(2.19)limsup⁡j→∞d(Tj+i(x,y),Tj+i(x′,y′))≤2(1-ρ̅k̅)1-2k̅dist⁡(A1,A2);∀x,y∈A1,∀x′,y′∈A2,
where ρ̅∶=min(k̅1,k̅2)/max(k̅1,k̅2)∈[0,1].Proof.
(i) First, note that the integer subsetẐ+ exists since the time-varying contraction sequences converge to real constants being less than unity. One gets from (2.5a) for k′∶=min(k1,k2)=ρk≤k<1/2 where ρ=k′/k≤min(k1,k2)/max(k1,k2)∈[0,1](2.20)d(Tj(x,y),Tj(x′,y′))≤kjmax⁡(d(x,x′)+d(y,y′))+(2-ρk)dist⁡(A1,A2)(∑i=0j-1ki)≤kjmax⁡(d(x,x′)+d(y,y′))+(2-ρk)(1-kj)1-kdist⁡(A1,A2)≤kjmax⁡(d(x,x′)+d(y,y′))+2-ρk1-kdist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,∀j∈Z+and (2.17) holds. Property (i) has been proven.
(ii) The proof of Property (ii) is close to that of Property (i) by using a close method to that of the proof of Corollary2.2(ii):
(2.21)d(Tj(x,y),Tj(x′,y′))≤(2k̅+ε1j)jmax⁡(d(x,x′)+d(y,y′))+2(1-ρ̅k̅)dist⁡(A1,A2)(∑i=1j∏l=i+1j[(2k̅+εij)l])≤(2k̅+ε1j)jmax⁡(d(x,x′)+d(y,y′))+2(1-ρ̅k̅)(1-(2k̅)j)1-2k̅dist⁡(A1,A2)+K(j,0)≤(2k̅+ε0j)jmax⁡(d(x,x′)+d(y,y′))+2(1-ρ̅k̅)1-2k̅dist⁡(A1,A2)+K(j,0),
for all x,y∈A1, for all x′,y′∈A2, where εij∶=(∏ℓ=i+1j[k1j+k2j-2k̅])1/(j-i)(∈R)→0 as j(>i)→∞, for alli∈Z0+, and {K(j,i)}j(>i)∈Z+;  are sequences of  positive finite real constants whose integer arguments  j and i are related to the iterate of the left and right hand sides of (2.20), respectively, defined by
(2.22)K(j+i,i)∶=(2(1-ρ̅k̅)(∑n=ij∏l=n+1j[(2k̅+εnj)l])-2(1-ρ̅k̅)(1-(2k̅)j)1-2k̅)dist⁡(A1,A2).
Since 0≤k1j+k2j≤1; for all j∈Z0+ then 0≤2k̅+εij≤1; for all i,j∈Z+ and since the limit k1j+k2j→k̅∶=max(k̅1,k̅2)<1/2 as j→∞, {K(j,i)}j(>i)∈Z+is uniformly bounded and limi→∞K(j+i,i)=0; for all j∈Z+. Thus, combining (2.20) and (2.21), one gets
(2.23)d(Tj+i(x,y),Tj+i(x′,y′))≤(2k̅+ε0j)jmax⁡(d(Tix,Tix′)+d(Tiy,Tiy′))+2(1-ρ̅k̅)1-2k̅dist⁡(A1,A2)+K(j+i,i),∀x,y∈A1,∀x′,y′∈A2,
and then
(2.24)limsup⁡j(>i)→∞,i→∞d(Tj+i(x,y),Tj+i(x′,y′))≤(2k̅+ε0j)jlimsup⁡i→∞max⁡(d(Tix,Tix′)+d(Tiy,Tiy′))+2(1-ρ̅k̅)1-2k̅dist⁡(A1,A2),limsup⁡j(>i)→∞,i→∞d(Tj+i(x,y),Tj+i(x′,y′))=limsup⁡j→∞d(Tj+i(x,y),Tj+i(x′,y′))≤2(1-ρ̅k̅)1-2k̅dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2.Some Krasnoselskii-type fixed point results follow for the mapT:(A1∪A2)×(A1∪A2)→X defined trough 2-cyclic binary self-mappings for the case when A1 and A2 intersect.Theorem 2.5.
