# On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

Authors: M. De la Sen
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-calledr-weaker Meir-Keeler or r,r0-stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed.


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## Body

## 1. Introduction

The background literature on best proximity points and associated convergence properties in cyclic contractions and proximal contractions in the framework of fixed point theory is abundant. See, for instance, [1–21] and references therein. The literature includes related studies on cyclic contractions and cyclic weak contractions and proximal contractions [1–14, 18–21] and proximal weak contractions [15–17]. See also [22–25] for related results. On the other hand, fixed point theory has a wide amount of applications, for instance, in the study of stability of dynamic systems and differential and difference equations. See, for instance, [21, 22, 26]. In this context, the relevance of cyclic contractions and cyclic nonexpansive mappings is also of interest when strips of the solutions of dynamic systems or difference equations have to lie in different time intervals or due to control actions or external events in distinct defined sets.The study of contractions in metric and quasi-metric spaces and in generalized metric and quasi-metric spaces has been focused on in a number of papers. See, for instance, [1–4] and references therein. A group of the obtained results are based on the existing background literature on Meir-Keeler contractive-type results. See, for instance, [5, 6]. In particular, the existence of periodic fixed point theorems of weak contractions in the setting of generalized quasi-metric spaces has been studied in [2], while the existence of fixed points for weak contraction mappings in complete generalized metric spaces has been investigated in [3]. The paper has a section of preliminaries where the concepts of Meir-Keeler functions, weaker Meir-Keeler functions, and stronger Meir-Keeler functions are generalized “ad hoc” to be used to define weak generalized contractive mappings involving subsets of a generalized metric space which do not intersect in general. In this context, appropriate nondecreasing functions φ:0,∞→0,∞, generalized weaker Meir-Keeler functions ϕ:D,∞→D,∞, and stronger Meir-Keeler functions ψ:0,∞→0,1 are used to define the generalized ϕ-φ- and ϕ-ψ-weak p-cyclic contraction mappings defined and studied in this paper. Section 3 gives and proves a set of main results on ϕ-φ-weak p-cyclic contraction mappings and on generalized ϕ-ψ-weak p-cyclic contraction mappings. Such results are related to boundedness and to convergence properties of generalized distances of sequences of points built through generalized ϕ-φ-weak and through generalized ϕ-ψ-weak contractive cyclic maps, either in adjacent subsets or in the same subset, and also on the convergences of sequences either to best proximity points or to fixed points in the case when the subsets of the cyclic disposal intersect. Some illustrative examples adapted to the stated and proved results are also discussed.


## 2. Preliminaries

LetZ and R be the sets of integer numbers and real numbers, respectively, and define their subsets: Z0+=Z+∪0, Z+=z∈Z:z>0, p-=1,2,…,p, R0+=R+∪0, and R+=z∈R:z>0.Definition 1.
For some givenr∈R0+, a mapping ϕ:r,∞→r,∞ is said to be a r-weaker Meir-Keeler function if, for each real number η>r∈R+, there exists a real number δ=δη∈R+ such that ϕn0t<η for some n0=n0η∈Z+, ∀t∈η,η+δ.Definition1 generalizes the two existing definitions below.Definition 2 (see [1, 2]).
A mappingϕ:0,∞→0,∞ which is a 0-weaker Meir-Keeler function is said to be a weaker Meir-Keeler function.Definition 3 (see [1]).
A mappingϕ:0,∞→0,∞ which is a weaker Meir-Keeler function for n0=1 is said to be a Meir-Keeler function.Definition 4.
For some givenr,r0<1∈R0+, a mapping ψ:r,∞→r0,1 is said to be a r,r0-stronger Meir-Keeler function if, for each real number η∈R+, there exist real numbers δ=δη∈R+ and γ=γη∈0,1 such that ψt<γ, ∀t∈η,η+δ.Definition4 generalizes the existing definition below.Definition 5 (see [1, 2]).
A mappingφ:0,∞→0,1 which is a 0-stronger Meir-Keeler function is said to be a stronger Meir-Keeler function.Through the paper, we will use the mappingsϕ, φ, and ψ which belong to the sets of functions defined below.Definition 6.
For some givenr∈R0+, the class Φr is the set of r-weaker Meir-Keeler functions ϕ:r,∞→r,∞ which satisfy the following:ϕ1
ϕt>r for t>r and ϕr=r.ϕ2
ϕnt is decreasing for all t∈r,∞.ϕ3
Fortn⊂r,∞, one has
ϕ31
limsupn→∞ϕtn<θ if  limn→∞tn=θ for any given real number θ>r, andϕ32
there existslimn→∞ϕtn=r if limn→∞tn=r.Definition 7.
The classΓr is the set of nondecreasing functions φ:0,∞→0,∞ which satisfy the following:φ1
φt>r for t>r and φt=t for t∈0,r.φ2
φ is subadditive; that is, for every α1,α2∈r,∞, φα1+α2≤φα1+φα2.φ3
For alltn⊂r,∞, limn→∞tn=r if and only if limn→∞ϕtn=r.Definition 8.
The classψ∈Ψrr0 is the set of r,r0-stronger Meir-Keeler functions ψ:r,∞→r0,1 for some real constant r0∈0,1 which satisfy the following:ψ1
ψt>r0 for t>r and ψr=r0.Definition 9 (see [2, 3]).
LetX be a nonempty set. A generalized metric (g.m.) is a mapping d:X×X→R which satisfies(1)
dx,y≥0, ∀x,y∈X, and dx,y=0 if and only if x=y;(2)
dx,y=dy,x, ∀x,y∈X;(3)
dx,y≤dy,ω+dω,z+dz,y, ∀x,y∈X, ∀ω,z≠ω∈X-x,y.Definition 10 (see [2, 3]).
LetX be a nonempty set and let d:X×X→R0+ be a g.m. on X. Then, X,d is said to be a generalized metric space (g.m.s.).Some basic considerations and properties on a g.m.s.X,d are now quoted from [2] to be then invoked in the body of this paper. Let X,d be a g.m.s. Then, xn⊂X is said to be g.m.s. convergent to x∈X if, for each given ε∈R+, ∃n0=n0ε∈Z0+ such that dxn,x<ε, ∀n>n0∈Z+, and this is denoted by limn→∞xn=x or xn→x as n→∞. If, for each given ε∈R+, ∃n0=n0ε∈Z0+ such that dxn,xn+m<ε, ∀n>n0∈Z+, ∀m∈Z0+, then xn is called a g.m.s. Cauchy sequence in X. If every g.m.s. Cauchy sequence in X is g.m.s. convergent in X, then X,d is called a complete g.m.s.. It has been pointed out in [2] that a g.m.s. Cauchy sequence is not necessarily a Cauchy sequence and that a g.m.s. convergent sequence is not necessarily either Cauchy or a convergent sequence.Example 11 (see [2]).
Consider the setX=jt:j∈5- for some given t∈R+ and define d:X×X→R0+ as follows for some given γ∈R+:(1)dx,x=0,∀x∈Xdx,y=dy,x,∀x,y∈Xdt,2t=3γ;dit,3t=γ,∀i∈2-dit,4t=2γ,∀i∈3-;dit,5t=32γ;∀i∈4-.Then, d:X×X→R0+ is a g.m. and then X,d is a g.m.s.. However, d:X×X→R0+ is not a metric, and then X,d is not a g.m.s., since dt,2t=3γ>dt,3t+d3t,2t=2γ.


## 3. Best Proximity Point and Fixed Point Theorems

We now get some results onϕ-φ-weak contraction mappings and some results on related best proximity points. Given a nonempty abstract set X with p≥2 nonempty subsets Xi, ∀i∈p-, we say that a self-mapping T:⋃i∈p-Xi→⋃i∈p-Xi is a p-cyclic self-mapping if TXi⊆Xi+1, ∀i∈p-, with the notation convention that Xi+np≡Xi, ∀i∈p-, ∀n∈Z0+.In fact, if we extend the above definition to the casep=1, we find trivially that T:X∣X1→X1 can be considered as an 1-cyclic mapping if ⌀≠TX1⊆X1⊆X.Definition 12.
