# On a General Contractive Condition for Cyclic Self-Mappings

Authors: M. De la Sen
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

This paper is concerned withp(≥2)-cyclic self-mappings T:⋃i∈p¯Ai→⋃i∈p¯Ai in a metric space (X, d),  with Ai⊂X, T(Ai)⊆Ai+1 for i=1,2,…,p, under a general contractive condition which includes as particular cases several of the existing ones in the literature. The existence and uniqueness of fixed points and best proximity points is discussed as well as the convergence to them of the iterates generated by the self-mapping from given initial points.


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## Body

## 1. Introduction

There are exhaustive results about fixed point theory concerning the use of general contractive conditions in Banach spaces or in complete metric spaces and in partially ordered metric spaces which include as particular cases previous ones in the background literature. See, for instance, [1, 2] and references therein. On the other hand, important attention is being paid to the study of fixed points and best proximity points of (p≥2)-cyclic contractive mappings and p-cyclic Meir-Keeler contractive mappings, [3–7]. Generally speaking, cyclic contract self-mappings T:⋃i∈p̅Ai→⋃i∈p̅Ai on the union of p nonempty closed convex subsets Ai of a complete metric space (X,d), subject to T(Ai)⊆Ai+1, have a unique fixed point located in the intersection of such p subsets if such subsets intersect, [3, 4]. If the p-subsets are disjoint convex closed nonempty subsets of a uniformly convex Banach space, then there is a unique best proximity point at each of the p subsets. The above properties also hold for cyclic Meir-Keeler contractions, [5–7]. In this paper, a contractive condition for p-cyclic self-mapping on the union of p subsets of a metric space which includes as particular cases a number of the existing ones in the background literature is proposed, and their basic associate properties are discussed. It is discussed the existence and uniqueness of fixed points if the metric space is complete and the set of subsets involved in the cyclic contractive condition have nonempty intersections and the existence and uniqueness of best proximity points within each of the subsets Ai⊂X if they are convex, closed and disjoint and X is a uniformly convex Banach space. The asymptotic convergence of the iterates from given initial point to best proximity points at each subset or to the unique fixed point if the subsets intersect is also discussed.


## 2. Main Results for a General Contractive Condition

This section contains the main results of the paper forp(≥2)-cyclic self-mapping on the union of a set of p nonempty subsets of a metric space (X,d) under a very general contractive condition which contains as particular cases several previous ones being known in the background of the literature for the noncyclic case (p=1).Theorem 2.1.
Let(X,d) be a metric space with p nonempty closed subsets Ai of X,∀i∈p̅:={1,2,…,p} such that Apj+ℓ≡Aℓ;∀ℓ∈p-1¯, and let T:⋃i∈p̅Ai→⋃i∈p̅Ai be a continuous p(≥2)-cyclic self-mapping subject to T(Ai)⊆Ai+1;∀i∈p̅ and satisfying the following contractive condition:
(2.1)d(Tx,Ty)≤αid(x,Tx)d(y,Ty)+d(x,Ty)d(y,Tx)d(x,y)+β1id(x,Tx)(d(x,Ty)+d(y,Ty))+β2id(y,Ty)(d(y,Tx)+d(x,Tx))d(x,y)+d(x,Ty)+d(y,Tx)+(γ1id(x,Tx)+γ2id(y,Ty))(d(y,Tx)+d(x,Ty))d(x,Tx)+d(y,Tx)+d(y,Ty)+d(x,Ty)+δ1id(x,Tx)+δ2id(y,Ty)+ηi(d(x,Ty)+d(y,Tx))+μid(x,y)+ωiDi,
for all x∈Ai,∀y(≠x)∈Ai+1;αi≥0,βji≥0,γji≥0,δji≥0,μi≥0,and ωi>0 if Di≠0;j=1,2,fo ralli∈p̅.Then, the following properties hold:(i)(2.2)Di≤d(Tp(n+1)+1x,Tp(n+1)x)≤(∏i=1p[Ki])d(Tpn+1x,Tpnx)+∑l=1p(∏j=i+1p[Kj])ωlDl,
for any x∈Ai and any given i∈p̅ provided that
(2.3){x∈Ai:(∏i=1p[Ki])d(Tpn+1x,Tpnx)+∑l=1p(∏j=i+1p+i-1[Kj])ωlDl≥Di}≠∅,
where
(2.4)Ki:=δ1i+μi+γ1i+β1i+ηi1-αi-β1i-β2i-δ2i-γ2i-ηi,∀i∈p̅.
If, in addition, the distances between any pairs of adjacent subsets Di=dist⁡(Ai,Ai+1)=D are identical; ∀i∈p̅, then T:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic self-mapping. If, furthermore,
(2.5)αi+β2i+γ1i+γ2i+δ1i+δ2i+μi+2(β1i+ηi)≤1,∀i∈p̅,∏i∈p̅[δ1i+μi+γ1i+β1i+ηi1-αi-β1i-β2i-δ2i-γ2i-ηi]<1,
then all the iterates d(Tn+1x,Tnx) fulfil the following constraints:
(2.6)d(Tn+1x,Tnx)≤Knd(Tx,x)+(1-K0)D1-K≤d(Tx,x)+(1-K0)D1-K,∀n∈N,limsup⁡n→∞d(Tpn+1x,Tnx)≤(1-K0)D1-K<∞,(2.7)d(Tnp+l+1x,Tnp+lx)≤(∏i=1p[Ki])pnmax⁡l1∈p̅(∏i=l1l1+l[Ki]i)d(Tx,x)+(1-K0)D1-K≤Md(Tx,x)+(1-K0)D1-K,∀n∈N,(2.8)d(Tnp+l+1x,Tnp+lx)≤(1-K0)D1-K<∞,
where ℓ∈N0,M∈R+,D:=max⁡i∈p̅⁡Di,K:=∏i=1p[Ki]<1 and K0=min⁡i∈p̅⁡Ki<1;∀n∈N for any given x∈⋃i∈p̅Ai, with ωi=1-Ki,∀i∈p̅.If (2.4) is replaced by Ki∈[0,1), that is,
(2.9)αi+β2i+γ1i+γ2i+δ1i+δ2i+μi+2(β1i+ηi)<1,∀i∈p̅,
then the inequalities (2.6)–(2.8) trivially hold.(ii) Assume that the contractive condition (2.1) satisfies (2.5). Then,
(2.10)Di≤limsup⁡n→∞d(Tpn+1x,Tpnx)=∑l=1p+i-1(∏j=i+1p+i-1[Kj])ωlDl1-∏i=1p[Ki]<∞,∀x∈Ai,∀i∈p̅ under the necessary condition (2.3).If, in particular, (2.9) holds, that is, Ki∈[0,1), then ωi:=((1-Ki)/Di)D̅ for any Di≠0 and some D̅≥D,∀i∈p̅, then
(2.11)Di≤lim⁡sup⁡n→∞d(Tpn+1x,Tpnx)≤D̅,∀x∈Ai,∀i∈p̅.