Assume thatX is a Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume that A1 and A2 are nonempty, convex, and closed subsets of X which intersect. Assume also that Ti:A1∪A2→A1∪A2 (i=1,2) are both 2-cyclic contractive self-mappings and T:(A1∪A2)×(A1∪A2)→X fulfilling T((A1∩A2)×(A1∩A2))⊆A1∩A2. Then, there is a unique fixed point z=z1+z2 of T:(A1∪A2)×(A1∪A2)→X in A1∩A2 which satisfies T1z+T2z=z, where zi∈Fix(Ti)⊆A1∩A2 are also the respective unique fixed points of Ti:A1∪A2→A1∪A2(i=1,2).Proof.
One gets from Theorem2.1(iii), (2.9)
(2.25)[0⟵d(Tj(x,y),Tj(x′,y′))≤d(T1jx,T1jx′)+d(T2jy,T2jy′)⟶0]⟹[d(T1jx,T1jx′)⟶0;d(T2jy,T2jy′)⟶0],∀x,y∈A1,∀x′,y′∈A2asj→∞,
since A1∩A2≠∅ implies dist(A1,A2)=0. Since X is a Banach space then (X,d) is a complete metric space, since Ti:A1∪A2→A1∪A2(i=1,2) are 2-cyclic contractive self-mappings then satisfying (1.1) with contraction constants ki∈[0,1) and dist(A1,A2)=0 (since A1∩A2≠∅) and subject Ti(Aj)⊆Aℓfor j, ℓ(≠j), i=1,2, and since T((A1∩A2)×(A1∩A2))⊆A1∩A2≠∅, the unique limits below for Cauchy sequences exist
(2.26)Tj(x,y)⟶T(z,z)=T1iz+T2iz=z∈A1∩A2,∀i∈Z+asj→∞,T1jx′⟶T1jx⟶T1iz1=z1∈A1∩A2,T2jy′⟶T2jx⟶T1i=z2∈A1∩A2,∀i∈Z+asj→∞,∀x,y∈A1;∀x′,y′∈A2,
where zi∈Fix(Ti)⊆A1∩A2(i=1,2) are unique in A1∩A2 since A1,A2 (and then A1∩A2) are nonempty, convex, and closed. It follows that z=z1+z2 by making i→∞ in T(z,z)=T1iz+T2iz=z∈Fix(T)⊆A1∩A2, for all i∈Z+. Since z1 and z2 then z is unique so that Fix(T)={z}.The following result follows from Corollary2.2 and Theorem 2.5.Corollary 2.6.
Theorem2.5 also holds if Ti:A1∪A2→A1∪A2 (i=1,2) are both 2-cyclic asymptotically contractive self-mappings with respective time-varying contraction sequences being in [0,1] with limits in [0,1).Theorem 2.7.
Assume thatX is a Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume that A1 and A2 are nonempty, convex, and closed subsets of X, which intersect and have a convex union. Assume that Ti:A1∪A2→A1∪A2   (i=1,2) are both 2-cyclic self-mappings, the first one being continuous, the second one being contractive, and, furthermore, the mapping T:(A1∪A2)×(A1∪A2)→Xsatisfies
(2.27)T((A1∪A2)×(A1∪A2))⊆A1∪A2,T((A1∩A2)×(A1∩A2))⊆A1∩A2.
Then, there is a (in general, nonunique) fixed point z∈A1∪A2of T:(A1∪A2)×(A1∪A2)→X which has the properties below:
(2.28)T1iz+T2iz=z,∀i∈Z+,∃lim⁡j→∞T1jz=z-y=T1z+T2z-y,
where y(=T2iy)∈Fix(T2)≡{y}⊆A1∩A2,  for all i∈Z+, is the unique fixed point of T2:A1∪A2→A1∪A2.Proof.