LetX,d be a g.m.s., let Xi be nonempty subsets of X, having a common distance in-between adjacent subsets dXi,Xi+1=D, ∀i∈p-, and let T:⋃i∈p-Xi→⋃i∈p-Xi be a p-cyclic self-mapping satisfying(2)φdTx,Ty≤ϕφdx,y;∀x,y∈Xi×Xi+1,∀i∈p-for some φ∈ΓD and some ϕ∈ΦD. Then, T is said to be a generalized ϕ-φ-weak p-cyclic contraction mapping.Theorem 13.
LetX,d be a g.m.s. and let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ϕ-φ-weak p-cyclic contraction mapping for some ϕ∈ΦD and some φ∈ΓD. Then, the following properties hold:
(i)(3)limn→∞⁡ϕnp+jφdx0,xmp+1=D;∀m∈Z0+,∀j∈p-1¯∪0,limm→∞⁡limn→∞⁡φdxmp+j,xm+np+j+1=limn→∞⁡φdxnp+j,xl+np+j+1=D;∀l∈Z0+,∀j∈p-1¯∪0,limn,m→∞⁡dxmp+j,xm+np+j+1=limn→∞⁡dxnp+j,xl+np+j+1=D;∀l∈Z0+,∀j∈p-1¯∪0,for any sequence xn constructed from xn+1=Txn, ∀n∈Z0+ for some given initial point x0∈⋃i∈p-Xi.
(ii) Any sequenceTxn built from any given initial point x0∈⋃i∈p-Xi is bounded.
(iii) Assume, in addition, thatX,d is a complete g.m.s. and that zi∈Xi has a best proximity point from Xi to Xi+1 (i.e., dzi,Xi+1=D) for some given i∈p- and that Xi+1 is approximatively compact with respect to Xi. Then,(4)Tnpx0⟶zi,Tnpx1⟶zi,Tnp+1x0⟶Tzi,Tnp+x1⟶Tzi,limn→∞φ⁡dTnpx0,Tm+npx1=limn→∞⁡dTnpx0,Tm+npx1=D,limn,l→∞⁡φdTnpx0,Tn+lpx1=limn,l→∞⁡dTnpx0,Tn+lpx1=D,limn→∞⁡φdTnpx0,Tm+np+1x1=limn→∞⁡dTnpx0,Tm+np+1x1=D,limn,m→∞⁡φdTnpx0,Tmp+1x1=limn,l→∞⁡dTnpx0,Tmp+1x1=D,limn,m→∞⁡φdTnpx0,Tmpx1=limn,m→∞⁡dTnpx0,Tmpx1=0,∀x0,x1∈Xj,∀j∈p-,∀m∈Z0+, and for any given ε∈R+, and there is N1=N1ε∈Z+ such that, for any positive integers m>n>N1, one has dTnpx0,Tmp+1x1<D+ε, dTnpx0,Tmpx1<ε, φdTnpx0,Tmp+1x1<D+ε, and φdTnpx0,Tmpx1<ε.
Also, all the best proximity pointszi+1=Tzi=Tjzi+1-j from Xi to Xi+1 are each them unique in Xi+1 if one of them zj, for some j∈p-, is unique in Xj and Xi is closed, ∀i∈p-. Also, each best proximity point is also a fixed point of the respective composite mapping Tp:⋃j∈p-Xj∣Xi→⋃j∈p-Xj,∀i∈p-, and then p-periodic fixed points of T:⋃i∈p-Xi→⋃i∈p-Xi.Proof.
Sincex0∈⋃i∈p-Xi, we can consider equivalently x0 to be an arbitrary point of Xi for some given arbitrary i∈p- and we can define the sequence xn inductively by xn+1=Txn, ∀n∈Z0+. Since T:⋃i∈p-Xi→⋃i∈p-Xi is a ϕ-φ-weak-cyclic p contraction mapping, one gets from (2) by induction for each n,m∈Z0+that(5)φdxnp+j,xn+mp+j+1=φdTxnp+j-1,Txn+mp+j≤ϕφdxnp+j-1,xn+mp+j≤ϕϕφdxnp+j-2,xn+mp+j-1=ϕ2φdxnp+j-2,xn+mp+j-1⋯≤ϕnp+jφdx0,xmp+1;∀m,n∈Z0+;∀j∈p-1¯∪0,where xnp,xn+mp∈Xi, xmp+1∈Xi+1, xnp+j,xn+mp+j∈Xi+j, and xnp-j∈Xi-j+p, ∀n,m∈Z0+, ∀j∈p-1¯∪0. Since ϕ∈ΦD, one has from property ϕ2 that ϕnt is decreasing for all t∈D,∞ so that ϕnp+jφdx0,xmp+1 is decreasing and converges to some limit η0j∈R0+, ∀j∈p-1¯∪0. It is now proved that all the limits η0j=D, ∀j∈p-1¯∪0. Since ϕ∈ΦD, it is also a D-weaker Meir-Keeler function ϕ:D,∞→D,∞ so that, for each real number η>D∈R+, there exists a real number δj=δjη∈R+ such that ϕnp+jt<η, ∀n≥n0j∈Z+ for some n0j=n0jη∈Z+, for any given j∈p-1¯∪0, ∀t∈η,η+δj. Thus, η=D+ε for any given arbitrary η>D∈R+ such that ε=η-D∈R+ is also arbitrary; one has for each given k∈p-1¯∪0 that if xk=Txk-1=Tkx0, then (6)D≤ϕnp+j-kφdxk,xmp+k+1<D+ε;∀j,k∈p-1¯∪0since φ:0,∞→0,∞ is nondecreasing, dxk,xmp+k+1≥D and φt≥D for t≥D with equality standing if and only if t=D. Thus, there exist the p identical limits limn→∞ϕnp+jφdxnp+k,xn+mp+k+1=D, ∀j,k∈p-1¯∪0. Then, one gets from (5) and the constraint ϕ32 of the class ΦD that(7)limm→∞⁡limn→∞⁡φdxmp+j,xm+np+j+1=limn→∞⁡φdxnp+j,xl+np+j+1=D;∀l∈Z0+,∀j∈p-1¯∪0.This also implies from the constraint φ3 of the class ΓD that limn,m→∞dxmp+j,xm+np+j+1=limn→∞dxnp+j,xl+np+j+1=D, ∀l∈Z0+, ∀j∈p-1¯∪0.
Property (i) has been proved.
To prove Property (ii), we use contradiction arguments by assuming that some sequenceTpnx0, generated by T:⋃i∈p-Xi→⋃i∈p-Xi from any x0∈⋃i∈p-Ai, is unbounded. Thus, for any given real constant A∈R+, there is N=NA∈Z0+ such that, for any n∈Z0+≥N, dx0,Tpnx0>A. There is also some B0=B0N,A>A∈R+ such that, for any B>B0∈R+, there is N1=N1A≥N∈Z0+ such that B≥dx0,Tpnx0>A for all n∈N,N1∩Z0+. This is trivial since, by defining α=dx0,TpNx0-A, we can choose any real constant B≥B0=A+α such that, for some nonempty interval of positive integers N,N1, B≥dx0,Tpnx0>A since the inequality holds by construction for the case N=N1. So if Tpnx0 is unbounded, then there is some subsequence Tpnkx0 which diverges so that there are some strictly increasing sequence nk⊆Z0+ and some strictly increasing sequence Ank⊆R0+ such that Ank+1≥dx0,Tpnkx0>Ank and dx0,Tpnk+1x0>Ank+1. Then, there is a strictly increasing sequence of natural numbers nk and a strictly increasing sequence of positive real numbers Ank such that(8)dx0,Tpnk+1x0>Ank+1>Ank,Ank+1≥dx0,Tpnkx0>Ank;∀k∈Z0+and then one gets(9)dx0,Tpnk+1x0Ank>dx0,Tpnk+1x0dx0,Tpnkx0>Ank+1dx0,Tpnkx0≥1;∀k∈Z0+.On the other hand, one has from the rectangular inequality of the g.m.s. X,d(10)dx0,Tpnk+1x0-dx0,Tpnkx0≤dTpnk+2x0,Tpnk+1x0+dTpnk+2x0,Tpnk+1x0,∀k∈Z0+ which leads to limk→∞dx0,Tpnk+1x0-dx0,Tpnkx0=0 from Property (i). Thus, either dx0,Tpnkx0→0 or there is some positive real sequence λnk→1 such that dx0,Tpnk+1x0=λnkdx0,Tpnkx0. If dx0,Tpnkx0→0, then Tpnkx0 is a bounded subsequence of Tpnx0, a contradiction to its claimed unboundedness. Otherwise, if dx0,Tpnkx0 does not converge to zero while λnk→1, with λnk=dx0,Tpnk+1x0/dx0,Tpnkx0, then this contradicts (10). As a result, Tpnx0 for any x0∈⋃i∈p-Xi and Property (ii) has been proved.
