# Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

Authors: M. De la Sen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.


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## Body

## 1. Introduction

A relevant  attention has been recently devoted to the research of existence and uniqueness of fixed points of self-mappings as well as to the investigation of associate relevant properties like, for instance, stability of the iterations [1–3]. The extension of such  topics to the existence of either fixed points of multivalued self-mappings [1, 4–19], in generalized metric spaces [20, 21], or to the existence of common fixed points of several multivalued mappings or operators is receiving an important attention, for example, [7, 8, 15–19, 22] and references therein. Relevant properties on the existence and uniqueness of fixed points and best proximity points for multivalued cyclic self-mappings have been studied under general contractive-type conditions based on the Hausdorff metric between subsets of a metric space. See, for instance, [4, 7–9], including as a relevant particular case the contractive condition on contractive single-valued self-mappings, [1, 4–10], as well as concerns related to their extension to cyclic self-mappings. See, for instance, [7, 8, 11] and references there in. The various related performed researches include the cases of strict contractive cyclic self-mappings and Meir-Keeler type cyclic contractions [23, 24]. Also, some of the existing background fixed point results on contractive single and multivalued self-mappings, [1, 4, 5, 9, 10, 25–28] and references therein, under some types of contractive conditions, have been revisited and extended in [4].  There is also a wide sample of fixed point type results available on fixed points and asymptotic properties of the iterations for self-mappings satisfying a number of contractive-type conditions while being endowed with partial order conditions. See [18, 19] and references therein.The main objective of this paper is the investigation of the properties of the distances as well as the existence and uniqueness of fixed points and best proximity points related to contractive so-called single-valued contractivep(≥2)-precyclic self-mappings T:⋃i∈p-Ai→⋃i∈p-Ai. Such a concept extends that of contractive p(≥2)-cyclic self-mappings.The concept of precyclic self-mapping generalizes that of cyclic self-mappings in the sense that a finite set of consecutive iterations are optionally allowed within a particular subset of the cyclic disposal of interest before a switching of the image of the self-mapping to the adjacent subset of its pre-image in the iterated sequence. It can also eventually happen that some sequence enters a certain subset and the solution remains permanent within such a subset. The precyclic self-mappings are contractive if they are subject to contractive conditions of similar types to those arising in contractive cyclic self-mappings.Precyclic contractive self-mappings allow the generation of iterated sequences under constraints of the formTj(Ai)⊆Ai∪Ai+1 for j=j(i,x) being less than a prescribed  positive integer number j-=j-(i,x); for all x∈⋃i∈p-Ai, for all i∈p- which can be set and point dependent, while  Tj-(Ai)⊆Ai+1; for all i∈p-. The ordering of the subsets for switching between pairs of adjacent subsets to perform the p-precyclic self-mapping is, so-called, in the sequel a p-precyclic disposal.LetR0+=R+∪{0} be the set of nonnegative real numbers and  Z0+=Z+∪{0} the set of nonnegative integer numbers. Consider a metric space (X,d) endowed with a metric d:X×X→R0+  and a finite set of nonempty subsets Ai; i∈p-={1,2,…p} of X and a so-called p(≥2)-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai such that T(Ai)⊆Ai∪Ai+1; for all i∈p-, where Ai=Aj for i,j∈p- under the congruence relation j≡i(modp), that is Apn+i=Ai; for all n∈Z0+, for all i∈p-. Note that the previous concept of precyclic self-mapping generalizes that of a p-cyclic self-mapping which satisfies the stronger constraint T(Ai)⊆Ai+1; for all i∈p-. Let Di,j=d(Ai,Aj) be the distance in-between any two subsets Ai,Aj⊂X; for all i,j∈p-. Note that, compared to a cyclic self-mapping, an iterated sequence from a precyclic self-mapping might contain iterated subsequences of finite or infinite cardinals in a single subset Ai⊂X(i∈p-) even if ⋂i∈p-Ai≠⌀.  Also, certain iterated sequences generated from p-precyclic self-mappings can converge to a fixed point, rather than oscillate in-between sets of distinct best proximity points, even if ⋂i∈p-Ai=⌀, provided that the iterated points all stay in the same subset  Ai⊂X,  for some i∈p-, after a finite number of iterations.For each givenx∈Ai, define the following nondecreasing strictly ordered (in general, point dependent) switching sequence of nonnegative integers:
(1)Si*(x)={ji-1*(x)=0,ji*(x),ji+1*(x),…,jp*(x),j1+p*(x),…,ji+j+ℓp*(x),…};∀i∈p-
containing the numbers of consecutive iterations within each individual subset Ai⊆X; for all i∈p-, before switching to the successive adjacent subsets Ai+1,Ai+2,…,Ai-1,Ai,Ai+1,Ai+2,…, and so forth for j∈p- of the iterated sequence
(2)Pi(x)={{Tji-1*(x)x(=x),Tji-1*+1(x)x(=Tx),…,Tji*(x)-1x}(⊆Ai),…,{Tji+j+ℓp*(x)x,…,Tji+j+ℓp+1*(x)-1x}(⊆Ai+j+1),…Tji*(x)-1}.