Note that the following properties hold(1)
By hypothesis,A1∪A2 is a nonempty convex set which is, furthermore, closed since A1 and A2 are both nonempty and closed.(2)
A1∩A2 is nonempty, convex, and closed since A1and A2 are nonempty, closed, and convex with nonempty intersection.(3)
(A1∪A2)×(A1∪A2) is invariant through the mapping T:(A1∪A2)×(A1∪A2)→X by hypothesis.(4)
A1∪A2 is a nonempty compact set which is trivially invariant under the 2-cyclic continuous self-mapping T1:A1∪A2→A1∪A2.(5)
T2:A1∪A2→A1∪A2 is a 2-cyclic contraction which has then a unique fixed point y∈Fix(T2)≡{y}⊆A1∩A2 from Theorem 2.1 since A1∩A2 is nonempty, convex, and closed.
Thus, from the standard Krasnoselskii fixed point theorem [20, 21], under the above Properties (1)–(4), it exists z∈Fix(T)⊆A1∪A2 which satisfies T1iz+T2iz=z=T(z,z); for all i∈Z+. Also, one gets from Property (5) that limj→∞T2jz=T2y=y∈Fix(T2)≡{y}⊆A1∩A2 are limits of Cauchy sequences so that T1jz+T2jz=z=T(z,z)→T1jz+y as j→∞ leading to ∃limj→∞T1jz=z-y=T1z+T2z-y.The following results follow from Krasnoselskii fixed point theorem extended to 2-cyclic self-mappings sinceA1∪A2 is a compact set [20].Theorem 2.8.
Theorem2.7 holds if  T2:A1∪A2→A1∪A2 is a 2-cyclic large contraction instead of being a contraction.Theorem 2.9.
Theorem2.7 holds if the hypothesis T((A1∪A2)×(A1∪A2))⊆A1∪A2 is replaced by (x=T1x+T2y;∀y∈A1∪A2)⇒x∈A1∪A2.Corollary 2.10.
Theorem2.7 also follows with the replacement of the 2-cyclic contractive self-mapping T2:A1∪A2→A1∪A2 by a 2-cyclic asymptotic contractive self-mapping T2:A1∪A2→A1∪A2 for i=1,2 with time-varying contraction sequences in [0,1] with limits in [0,1).Corollary2.10 follows by using Corollary 2.2 instead of Theorem 2.1. Also, Theorems 2.8 and 2.9 can be extended for T2:A1∪A2→A1∪A2 being asymptotically contractive closely to the extension Corollary 2.10 to Theorem 2.7.On the other hand, note that ifA1∩A2=∅ then the convergence through the mapping T:(A1∪A2)×(A1∪A2)→X to a fixed point under the conditions of Theorem 2.5, or those of Corollary 2.6, cannot be achieved since fixed points, if any, are in A1∩A2 since A1∩A2=∅. See Theorem 2.1, ((2.6)), Corollary 2.2, (2.13), Corollary 2.4, (2.18), or (2.19). It is not either guaranteed the convergence to best proximity points because the guaranteed upper-bounds for the limit superiors of distances of the iterates exceed the distance between the adjacent subsets A1 and A2 since the lower upper-bounds of the above respective referred to limit superiors are of the form λdist(A1,A2) for some real constant λ>1 defined directly by inspection of ((2.6)), (2.13), or (2.18), or (2.19). Then, the following result follows instead of Theorem 2.5.Theorem 2.11.
Assume thatX is a Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume also that A1 and A2 are nonempty subsets of X, which do not intersect. Assume that Ti:A1∪A2→A1∪A2 (i=1,2) are both 2-cyclic either contractive under Theorem 2.1 or under Corollary 2.4 (i) (or asymptotically contractive under Corollary 2.2 or under Corollary 2.4(ii)) self-mappings, and, furthermore,
(2.29)T:(A1∪A2)×(A1∪A2)⟶A1∪A2.
Then, Tj(x,y) is asymptotically permanent as j→∞; for all x∈A1, for all y∈A2 entering a compact subset Â1∪Â2 of cl(A1∪A2), where:
(2.30)Â1∶={z∈A1:dist⁡(A1,A2)≤dist⁡(z,A2)≤λdist⁡(A1,A2)};Â2∶={z∈A2:dist⁡(A1,A2)≤dist⁡(z,A1)≤λdist⁡(A1,A2)},
and λ>1 is defined directly from (2.6), under Theorem 2.1; from (2.13), under Corollary 2.2, or from either (2.18) or (2.19), under Corollary 2.4.Proof.