If (2.9) holds, Di=D and
(2.12)ωi=1-Ki=αi+β2i+γ1i+γ2i+δ1i+δ2i+μi+2(β1i+ηi),∀i∈p̅,
then the limit below exists:
(2.13)lim⁡n→∞d(Tpn+1x,Tpnx)=D;∀x∈⋃i∈p̅Ai,∀i∈p̅
which is guaranteed by the condition (2.3).(iii) If the constants of (2.4) fulfil K=∏i=1p[Ki]<1, then the p subsets Ai of X, ∀i∈p̅ have nonempty intersection (i.e., Di=D=0,∀i∈p̅), and, if furthermore, the metric space (X,d) is complete, then lim⁡n→∞⁡d(Tpn+1x,Tpnx)=0,∀x∈⋃i∈p̅Ai and T:⋃i∈p̅Ai→⋃i∈p̅Ai has a unique fixed point in ⋂i∈p̅Ai  to which all the sequences {Tnx}n∈N0, which are then bounded, converge, ∀x∈⋃i∈p̅Ai.IfAi are disjoint, closed, and convex, ∀i∈p̅,X is uniformly convex and ∏i=1p[Ki]<1 and (2.12) holds with Di=D>0,∀i∈p̅, then all sequences {Tpnx}n∈N0,∀x∈Ai converge to a best proximity point of Ai,∀i∈p̅.Proof.
Letx0 be an arbitrary point in ⋃i∈p̅Ai⊂X and take x=x(n,x0)≡xn=Tnx0∈Aj=Aj(n,i) and y≡xn+1(n+1,x0)=Txn=Tn+1x0∈Aj+1,∀n∈N0(:=N∪{0}), where T0=id, and j=ĵ:=[(i+n)/p] if (i+n)/p∈N and j=ĵ+1, otherwise. Then, from (2.1),
(2.14)d(xn+1,xn+2)=d(Txn,Txn+1)≤αid(xn+1,xn+2)+β1id(xn+1,xn+2)+β2id(xn+1,xn+2)d(xn,xn+1)+d(xn,xn+2)d(xn,xn+1)+(β1id(xn,xn+1)d(xn,xn+1)+d(xn,xn+2)+ηi)d(xn,xn+2)+γ1id(xn,xn+1)+γ2id(xn+1,xn+2)d(xn,xn+1)+d(xn+1,xn+2)+d(xn,xn+2)d(xn,xn+2)+(δ1i+μi)d(xn,xn+1)+δ2id(xn+1,xn+2)+ωiDi≤αid(xn+1,xn+2)+ωiDi+((β1i+β2id(xn,xn+1)+d(xn,xn+2)+δ1i+μi)d(xn,xn+1)+δ2i)d(xn+1,xn+2)+(γ1id(xn,xn+1)+γ2id(xn+1,xn+2)d(xn,xn+1)+d(xn+1,xn+2)+d(xn,xn+2)+β1id(xn,xn+1)d(xn,xn+1)+d(xn,xn+2)+ηi)×d(xn,xn+2)≤αid(xn+1,xn+2)+(δ1i+μi)d(xn,xn+1)+(β1i+β2i+δ2i)d(xn+1,xn+2)+ωiDi+ηid(xn,xn+2)+γ1id(xn,xn+1)+γ2id(xn+1,xn+2),
and one gets
(2.15)(1-αi-β1i-β2i-δ2i-γ2i-ηi)d(xn+1,xn+2)≤(δ1i+μi+γ1i+β1i)d(xn,xn+1)+ηid(xn,xn+2)+ωiDi≤(δ1i+μi+γ1i+β1i+ηi)d(xn,xn+1)+ωiDi,
so that, since
(2.16)(β1id(xn,xn+1)d(xn,xn+1)+d(xn,xn+2)+ηi)d(xn,xn+2)≤β1id(xn,xn+1)+ηi(d(xn,xn+1)+d(xn+1,xn+2)).
Thus, it follows proceeding recursively with (2.14) subject to (2.9), (2.16), and ωi=1-Ki,∀i∈p̅(2.17)d(Tpn+j+1x,Tpn+jx)≤(∏j=i+1p+i-1[Kj])nd(Tx,x)+∑l=ip+i-1(∏j=i+1p+i-1[Kj])ωlDl≤Kpnd(Tx,x)+(1-K0)D1-K,
where the first inequality holds irrespective of the identities ωi=1-Ki;∀ip̅ and it implies directly (2.2) since ωi+jp≡ωi and Ki+jp≡Ki and Di+jp≡Di;∀i∈p̅,∀j∈N0, so that ∑ℓ=1p(∏j=i+1p[Kj])ωiDi=∑ℓ=ip+i-1(∏j=i+1p+i-1[Kj])ωℓDℓ. The second inequality follows in the case that ωi=1-Ki, if (2.9) holds so that Kp<K=∏i=1p[Ki]<1,∀i∈p̅ if (2.3) leading directly to (2.5)–(2.7). Property (i) has been proven.