For all i∈p- such that(3)ji+j+ℓp*(x)={min(k(≥j+ℓp)∈Z0+:Tkx∈Ai+j+1∖Ai+j);∀i∈p-,∀ℓ∈Z0+,if0≤j≤p-i-1min(k(≥j+ℓp)∈Z0+:Tkx∈Ai+j+1-p∖Ai+j-p);∀i∈p-,∀ℓ∈Z0+,ifp≥j>p-i-1for any givenx∈Ai and,  either jk+1*(x)-jk*(x)<∞; for all k∈Z0+ with cardSi*(x)=χ0 (i.e., infinite cardinal of a numerable set) or there is jk*(x)*(x)<∞ for some existing finite k*=k*(x)∈Z0+ and then Si*(x)={ji-1*(x)=0,ji*(x),ji+1*(x),…,jk*(x)*(x)<∞} is a finite set, that is, cardSi*(x)=k*(x)-i(x)+2<χ0 with Tjk*(x)*+kx being in the same subset Ai⊂X as Tjk*(x)*x; for all k(≤cardSi*(x))∈Z0+; for all i∈p-. If cardSi*(x)=1 for any x∈⋃i∈p-Ai, for all i∈p-, then  only one iteration stays at each subset  before switching to the adjacent one so that the p-precyclic self-mapping  is a standard p-cyclic one. Note, for instance, that if ji*(x)=1 in (2) and x∈Ai, then Tx∈Ai+1. If this occurs for each x∈⋃i∈p-Ai, then T on ⋃i∈p-Ai is a usual p-precyclic self-mapping.We will establish the formulation in metric spaces(X,d). It might be pointed out that it is usual to state formulations related to differential or dynamic systems and their stability, including those  being formulated in a fractional calculus framework, in normed or Banach spaces since their dynamics evolve  through time described by their state vectors [14, 29–39]. A possibility to focus on the study of their equilibrium points in a formal and structured fashion as well as their limit solutions, provided that they exist, (for instance, the presence of possible limit cycles) is through fixed point theory since the equilibrium points are fixed points of certain mappings and the limit cycles are  repeated portions of limit state  space trajectories. See, for instance, [33] and references therein. In the subsequently  formulated and proved results, where the convexity of sets of X is required, it will  be assumed that (X,∥∥) is a normed space with  associated metric space (X,d) for a norm-induced metric d:X×X→R0+, [29]. If (X,∥∥) is a Banch space, then (X,d) is a complete metric space. The converses are not true without invoking additional assumptions. For instance, if (X,d) is a metric space (resp., a complete metric space) endowed with a homogeneous and translation -invariant metric d:X×X→R0+, then the metric-induced norm ∥∥ exists so that (X,∥∥) is a normed (resp., Banach) space endowed with such a metric-induced norm.Example 1.
For some givenε∈R0+, define the real intervals
(4)A1=Rε+={z∈R:z≥ε};A2=Rε-={z∈R:z≤-ε}
and consider the scalar sequence
(5)xn+1={x^n+1,if|x^n+1|≥ε,-εsgnxn,otherwise,
where
(6)x^n+1=(-1)n+1anxn;∀n∈Z0+withinitialconditionsatisfying|x0|≥ε
with {an}⊂R0+. Note that, if ε=0, then A1∩A2={0} and {0} can be a candidate for fixed point depending on certain simple contractive or, at least, nonexpansive conditions on the sequence {an}. If ε≠0, then the convergence of the sequences {x2n} and {x2n+1} (but not that of {xn} which is not convergent) can be possible only to best proximity points ±ε. The self-mapping  T on A1∪A2 defining the solution sequence is a 2-cyclic one since the solution points are alternately in A1 and A2. However, the modification
(7)xn+1+i={x^n+1+i,if|x^n+1|≥ε,-ενn+isgnxn,otherwise,
where
(8)x^n+1+i=(-1)n+1νn+ianxnνn+i={(-1)n+1,if0≤i≤j*=2,1,ifi=j*.
For all n∈Z0+ with initial value |x0|≥ε is a 2-precyclic (but no a 2-cyclic) self-mapping  which generates two consecutive iterations in both A1 and A2 before switching to the other subset. Several extended variants are possible; that is, j*=j0*=j1*=2 is constant in this case. For instance, j* can be dependent on the solution point x(n) or on the initial condition. If j*(x(n0)) is infinity, then the trajectory solution remains in either A1 (resp., in A2) for n≥n0 if x(n0)∈A1  (resp., x(n0)∈A2). In this case, depending on conditions on the parameterization sequence {an}, the convergence of the solution in one of the subsets could be possible, even if ε≠0, when j* is infinity in at least one of the sets A1 and A2 for some subset of values of the solution so that the solution enters such a set and remains in it for all later iterations. Ifj*=1 for any point of the solution sequence at any iteration,  then the solution trajectory switches in-between the subsets A1 and  A2 so that the 2-precyclic self-mapping is also a 2-cyclic one.


## 2. Convergence of Iterated Sequences to Fixed Points

The following assumptions are made.(1) There arep bounded real functions  Ki:Ai→Ki∈(0,K-i]; for all i∈p-  fulfilling Ki(x)=Kj(x) in ⋃i∈p-Ai under the  congruence relation j≡i(modp) for some i=i(j)∈p-  and any given j∈Z+ such that
(9)d(Tx,T2x)≤Ki(x)d(x,Tx)+(1-Ki(x))νi(x)Di
for x∈Ai where νi:Ai→{0,1} are binary functions; for all i∈p- such that νi(x)=0 if  Tx∈Ai and νi(x)=1 if Tx∈Ai+1∖Ai; for all x∈Ai, for all i∈p-. The notation to be used does not distinguish explicitly the cases when the contractive-like functions are real constants or point-dependent functions, but this can be easily deduced from context.(2) IfDi>0, that is,  if  Ai∩Ai+1=⌀, then K-i<1;  for all i∈p-.(3) IfcardS*(x)<χ0, then Kk*(x)≤1, where k*(x)≡i+j(modp); for all x∈Ai, for all i∈p-, provided that Ai+j  is closed(4)(10)limsupk→∞(∑j=0k[Ki+jji+j*(x)])≤1;x∈⋃i∈p-Ai,∀i,j∈p-.If(x,y≠Tx)∈Ai×Ai+1 for some i∈p-, then the constraint (9) is replaced with its following standard counterpart stated for p-cyclic self-mappings:
(11)d(Tx,Ty)≤Ki(x)d(x,y)+(1-Ki(x))Di.Note that the previous condition holds ifTx∈Ai+1 and Ty∈Ai+2 but also if  Tx,Ty∈Ai+1 in its particular version d(Tx,Ty)≤Ki(x)d(x,y), since Di≥0, and that it  contains (9)  for iterated sequences  generated from T:⋃i∈p-Ai→⋃i∈p-Ai as a particular case. Note also the following. (a)
A particular pair(x,Tx) can satisfy simultaneously several constraints (9). For instance, assume that  x,Tx∈Ai∩Ai+1(≠⌀) for some  i∈p-. Then
(12)d(Tx,T2x)≤min(Ki(x),Ki+1(x))d(x,Tx).(b)
IfcardS*(x)<χ0, then there is some set  Aj(j∈p-) such that Tkx⊆Aj; for all k≥k0 and some finite k0=k0(x)∈Z0+ for each given x∈⋃i∈p-Ai. Then, T:⋃i∈p-Ai→⋃i∈p-Ai is nonexpansive and {Tkx}k∈Z0+is bounded. Note that Assumption 4 is guaranteed directly by Assumption 3 if cardS*(x)<χ0. If cardS*(x)=χ0, then Assumption 3 is not invoked; however, Assumption 4 guarantees that {Tkx}k∈Z0+ is bounded  with Tkx∈⋃i∈p-Ai.(c)
Assumption 4 implies that thep-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is asymptotically nonexpansive.(d)
Anyp-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is also a p-cyclic self-mapping.In the following,fix(G) denotes the set of fixed points of the self-mapping G:X→X. The following results hold.Theorem 2.