It remains to prove Property (iii). SinceXi+1 is approximatively compact with respect to Xi if dx,yn→dx,Xi+1 as n→∞ for some x∈Xi and yn⊂Xi+1, then yn has a convergent subsequence ynk⊆yn [6]. Since zi∈Xi is a best proximity point from Xi to Xi+1, dzi,Tzi=dzi,Xi+1=D. If zi∈Xi is the unique best proximity point from Xi to Xi+1, then it is a p-periodic point of T:⋃j∈p-Xj→⋃j∈p-Xj and a fixed point of Tp:⋃j∈p-∣Xi→⋃j∈p-Xj. This follows by reformulating (5) with initial points Tpzi∈Xi and Tp+1zi∈Xi+1 with zi∈Xi being the unique best proximity point from Xi to Xi+1. One concludes from the properties of the classes ΦD and ΓD that dTnpzi,Tnp+1zi→dTpzi,Xi+1=D and dTnpzi,Tnp+1zi→dzi,Xi+1=D. Since Xi has a unique best proximity point to Xi+1, one concludes that Tpzi=zi. Next, we prove that dTpzi,Tp+1zi=dzi,Xi+1=D. Assume that this property is not true. Thus, there is some ε=εn∈R+ for n∈Z0+ with ε0=liminfn→∞εn>0 such that one gets from (5) and the constraints ϕ2 and ϕ32 for the class ΦD and the constraint φ3 for the class ΓD that(11)φdTnpzi,Tnp+1zi≤ϕn-1pφdTpzi,Tp+1zi=D+εn≤ϕnpφdzi,Tziwhich leads by taking limits as n→∞ to the contradiction D+ε0≤D. Then, dTnpzi,Tnp+1zi=dTpzi,Tp+1zi=D, ∀n∈Z0+. We now use again (5) with x0=zi=Tpzi=Tnpzi∈Xi, ∀n∈Z0+ and an arbitrary x1∈Xi+1 to yield(12)φdzi,Tnp+1x1≤ϕφdzi,Tnpx1=ϕpφdzi,Tn-1px1⋯≤ϕnpφdzi,x1;∀m,n∈Z0+and one concludes that ϕnpφdzi,x1→D, limsupn→∞φdzi,Tnp+1x1≤D, and D≤φdzi,Tnp+1x1≤limsupn→∞φdzi,Tnp+1x1≤D since dzi,Tnp+1x1≥D, ∀n∈Z0+, and then φdzi,Tnp+1x1→D as n→∞ and dzi,Tnp+1x1→dzi,Xi+1=D as n→∞ since ϕ∈ΦD and φ∈ΓD. Since Xi+1 is approximatively compact with respect to Xi, there is a convergent subsequence Tnkp+1x1→zi+1⊆Tnp+1x1⊂Xi+1 for the arbitrary given i∈p-.
It is now proved thatTnp+1x1→zi+1 with zi+1∈Xi+1 for any x1∈Xi. Assume that this is not the case. Then, there exists a sequence mk with mk∈Z0+>nk≥n0k and some n0k∈Z0+such that the subsequence Tmkp+1x1⊆Tnp+1x1⊂Xi+1 does not converge to zi+1. Since T is single-valued, if Tmkp+1x1 does not converge to zi+1, then it does not converge. It cannot have either a convergent subsequence Tmkjp+1x1→z^i+1≠zi+1⊆Tmkp+1x1⊆Tnp+1x1⊂Xi+1 and since then zi would have two distinct images in Xi+1 which is impossible.
Thus, one gets from (12) that(13)D+ε≤φdzi,Tmkp+1x1≤ϕnkpφdzi,Tmk-nkp+1x1;∀m,n∈Z0+for the given i∈p- and some subsequences mk, nk in Z0+ with mk>nk>N0=N0ε; since ϕ∈ΦD is a D-weaker Meir-Keeler function which satisfies ϕ2 and ϕ32, φ∈ΓD is defined from 0,∞ to 0,∞, nondecreasing, and satisfies φ1 and φ3. Thus, one gets from (13) the following contradiction:(14)D+ε≤limk→∞⁡ϕnkpφdzi,Tmk-nkpx1=Dso that Tmkp+1x1→zi+1=Tzi, irrespective of the initial point x1∈Xi, and zi+1 is unique if zi is unique. Since the same contradiction arguments can be used for any claimed nonconvergent subsequence of Tnp+1x1, one concludes that any such a sequence as well as the whole sequence converges Tnp+1x1→zi+1 for the given i∈p-. The sequence is a g.m.s. Cauchy sequence since it is convergent and X,d is a complete g.m.s.. Since Xi+1 is closed, then zi+1∈Xi+1 for the given i∈p- and it is unique if zi is unique. Thus, the set of best proximity points zi∈Xi:i∈p- is unique if any of them is unique.