Property (ii) is proven by takingx∈Ai for any i∈p̅ and proceeding recursively with the first inequality of (2.17) to obtain directly (2.10) and (2.11) since ∏i=1p[Ki]<1 and (2.13) if, in addition, (2.12) holds. To prove Property (iii), note from (2.8), (2.9), (2.12), and (2.13) that 0=lim⁡n→∞⁡d(Tpn+jx,Tpn+j-1x)≤lim⁡n→∞⁡d(Tpn+1x,Tpnx)=0,∀x∈⋃i∈p̅Ai,if Di=D=0 and ∏i=1p[Ki]<1;∀i,j∈p̅. If, furthermore, (X,d) is a complete metric space, then each sequence {Tnx}n∈N0;∀x∈⋃i∈p̅Ai is a Cauchy sequence with a limit x̅=lim⁡n→∞Tnx=lim⁡n→∞⁡Tpnx in ⋂i∈p̅Ai since this set intersection is nonempty and closed since all the intersected sets are nonempty and closed. Since the sequences {Tnx}n∈N0 are convergent to a limit x̅, they are bounded. Also, since T:⋃i∈p̅Ai→⋃i∈p̅Ai is continuous in ⋂i∈p̅Ai, then Tx̅=T(lim⁡n→∞⁡Tpnx)≤lim⁡n→∞⁡Tpn+1x=x̅ so that x̅∈Fix⁡(T)⊂⋂i∈p̅Ai. It is proven by contradiction that there exists a unique fixed point. Assume that there exist u=x̅ and v subject to u=Tu≠v=Tv in Fix⁡(T), the set of fixed points of T. Then, the subsequent contradiction follows from (2.1) for Di=D=0,∀i∈p̅, by using d(u,Tu)=d(v,Tv)=0,d(u,Tv)=d(v,Tu)=d(u,v)≠0 and (2.4):
(2.18)d(u,v)=d(Tu,Tv)≤αid(u,Tu)d(v,Tv)+d(u,Tv)d(v,Tu)d(u,v)+β1id(u,Tu)(d(u,Tv)+d(y,Ty))+β2id(v,Tv)(d(v,Tu)+d(u,Tu))d(u,v)+d(u,Tv)+d(v,Tu)+(γ1id(u,Tu)+γ2id(v,Tv))(d(v,Tu)+d(u,Tv))d(u,Tu)+d(v,Tu)+d(v,Tv)+d(u,Tv)+δ1id(u,Tu)+δ2id(v,Tv)+ηi(d(u,Tv)+d(v,Tu))+μid(u,v)≤max⁡i∈p̅(αi+2ηi+μi)d(u,v)<d(u,v),
so that u=v. Property (iii) has been proven.
Note that it cannot be concluded from Theorem2.1 that T:⋃i∈p̅Ai→⋃i∈p̅Ai under the contractive condition (2.1) is either a p-cyclic nonexpansive self-mapping from Theorem 2.1(i), even if all the contractive constants in (2.7) are identical and all the distances between adjacent subsets of X are also identical, or a p-cyclic contraction under Theorem 2.1(ii) since (2.6)–(2.8), or its respective versions with strict inequalities, are only guaranteed for the iterates {Tnx}n∈N0 and {Tpnx}n∈N0 for any x∈⋃i∈p̅Ai. Assume that the norm ∥∥ of the uniformly convex (Banach) space (X,∥∥) induces the metric d:X×X→R0+ being used. Otherwise, any alternative equivalent metric d1:X×X→R0+ may be used to conclude the result. If (2.9) and (2.12) hold with distances between adjacent subsets Di=D>0;∀i∈p̅, then all sequences {Tpnx}n∈N0 are Cauchy sequences, ∀x∈Ai converge to a best proximity point x̅i∈Ai of Ai for any given i∈p̅ from (2.13) and the continuity of Tp:⋃i∈p̅Ai→⋃i∈p̅Ai ensured by that of T:⋃i∈p̅Ai→⋃i∈p̅Ai.Remark 2.2.
If∏i=1q[Ki]p∈[0,1], with some of the Ki being eventually larger than one, then d(Tn+1x,Tnx),∀n∈N is bounded provided that d(Tx,x) is finite. If, furthermore, ∏i=1q[Ki]p∈[0,1) and D=0, then all the composed mappings T̂i:Ai→Im⁡T̂i⊂Ai defined by T̂i(Ai)=(T)p(Ai)=(T∘Tp-1)(Ai);∀i∈p̅ satisfies limsup⁡n→∞⁡d(T̂in+1x,T̂inx)=0;∀x∈Ai,∀i∈p̅, so that T:⋃i∈p̅Ai→⋃i∈p̅Ai satisfies and (2.13) holds with some of the Ki being eventually not less than one. The last part of the proof of Theorem 2.1(iii) leads to the conclusion that ifT:⋃i∈p̅Ai→⋃i∈p̅Ai is not continuous while the composed self-mapping Tp:⋃i∈p̅Ai→⋃i∈p̅Ai is continuous, then the convergence of the iterates to a best proximity point in each adjacent subset Ai(i∈p̅) is still ensured.The existence of a unique fixed point is guaranteed if thep subsets are closed and intersect even if T:⋃i∈p̅Ai→⋃i∈p̅Ai is not continuous and all the constants (2.4) are not less than unity provided that (X,d) is complete and ∏i=1p[Ki]<1 as follows.Theorem 2.3.