LetT:⋃i∈p-Ai→⋃i∈p-Ai be a p-precyclic self-mapping. Assume also that the  constraint (9) is extended  for any x∈Ai and y∈Aj; for all i,j∈p- as follows:
(13)d(Tx,Ty)≤max(Ki(x),Kj(y))d(x,y)+(1-max(Ki(x),Kj(y)))νij(x)Dij,
where Dij=d(Ai,Aj) and νij(x)=1 if  Tx∈Ai+1∖Ai  and Ty∈Aj+1∖Aj and  νij(x)=0, otherwise; for all x∈Ai, for all y∈Aj.
Then, the following properties hold.(i)
T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic self-mapping if and only if  T(Ai)⊆Ai+1; for all i∈p-.(ii)
IfDi>0; for all i∈p- and cardS*(x)=χ0; for all x∈⋃i∈p-Ai, then T:⋃i∈p-Ai→⋃i∈p-Ai has no fixed point in ⋃i∈p-Ai.(iii)
IfcardS*(x)<χ0 for some x∈⋃i∈p-Ai, then T:⋃i∈p-Ai→⋃i∈p-Ai has a fixed point in a subset  Aj⊂X,   for some j∈p-, to which the iterated sequence {x,Tx,…,Tkx,…} converges if (X,d) is complete, Kj<1 and Aj is closed. If x∈Ai  for some i∈p-, then the iterated sequence {x,Tx,…,Tkx,…} converges to a fixed point xi+j*∈Ai+j of  T:⋃i∈p-Ai→⋃i∈p-Ai for some j∈p-i¯ with Ki+j<1 such that k*(x)≡i+j(modp), provided that Ai+j is closed, or to a fixed point xj*∈Aj  for some j∈i-1¯with Kj<1 and k*(x)≡i+j(modp) provided that Aj is closed. If, furthermore, xi+j*∈Ai+j  for j∈p-i¯, respectively, xj*∈Aj  for j∈i¯are convex then  the corresponding fixed point is unique.Proof.
T
(
A
i
)
∩
A
i
≠
⌀ for some i∈p-; then ∃x∈Ai such that Tx∉Ai+1. Then, T(Ai)⊆Ai∪Ai+1 is not a cyclic self-mapping. Hence, Property (i) follows.
Note thatDi>0; for all i∈p-⇔Ai∩Ai+1=Ø; for all i∈p-. If cardS*(x)=χ0, then S*(x) is an infinite sequence of switches in-between adjacent subsets of X which are never intersecting. Thus,   T:⋃i∈p-Ai→⋃i∈p-Ai has no fixed point in ⋃i∈p-Ai. Hence,  Property (ii) follows.
To prove Property (iii), first note that(14)Ai+j+ℓp=Ai+j=Ak;∀ℓ∈Z0+;∀i,j∈p-
such that
(15)i+j+ℓp≡i+j(modp)≡k(=i+j-p;k∈Z+)(modp).
Note also that, ifcardS*(x)<χ0 for some x∈⋃i∈p-Ai, then the iterated sequence  S(x)={x,Tx,…,Tkx,…} built from such an initial point x∈⋃i∈p-Ai through T:⋃i∈p-Ai→⋃i∈p-Ai remains in Aj⊂X, for such a j∈p-, for all k≥jk*(x)* and some finite integer jk*(x)*. Then, the asymptotically nonexpansive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is asymptotically contractive from Assumption 4 and also contractive after a finite number of iterations since Kj<1. Thus, S(x) is bounded and has a Cauchy convergent subsequence since (X,d) is complete. Since the subset Aj⊂X is nonempty and closed for such j∈p-, there is a fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai from Banach contraction principle and such a fixed point is unique if the subset is, furthermore, convex.Now, the following result is proven for a class of contractivep-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai.Theorem 3.
Assume that(X,∥∥) is a normed space with  associated metric space (X,d) for a norm-induced metric d:X×X→R0+. Let T:⋃i∈p-Ai→⋃i∈p-Ai be a p-precyclic self-mapping and Di=0; for all i∈p-. If Ai⊂X are nonempty, bounded, closed, and convex; for all i∈p- and K-=∏i∈p-[K-ij-i*]<1, where j-i*=supx∈Aiji*(x). Then T:⋃i∈p-Ai→⋃i∈p-Ai has a unique fixed point x*∈⋂i∈p-Ai. If Ai⊂X is not closed for some i∈p- while (X,∥∥) is a Banach space,  and then (X,d) is a complete metric space,  then T:⋃i∈p-Ai→⋃i∈p-Ai has a unique fixed point in ⋂i∈p-clAi.Proof.