It is now proved thatlimn,m→∞φdTnpx0,Tmpx1=D, ∀x0∈Xi, ∀x1∈Xi+1, and ∀i∈p-. Proceed by contradiction by assuming that there is ε∈R+ and some subsequences of nonnegative integers mk, nk, and lk such that one gets for any two distinct initial points x0,x1∈Xi from the subadditivity property φ2 of the class ΓD and the rectangular inequality of the generalized metric(15)D+ε≤φdTnkpx0,Tmk+nkp+1x1≤φdTnkpx0,Tmk+nk+lkp+1x0+φdTmk+nk+lkp+1x0,Tmk+nk+lk+1p+1x1+φdTmk+nk+lk+1p+1x1,Tmk+nkp+1x1.Note that φdTnkpx0,Tmk+nk+lkp+1x0→D as k→∞ from Property (i) and also(16)Tmk+nkp+1x1⟶Tzi,Tmk+nk+lkp+1x0⟶Tzi,Tmk+nk+lk+1p+1x1⟶Tziask⟶∞.If the convergence of these subsequences fails, then the conclusion got from (12) fails and then Property (i) is not true. For instance, if Tmk+nkp+1x1→Tzi is false, then dzi,Tn+mp+1x1→D as n→∞ fails. Then,(17)φdTnkpx0,Tmk+nk+lkp+1x0⟶D,dTnkpx0,Tmk+nk+lkp+1x0⟶Dask⟶∞,φdTmk+nk+lkp+1x0,Tmk+nk+lk+1p+1x1⟶0,dTmk+nk+lkp+1x0,Tmk+nk+lk+1p+1x1⟶0as k→∞ since zi+1=Tzi is the unique best proximity point in Xi+1 (since zi is the unique best proximity point in Xi) and it has been already proved that any sequence in each Xi has to converge to its best proximity point if unique. Also (18)φdTmk+nk+lk+1p+1x1,Tmk+nkp+1x1⟶0,dTmk+nk+lk+1p+1x1,Tmk+nkp+1x1ask⟶∞.Thus, the subsequent contradiction is got if φdTnkpx0,Tmk+nkp+1x1→D as k→∞ is not true for any x0,x1≠x0∈Xi:(19)D+ε≤liminfk→∞⁡φdTnkpx0,Tmk+nkp+1x1≤limsupk→∞⁡φdTnkpx0,Tmk+nkp+1x0+limk→∞⁡φdTmk+nk+lkp+1x0,Tmk+nk+lk+1p+1x1+limk→∞⁡φdTmk+nk+lk+1p+1x1,Tmk+nkp+1x1=limsupk→∞⁡φdTnkpx0,Tmk+nkp+1x1=D.Then, one gets(20)limk→∞⁡φdTnkpx0,Tmk+nkp+1x1=limk→∞⁡dTnkpx0,Tmk+nkp+1x1=Dask⟶∞limk→∞⁡φdTnkpx0,Tnk+np+1x1=limk→∞⁡dTnkpx0,Tnk+np+1x1=D,∀m∈Z+ask⟶∞and the convergence properties of all subsequences distances and points also hold for the whole sequence so that(21)Tnpx0⟶ziTn+mp+1x1⟶Tzi,Tn+m+lp+1x0⟶Tzi,Tm+n+l+1p+1x1⟶Tzi;∀m,l∈Z0+asn⟶∞,limn,m→∞⁡φdTnpx0,Tm+np+1x1=limn,m→∞⁡dTnpx0,Tm+np+1x1=D,limn→∞⁡φdTnpx0,Tm+np+1x1=limn→∞dTnpx0,Tm+np+1x1=D,∀m∈Z+,∀x0,x1∈Xi for i∈p- such that Xi possesses a unique best proximity point zi∈Xi. The two above second limit identities follow since the sequences mk⊂Z0+ and lk⊂Z0+ may be replaced by any positive integers without altering the got final conclusion. Since T:⋃i∈p-Xi→⋃i∈p-Xi is a ϕ-φ-weak-cyclic p contraction single-valued mapping, then zi+j=Tjzi∈Xi+j for 1≥i+j≤p are unique best proximity points, so equivalently zi∈Xi, ∀i∈p-, are all unique so that the above limit properties can be extended for any sequences with given initial points x0,x1∈Xj, ∀j∈p-. Since mk⊂Z0+, nk⊂Z0+, and lk⊂Z0+ are strictly increasing sequences which can be chosen independently of each other, the above conclusion is equivalently enounced as follows: for each given ε∈R+, there is some N1=N1ε∈Z0+ such that if nk>mk>N1, then dTnkpx0,Tmkp+1x1<D+ε.
It is now proved thatdzi+1,Tzi+1=D. Define xnp+1=Txnp, ∀n∈Z0+ with xpn⊂Xi+1 having a convergent subsequence xpnk→zi+1 as it has been proved above and pick up any arbitrary initial point x0∈Xi such that T2x0∈Xi+2 for the given arbitrary i∈p-. Then, one gets by using the rectangular inequality of the g.m.s. X,d that (22)D≤dTzi+1,xpnk≤dTzi+1,xpnk+1+dxpnk+l+1,xpnk+1+dxpnk+l+1,xpnk≤dzi+1,xpnk+dxpnk+l+1,xpnk+1+dxpnk+l+1,xpnk;∀l∈Z0+.Since limk→∞dxpnk+l+1,xpnk+1=limk→∞dxpnk+l,xpnk=0, it follows directly that(23)D=dTzi+1,zi=limk→∞⁡dTzi+1,xpnkand Tzi+1 is unique since zi∈Xi is unique and T:⋃i∈p-Xi→⋃i∈p-Xi is single-valued with Tzi+1∈Xi+1 since Xi+1 is closed. As a result, since i∈p- is arbitrary, all the best proximity points in-between adjacent subsets are unique if any of them is unique and satisfy zi+j=Tjzi for any i,j∈p-. It also turns out that Tpzi=zi; then it is a fixed point of the composite mapping Tp:⋃j∈p-Xj∣Xi→⋃j∈p-Xj, ∀i∈p-, and then a p-periodic fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi. Property (iii) has been proved.Note that the classesΦD and ΨDr0 (Definitions 6 and 8) can be redefined as sets of functions from 0,∞ to 0,∞ which are identically zero in 0,D. This generalization is irrelevant in practice since the domains of the functions of the classes ΦD and ΨDr0 used to define the weak contractive mappings involve distances of points in-between adjacent subsets of the cyclic disposal, so distances are not smaller than D. Theorem 13 and also the relevant subsequent results of the paper can be also got with those extended definitions since dx,y≥D, ∀x,y∈Xi×Xi+1, ∀i∈p-, implies that (24)φdx,y≥D,ϕφdx,y≥D;∀x,y∈Xi×Xi+1,∀i∈p-,minx,y∈Xi×Xi+1dx,y=D.In the case that the adjacent subsets Xi are nonempty, closed, and intersecting, we can obtain the subsequent result on the existence of a unique fixed point allocated in their intersection.Theorem 14.
LetX,d be a complete g.m.s. and let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ϕ-φ-weak p-cyclic contraction mapping such that that the sets Xi, ∀i∈p-, are closed and intersect for some ϕ∈Φ0 and φ∈Γ0. Then, there is a unique fixed point z∈⋂i∈p-Xi.Proof.
From Theorem13(i) for D=0, it follows for any x0∈Xi and any i∈p- that(25)limn,m→∞dxmp+j,xm+np+j+1=limn→∞dxnp+j,xl+np+j+1=0;∀l∈Z0+,∀j∈p-1¯∪0for any given x0∈Xi and any given i∈p-. Thus, for any given ε∈R+, there is n0=n0ε∈Z0+ such that dxnp+k,xmp+k<ε∀n≥n0∈Z0+, ∀m≥n+1∈Z0+, and ∀k∈Z0+ and xn is g.m.s. Cauchy sequence. Assume on the contrary that there are subsequences xpnj+k⊆xpn and xpmj+k⊆xpm such that dxpnj+k,xpmj+k≥ε. But this contradicts (25) so that xpn is g.m.s. Cauchy sequence for any given initial point x0∈⋃i∈p-Xi. Since X,d is g.m.s. complete, any xpn+k, ∀k∈Z0+, which is g.m.s. Cauchy sequence is convergent to some zi+k∈Xi+k if x0∈Xi for any given i∈p-, with k≤p-i. It suffices to notice that Xi+k=Xni+k=Xni+k-mp,∀i∈p-, ∀k∈p-1¯∪0, and ∀n,m∈Z+ with m satisfying the constraint ni+k-p/p≤m≤ni+k-1/p (equivalently, p≥ni+k-mp≥1). Since Xi is closed, ∀i∈p-⋂i∈p-Xi≠⌀, xpn+k⊆⋂i∈p-Xi→zi+k,∀i∈p-, ∀k∈p-1¯∪0, and it follows that zj∈⋂i∈p-Xi, ∀j∈p-. Also zj is a fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi, ∀j∈p-. Otherwise, we would get the contradiction 0←dxpn+j,Txpn+j→dzi+j,Tzi+j>0. Since zi=Tk+1zi=TTkzi, ∀i∈p-, ∀k∈Z0+, and T:⋃i∈p-Xi→⋃i∈p-Xi is single-valued, then zj=z∈⋂i∈p-Xi, ∀j∈p-, for the given x0∈Xi and i∈p-.