It follows from either Theorem2.1, Corollary 2.2 or Corollary 2.4 since (2.29) holds.Note that (2.29) which restricts the image of T:(A1∪A2)×(A1∪A2)→X, used  in Theorem 2.5  and in Corollary 2.6 is essential for the existence of the residual set where the  iterates Tj(x,y) enter asymptotically as j→∞. It is now proven how Corollary 2.4 may be improved to obtain λ=1 in Theorem 2.11 so that the convergence of the iterates Tj:((A1∪A2)×(A1∪A2))→A1∪A2 converge to a best proximity point if A1 and A2 are nonempty, disjoint, convex, and closed. The part of Theorem 2.11 referred to the fulfilment of Corollary 2.4 (i.e., either the sum of both contraction constants or the sum of their limits if they are time-varying sequences is less than unity) is improved as follows.Corollary 2.12.
Assume thatX is a Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume that A1 and A2 are nonempty, disjoint, convex, and closed subsets of X. Assume also that Ti:A1∪A2→A1∪A2 (i=1,2) are both 2-cyclic either contractive under Corollary 2.4(i) with (or asymptotically contractive under Corollary 2.4(ii)) self-mappings and, furthermore, (see (2.29)). Then, any iterates Tj(x,y), respectively, Tj+1(x,y) converge to a best proximity point of either A1, respectively, A2, or conversely, as j→∞ for any given x∈A1, y∈A2.Proof.
Take some real constantk∈[k1+k2,1) and note from (1.1) that (2.8) is modified as follows:
(2.31)d(T(x,y),T(x′,y′))≤(k1+k2)max⁡(d(x,x′),d(y,y′))+(2-k1-k2)dist⁡(A1,A2)≤kmax⁡(d(x,x′),d(y,y′))+(1-k)dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,
which holds since
(2.32)(k-k1-k2)max⁡(d(x,x′),d(y,y′))-(k-k1-k2)dist⁡(A1,A2)≥0⟺max⁡(d(x,x′),d(y,y′))≥dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2.
Thus,
(2.33)d(Tj(x,y),Tj(x′,y′))≤(k1+k2)max⁡(d(x,x′),d(y,y′))+(2-k1-k2)dist⁡(A1,A2)≤kjmax⁡(d(x,x′),d(y,y′))+(1-k)dist⁡(A1,A2)(∑l=0jkl)=kjmax⁡(d(x,x′),d(y,y′))+(1-kj)dist⁡(A1,A2),∀x,y∈A1,∀x′,y′∈A2,
so that the limit limj→∞d(Tj(x,y),Tj(x′,y′))=dist(A1,A2) exists modifying (2.18) in Corollary 2.4(i). Then, Theorem 2.11 holds with λ=1 and Âi is a subset of the boundary of Ai containing the best proximity points of Ai to the set Ai+1 (i=1,2) with A3=A1. Thus, since {Tj(x,y)}j∈Z+ is a Cauchy sequence since (X,d) is complete, then Tj(x,y) converges to a best proximity point of A2 (resp., A1) if j→∞ and is odd (resp., even). A  close proof leading to a similar modified limit follows by modifying (2.19) in Corollary 2.4(ii)  by using the limits k̅i of the time-varying asymptotically contractive sequences for i=1,2, instead of the constants ki, satisfying k̅i∈[0,1/2) implying k̅1+k̅2<1 for k̅∈[k̅1+k̅2,1).Note that Corollary2.4 and its referred to part of Corollary 2.12 also hold in a more general version for limits k̅i∈[0,1) with k̅1+k̅2<1 and k̅∈[k̅1+k̅2,1).Some illustrative examples follow.Example 2.13.