It follows that(16)limn→∞d(Tpnj-*x,Tpnj-*-1x)≤limn→∞(K-n)d(Tx,x)=0;∀x∈⋃i∈p-Ai
since d(x,Tx)<∞ for any finite x∈⋃i∈p-Ai, since ⋃i∈p-Ai is bounded, where
(17)j-*=∑i=1pj-i*=maxi∈p-supx∈Ai(ji*(x)+ji+1*(Tji*+1x)+⋯+jp*(T∑j=ip-1jj*+1x)+j1*(T∑j=ipjj*+1x)+⋯+ji-1*(T∑j=ipjj*+∑j=p-i+1p-1jj*+1x));
that is, the distance between any two consecutive elements of any such a sequence converges asymptotically to zero. Furthermore, since the subsets Ai⊂X are nonempty, closed,  all of them intersect and the composite self-mapping  Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai is uniformly Lipschitz -continuous in ⋃i∈p-Ai, since it is contractive with constant K-<1, the sequence {Tpnj-*x} converges to xi*=xi*(x)∈Ai for some i∈p-, which is a unique fixed point of the composite Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai in Ai; for all i∈p-.  To prove uniqueness, proceed by contradiction. Assume that there are two fixed points xi*=Tpj-*xi*, yi*(≠xi*)=Tpj-*yi* in Ai of Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai.
Then,(18)d(xi*,yi*)=d(Tpnj-*xi*,Tpnj-*yi*)≤d(Tpnj-*+1xi*,Tpnj-*xi*)+d(Tpnj-*+1xi*,Tpnj-*+1yi*)+d(Tpnj-*+1yi*,Tpnj-*yi*)≤K-n(d(Txi*,xi*)+d(Txi*,Tyi*)+d(Tyi*,yi*))=K-nd(xi*,yi*)<d(xi*,yi*);∀n∈Z+
which leads to the contradiction that xi*=yi*. Thus, there is a unique fixed point of Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai in Ai. Also, one gets from (16) that
(19)limn→∞d(Tpnj-*+jx,Tpnj-*+j-1x)≤limn→∞(K-n)(∏ℓ=1j[K-ℓ])d(Tx,x)=0;∀j∈p-
for any j∈p-. As a result,
lim
n
→
∞
d
(
T
p
n
j
-
*
+
j
x
,
T
p
n
j
-
*
+
j
-
1
x
)
=
0 and {Tpnj-*+jx}→xi*=Tpj-*xi* with some unique xi*∈Ai for some i∈p-;  for all j∈p-. Now, assume that there are two distinct fixed points xi*∈Ai and xj*(≠xi*)∈Aj(≠Ai) of for some i,j(≠i)∈p-. Since Ai⊆X  is nonempty, closed, and convex for any i∈p-, then Ai∩Aj is nonempty, closed, and convex; for all i,j(≠i)∈p-. From the convexity of Ai∩Aj, there is z(≠xi*,xj*)∈Ai∩Aj such that d(xi*,z)=λd(xi*,xj*) with some real constant λ∈(0,1) and xi*,z∈Ai. Then,
(20)d(xi*,Tpnj-*z)=d(Tpnj-*xi*,Tpnj-*z)≤K-nd(xi*,z)=λK-nd(xi*,xj*)
so that d(xi*,Tpnj-*z)→0 as n→∞ and,  since Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai is uniformly Lipschitz continuous in ⋃i∈p-Ai, the sequence {Tpnj-*z}n∈Z0+ converges to xi*∈Ai. But z∈Aj so that we can repeat the previous reasoning with d(xj*,z)=λ′d(xi*,xj*), xj*, z∈Aj and some real constant λ′∈(0,1) to conclude that d(xj*,Tpnj-*z)→0 as n→∞ and {Tpnj-*z}n∈Z0+converges to xj*(≠xi*)∈Aj which is a contradiction to its convergence to xi*∈Ai. Then, there is a unique fixed point in the nonempty, closed, and convex set Ai∩Aj. By extending the same reasoning to any pair of subsets Ai and Aj of X, one concludes that the composite self-mapping Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai has a unique fixed point x*∈{x*}=Fix(Tpj-*)⊆⋂i∈p-Ai.
It remains to be proved that the unique fixed point of the composite mapping is also the unique fixed point ofT:⋃i∈p-Ai→⋃i∈p-Ai. Since the subsets Ai⊆X intersect, one gets from (16) that
(21)(d(T2pj-*+1x*,T2pj-*x*)≤K′K-d(Tpj-*x*,x*)=0)⟹(d(T(T2pj-*)x*,x*)=d(Tx*,x*)=0)⟹(Tx*=x*)⟺(x*∈Fix(Tpj-*))⟹(x*∈Fix(T))
since Tpj-*x*=x*, where K′=maxi∈p-Ki. Also,
(22)(x*∈Fix(T))⟹(d(Tx*,x*)=d(x*,x*)=d(T2x*,Tx*)=⋯=d(Tn+1x*,Tnx*)=0;∀n∈Z)⟺(x*∈Fix(Tn);∀n∈Z+)⟹(x*∈Fix(Tpj-*))
so that x*∈Fix(T) and it is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai. If some Aii∈p- is not closed, then  all Cauchy sequences have a limit in X if (X,d) is complete so that there is still a unique fixed point in ⋂i∈p-clAi of T:⋃i∈p-Ai→⋃i∈p-Ai and  Tpj-*:⋃i∈p-Ai→⋃i∈p-Ai.The following result is now proven for a class of nonexpansivep-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai.Corollary 4.
Assume the hypothesis of Theorem3 with the modified weaker condition K-=∏i∈p-[K-ij-i*]≤1 of nonexpansive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai. Then, {Tnx}n∈Z0+ is bounded; for all x∈⋃i∈p-Ai there is a subsequence {Tpj-*nkx}nk∈Zs⊆Z0+⊆{Tnx}n∈Z0+, for some strictly ordered subset  Zs of  Z0+, such that  d(Tpj-*nkx,Tpj-*nk+1x)→C=C(x) as k→∞; for all x∈⋃i∈p-Ai for some real C∈R0+. If, in addition, K-i≤1; for all i∈p-, then d(Tn+1x,Tnx)→C=C(x) as n→∞; for all x∈⋃i∈p-Ai.Proof.