It remains to prove that the fixed pointz to which the sequences converge is independent on the initial point x0∈⋃i∈p-Xi so that it is unique. Assume that z=Tz,z^=Tz^≠z∈⋂i∈p-Xi. We consider them to be initial conditions of two distinct sequences in some pair of adjacent subsets Xi and Xi+1 for some i∈p- so as to use the generalized ϕ-φ-weak contractive condition (2), and one gets(26)φdTznp+j,Tz^n+mp+j+1≤ϕnp+jφdx0,xmp+1;∀m,n∈Z0+;∀j∈p-1¯∪0which leads to (27)limn→∞ϕnp+jφdTznp+j,Tz^n+mp+j+1=limn→∞φdTznp+j,Tz^n+mp+j+1=dz,z^=limn→∞dTznp+j,Tz^n+mp+j+1=0;∀j,m∈p-1¯∪0which contradicts z≠z^.Note that Theorem14 does not require the sets Xi, ∀i∈p-, to be convex sets so as to prove the uniqueness of the fixed point. Simply, the weak contractive condition (2) is used by considering that two claimed distinct fixed points which are in the nonempty intersection of Xi, ∀i∈p-, belong in any case to two adjacent subsets Xi, while the weak contractive condition guarantees that they are identical. We can extend easily Definition 12 and Theorem 13 to the case when the distances in-between adjacent subsets are distinct as follows. The existence and uniqueness of fixed points for generalized ϕ-φ-weak (noncyclic) contraction mappings of the type (2) on a g.m.s. X,d have been proved in [3].Sufficient conditions for the uniqueness of uniqueness of the best proximity points in each of the subsetsXi⊆X, ∀i∈p-, are discussed in the subsequent result which is related to the consideration of a complex structure on the metric space.Theorem 15.
LetX,d be a complete g.m.s. admitting midpoints which has p nonempty closed subsets Xi⊆X, with dXj,Xj+1=D, ∀j∈p-, where X is an abstract set, with one of the subsets Xi, for some i∈p-, being strictly convex. Assume that there is some strictly increasing function δ:0,1→0,2 such that X has the following uniform convexity property for all x,y∈Xi, p∈X, R∈R+, and r∈0,2R:(28)max⁡dx,p,dy,p≤R∧dx,y≥r⟹dx+y2,p≤1-δrRR.Let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ϕ-φ-weak p-cyclic contraction mapping for some ϕ∈Φ0 and some φ∈Γ0. Then, T:⋃i∈p-Xi→⋃i∈p-Xi has a unique best proximity point zj∈Xj, ∀j∈p-, which is also a p-periodic point of T:⋃i∈p-Xi→⋃i∈p-Xi and a fixed point of the composite mapping Tp:⋃j∈p-Xj∣Xk→⋃j∈p-Xj, ∀k∈p-.
If⋂i∈p-Xi≠⌀, then zk=z∈⋂i∈p-Xi, ∀k∈p-, is the unique fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi.Proof.
Assume thatXi for some i∈p- is a strictly convex set. Since T:⋃i∈p-Xi→⋃i∈p-Xi is a generalized ϕ-φ-weak p-cyclic contraction mapping, it follows from Theorem 13 that dTpn+jx,Tpn+j+1x→D as n→∞, ∀j∈p-, for any given x∈⋃j∈p-Xj. Note that there is some integer ji=ji,x0∈p-1¯∪0 such that x=xx0=Tjix0∈Xi for any given x0∈⋃j∈p-Xj (i.e., x=xx0 belongs to the strictly convex subset Xi⊂X) for the given initial point x0∈⋃j∈p-Xj and dTpnx,Tpn+1x=dTpn+jix0,Tpn+ji+1x0→D with x∈Xi, Tpnx⊂Xi, Tpn+1x⊂Xi+1, and some x=xx0 for any given x0∈⋃j∈p-Xj. We now take initial conditions x∈Xi for sequences built T:⋃i∈p-Xi→⋃i∈p-Xi with x=xx0 for any given x0∈⋃j∈p-Xj such that x=xx0=Tjix0∈Xi and Tpn+jix⊂Xi for some ji∈p-1¯∪0. Then, it follows that(29)dTpn+jx,Tpn+j+1x⟶Dasn⟶∞,dTp+1n+jx,Tpn+j+1x⟶Dasn⟶∞with Tpn+jx⊂Xi+j, Tjx∈Xi+j, Tpn+j+1x⊂Xi+j, and Tj+1x∈Xi+j. Then, for every given ε∈R+, there exists an integer N0=N0ε,x such that dTpn+jx,Tpn+j+1x≤D+ε, ∀n≥N0∈Z0+.
One now firstly proves thatdTn+mp+jx,Tpn+jz→0 as n→∞, ∀m∈Z0+, ∀x,z∈Xi; equivalently, dTpn+jx,Tpm+jz→0 as n,m→∞, so that, for every ε∈R+, there is N1=N1ε,x,z such that, for all integer numbers m>n≥N1, dTpn+jx,Tpm+jz≤ε, ∀x,z∈Xi. Assume that the property dTpn+jx,Tpm+jz→0 as n,m→∞, ∀x,y∈Xi, does not hold. Thus, there exists some ε0∈R+ such that, for each k∈Z0+, there exist integers mk>nk≥k for which dTpnk+jx,Tpmk+jz≥ε0. Choose γ∈0,1 such that ε0/γ>D and choose ε such that 0<ε<minε0/γ-D,δγD/1-δγ. For such a chosen arbitrary constant ε∈R+, there exists N0=N0ε,x∈Z0+ such that for all integer numbers mk>nk≥N0, dTpnk+jx,Tpmk+j+1x≤D+ε for all x∈Xi, and also there exists N2=N2ε,z∈Z0+ such that for mk>nk≥N2 and all z∈Xi, dTpnk+jz,Tpmk+j+1z≤D+ε, ∀j∈p-1¯∪0, from Theorem 13(i). Similar distance constraints also hold from Theorem 13(i) for appropriate subsequences of nonnegative integers mk and nk (with mk>nk) exceeding certain finite thresholds, dependent on ε, for each initial conditions x,z∈Xi (Xi being a strictly convex subset of X) and y∈Xi+1; that is, dTpnk+jx,Tpmk+jy≤D+ε for all x∈Xi,y∈Xi+1 for mk>nk≥N0 and some n0=n0ε,x,y∈Z0+, ∀j∈p-1¯∪0, and dTpnk+jz,Tpmk+jy≤D+ε for all z∈Xi, some n2=n2ε,z,y∈Z0+, ∀j∈p-1¯∪0, and all mk>nk≥n2. Now, note that since the set Xi is strictly convex, xmk⊂Xi and zmk⊂Xi, it follows that(30)xmk+znk2⊂Xi,dxmk+znk2,Xi+1≥dXi,Xi+1≥D;∀k∈Z0+.Now, choose n1=maxn0,n2 so that one gets from (28) for all integers mk>nk≥n1 that the following contradiction holds:(31)dxmk+znk2,ynk≤1-δε0D+εD+ε=D+ε-D+εδε0D+ε<D.Since ε is arbitrary, it can be chosen so that δε0/D+ε>δγD/D+ε>ε/D+ε since δε is strictly increasing in 0,1. Then, for each given ε∈R+, there is some n3=n3ε,x,z such that, for all integer numbers m>n≥n3, dTpn+jx,Tpm+jz≤ε, ∀x,z∈Xi, such that the set Xi is strictly convex for some i∈p-, and then (32)limn,l→∞dTpn+jx,Tpl+jz=limn→∞dTpn+jx,Tpm+n+j+kw=0,∀x,z∈Xi, ∀m∈Z0+, ∀j∈p-1¯∪0, and w∈Xi-k for any given k∈p-1¯∪0 implying that Tpn+jx, Tpm+n+j+kw are sequences in Xi+j. From Theorem 13(i) (see (5)), it follows that dTpnx,Tpn+1z→D as n→∞, ∀x,z∈Xi. From Theorem 13(ii), it follows that Tpnx, Tpnz are bounded sequences in the strictly convex set Xi. From the above discussion in this proof, it follows that dTpnx,Tpn1z→0 as n→∞. Since d is a generalized metric and X,d is a complete g.m.s., it follows that Tpnx and Tpnz are bounded and g.m.s. convergent sequences in Xi to xi∈Xi and zi∈Xi, respectively, and then g.m.s. Cauchy sequences and dxi,Xi+1=dzi,Xi+1=D. Finally, since Xi is strictly convex, there cannot exit two distinct xi,zi∈Xi with the minimum distance D to Xi+1, and thus xi=zi. As a result, there exists a unique best proximity point zi∈Xi such that Xi is strictly convex and closed. Since T:⋃i∈p-Xi→⋃i∈p-Xi is single-valued, then Tjzi=zi+j is the unique best proximity point in Xi+j, ∀j∈p-, which satisfies dTjzi,Tj+1zi=dXi+j,Xi+j+1=D, ∀j∈p-. Then, T:⋃i∈p-Xi→⋃i∈p-Xi has a unique best proximity point zk∈Xk, ∀k∈p- (since Xk is closed, ∀k∈p-), which is also, by construction, a p-periodic point of T:⋃i∈p-Xi→⋃i∈p-Xi and a fixed point of the composite mapping Tp:⋃j∈p-Xj∣Xk→⋃j∈p-Xj, ∀k∈p-. It turns out that if ⋂i∈p-Xi≠⌀, then zk=z∈⋂i∈p-Xi, ∀k∈p- is the unique fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi.Remark 16.