Consider the scalar differential equations(2.34)ẋi(t)=aixi(t)+ri,xi(0)=xi0,i=1,2,
with ai<0 for  i=1,2. The solution trajectories are defined by contractive self-mappings Ti:R→R for i=1,2 as follows:
(2.35)xi(t,xi0)=(Tixi0)(t)=eait(xi0+ri∫0te-aiτdτ),i=1,2;∀xi0∈R,∀t∈R0+.
Both solutions converge to respective unique fixed points zi=ri/|ai| as t→∞ which are also stable equilibrium points, for i=1,2 since by using the Euclidean metric, it follows trivially for any two initial conditions xi0,x̅i0(i=1,2)(2.36)|xi(t,xi0)-xi(t,x̅i0)|≤e-|ai|t|xi0-x̅i0|→0ast→∞fori=1,2,
and xi(t,xi0)→zi with xi0∈R as t→∞ for i=1,2. Note that the same property holds by redefining the self-mappings T̅ih:R→R for i=1,2 so as to pick up by successive iterates starting from initial conditions the sequence of points of the solutions {xi(kh,xi0)}k∈Z0+for some h∈R+ as
(2.37)xi((k+1)h,xi0)=T̅ihxi(kh)=eaih(xi(kh)+ri∫0he-aiτdτ),i=1,2;∀xi0∈R,∀t∈R0+,
which are both contractive since |xi((k+1)h,xi0)-xi((k+1)h,x̅i0)|≤e-|ai|h|xi(kh)-x̅i(kh)|→0 as Z0+∋k→∞ for i=1,2 and xi(kh,xi0)→zi with xi0∈R as Z0+∋k→∞ for i=1,2. Define the self-mapping T:R→R by T(x,y)(t)=(T1x)(t)+(T2y)(t) for any k∈Z+; for all x,y∈R. Note that T(x,y)(t)→T(z,z) and T̅hk(x,y)→T(z,z)Z0+∋k→∞ for any x,y∈R with z=z1+z2=r1/|a1|+r2/|a2|. Now, assume that the initial conditions are restricted to fulfil the constraint |xi0|≤Mi for i=1,2.  Consider a convex closed real interval A=B≡[-(M1+M2+r1/|a1|+r2/|a2|),M1+M2+r1/|a1|+r2/|a2|]. Then, T(x,y)(t) and T̅hk(x,y) are in such an interval and converge to the fixed point z for any initial conditions x,y∈A satisfying the constraint. Note that Ti(A),Ti(B)⊆A≡B and T̅ih(A),T̅ih(B)⊆A≡B are trivially 2-cyclic contractive self-mappings from A to A (Theorem 2.5).Example 2.14.
Now, assume the replacementr1→r1(t) in Example 2.13 where it exists the integral ∫0te|a1|τr1(τ)dτ=O(e|a1|t);forallt∈R0+. Then, the above results still hold with the individual and combined resulting mappings still being contractive leading to respective unique fixed points z1=r̅1,z2=r2/|a2|, and z=r̅1+(r2/|a2|) with r̅1∶=∫0te-|a1|(t-τ)r1(τ)dτ, subject to r̂1∶=supt∈R+(∫0te-|a1|(t-τ)r1(τ)dτ)<∞, and A≡B=[-(M1+M2+r̂1+(r2/|a2|)),M1+M2+r̂1+r2/|a2|].Example 2.15.