Let(X,d) be a complete metric space with p nonempty closed subsets Ai of X,∀i∈p̅, with nonempty intersection satisfying Apj+ℓ≡Aℓ;∀ℓ∈p-1¯, and let T:⋃i∈p̅Ai→⋃i∈p̅Ai be a p(≥2)-cyclic self-mapping subject to T(Ai)⊆Ai+1; ∀i∈p̅ and satisfying the contractive condition (2.1) subject to ∏i=1p[Ki]<1 for constants defined in (2.8). Then, all the sequences {Tpn+jx}n∈N0 are bounded and converge to a unique fixed point in ⋂i∈p̅Ai;∀i∈p̅ of T:⋃i∈p̅Ai→⋃i∈p̅Ai.Proof.
SinceDi=0 (since ⋂i∈p̅Ai≠∅), ∀i∈p̅,∏i=1p[Ki]<1,T(Ai)⊆Ai+1, ⋂i∈p̅Ai is nonempty and closed and (X,d) is complete, it follows that
(2.19)0=lim⁡n→∞d(Tpn+j+1x,Tpn+jx)≤(∏j=i+1p[Kj])lim⁡n→∞d(Tpn+1x,Tpnx)=0,
for any x∈Ai, and then for any x∈⋃i∈p̅Ai, from Theorem 2.1(iii) irrespective of T:⋃i∈p̅Ai→⋃i∈p̅Ai being continuous or not. Thus, for any ε∈R+,∃n0∈N0 such that from triangle inequality for distances, one has
(2.20)d(Tpn+jx,Tpn+j-1x)<εp<ε;∀j∈p̅,d(Tpn+jx,Tpn+jx)≤∑j=1pd(Tpn+jx,Tpn+j-1x)<ε,∀x∈⋃i∈p̅Ai,∀n>n0 so that each {Tpn+jx}n∈N0 is a Cauchy sequence with a limit in the closed nonempty set ⋂i∈p̅Ai which is also bounded since it is convergent. Hence, it follows the existence of fixed points in ⋂i∈p̅Ai being each of those limits. The uniqueness of the fixed point follows from its counterpart in the proof of Theorem 2.1(iii) where the continuity of T:⋃i∈p̅Ai→⋃i∈p̅Ai has not been used. Hence, the theorem.The following result allows to fulfil (2.10) in Theorem 2.1 under some negative scalars ωi and constants Ki exceeding unity provided that a set of necessary conditions involving distances between the adjacent subsets and such scalars are satisfied.Corollary 2.4.
Assume that (2.1) holds and ∏i=1p[Ki]<1 and Di≠0,∀i∈p̅. Then, (2.3), and thus (2.2), equivalently (2.17), holds if αi≥0,βji≥0,γji≥0,δji≥0,μi≥0,∀i∈p̅ provided that the distances between adjacent subsets and the real scalars ωi satisfy the joint constraints:
(2.21)Di≤∑l(≠i)=1p+i-1(∏j=i+1p+i-1[Kj])ωlDl1-∏i=1p[Ki]-∏j=i+1p+i-1[Kj]ωi,∀i∈p̅.Proof.
A necessary condition for (2.2)-(2.3) to hold, with a nonnegative second right-hand side term in (2.2) is that the constraints (2.21) hold.It is wellknown thatp-cyclic nonexpansive self-maps require that the adjacent subsets have all the same pairwise distances. In the case that the relevant self-mappings are contractive are Meir-Keeler contractions, they have a unique fixed point if all the subsets intersect and the metric space is complete. In the case that the subsets do not intersect, there is a unique convergence best proximity point at each subset, to which the iterates through the self-mapping T converge asymptotically, provided that the subsets are nonempty convex and closed and the vector space X defining the metric space is uniformly convex, then also being reflexive and strictly convex, [8]. It is still required that all the distances between adjacent subsets be identical so that, otherwise, the self-mappings from the union of the subsets to itself cannot be nonexpansive [9, 10]; hence, they cannot be contractive. In the following, the condition of all the distances between the adjacent subsets being identical is not longer being required. The price to be paid is that the convergence of the iterates through the self-mapping do not necessarily converge to best proximity points located in the boundaries of the sets but to best proximity points located at the boundaries of appropriate nonempty closed convex subsets of the original subsets of X. In order to facilitate the formalism for the case of distinct distances between adjacent subsets, the maps of interest are restricting their images as the iterations progress in order to asymptotically reach a new set of adjacent subsets all possessing identical pairwise distances. For such a subsequent analysis, first proceed as follows by first introducing the following hypotheses.Hypotheses
(H1) Assume that a sequence of nonempty closed sets{Âij}j∈N0exists such that Âi0≡Aiand Âi,j+1⊆Âij;∀i∈p̅,j∈N and that ∃D:=lim⁡j→∞⁡Dij=lim⁡j→∞⁡dist⁡(Âij,Âi+1,j)>0;∀i∈p̅ satisfying D:=lim⁡j→∞⁡dist⁡(Âij,Âi+1,j)≥max⁡i∈p̅⁡Di,∀i∈p̅.
(H2) Assume that any sequence ofp(≥2)-cyclic self-mappings Tj:⋃i∈p̅Âij→⋃i∈p̅Âij is subject to Tj(Âij)⊆Âi+1,j;∀i∈p̅,∀j∈N0 while all its elements satisfy the contractive condition (2.1), satisfying (2.8)-(2.9), with Kij=Ki with ∏i=1p[Ki]<1 andωij≡ωi:=D/Di-Ki;∀i∈p̅,∀j∈N0.