x
∈
⋃
i
∈
p
-
A
i is always finite since ⋃i∈p-Ai is bounded and then {Tnx}n∈Z0+⊆⋃i∈p-Ai is a bounded sequence; for all x∈⋃i∈p-Ai. Also, d(Tpj-*nx,Tpj-*n+1x)≤d(x,Tx); for all n∈Z0+, for all x∈⋃i∈p-Ai since K-=∏i∈p-[K-ij-i*]≤1. Thus, limsupn→∞d(x,Tn+1x)<∞;  for all x∈⋃i∈p-Ai since {Tnx}n∈Z0+ is bounded and d(Tnx,Tn+1x)<∞;  for all n∈Z0+ from the properties of distances since x∈⋃i∈p-Ai is finite. Thus, there is a subsequence {Tpj-*nkx}nk∈Zs⊆Z0+⊆{Tkx}k∈Z0+ for which d(Tpj-*nkx,Tpj-*nk+1x) converges as k→∞. If, in addition, K-i≤1; for all i∈p-, then d(Tn+1x,Tnx)≤(maxi∈p-Ki)d(x,Tx)≤d(x,Tx); for all x∈⋃i∈p-Ai, so that d(Tn+1x,Tnx)→C=C(x) as n→∞; for all x∈⋃i∈p-Ai.The subsequent result is related to convergence to a unique fixed point in one of the subsetsAi⊆X(i∈p-) of the precyclic disposal even if the subsets do not intersect, provided that the self-mapping is asymptotically contractive in one of the subsets.Corollary 5.
Assume the hypothesis of Theorem3 with the subsequent further hypothesis on the p-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai:  (1)
⋂i∈p-Ai=⌀; cardSi*(x)<χ0(2)
∃j∈J0=J0(x)⊆p- (nonnecessarily unique; that is, J0 can have a cardinal greater than one) such that limsupℓ→∞Kj(Ti+j+ℓp+1x)<1 for some given initial point x∈Ai.Then, the iterated sequence Pi(x), (2), converges to a fixed point in Ak⊂X  for a unique k=min(j∈Ji≺:j∈J0)∈p-, where
(23)Ji≺=Ji≺(x)={Ji≺(x)i(≺i+1),i+1(≺i+2),i+2(≺i+3),…,p-1(≺p),p(≺,1),1(≺2),…,i-1≺(i)}
is a strictly ordered finite set of cardJi≺=p, containing all the p elements of the set p-, with the strict order relation ≺ defined by its enumeration order  defined by a≺b for any a,b∈J if  a precedes b in the previous enumeration definition of the set J.Proof.
SincecardSi*(x)<χ0; ∃j∈p- such that for the given x∈Aiji+j+ℓp+1*(x)→∞ as ℓ→∞ if 0≤j≤p-i-1, or jk+ℓp*(x)→∞ as ℓ→∞ if 2p≥j>p-i-1 for  any nonnegative integer k>i-1 such that j≡k-i-1(modp). This is obvious since, if such aj∈p- does not exist, then the iterated sequence Pi(x), (2), with starting point x∈Ai for the given i∈p- has infinitely many switches in-between consecutive adjacent subsets Ai⊂X; then the switching sequence Si*(x) associated with such an iterated sequence Pi(x) is finite so that cardSi*(x)=χ0.
From the extra previous hypothesis 2, there is a nonempty setJ0(x)⊆p- for which limsupℓ→∞Kj(Ti+j+ℓp+1x)<1 for the given x∈Ai. Note that the p-precyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is asymptotically contractive, for the given x∈Ai on Ak⊂X, where k=min(j∈Ji≺(x):j∈J0(x))∈p- is unique (even if J0(x) is of cardinal greater than one)  from the fact that Ji≺(x) in (23) is of finite cardinal, strictly ordered, then with unique minimal and maximal elements which are then a unique i∈p- minimum and maximum i-1∈p-1¯∪{0}, respectively. From the previous part of this proof, it also turns out that T:⋃i∈p-Ai→⋃i∈p-Ai  is, furthermore, (strictly) contractive on such a subset Ak⊂X for any infinite subsequence Pi(Tji+j+ℓp*(x)x)⊆Pi(x). Therefore, such a subsequence Pi(Tji+j+ℓp*(x)x) is itself a Cauchy sequence fulfilling limn→∞d(Tnx,Tn+1x)=0and then being bounded and convergent in Ak (since {Tnx}n≥ji+j+ℓp*(x) satisfies a  Lipschitz-continuous property) to a fixed point x*∈Ak, since Ak is nonempty, bounded (then x is  bounded), and closed, which is unique since Ak is convex. Since the sequence Pi(x) contains, by construction, all the elements of the subsequence alter a finite number of iterations, it also converges to the same unique fixed point x*.Remark 6.
(1) Note that Corollary5 is stated for a certain iterated sequence being built from a starting point since the contractive conditions a point-to-point condition. Point-to-point contractivity-type  conditions have been also used in the literature for the characterization of fixed point properties of contractive self-mappings. See, for instance, [40, 41] and references therein. It can be generalized directly under generalization for any starting point in any of the subsets or in some subset of its union. It can be reformulated, in particular, if limsupℓ→∞Kj(Ti+j+ℓp+1x)<1; for all x∈Ai. In such a case, it follows the convergence of the sequence of iterates to the same unique fixed point in Ak built from any initial point x∈Ai.
(2) Note that the uniqueness of the final setAk from the initial set Ai, such that x∈Ai,  arises from the fact that the first subset where the iterations remain after a finite number of them is the relevant one for the final  reached limit if the precyclic self-mapping stops to transfer the iterated sequence to the next adjacent subset.
(3) Note that, if the convexity assumption is only made on the subsetAk, then Corollary 5 still holds.
(4) Note also that, if the convexity assumption on the subsets is removed in Corollary5, then the existence of the fixed point is still proven although that one is not necessarily unique.The subsequent result is a consequence of direct proof of Corollaries4 and 5.Corollary 7.
Assume the hypothesis of Corollary5 under the weaker limsupℓ→∞Kj(Ti+j+ℓp+1x)≤1 condition of asymptotic nonexpansivness of the self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai in some of the subsets Ak⊆X for some given initial point x∈Ai. Then, d(Tn+1x,Tnx)→C=C(x) and {Tnx}n≥n0⊆Ak  for some finite n0∈Z0+ and  a unique k∈p- as defined in Corollary 5.Some extensions of Corollary7 can be directly obtained from Remark 6 (3)-(4).