Consider self-mappingsTi:A̅∪B→A̅∪B (i=1,2) defined  for any given h,ε∈R+, ai∈R-, and ri∈R (i=1,2)  by
(2.38)Tixi(kh)=xi((k+1)h,xi0),xi((k+1)h,xi0)={x̅i((k+1)h,xi0)=e-|ai|hxi(kh)+ri(kh)|ai|(e|ai|h-1),if|x̅i((k+1)h,xi0)|≥ε,εsign⁡x̅i((k+1)h,xi0),otherwise,ri(kh)=|ai|e|ai|h-1((-1)k+iki-e-|ai|h)xi(kh),x̅i((k+1)h,xi0)=e-|ai|h(xi(kh)+ri∫0he-aiτdτ);i=1,2;∀x10∈A;∀x20∈B,∀t∈R0+,
for all k∈Z0+, ki∈[0,1) being given contraction constants for i=1,2 for disjoint sets A≡R2ε-⊂A̅≡Rε-, B≡Rε+≡-A≡-Rε-∶={z∈R+:z≥ε} for some given ε∈R+. The above equations are interpreted as the solution of the discretized versions of the differential equations (2.34) discussed in Examples 2.13 and 2.14 subject to saturated values on the near boundaries of the real subintervals A̅ and B.  Note that Ti(A)⊆B and Ti(B)⊆A for i=1,2 so that Ti:A∪B→A∪B (i=1,2) are 2-cyclic self-mappings for which any succession of iterates converges to best proximity points -2ε∈A̅ and ε∈B for any bounded initial conditions which restricts the iterates to be real successions on bounded convex disjoint subsets of R. The composed self-mapping T:(A∪B)×(A∪B)→A∪B restricted to T:(A̅∪B)×(A̅∪B)→A∪B by a domain restriction converges to best proximity points -ε∈A and ε∈B (Corollary 2.12). If now a1=ε=x10=0 then A∩B={0} and z=0 is a fixed point of T:(A∪B)×(A∪B)→A∪B. Note that the mapping T1 is continuous although it is noncontractive while some Krasnoselskii-type theorems still apply.The above examples are easily extended for nonscalar cases by using the same tools.


## 3. Results for Mappings Defined byP(≥2)-Cyclic Self-Mappings

Definen̅∶={1,2,…,n} for any n∈Z+. Then, the above results are easily extended to the case of ℓ(≥2) p(≥2) p-cyclic Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai;forallℓ∈n̅ self-mappings which define a mapping T:((⋃i∈p̅Ai)×(⋃i∈p̅Ai))→X defined by T(x1,x2,…,xn)=∑i=1nTixi, for any xi∈Aj and any given j∈p̅Tj:((⋃i∈p̅Ai)×(⋃(i∈p̅)Ai))→⋃(i∈p̅)Ai. To facilitate the exposition, assume that(1)
Tℓ:⋃i∈p̅Ai→⋃i∈p̅Aiare given p-cyclic noncontractive self-mappings for ℓ∈n̅1⊂n̅ where cardn̅-cardn̅2≥cardn̅1≥0,(2)
Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai are p-cyclic contractive self-mappings for ℓ∈n̅2∶=n̅∖n̅1⊆n̅, where cardn̅2≥1, with contraction constants kℓ∈[0,1); for all ℓ∈n̅2.That is, we allocate, with no loss of generality, then-cyclic self-mappings which are not contractive, while they are typically nonexpansive or continuous, all in the first strictly ordered set of integers and those being contractive in the second strictly ordered one. The identities of sets Ai+p≡Ai; for all i∈p̅ are assumed in all the necessary notations as associated with p-cyclic self-mappings. It is also used that any p-cyclic nonexpansive self-mappings have identical distances d̅∶=dist(Ai,Ai+1)>0; for all i∈p̅ between disjoint adjacent sets [6]. The following three results can be proven in a similar way as those counterparts of Section 2 for p=q=2.Theorem 3.1.
Assume thatTi:(⋃ℓ∈p̅Ai)→(⋃ℓ∈p̅Ai) for i∈n̅ are p-cyclic nonexpansive for Part (i), with at least one of them is being contractive (p-1≥p2≥1) for Part (ii). Then, Theorem 2.1 holds with the subsequent replacements:
(3.1)2⟶n=n2≥2;(diam⁡A1+diam⁡A2)⟶(∑i=1pdiam⁡Ai),(k1+k2)⟶(∑i=1pki),(max⁡(min⁡)(k1,k2))⟶(max⁡(min⁡)(ki:i∈p̅)),dist⁡(A1,A2)⟶(dist⁡(Ai,Ai+1),∀i∈p̅),(∀x,y∈A1,∀x′,y′∈A2)⟶(∀x,y∈Ai,∀x′,y′∈Ai+1;∀i∈p̅).Corollary 3.2.