(H3) Define composed self-mappingT{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅Ai as follows (with a certain abuse of notation):
(2.22)T{jl}pn:=(Tjn2∘∘⋯∘∘p︸Tjn1)∘⋯∘n︸(Tj2∘∘⋯∘∘p︸T1),T{jl}pn+k:=(Tjn̅2∘∘⋯∘∘i︸Tjn̅1)∘(Tjn2∘∘⋯∘∘p︸Tjn1)∘⋯∘n︸(Tj2∘∘⋯∘∘p︸T1),∀k∈p-1¯∪{0},
where {jℓ}ℓ∈N is a (nonnecessarily monotone) increasing sequence of natural numbers fulfilling j1=1,p=jn2-jn1+1 for the given jn1,2∈N, for some n̅2≥n̅1≥n2 if k≥1, with the sequence of natural numbersjℓsatisfying 1≤jℓ≤jn̅2,n1,2,k∈p-1¯∪{0}. Note that in fact T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij) since its image is always restricted to be contained in ⋃i∈p̅(⋂j=1jn̅2Âij) by construction.The reason of the abuse of notation when defining the composed mapping is useful since it is explicitly indicated that we are dealing with the composition ofn times groups of p compositions of the sequence of self-mappings Tj:⋃i∈p̅Âij→⋃i∈p̅Âij. The following result holds.Theorem 2.5.
Any composed mappingT{jℓ}pn+k from ⋃i∈p̅Ai to ⋃i∈p̅(⋂j=1jn̅2Âij), defined by (2.22) under the Hypothesis (H1)–(H3), has the following properties.
(i) If the sequence{jℓ}ℓ∈Nα, defined for some Nα⊂N, is unbounded then,
(2.23)∃lim⁡n→∞d(T{jl}pn+kx,T{jl}pn+k+1x)=D;∀x∈⋃i∈p̅Ai,∀k∈p-1¯∪{0}.
(ii) Assume that the sequenceTj:⋃i∈p̅Âij→⋃i∈p̅Âij, with Tj(Âij)⊆Âi+1,j, converges uniformly on some nonempty subset S⊂⋃i∈p̅Ai (under proper set inclusion) as j→∞. Then, any sequence T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij) converges uniformly to a limit self-mapping, dependent on the sequence {jℓ}ℓ∈Nα,T̂pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1∞Âij) as jℓ,n→∞ provided that S⊇⋃i∈p̅(⋂j=1jn̅2Âij). Also,
(2.24)∃lim⁡n→∞d(T̂pn+kx,T̂pn+k+1x)=lim⁡n→∞d(T{jl}pn+kx,T{jl}pn+k+1x)=D;∀x∈⋃i∈p̅Ai,∀k∈p-1¯∪{0}.
(iii) The limitT̂pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1∞Âij) of any composed sequence of mappings T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij),∀x∈⋃i∈p̅Ai generated by some unbounded sequence {jℓ}ℓ∈Nα has a best proximity point x̅i between the adjacent sets(⋂j=1∞Âij) and (⋂j=1∞Âi+1,j)at each ∂(⋂j=1∞Âij) which is also in Ai,∀i∈p̅ to which all the sequences {T̂pnx}n∈N0, which are then bounded, converge; ∀x∈Ai,∀i∈p̅.
IfDi=D>0,∀i∈p̅, then all sequences{Tpnx}n∈N0;∀x∈Ai converges to a best proximity point of Ai,∀i∈p̅.
If, in addition, the metric space(X,d) is complete andDi=D=0,∀i∈p̅ then all sequences {Tpnx}n∈N0,∀x∈Ai converge to a unique fixed point of Ai,∀i∈p̅.
(iv) Assume thatX is a uniformly convex space and that the subsets Ai are convex and closed. Then, the best proximity points x̅i∈(⋂j=1∞Âij) of the adjacent subsets (⋂j=1∞Âij)⊆Ai and (⋂j=1∞Âi,j+1)⊆Ai+1 are unique for each i∈p̅. Furthermore, x̅i+j=Tjx̅i and d(x̅i,T̂x̅i)=d(T̂jx̅i,T̂j+1x̅i)=d(x̅i+j,x̅i+j+1)=D;∀i,j∈p̅ with x̅i+np=x̅i;∀n∈N0,∀i∈p̅.Proof.
Note that the necessary condition (2.3) for Property (i) holds by construction and take, with no loss of generality, n=(m1jn̅2+m2-k)/p for k∈p̅ and n:=m1jn2+m2/p for k=0 for any given finite positive natural numbers m1,2, dependent on n and k, so that jn̅1≔max⁡1≤ℓ≤n̅2⁡jℓ→∞, if k=0 in (2.22), and also jn̅2:=max⁡1≤ℓ≤n̅2⁡jℓ→∞, if k∈p-1¯, as n→∞ in (2.22) for {jℓ}ℓ∈Nα being some unbounded sequence of natural numbers as ℓ,jℓ→∞. Since Kij≡Ki with ∏i=1p[Ki]<1 and  ωij≡ωi≔D/Di-Ki;∀i∈p̅,∀j∈N0,D≔lim⁡j→∞⁡dist⁡(Âij,Âi+1,j)≥max⁡i∈p̅⁡Di and Tj:⋃i∈p̅Âij→⋃i∈p̅Âij subject to Tj(Âij)⊆Âi+1,j;∀i∈p̅,∀j∈N0, it follows from Cantor’s intersection theorem that any arbitrary intersection ⋂j∈N̂0Âij is nonempty and closed; ∀i∈p̅ andN̂0⊆N0 since Âi0≡Aiand Âi,j+1⊆Âij;∀i∈p̅,∀j∈N0. Then, the composed ∀k∈p-1¯∪{0} mapping T{jℓ}pn from ⋃i∈p̅Ai to ⋃i∈p̅Ai, whose nonempty image is restricted by construction to the subset ⋃i∈p̅(⋂j=1jn̅2Âij) of ⋃i∈p̅Ai, is well-posed for any arbitrary sequence{jℓ} and for any x∈⋃i∈p̅Ai. Thus, one gets from Theorem 2.1, (2.10), that for any given ε∈R+,∃n0=n0(ε)∈N such that:
(2.25)D≤d(T{jl}pn+kx,T{jl}pn+k+1x)≤D+ε;∀x∈⋃i∈p̅Ai,∀n(∈N0)≥n0,∀x∈⋃i∈p̅Ai,∀k∈p-1¯∪{0},
where the sequence of natural numbers {jℓ}ℓ∈N being subject to n=(m1jn̅2+m2-k)/p for k∈p̅ and n:=m1jn2+m2/p for k=0. Thus, it follows that (2.23) holds from (2.25) since m1jn̅2+m2=np+k→∞ that implies jn2,jn̅2→∞ for k∈p̅, and jn2→∞for k=0 so that {jℓ}ℓ∈N̂0diverges to infinity as ℓ→∞ taking values in a subset N̂0 of N of infinite cardinal. Hence, Property (i).