## 3. Convergence to Best Proximity Points

The following preliminary technical result concerning the convergence for distances in-between consecutive iterates through the self-mappingT:⋃i∈p-Ai→⋃i∈p-Ai is now given in the case that the subsets Ai⊂X do not necessarily intersect.Theorem 8.
Assume that the following constraints hold.(1)
Di≠0; for all i∈p-.(2)
Ki∈(0,∞) is constant within each subset Ai⊂X and the switching sequences
(24)Si*={ji-1*=0,ji*,ji+1*,…,jp*,j1+p*,…,ji+j+ℓp*,…}={ji-1*=0,ji*,Si+1*∖ji*};∀i∈p-are not point dependent on the given x∈Ai;for all i∈p- so that, in addition,
(25)j-i*=j-i+np*=supx∈Aiji*(x)=ji+np*∀n∈Z0+.(3)
(26)K=∏i∈p-[Kij-i*]<1.(4)
(27)D0:=∑j=1p(∏i=j+1p[Kiji*])(1-Kj)Dj1-∏i=1p[Kiji*].
Then, the following properties hold for anyx∈Ai:
(28)limsupn→∞d(Tn(∑i=1pji*)+jx,Tn(∑i=1pji*)+j-1x)≤Kij-1D0forj=1,2,…,ji*(29)Di+k≤limsupn→∞d(Tn(∑i=1pji*)+∑ℓ=ikjℓ*+j+1x,Tn(∑i=1pji*)+ji*+jx)≤(∏ℓ=ik[Kℓ])D0+∑j=ik(∏ℓ=j+1k[Kℓjℓ*])(1-Kj)Dj;
for all x∈⋃i∈p-Ai, j=0,1,…,jk+1*-1, for all k∈p-.
If, in addition, Assumption 5(5)
(30)D1=K1j1*-1D0;Dk=Kkjk*-1(∏i=1k-1[Kiji*])D0≥∏i=1k-1[Kiji*]Dkfork≥2Holds; then,
(31)limsupn→∞d(Tn(∑i=1pji*)+∑i=1kji*x,Tn(∑i=1pji*)+∑i=1kji*-1x)≤Dk(32)∃limn→∞d(Tn(∑i=1pji*)+∑i=1kji*+1x,Tn(∑i=1pji*)+∑i=1kji*x)=Dk;∀k∈p-(33)limsupn→∞d(Tn(∑i=1pji*)+∑i=1k+1ji*+j+1x,Tn(∑i=1pji*)+∑i=1k+1ji*+jx)=Kk1+j-jk*Dk;j=1,…,jk+1*-1,∀k∈p-;∀x∈⋃i∈p-Ai.Proof.
Take anyx∈A1. Thus, one gets from (9)
(34)d(T2x,Tx)≤K1d(x,Tx)d(Tj1*x,Tj1*-1x)≤K1j1*-1d(x,Tx)D1≤d(Tj1*+1x,Tj1*x)≤K1j1*d(x,Tx)+(1-K1)D1d(Tj1*+2x,Tj1*+1x)≤K2d(Tj1*+1x,Tj1*x)≤K2[K1j1*d(x,Tx)+(1-K1)D1]d(Tj1*+j2*x,Tj1*+j2*-1x)≤K2j2*-1[K1j1*d(x,Tx)+(1-K1)D1]D2≤d(Tj1*+j2*+1x,Tj1*+j2*x)≤K2j2*[K1j1*d(x,Tx)+(1-K1)D1]+(1-K2)D2d(T∑i=1pji*x,T∑i=1pji*-1x)≤Kpjp*-1((∏i=1p-1[Kiji*])d(x,Tx)+∑j=1p-1(∏i=j+1p-1[Kiji*])(1-Kj)Dj)Dp≤d(T∑i=1pji*+1x,T∑i=1pji*x)≤Kd(x,Tx)+M,
where
(35)K=∏i=1p[Kiji*]<1;M=∑j=1p(∏i=j+1p[Kiji*])(1-Kj)Dj.
Then, sinceK<1,
(36)Dp≤d(Tn(∑i=1pji*)+1x,Tn(∑i=1pji*)x)≤Knd(x,Tx)+1-Kn1-KM,d(Tn(∑i=1pji*)+j1*+1x,Tn(∑i=1pji*)+j1*x)≤K1j1*(Knd(x,Tx)+1-Kn1-KM)+(1-K1)D1
with Tn(∑i=1pji*)+j1*+1x∈A2. Now, since
(37)D0=∑j=1p(∏i=j+1p[Kiji*])(1-Kj)Dj1-∏i=1p[Kiji*]=M1-K,
it follows that(38)Dp≤limsupn→∞d(Tn(∑i=1pji*)+1x,Tn(∑i=1pji*)x)≤D0limsupn→∞d(Tn(∑i=1pji*)+2x,Tn(∑i=1pji*)+1x)≤K1D0limsupn→∞d(Tn(∑i=1pji*)+j1*-1x,Tn(∑i=1pji*)+j1*-2x)≤K1j1*-2D0limsupn→∞d(Tn(∑i=1pji*)+j1*x,Tn(∑i=1pji*)+j1*-1x)≤K1j1*-1D0D1≤limsupn→∞d(Tn(∑i=1pji*)+j1*+1x,Tn(∑i=1pji*)+j1*x)≤K1j1*D0+(1-K1)D1.
Thus, ifD1=K1j1*-1D0, then  one gets from the previous relationships
(39)∃limn→∞d(Tn(∑i=1pji*)+j1*+1x,Tn(∑i=1pji*)+j1*x)=D1limsupn→∞d(Tn(∑i=1pji*)+j1*x,Tn(∑i=1pji*)+j1*-1x)≤D1∃limn→∞d(Tn(∑i=1pji*)+j1*+1x,Tn(∑i=1pji*)+j1*x)≤K1j1*D0+(1-K1)D1=D1limsupn→∞d(Tn(∑i=1pji*)+1+j1*+jx,Tn(∑i=1pji*)+j1*+jx)≤K1j+j1*D0=K11+jD1;j=1,…,j2*-1.