(
1
) Note that X,d in Theorem 15 is a uniformly convex metric space (not specifically a uniformly convex Banach space) and X might be an abstract set (nonnecessarily being a linear space). However, it is obvious that any uniformly convex Banach space is also a uniformly convex metric space. In this case, it is assumed that one of the subsets of the cyclic disposal, for which the generalized ϕ-φ-weak p-cyclic contraction mapping is defined, is strictly convex and subject to a uniform convexity-type condition (28) for the elements of such a set with respect to the points of X. That condition is related to the uniformly convex structure of the metric space and, equivalently, to the existence of midpoints in the sense that if (28) fails to hold, then the metric space is not uniformly convex. Such a condition, supported by the existence of midpoints, is close to the one satisfied on uniformly convex Banach spaces X,d [27] and used in Lemmas 3.7-3.8 and Theorem 3.10 of [6] to prove the existence and uniqueness of fixed points in a 2-cyclic contractive condition on the union of two nonempty closed and convex subsets of X. See also [23–25]. In particular, X could be endowed with a simpler structure than a linear space as, for instance, a group endowed with a composition law, provided that the subsets of the cyclic disposal are closed while just one of them is strictly convex.
(
2
) Note also that X could even be just an abstract set with nonempty closed subsets with one of them being strictly convex.
(
3
) Note that Theorem 15 is applicable to the case that X, is a uniformly convex Banach space with a norm-induced metric d so that X,≡X,d with nonempty closed convex subsets Xi⊂X, ∀i∈p-. This is a direct consequence of the fact that if X,≡X,d is a uniformly convex Banach space, then it is also complete and the norm triangle inequality can be trivially upper-bounded by quadrangular-type sum of norms. This allows adding distances related to distinct subsequences converging to zero to the usual triangle inequality leading to be able to characterize the complete X,d also as being a complete g.m.s. [28].
(
4
) Finally, note that the assumption of the existence of midpoints [27] can also be focused on under the framework of convex structures perhaps at the expenses of a more involved presentation. It is said that a mapping WX,X,0,1→X is a convex structure on X if, for each x,y,z∈X, dz,Wx,y,λ≤λdx,z+1-λdy,z, ∀λ∈0,1. See, for instance, [29–33]. Note that x=Wx,x,λ for any λ∈0,1 and dy,Wx,x,λ=dx,y for any x,y∈X. If z=x and λ=1/2, then z1=Wx,y,1/2 satisfying 1/2dx,y=dx,z1=dx,Wx,y,1/2≤1/2dx,y is the midpoint of the segment x,y. The convex structure allows characterizing uniformly convex metric spaces as triples X,d,W associated with metric spaces X,d [32]. See also [28, 34–36]. For a generalized metric and its associate quadrangular constraint of distances, we can define in the same way a generalized convex structure WX,X,X,0,1×0,1→X such that, for w,x,y,z∈Z, one has(33)dw,Wx,y,z,λ1,λ2≤λ1dw,x+λ2dw,y+1-λ1-λ2dw,z;∀λ1,λ2∈0,1and the choice λ1=λ2=1/3, w=x, and ω=Wx,y,z,1/3,1/3 leads to (34)3dx,ω=3dx,x+13dx,y+13dx,z=dx,y+dx,z.Thus, if y is the midpoint of x,z, then ω is located at 1/3 from x and 2/3 from y of the length x,z.Example 17.
Consider the set of real numbersX=jt:±j∈5- for some given t∈R+ and consider nonempty closed subsets X1=jt:j∈5-∪0 and X2=-X1∖0 of X. For some given real constants γ∈R+,γ1∈0,3/5γ, and γ2∈max⁡γ,3γ-2γ1,1, a g.m. d:X×X→R is defined as follows:(35)dx,x=0,∀x∈Xdx,y=dy,x,∀x,y∈Xd0,jt=d-jt,0=jγ1,j∈5-dt,2t=d-t,-2t=3γdjt,3t=d-jt,-3t=γ,j∈2-djt,4t=d-jt,-4t=2γ,j∈3-djt,5t=d-jt,-5t=32γ,j∈4-d-it,jt=dit,-jt=γ2;i,j∈5-.Note that dX1,X2=γ1 and that d:X×X→R is not a metric since d±t,±2t=3γ>d±t,±3t+d±3t,±2t=2γ, and then X,d is not a metric space, while it is a g.m.s. since (36)maxx,y∈X⁡dx,y=3γ=dt,2t≤dt,x+dx,y+dy,2t=dt,0+d0,-t+d-t,2t=2γ1+γ2,where x∈X-t,2t is such that dt,x=mindt,z:z∈X-t,2t. For instance, take x=0. y∈X-t,2t,x, for the above x∈X-t,2t, is such that dx,y=mindx,z:z∈X-t,2t,x.
For instance, takex=-t. Define a 2-cyclic mapping T:X1∪X2→X1∪X2 as follows:(37)T±it=∓i-1tfori∈5-≥2T-t=0;Tt=T0=-twhich is a generalized ϕ-φ-weak p-cyclic contraction for some φ∈Γγ1, for instance, it can be defined so as to satisfy the constraints φγ1=γ1 and φx+φy≥φz≥z>γ1 for x,yz=x+y∈γ1,γ2, and some ϕ∈Φγ1 chosen to satisfy ϕγ1=γ1 and ϕx=ργ2 for x∈γ1,γ2 for some ρ∈R+≥1 so as to be nondecreasing and subadditive. Then, T is a generalized ϕ-φ-weak 2-cyclic contraction which converges to a limit cycle -1,0 with the two best proximity points 0∈X1 and -1∈X2. Theorem 13 leads to the above conclusion.
Note that aϕ-φ-weak 1-cyclic contraction on X1 is defined by the sequence {t,2t,4t,5t,t,0}, with φ∈Γγ1 and ϕ∈Φγ1 defined under the same above constraints, leading to the distances in-between adjacent subsets given by the sequence {3γ,2γ,3/2γ,3/2γ,γ1}.Example 18.
Reconsider Example17 with the same set X and the same metric with X1=jt:j∈5-∪0 and X2=-X1. Then, X1∩X2=0 and dX1,X2=0. The map T:X1∪X2→X1∪X2 is defined by T±it=∓it for i∈5-≥2, Tt=T-t=0, and T0=0. Then, T is a generalized 2-cyclic contraction which converges to its fixed point 0∈X1∩X2. Theorem 14 leads to that conclusion.Example 19.