To prove Property (ii), note first that the composed self-mapping mapT{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅Ai has an image restricted to ⋃i∈p̅(⋂j=1jn̅2Âij) by construction. If Tj:⋃i∈p̅Âij→⋃i∈p̅Âij, satisfying Tj(Âij)⊆Âi+1,j, converges uniformly as j→∞ on some subset S of X satisfying ⋃i∈p̅Ai⊃S⊇⋃i∈p̅(⋂j=1jn̅2Âij), then T{jℓ}pn+k can converge uniformly in ⋃i∈p̅(⋂j=1jn̅2Âij) as jℓ→∞. Proceed by contradiction. If T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij) do not converge uniformly, while Tpn+k converges does, then, for any given ε∈R+, there exist some ε0∈R+, some x∈⋃i∈p̅Ai and some n0=n0(ε0),n̅0=n̅0(ε̅)∈N0 and some sequences of nonnegative ordered integers {nt} and {mt} of minimal element n̂0=pmax⁡(n0,n̅0) such that:
(2.26)ε+g≥d(T{nt}pn+kx,Tpm+k+1x)+d(Tpm+k+1x,Tpn+kx)+d(Tpn+kx,T{mt}pn+k+1x)≥d(T{nt}pn+kx,T{mt}pn+k+1x)≥ε0,
after using the triangle property for distances, for some g=g(x)∈R0+ and n,m∈N satisfying
(2.27)nt≥pn+k≥n̂0,mt≥pm+k≥n̂0,∀k∈p-1¯∪{0}.
The chained inequalities (2.25) contradict a choice of ε∈R+ satisfying ε<ε0. Since ε∈R+ is arbitrary, T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij) converges uniformly in (⋂j=1jn̅2Âij) as jn̅1→∞, or trivially, if jn̅1is finite, for all k∈p-1¯∪{0}. On the other hand, the triangle inequality yields
(2.28)d(T̂pn+kx,T̂pn+k+1x)≤d(T̂pn+kx,T{jl}pn+kx)+d(T{jl}pn+kx,T̂pn+k+1x)⟶0asjl,n⟶∞,
∀x∈⋃i∈p̅Ai,∀k∈p-1¯∪{0}, since a uniform convergence of the sequence T{jℓ}pn+kx→T̂pn+kx as jℓ,n→∞, ∀x∈⋃i∈p̅Ai has been proven, so that (2.24) follows. Hence, Property (ii).
To prove Property (iii), remember that form Property (ii),Tj:⋃i∈p̅Âij→⋃i∈p̅Âij and the composed T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij) fulfil:
(2.29)Tjx⟶T̂xasj⟶∞⟹T{jl}pn+kx⟶T̂pn+kx,∀k∈p-1¯∪{0}asjl,n⟶∞,∀x∈⋃i∈p̅Ai.
From (2.24), ∀ε∈R+⇒∃n1=n1(ε) such that for any integers n,m≥n1,
(2.30)d(T̂pmx,T̂pnx)≤ε;d(T̂pnx,T̂pn+1x)=d(T{jl}pnx,T{jl}pn+1x)<D+ε,∀x∈⋃i∈p̅Ai.
Proceed by contradiction to prove the convergence of the iterates to best proximity points of ⋂j=1∞Âij;∀i∈p̅. Assume that there is no x̅i∈Ai such that x̅i=T̂npx̅i∈⋂j=1∞Âij,andD=d(x̅i,T̂x̅i)=d(x̅i+1,T̂x̅i+1);∀i∈p̅ so that there exist some integers n̅,m̅≥max⁡(n1,n2),ε1∈R+ and x∈Ai such that
(2.31)D+2ε>d(T̂pm̅x,T̂pn̅x)+d(T̂pn̅x,T̂pn̅+1x)≥d(T̂pm̅x,T̂pn̅+1x)≥D+ε1.
Since ε is arbitrary, one can choose ε<ε1/2 which yields a contradiction in (2.31). Then, ∃x̅i=T̂npx̅i∈⋂j=1∞Âij such that D=d(x̅i,T̂x̅i)=d(x̅i+1,T̂x̅i+1),∀i∈p̅. Furthermore, x̅i∈Ai since Tj(Âij)⊆Âi+1,j⊆Âi+1,0≡Ai+1,∀i∈p̅,∀j∈N0. If all the distances are nonzero and identical then the best proximity points are at the boundaries of the closed subsets Ai,∀i∈p̅. If, in addition, the metric space is complete and the subsets intersect (i.e., all the distances between adjacent subsets are zero), then ωi=1-Ki and the best proximity points are also coincident in a unique fixed point (Theorem 2.1(iii)) in the nonempty intersection of all such subsets. Property (iii) has been proven.