In the same way, if  the constraintD1=K1j1*-1D0 is extended to D1=Kkj1*-1D0; Dk=Kkjk*-1(∏i=1k-1[Kiji*])D0;  for all k(≥2)∈p-, then  one gets from (34) and (38) that
(40)d(Tn(∑i=1pji*)+∑i=1kji*+1x,Tn(∑i=1pji*)+∑i=1kji*x)≤∏i=1k[Kiji*](Knd(x,Tx)+1-Kn1-KM)+(1-Kk)Dk;
for all k∈p-. Thus,
(41)Dk≤limsupn→∞d(Tn(∑i=1pji*)+∑i=1kji*+1x,Tn(∑i=1pji*)+∑i=1kji*x)≤limn→∞(∏i=1k[Kiji*](Knd(x,Tx)+1-Kn1-KM))+(1-Kk)Dk=∏i=1k[Kiji*]1-KM+(1-Kk)Dk=D0(∏i=1k[Kiji*])+(1-Kk)Dk=KkDk+(1-Kk)Dk=Dk;∀k∈p-
with Tn(∑i=1pji*)+∑i=1kji*+1x∈Ak+1. Then, (31)–(33) follow and the result is proven for x∈A1.  Such a choice can be made with no loss in generality since, if, instead, x∈Ai for any given i∈p-, then the previous result still holds with Tn(∑i=1pji*)+∑i=1kji*+1x∈Ak+1 with k+1+i≡j(modp) for a unique integer j∈p-.   On the other hand (28)-(29) are a consequence of (36) which is independent of the constraints Dk=Kkjk*-1(∏i=1k-1[Kiji*])D0;  for all k∈p-.Remark 9.
(1) Note that Theorem8 requires a set of necessary constraints on the Ki and ji*; for all i∈p- which are induced by Assumptions 4-5 as follows:
(42)σi=Di+1Di=Ki+1ji+1*-1Ki(i=2,3,…,p)(43)(1-K)D0=[∑j=2p{(∏i=j+1p[Kiji*])(1-Kj)(∏k=1j-1[σk])}+∏i=2p[Kiji*](1-K1)∑j=2p{(∏i=j+1p[Kiji*])(1-Kj)(∏k=1j-1[σk])}]D1=[∑j=2p{(∏i=j+1p[Kiji*])(1-Kj)×(∏k=1j-1[Kk+1jk+1*-1Kk])(∏i=j+1p[Kiji*])(1-Kj)}+∏i=2p[Kiji*](1-K1)∑j=2p{(∏i=j+1p[Kiji*])(1-Kj)]D1
with K=∏i=1p[Kiji*].
(2) Note that ifKi=K<1, ji*=1 and  Di=D; for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is a contractive p-cyclic and
(44)D0=∑i=0p-1Ki1-Kp(1-K)D=11-Kp1-Kp1-K(1-K)D=D.
If (29) is evaluated in this case, then the limit superior is also the limit inferior of the obtained expression leading to the existence of the limit being equal to D, the distance in-between al, the adjacent subsets Di⊂X; for all i∈p-. This is a well- known result for contractive p-cyclic self-mappings T:⋃i∈p-Ai→⋃i∈p-Ai, [23, 24, 42].
(3) Note that the first property of (32) is a convergence of the iterated sequences of distances to best proximity points of adjacent subsets provided that Assumptions 1–5 of Theorem 8 hold implying, in particular, a strict contractive  constant on a composite self-mapping T∑i=1pji*:⋃i∈p-Ai→⋃i∈p-Ai (see Assumption 3) and a set of further constraints on the distances in-between adjacent subsets even if such distances are not identical (see Assumptions 4-5).  The following result is supported by parallel results in [42] (see also [43]) for cyclic self-mappings. It establishes the asymptotic convergence of the iterated sequences to cycles containing best proximity points in-between adjacent subsets in uniformly convex Banach spaces.Theorem 10.
Assume that(X,∥∥) is a uniformly convex Banach space with the subsets Ai⊂X being all disjoint, bounded, closed, and convex for all i∈p- and that T:⋃i∈p-Ai→⋃i∈p-Ai is a p-precyclic self-mapping  subject to Assumptions 1–5 of Theorem 8.
Then, the subsequences{Tn(∑i=1pji*)+∑i=1kji*x}n∈Z0+⊆Ak and {Tn(∑i=1pji*)+∑i=1kji*+1x}n∈Z0+⊆Ak+1 converge to  respective unique best proximity points xk* in Ak and xk+1*=Txk* in Ak+1 for each  starting point x∈Ai; for all i,k∈p-. The sequence {Tnx}n∈Z0+converges asymptotically to a  unique cycle of q=∑i=1pxi* points:
(45)x^*:=(x-11,x-12=Tx-11,…,x-1j1*=Tj1*-1x-11=x1*,x-21=Tx1*,…,xp-1*,x-p1=Txp-1*,…,x-pjp*=xp*)
which contains the p best proximity points.Proof (Outline of Proof).
Note that(X,d) is a complete metric space for a ∥∥ (norm-) induced metric  d:X×X→R0+ since (X,∥∥) is a Banach space. Thus, Theorem 8 remains true for such a metric. Since the subsets Ai⊂X are nonempty, bounded, and closed (then compact and also boundedly compact), there exist xi*∈Ai,Txi*Ai+1 such that d(xi*,Txi*)=∥xi*-Txi*∥=Di for each i∈p- so that xi* and Txi* are the best proximity points in Ai to Ai+1 and Ai+1 from to Ai, respectively [42]. If Di=0 and ji*=1, then both of them are confluent in the fixed point xi*=Txi*∈Ai∩Ai+1; i∈p-. From  the relation (32) of Theorem 8,  it follows that
(46)∃limn→∞d(Tn(∑i=1pji*)+∑i=1kji*+1x,Tn(∑i=1pji*)+∑i=1kji*x)=Dk;∀x∈Ai;∀i,k∈p-.