Assume thatTi:(⋃ℓ∈p̅Ai)→(⋃(ℓ∈p̅)Ai) for i∈n̅ are p-cyclic asymptotically contractive self-mappings. Then, Corollary 2.2 holds with the replacements:
(3.2)(ki←∞←jkij∈[0,1],i=1,2,∀j∈Z+)⟶(ki⟵∞←jkij∈[0,1],i∈p̅,∀j∈Z+)
and all further necessary replacements borrowed from Theorem 3.1.Corollary 3.3.
Assume thatTi:(⋃ℓ∈p̅Ai)→(⋃ℓ∈p̅Ai) for i∈n̅ are all p-cyclic contractive for Part (i) and p-cyclic nonexpansive and asymptotically contractive for Part (ii). Thus, Corollary 2.4 holds by replacing:
(3.3){k1j+k2j}j∈Z+⟶{∑i=1nkij}j∈Z+,(k≤max⁡j∈Ẑ+(k1j+k2j))⟶(k≥max⁡j∈Ẑ+{∑i=1nkij}j∈Z+),(k′≔min⁡j∈Z+(k1j+k2j)≥0)⟶(k′≔min⁡j∈Ẑ+{∑i=1nkij}j∈Z+),(k̅≔max⁡(k̅1,k̅2)<12)⟶(k̅≔max⁡(k̅i:i∈n̅)<1n),(k′̅≔min⁡(k̅1,k̅2)≥0)⟶(k′̅≔min⁡(k̅i:i∈n̅)≥0),
and all further necessary replacements borrowed from Theorem 3.1 or Corollary 2.2.The following results based on Krasnoselskii theorems follow for fixed and best proximity points in a Banach space endowed with a translation-invariant and homogeneous metric. The results are a direct   extension from the results for mappingsT:(A1∪A2)×(A1∪A2)→A1∪A2 built with mixed continuous and contractive 2-cyclic self-mappings of Section 2 to mappings  T:((⋃i∈p̅Ai)×(⋃i∈p̅Ai))→⋃i∈p̅Ai built with p(≥2)-cyclic self-mappings, some of them being continuous while the remaining ones are contractive. The underlying ideas are (a) a mapping which is a sum of continuous mappings is continuous, (b) a set of contractive p-cyclic self-mappings for the same set of convex and closed subsets of X has each a unique fixed point if the p subsets involved in the p-cyclic contractive self-mappings intersect. The sum of all those contractive self-mappings has a unique  fixed point which is the sum of the whole set of contractive self-mappings, (c) the combinations of both results yields the existence of a fixed point  from Krasnoselskii fixed point theorems of T:((⋃i∈p̅Ai)×(⋃i∈p̅Ai))→⋃i∈p̅Ai if all the subsets  of the p-cyclic structure intersect. If those sets do not intersect, there is a limiting best proximity point at any of the p subsets of X for any p-cyclic iterates for initial points in between adjacent subsets.Theorem 3.4.
Assume thatX is a Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume also that Ai; for all i∈p̅ are nonempty, convex, and closed subsets of X with ⋂i∈p̅Ai≠∅. Assume also that Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai; for all ℓ∈n̅ are p-cyclic contractive self-mappings fulfilling T((⋂i∈p̅Ai)×(⋂i∈p̅Ai))⊆⋂i∈p̅Ai. Then, there is a unique fixed point z=∑i=1nzi of T:⋃i∈p̅Ai→X in ⋂i∈p̅Ai which satisfies ∑i=1nTiz=z, where zℓ∈Fix(Tℓ)⊆⋂i∈p̅Ai; for all ℓ∈n̅ are also the respective unique fixed points of  Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai; for all ℓ∈n̅.Theorem3.4 above and Corollary 3.5 below extend, respectively, Theorem 2.5 and Corollary 2.6 to the existence of fixed points for a mapping built from a set of n contractive, respectively,  asymptotically contractive,  p(≥2)-cyclic self-mappings.Corollary 3.5.
Theorem3.4 also holds if  Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai; for all ℓ∈n̅ are p-cyclic asymptotically contractive self-mappings with respective time-varying contraction sequences being in [0,1] with limits in [0,1).Theorem2.7 extends directly as followsTheorem 3.6.