Property (iv) is proven by contradiction. Assume that there are two distinct best proximity pointsx̅i,y̅i∈(⋂j=1∞Âij) for any given i∈p̅. Define the following real sequences {xn}n∈N, {yn}n∈N, and {zn}n∈N by its general terms as follows:
(2.32)xn=T{jl}pn+1x⟶T̂pn+1x⟶T̂x̅i=T̂pnx̅iasn⟶∞,yn=T{jl}pnx⟶T̂pnx⟶x̅i=T̂pnx̅iasn⟶∞,zn=T{jl}pn+1y⟶T̂pny̅i=y̅i,
generated by a uniformly convergent mapping T{jℓ}pn+k:⋃i∈p̅Ai→⋃i∈p̅(⋂j=1jn̅2Âij),k∈p-1¯∪{0} for some x,y(≠x)∈Ai, some i∈p̅ as the integer sequencesjℓ,jn̅2→∞. The limits above exist from Property (iii). Also, one gets from Property (i), (2.23), and Lemma 3.8 of [4], since X is uniformly convex and (⋂j=1∞Âij) are nonempty and closed ∀i∈p̅, that
(2.33)(d(T{jl}pn+1x,T{jl}pnx)⟶d(T̂pn+1x,T̂pnx)⟶d(x̅i,T̂x̅i)=Dasn⟶∞)∧(d(T{jl}pn+1y,T{jl}pnx)⟶d(T̂pn+1y,T̂pnx)⟶d(x̅i,T̂y̅i)=Dasn⟶∞)(2.34)⟹(d(T{jl}pnx,T{jl}pny)⟶d(T̂pnx,T̂pny)⟶d(x̅i,y̅i)=0),
since the uniformly convex space (X,∥·∥) with a norm ∥·∥ is a Banach space so that we could rewrite the above constraints by replacing the metric d:X×X→R0+ by a metric d1:X×X→R0+, induced by ∥·∥, which is always equivalent to d:X×X→R0+ even in the case that both metric functions are not coincident. Equation (2.34) contradicts x̅i≠y̅i, since there is no n0=n0(ε) for any given ε∈R+ such that d(T{jℓ}pnx,T{jℓ}pny)≤ε,∀n≥n0 implying that d(T{jℓ}pmx,T{jℓ}pmy)>ε for some m≥n0. Thus, (2.33) is false unless x̅i=y̅i;∀i∈p̅. It follows also as a result that the p best proximity points x̅i∈(⋂j=1∞Âij),∀i∈p̅ satisfy x̅i+j=Tjx̅iand d(x̅i,T̂x̅i)=d(T̂jx̅i,T̂j+1x̅i)=d(x̅i+j,x̅i+j+1)=D, ∀i,j∈p̅ with x̅i+np=x̅i, ∀n∈N0, ∀i∈p̅.Note that the existence of nonempty subsetsÂi⊆Ai,∀i∈p̅, in Theorem 2.5 are guaranteed if Ai are bounded (although nonnecessarily closed) and Di≤D≤Di+diam⁡Ai; ∀i∈p̅. Note also that ∏i=1p[Ki+ωi]≥1 is guaranteed since ωi:=D/Di-Ki≥1-Ki,∀i∈p̅.


## 3. Links with General Kannan Mappings andp-Cyclic Kannan Mappings

The next result is concerned withp-cyclic Kannan self-mappings [7, 11, 12] which can eventually satisfy contractive conditions within the class (2.1).Theorem 3.1.
Assume that ap(≥2)-cyclic self-mapping T:⋃i∈p̅Ai→⋃i∈p̅Ai satisfies the contractive condition (2.1) with αi=0,Di=D;∀i∈p̅. Then, the following properties hold.
(i) If(3.1)γi:=11-2ηi-μi[ηi+μi+β1i+β2i2+max⁡(β1i+γ1i+δ1i,β2i+γ2i+δ2i)]<12,∀i∈p̅,
and, furthermore, ωi=(2ηi+μi)(1-Ki) with the constants Ki defined in (2.4), ∀i∈p̅ then, T:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic Kannan self-mapping.
(ii) If, in addition, (2.9) is  replaced by the constraints
(3.2)β2i+γ1i+γ2i+3(δ1i+δ2i+μi+2(β1i+ηi))<1,∀i∈p̅,
and, furthermore,
(3.3)2ηi+μi<1,∀i∈p̅,δ1i+μi+γ1i+β1i+ηi1-2β1i-β2i-δ1i-δ2i-γ2i-2ηi-μi-γ1i≤γi:=11-2ηi-μi[ηi+μi+β1i+β2i2+max⁡⁡(β1i+γ1i+δ1i,β2i+γ2i+δ2i)]<12,∀i∈p̅
then
(3.4)lim⁡n→∞d(Tpn+1x,Tpnx)=D,∀x∈Ai,∀i∈p̅,
which is guaranteed by the condition (2.3). If the p subsets Ai of X,∀i∈p̅, have nonempty intersection and if the metric space (X,d) is complete, then lim⁡n→∞⁡d(Tpn+1x,Tpnx)=0;∀x∈⋃i∈p̅Ai and T:⋃i∈p̅Ai→⋃i∈p̅Ai have a unique fixed point in ⋂i∈p̅Ai to which all the sequences {Tnx}n∈N0, which are then bounded, converge; ∀x∈⋃i∈p̅Ai. If the above p subsets of X are disjoint, closed, and convex, ∀i∈p̅, X is uniformly convex and D>0 then all sequences {Tpnx}n∈N0,∀x∈Ai converge to a best proximity point x̅i of Ai and x̅i+1=Tx̅i=Tpx̅i,∀i∈p̅.Proof.
Ifαi=0,∀i∈p̅, then the contractive condition (2.1) may be upper-bounded as follows for all ∀x,y(≠x)∈Ai; anyi∈p̅ by using the triangle inequality where necessary and upper-bounding the necessary fractions by unity when the denominator is not less than the numerator in the right-hand side of (2.1):
(3.5)d(Tx,Ty)≤(2ηi+μi)d(Tx,Ty)+(β1i+β1i+β2i2+γ1i+ηi+μi)d(x,Tx)+(β2i+β1i+β2i2+γ2i+δ2i+ηi+μi)d(y,Ty)+ωiD,
and, since 2ηi+μi<1, one gets
(3.6)d(Tx,Ty)≤γi(d(x,Tx)+d(y,Ty))+ωiDi,∀x,y(≠x)∈Ai,∀i∈p̅,
with γi∈[0,1/2),∀i∈p̅. Thus, T:⋃i∈p̅Ai→⋃i∈p̅Ai is a p-cyclic Kannan self-mapping. Hence, Property (i).