The composite self-mappingT∑i=1pji*:⋃i∈p-Ai→⋃i∈p-Ai Lipschitz with constant K=∏i∈p-[Kij-i*] everywhere in its definition domain ⋃i∈p-Ai. Thus, the limit of the distance and the  distance of limits of the sequences {Tn(∑i=1pji*)+∑i=1kji*x}n∈Z0+⊆Ak and {Tn(∑i=1pji*)+∑i=1kji*+1x}n∈Z0+⊆Ak+1 can be interchanged in (45) if the limits of such sequences exist. But such limits exist for any initial iteration point x∈Ai since both sequences are Cauchy sequences and convergent since (X,d)≡(X,∥∥) is a uniformly convex Banach space (Lemmas 3.7, 3.8  of [42]) The whole sequences and their limits are within the corresponding subsets since such subsets are closed and  (X,d)≡(X,∥∥) is complete.  The uniqueness of the best proximity points in each of the subsets follows from the fact that the subsets Ai⊂X; for all i∈p- are convex. The convergence of the sequences {Tnx}n∈Z0+ to a unique cycle x^* of the form (45), containing q=∑i=1pxi* points, follows from the convergence of the above subsequences to unique best proximity points taking part of such a cycle and the fact that T:⋃i∈p-Ai→⋃i∈p-Ai is  a single-valued self-mapping.Example 11.
Consider the following nonlinear difference sequence:(47)x2n+i-1=(1-α2n+i-2)x2n+i-2+ω2n+i-2,x2n+j*=x2mn+1=-β2mnx2mn;∀n∈Z+ifx1∈A1(≡R0+)x2n+j*+1=x2mn+2=-β2mn+1x2mn+1,x2n+i=(1-α2n+i-1)x2n+i-1+ω2n+i-1;∀n∈Z+ifx1∈R0-(≡-R0+≡A2)
for i∈j*¯ with j*=j1*=jn*=jn*(m)=2(m-1)n+1, when xn∈A1≡R0-,  and j2*=1 (that is, only one iteration remains in A2≡R0- before each switching to R0+) defined for some positive integer m=m(n) under the following constraints:{ωn} is a nonnegative real summable sequence{αn}⊂[0,1] is a real sequence, ∑n=0∞αn=∞ and αn→0 as n→∞{ωn}⊂R0+ is a real sequence and ∑n=0∞ωn<∞{βn}⊂[0,β-] is a bounded real sequence fulfilling β2mn+2≤β2nm+1x2nm+1/x2mn+2,  for all n∈Z+ if x1∈R0+ and β2mn+1≤β2mn-1x2mn-1/x2mn+1; for all n∈Z+, otherwise.Note that the difference equation is generated by a 2-precyclic self-mapping on R=R0+∪R0- defined by the solution got from any initial condition. If m=jn*=1, then the solution point simply alternates in-between the subsets R0+ and R0- of R and the mapping becomes a 2-cyclic self-mapping. If m>1, the solution remains jn*=2(m-1)n+1 consecutive iterations in R0+ after entering it before the next switching to R- if m(n) is infinity; for all n≥n0 and some finite n0∈Z0+, then the solution remains  in R0+ after a given finite iteration. The following nonexpansive condition holds:
(48)|x2n+j*(n)+1|≤β2m(n)n+1(∏i=2n-12m(n)[1-αi])|x2n|+∑i=2n-12m(n)∏j=i+12m(n)[1-αj]ω2i;
for all n≥n1 since {ωn} is nonnegative, summable, and then converges to zero,  so that it has  some strictly decreasing subsequence {ωn} and then ω2n-ω2n-1<0; for all n≥n1 for some finite positive integer n1. It follows, since  αn→0 as n→∞, that
(a) ifβn→1 as n→∞, then the solution is weakly 2-precyclic asymptotically nonexpansive since αn→0 and ωn→0 as n→∞ and then limsupn→∞(|x2n+j*(n)+1|-|x2n|)≤0,  liminfn→∞(|x2n+j*(n)+1|-|x2n|)≥0 and {xn} converges.
Ifjn*(x)=1 for any n∈Z0+ and x∈R, then the mapping defining the solution is also 2-cyclic asymptotically nonexpansive.
(b) Ifβn→1 as n→∞ and there are no subsequences of {ωn} and {αn} being simultaneously zero, then the solution is 2-precyclic  (2-cyclic if jn* is identically unity) weakly  asymptotically contractive so that it converges to zero which is the fixed point being a confluent best proximity point of R0+ and R0-.
Ifαn→α∈(0,1) or if limsupn→∞αn≤α-<1, then the solution is 2-precyclic (2-cyclic if jn* is identically unity) strongly asymptotically contractive so that it converges to zero. If there is some finite positive integer n0 such that jn0* is infinite, then the solution is permanent and nonnegative in R0+ after a finite number of iterations and converges asymptotically to zero.
It is now proven that, in the most general considered case when the solution mapping is weakly2-precyclic asymptotically nonexpansive, the fixed point (which is also stable equilibrium point and best proximity point on both subsets) is x=0.
Definex-1=x1, if x1∈R0+ and  x-1=x2, then  x-0∈R0+, if x1∈R- and build the sequence {x-n} by x-n=xn if xn≥0 and x-n=0, otherwise. Then, one gets
(49)-∞<-x-1=limn→∞(x-n+1-x-1)=limn→∞(∑i=1n(ωi-αix-i))
so that ∑n=1∞(αnx-n-ωn)=x-1<∞. Since {ωn} is summable and ∑i=1∞αn=∞, one gets
(50)(minn∈Z+x-n)(∑i=1∞αn)≤∑i=1∞αnx-n=x-1+∑n=1∞ωn<∞⟹minn∈Z+x-n=minn∈Z+xn=0.
Also,maxn∈Z-(|xn|)≤limn→∞(βn)minn∈Z0+xn≤minn∈Z0+xn=0. As a result, the fixed point, and also equilibrium point, is x=0 and xn→0 as n→∞ for any initial condition x1∈R.



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*Source: 310106-2013-08-04.xml*