Assume thatX is a Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume that A1 and A2 are nonempty, convex, and closed subsets of X, which intersect and have convex union. Assume that Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai; for all ℓ∈n̅ are p-cyclic self-mappings, of which n1 are continuous and n2 are contractive subject to 1≤n1=n-n2 and n2≥1. Assume, furthermore, that the mapping T:⋃i∈p̅Ai→X satisfies
(3.4)T((⋃i∈p̅Ai)×(⋃i∈p̅Ai))⊆⋃i∈p̅Ai,T((⋂i∈p̅Ai)×(⋂i∈p̅Ai))⊆⋃i∈p̅Ai.
Then, there exists a (in general, nonunique) fixed point z∈⋃i∈p̅Ai of T:((⋃i∈p̅Ai)×(⋃i∈p̅Ai))→X which has the properties below:
(3.5)∑l=1nTliz=z,∀i∈Z+,∃lim⁡j→∞Tijz=z-y=∑l=1nTliz-y,∀i∈n̅1,
where y(=∑ℓ=n1+1nTℓiyℓ=∑ℓ=n1+1nyℓ), yℓ∈Fix(Tℓ)≡{yℓ}⊆⋂i∈p̅Ai; for all ℓ∈n̅2, for all i∈Z+,  is the unique fixed point of  the p-cyclic self-mapping Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai; for all ℓ∈n̅.Theorem2.7 extends directly as followsTheorem 3.7.
Theorem3.6 holds if Tℓ:⋃i∈p̅Ai→⋃i∈p̅Ai; for all ℓ∈n̅2 is a p-cyclic large contraction instead of being a contraction, for all ℓ∈n̅.Theorem 3.8.
Theorem3.6 holds if the hypothesis T((⋃i∈p̅Ai)×(⋃i∈p̅Ai))⊆⋃i∈p̅Ai is replaced by (x=∑ℓ=1n1Tℓix+∑ℓ=n1+1nTℓiy;∀y∈⋃i∈p̅Ai)⇒x∈⋃i∈p̅Ai.One gets by extending Corollary2.12 based on the ad hoc extensions of Theorem 2.20 and Corollary 2.4 for the composition of p-cyclic self-mappings.Theorem 3.9.
Assume thatX is a uniformly convex Banach space which has an associate complete metric space (X,d) with the metric d:X×X→R0+ being translation-invariant and homogeneous. Assume also that Ai for i∈p̅are nonempty, disjoint, convex and closed subsets of X. Furthermore, assume that T:((⋃i∈p̅Ai)×(⋃i∈p̅Ai))→⋃i∈p̅Ai. Then, the following properties hold.(i)
Assume thatTi:⋃i∈p̅Ai→⋃i∈p̅Ai are p-cyclic contractive self-mappings with time-varying real sequences {kij}j∈Z; for all i∈p̅ such that the sum sequence {∑i=1nkij}j∈Z+is in [0,1] with k≤maxj∈Ẑ+(∑i=1nkij)∈[0,1) and k′∶=minj∈Z+(∑i=1nkij)≥0 where Ẑ+ is some infinite subset of Z+. Then, any iterates Tj+i(x,y) converges as j→∞ to a best proximity point of Ai+ℓ≡Ai+ℓ-p for some integer p subject to 2p-i≥ℓ≥p-i-1 for any given x∈Aμ, y∈Aμ+1; for all μ∈p̅.(ii)
Assume thatTi:⋃i∈p̅Ai→⋃i∈p̅Ai are p-cyclic asymptotically contractive self-mappings, and, furthermore, with the sequences {∑i=1nkij}j∈Z+in [0,1] and having limits k̅i for i∈n̅ satisfying k̅∶=max(k̅i:i∈n̅)<1/2, k′̅∶=min(k̅i:i∈n̅)≥0. Then, any iterates Tj+i(x,y) converges as j→∞ to a best proximity point of Ai+ℓ≡Ai+ℓ-p for some integer p subject to 2p-i≥ℓ≥p-i-1 for any given x∈Aμ, y∈Aμ+1; for all μ∈p̅.



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*Source: 568072-2011-09-04.xml*