How to combine Examples17 and 18 with the constraint (28) is now checked, assumed as hypothesis in Theorem 15. Such a constraint is close to the property of uniformly convexity of Banach spaces 8 (see also Remark 16). Now, for all real constant ε∈0,2 such that dx,y≥ε, there is some real constant δ∈0,1 such that dx,-y≤21-δ. Several cases can occur:(a)
2≥jγ1=d0,x=d0,-x≤21-δj for x∈X-0, j∈5-.This is solved with strictly increasingδj=δj for j∈5- if 0<δj=jσ≤1-jγ1/2; j∈5-, that is, if σ∈0,2-5γ1/10, γ1∈0,2/5 and δj=jσ, j∈5-.(b)
2≥dx,y=λγ⇒dx,-y=γ2≤21-δλγ,forx,y∈X-0 and for x,y∈X2 with δλγ=λγ≤1-γ2/2 for λ=1,2,3,3/2 which leads to γ2≤21-λγ.(c)
2≥dx,-y=γ2⇒dx,y=λγ≤21-δγ2,forx,y∈X1-0 (and, equivalently, for x,y∈X2) with δλγ=λγ≤1-γ2/2, with the strictly increasing δγ2=ργ2≤1-λγ/2, for λ=1,2,3,3/2 and some real constant ρ>0.The various above parametrical constraints are compatible with those in Example 17, and the constraint (28) of Theorem 15 holds, if (38)σ∈1,2-5γ110,γ1∈0,max⁡25,32γ,ρ>0,γ2∈max⁡5σ,γ,3γ-2γ1,min⁡2-6γ,1ρ1-32γ.However, due to the discrete nature of the set X, the assumption of Theorem 15 on strict convexity of sets towards the proof of existence of unique best proximity points is not fulfilled by this example.Definition 20.
LetX,d be a g.m.s., let Xi be a set of nonempty p≥2 subsets of X distances in-between adjacent subsets dXi,Xi+1=Di, ∀i∈p-, and let T:⋃i∈p-Xi→⋃i∈p-Xi be a p-cyclic self-mapping satisfying(39)φi+1dTx,Ty≤ϕiφidx,y;∀x,y∈Xi×Xi+1,∀i∈p-for some φi∈ΓDi and some ϕi∈ΦDi and each i∈p-, where the classes ΓDi and ΦDi are defined with functions φi:0,∞→0,∞ and ϕi:Di,∞→Di,∞, ∀i∈p-.
Then,T is said to be a generalized ϕ-φ-weak p-cyclic contraction mapping.Theorem 21.
LetX,d be a g.m.s. and let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ϕ-φ-weak p-cyclic contraction mapping. Then, the following properties hold:
(i)(40)limn→∞⁡ϕni,i-pφi-p-1dx0,xmp+1=Dilimm→∞⁡limn→∞⁡φi-p-1dxmp,xm+np+1=limn→∞⁡φi-p-1dxnp+j,xl+np+j+1=Di,∀l,m∈Z0+, ∀j∈p-1¯∪0, ∀i∈p-, and for any given x0∈Xi and any given arbitrary i∈p-, where ϕi,i-p=ϕi-p·…·ϕi.-1·ϕi is a composite mapping of the p functions ϕi∈ΦDi for i∈p-.
(ii) Assume thatX,d is a complete g.m.s. and that zi∈Xi has a best proximity point from Xi to Xi+1 (i.e., dzi,Xi+1=Di) for some given i∈p- and that Xi+1 is approximatively compact with respect to Xi. Then, there is a best proximity point zi+1=Tzi from Xi+1 to Xi which is unique and in Xi+1 if zi is unique and Xi+1 is closed.
Also, if all the subsetsXi⊆X, ∀i∈p-, are closed, then there are p best proximity points zi+j=Tjzi∈Xi+j, ∀i,j∈p-, which are unique if anyone of them is unique. Each zi∈Xi is also a fixed point of the composite mapping Tp:⋃j∈p-Xj∣Xi→⋃j∈p-Xj, ∀i∈p-, and then a p-periodic fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi.Proof.
The proof is close to that of Theorem13. Let x0 be an arbitrary point of Xi for some given arbitrary i∈p- and define the sequence xn inductively by xn+1=Txn, ∀n∈Z0+. Since T:⋃i∈p-Xi→⋃i∈p-Xi is a ϕ-φ-weak p-cyclic contraction mapping, one gets from (39) by induction for each n,m∈Z0+ that(41)φi+1dxnp,xn+mp+1=φi+1dTxnp-1,Txn+mp≤ϕiφidxnp-1,xn+mp+≤ϕi-1ϕiφi-1dxnp-2,xn+mp-1=ϕi-1ϕiφi-1dxnp-2,xn+mp-1⋯≤ϕi,i-pφi-pdxn-1p,xn+m-1p-1⋯≤ϕni,i-pφi-pdx0,xmp+1;=ϕni,i-pφidx0,xmp+1;∀m,n∈Z0+since ϕi≡ϕi-p and φi≡φi-p, ∀i∈p-, where ϕi,i-p=ϕi-p,…,ϕi.-1·ϕi, ∀i∈p-. Since ϕi∈ΦDi,∀i∈p-, one has from property ϕ2 and the composite mapping structure that ϕni,i-pt is decreasing for all t∈Di,∞ so that ϕnpφdx0,xmp+1 is decreasing and converges to some limit η0i∈R0+. It is now proved that all the limits η0j=D, ∀j∈p-1¯∪0. Since ϕ∈ΦD, it is also a D-weaker Meir-Keeler function ϕ:D,∞→D,∞ so that, for each real number η>D∈R+, there exist a real number δj=δjη∈R+ such that ϕnp+jt<η, ∀n≥n0j∈Z+ for some n0j=n0jη∈Z+, for any given j∈p-1¯∪0, ∀t∈η,η+δj. Thus, η=D+ε for any given arbitrary η>D∈R+ such that ε=η-D∈R+ is also arbitrary; one has for each given k∈p-1¯∪0 that if xk=Txk-1=Tkx0, then (42)ϕi-1,…,ϕi-jϕni,i-pφi-jdxk,xmp+k+1<Di-j+ε,∀m∈Z0+,∀j,k∈p-1¯∪0, and ∀n≥n0j∈Z+, and then there exist the p identical limits limn→∞ϕi-1,…,ϕi-jϕni,i-pφi-jdxk,xmp+k+1=Di-j, ∀m∈Z0+,∀j,k∈p-1¯∪0. Then, one gets from (42) that(43)limm→∞⁡limn→∞⁡φi+jdxmp+j,xm+np+j+1=limn→∞⁡φi+jdxlp+j,xl+np+j+1=Di+j;∀l∈Z0+,∀j∈p-1¯∪0.The notion ofψ-φ-weak p-cyclic contraction mapping is given below.Definition 22.
LetX,d be a g.m.s., let Xi be nonempty subsets of X, having a common distance in-between adjacent subsets dXi,Xi+1=D, ∀i∈p-, and let T:⋃i∈p-Xi→⋃i∈p-Xi be a p-cyclic self-mapping satisfying(44)φdTx,Ty≤ψφdx,yφdx,y;∀x,y∈Xi×Xi+1,∀i∈p-for some φ∈ΓD and some ψ∈ΨDr0. Then, T is said to be a generalized ψ-φ-weak p-cyclic contraction mapping.Theorem 23.
LetX,d be a g.m.s. and let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ψ-φ-weak p-cyclic contraction mapping for some function ψ∈ΨDr0, some r0∈0,1, and some φ∈ΓD. Then, the following properties hold:
(i)(45)limn→∞⁡ϕnp+jφdx0,xmp+1=D;∀m∈Z0+,∀j∈p-1¯∪0,limm→∞⁡limn→∞⁡φdxmp+j,xm+np+j+1=limn→∞⁡φdxnp+j,xl+np+j+1=D;∀l∈Z0+,∀j∈p-1¯∪0,limn,m→∞⁡dxmp+j,xm+np+j+1=limn→∞⁡dxnp+j,xl+np+j+1=D;∀l∈Z0+,∀j∈p-1¯∪0for any sequence xn constructed from xn+1=Txn, ∀n∈Z0+ for any given initial point x0∈⋃i∈p-Xi.
(ii) Any sequenceTxn built from any given initial point x0∈⋃i∈p-Xi is bounded.