Furthermore, since (2.9) is replaced by (3.1) for αi=0 so that if Ki∈[0,1/3) in (2.4) and ωi=(2ηi+μi)(1-Ki),∀i∈p̅ in (2.1), it follows from (3.5) that (3.1)–(3.3) guarantee that Ki/(1-Ki)≤γi<1/2 under the necessary condition Ki∈[0,1/3),∀i∈p̅, [11, 12]. Then, the self-mapping T:⋃i∈p̅Ai→⋃i∈p̅Ai satisfies the following contractive condition for y=Tx(≠x)∈Ai and any i∈p̅:
(3.7)d(Tx,Ty)≤γi(d(x,Tx)+d(y,Ty))d(Tx,Ty)≤Kid(x,y)≤Kid(Tx,Ty)+Ki(d(x,Tx)+d(y,Ty))≤γi(d(x,Tx)+d(y,Ty));∀x,y(≠x)∈Ai;∀i∈p̅⟹d(Tx,Ty)≤Ki1-Ki(d(x,Tx)+d(y,Ty)).
Hence, Property (ii) follows directly from Theorem 2.1.Remark 3.2.
Note that there are several contractive conditions discussed in the literature for the noncyclic(p=1) case and eventually for the cyclic (p≥2) case which are particular cases of (2.1), under (2.9), as follows.(1)
Ifμ1∈[0,1) while all the remaining constants are zero, the contractive condition reduces to that of Banach contraction principle, [13]. A similar extended condition for i∈p̅, p≥2 relies on Banach contraction principle for p-cyclic self-mappings.(2)
Ifα1≠0 and μ1∈[0,1), then the contractive condition (2.1) is of the same type as that of [2] and generalizes it.(3)
Ifδ=δi1≠0(i=1,2),μ1≠0 with appropriate constraints and the remaining constants being zero then the contractive condition is that proposed by Chaterjee [14].(4)
Ifη1≠0,μ1≠0, with appropriate constraints and the remaining constants being zero, then the contractive condition is that proposed by Fisher [15].(5)
If1/2>δ=δi1≠0(i=1,2) with the remaining constants being zero, then the contractive condition is that proposed by Kannan, [16, 17], recently generalized in [12]. See also [7, 11]. A similar extended condition for i∈p̅,p≥2 relies on Banach contraction principle for p-cyclic Kannan self-mappings which can be simultaneously contractive [7].(6)
δ1≠0,η1≠0,μ1≠0 with appropriate constraints and the remaining constants being zero, then the contractive condition is that proposed by Reich [18].(7)
γi=β2=0 for i=1,2 and the remaining constants are either zero or nonzero then the contractive condition is that of Bhardwaj et al. [1].(8)
The contractive condition also can include a generalization of Kannan’s fixed point theorem due to Kikkawa and Syuzuki [19], reported in Enjouji et al. [12] if an implication is established to get Kannan’s condition supported by the use of a nonincreasing real function φ:[0,1/2)→(1/2,1] defined as:
(3.8)φ(δ):={1,if0≤δ<2-1,1-α,if2-1<δ≤12,
so that
(3.9)φ(δ)d(x,Tx)≤d(x,y)⟹d(Tx,Ty)≤δ(d(x,Tx)+d(y,Ty)),∀x,y∈X.
This can be addressed with (2.1) as follows. Take α≠0 and 1/2>δ=δi1≠0(i=1,2) and all the remaining constants in (2.1) being zero by defining
(3.10)φ(δ)=δ1-α(ε)-δ={1,if0≤δ<δ0,1-δ+ε,ifδ0<δ≤12,
where δ0=2-1 subject to the constraints δ∈(0,1/2), and
(3.11)α=1-2δif0≤δ<2-1,(3.12)α=α(ε)=1-δ0+δ0+ε-1δ0ifδ0<δ≤12withε≥1-δ0(2-δ0).(9)
Ifp≥2,μi∈[0,1) and ωi=1-μi,∀i∈p̅, then the contractive condition is a p-cyclic contractive condition, [3–7].Remark 3.3.
An important property of Kannan mappings on a metric space(X,d) is that it is complete if and only if every Kannan mapping on X has a fixed point, [12, 19, 20].The first property of the following result follows directly from the contractive condition (2.1) and Remark 3.2.7 by using the fixed point result of [19]. The second part follows directly from Theorem 3.1 by taking the particular condition for Kannan p(≥2)p-cyclic mappings which are simultaneously contractive (see Remarks 3.2.4 and 3.2.8).Theorem 3.4.
The following properties hold.
(i) Let T  be a self-mapping on a complete metric space(X,d) and that (3.9)–(3.12) hold for all x,y∈X with α>0 and 1/2>δ>0(i=1,2). Then, T has a fixed point x̅∈X satisfying x̅=lim⁡n→∞⁡Tnx,∀x∈X.
(ii) LetT:⋃i∈p̅Ai→⋃i∈p̅Ai  be a p(≥2)-cyclic self-mapping, satisfying T(Ai)⊆Ai+1,∀i∈p̅, with p being nonempty closed subsets Ai of X, such that (X,d) is a complete metric space and Apj+ℓ≡Aℓ,∀ℓ∈p-1¯, which all intersect, ∀i∈p̅ satisfying the following particular contractive condition in (2.1): αi>0;1/2>δi=δji≠0(j=1,2;i∈p̅) and the remaining constants are zero with Ki:=2δi/(1-αi-δi)<1/3. Then, T:⋃i∈p̅Ai→⋃i∈p̅Ai is simultaneously a contractive and Kannan p-cyclic self-mapping which has a unique fixed point x̅=lim⁡n→∞⁡Tnx in ⋂i∈p̅Ai,∀x∈⋃i∈p̅Ai.



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*Source: 542941-2011-12-12.xml*