(iii) Assume, in addition, thatX,d is a complete g.m.s. and that zi∈Xi has a best proximity point from Xi to Xi+1 (i.e., dzi,Xi+1=D) for some given i∈p- and that Xi+1 is approximatively compact with respect to Xi. Then, there is a best proximity point zi+1=Tzi from Xi+1 to Xi which is unique in Xi+1 if zi is unique in Xi which is closed for the given i∈p-.
Also, if all the subsetsXi of X; ∀i∈p- are closed, then there are p best proximity points zi+j=Tjzi∈Xi+j, ∀i,j∈p-, which are unique if one of them is unique, with each of them being also a fixed point of the respective composite mapping Tp:⋃j∈p-Xj∣Xi→⋃j∈p-Xj, ∀i∈p-, and then p-periodic fixed points of T:⋃i∈p-Xi→⋃i∈p-Xi.Proof.
Sincex0∈⋃i∈p-Xi, we can consider equivalently x0 to be an arbitrary point of Xi for some given arbitrary i∈p- and we can define the sequence xn inductively by xn+1=Txn, ∀n∈Z0+. Since T:⋃i∈p-Xi→⋃i∈p-Xi is a ψ-φ-weak-cyclic p contraction mapping, one gets from (44) that(46)φdxnp+j,xn+mp+j+1=φdTxnp+j-1,Txn+mp+j≤ψφdxnp+j-1,xn+mp+j·φdxnp+j-1,xn+mp+j<φdxnp+j-2,xn+mp+j-1,∀n,m∈Z0+, ∀j∈p-1¯∪0, provided that dxnp+j-2,xn+mp+j-1>D, and then the sequences φdxnp+j-2,xn+mp+j-1, ∀j∈p-1¯∪0, bounded from below are strictly decreasing, so convergent to some η∈R+≥D, since φ∈ΓD, 1>ψt>r0 for t>D and ψD=r0<1 since ψ∈ΨDr0 and the property ψ1 of the class ΨDr0. Note that xnp,xn+mp∈Xi, xmp+1∈Xi+1, xnp+j,xn+mp+j∈Xi+j, and xnp-j∈Xi-j+p, ∀n,∀m∈Z0+. It is be proved from (46), since φD=D, that η=D so that(47)limn→∞⁡φdxnp+j,xn+mp+j+1=limn→∞⁡dxnp+j,xn+mp+j+1=D;∀j∈p-1¯∪0.Assume that this is not the case, so that η>D, and proceed by contradiction to conclude that η=D. Suppose that limn→∞φdxnp+j,xn+mp+j+1=η>D for some j∈p-1¯∪0. Thus, for each δ∈R+, there is nδ=nδη∈Z0+ and γη=γηη∈0,1 such that for all n∈Z0+≥nδ(48)η≤φdxnp+j,xn+mp+j+1<η+δ;ψφdxnp+j,xn+mp+j+1<γη.∀j∈p-1¯∪0, ∀n∈Z0+ since φdxnp+j,xn+mp+j+1→η and since ψ∈ΨDr0 is a stronger Meir-Keeler function. Then, one gets from (46) since dxnp+j,xn+mp+j+1≥D and φdxnp+j,xn+mp+j+1≥φD≥D, ∀j∈p-1¯∪0, ∀n,m∈Z0+, that(49)D<η=limn→∞⁡φdxnp+j,xn+mp+j+1=φdTxnp+j-1,Txn+mp+j≤max⁡D,limsupn→∞⁡ψφdxnp+j-1,xn+mp+j·φdxnp+j-1,xn+mp+j≤max⁡D,limsupn→∞⁡γnp+jφdxnp+j-1,xn+mp+j⋯≤max⁡D,limsupn→∞⁡γηn-nδp+jφdxnδp,xnδ+mp+1=D;∀j∈p-1¯∪0,∀m∈Z0+since γη<1, a contradiction. Then, for any m∈Z0+,(50)limn→∞⁡φdxnp+j,xn+mp+j+1=limn→∞⁡dxnp+j,xn+mp+j+1=D;∀j∈p-1¯∪0.Property (i) has been proved. Property (ii) is proved in the same ways as its counterpart in Theorem 13 since it is based on Theorem 13(i) which still holds for this theorem and from the rectangular property of distances of the g.m.s. X,d. Finally, Property (iii) follows from the given assumptions, similar to those of Theorem 13(ii) since (50) has been got from (46) which is a similar property to that used in the proof of Theorem 13(iii).Based on (50) and (46) obtained for the proof of Theorem 23, it is direct to prove in a similar way as it has been done for Theorem 14, Theorem 15 (see also Remark 16), and Theorem 21, the following results.Theorem 24.
LetX,d be a complete g.m.s. and let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ψ-φ-weak p-cyclic contraction mapping such that that the sets Xi; ∀i∈p- are closed and intersect for some ψ∈Ψ00 and φ∈Γ0. Then, there is a unique fixed point z∈⋂i∈p-Xi.Theorem 25.
LetX,d be a complete g.m.s. admitting midpoints which has p nonempty closed subsets Xi⊆X, with dXj,Xj+1=D, ∀j∈p-, where X is an abstract set, with one of the subsets Xi, for some i∈p-, being strictly convex. Assume that there is some strictly increasing function δ:0,1→0,2 such that X has the property (28) for all x,y∈Xi, p∈X, and R∈R+,r∈0,2R. Let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ψ-φ-weak p-cyclic contraction mapping for some ψ∈ΨDr0, some r0∈0,1, and some φ∈ΓD. Then, T:⋃i∈p-Xi→⋃i∈p-Xi has a unique best proximity point zj∈Xj, ∀j∈p-, which is also a p-periodic point of T:⋃i∈p-Xi→⋃i∈p-Xi and a fixed point of the composite mapping Tp:⋃j∈p-Xj∣Xk→⋃j∈p-Xj, ∀k∈p-.
If⋂i∈p-Xi≠⌀, then zk=z∈⋂i∈p-Xi, ∀k∈p-, is the unique fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi.Theorem 26.
LetX,d be a g.m.s. and let T:⋃i∈p-Xi→⋃i∈p-Xi be a generalized ψ-φ-weak p-cyclic contraction mapping. Then, the following properties hold:
(i)(51)limn→∞⁡ψni,i-pφi-p-1dx0,xmp+1=Dilimm→∞⁡limn→∞⁡φi-p-1dxmp,xm+np+1=limn→∞⁡φi-p-1dxnp+j,xl+np+j+1=Di,∀l,m∈Z0+, ∀j∈p-1¯∪0,  ∀i∈p-, and for any given x0∈Xi and any given arbitrary i∈p- for some set of functions ψi∈ΨDir0i, ∀i∈p-, some real constants r0i∈0,1, ∀i∈p-, where ψi,i-p=ψi-p·…·ψi.-1·ψi is a composite mapping of p such functions.
(ii) Assume thatX,d is a complete g.m.s. and that zi∈Xi has a best proximity point from Xi to Xi+1 (i.e., dzi,Xi+1=Di) for some given i∈p- and that Xi+1 is approximatively compact with respect to Xi. Then, there is a best proximity point zi+1=Tzi from Xi+1 to Xi which is unique and in Xi+1 if zi is unique and Xi+1 is closed.
Also, if all the subsetsXi⊆X, ∀i∈p-, are closed, then there are p best proximity points zi+j=Tjzi∈Xi+j, ∀i,j∈p-, which are unique if anyone of them is unique. Each zi∈Xi is also a fixed point of the composite mapping Tp:⋃j∈p-Xj∣Xi→⋃j∈p-Xj, ∀i∈p-, and then a p-periodic fixed point of T:⋃i∈p-Xi→⋃i∈p-Xi.Example 27.
Examples17–19 can be reformulated for generalized ψ-φ-weak p-cyclic contraction mappings with the replacement ϕ∈ΦD→ψ∈ΨDr0 with some r0∈γ2,1 since γ2∈0,1 and D=dX1,X2<1.



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*Source: 4186960-2016-04-19.xml*