# About Fixed Points in CAT(0) Spaces under a Combined Structure of Two Self-Mappings

Authors: Manuel De la Sen
Journal: Journal of Mathematics
 (2017)

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## Abstract

This paper investigates some fixed point-related questions including the sequence boundedness and convergence properties of mappingsQtSx,Ty=tSx⊕1-tTy defined in CAT0 spaces, which are parameterized by a scalar t∈0,1, where S,T:  X→X are nonexpansive Lipschitz-continuous mappings and X,d is a metric space which is a CAT0 space.


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## Body

## 1. Introduction

ACATk space, where k is a real number related to the curvature, is a type of metric space where triangles of potential vertices being each set of three points are thinner, that is, of length being less than or equal to the corresponding so-called comparison triangles (namely, those whose sides have the same lengths as the sides of the original triangle) in the model spaces. The curvature in a CATk space is bounded from above by k. A particular case of CATk spaces [1–3] is that arising when the curvature is bounded from above by 0 (CAT0 spaces. See, for instance, [4–10]). Complete CAT0 spaces, being often referred to as Hadamard spaces (in honor to Jacques Hadamard), generalize Hilbert spaces to the nonlinear framework. In Hadamard spaces, there is a unique geodesic path joining each pair of given points. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space. It is well-known that Hadamard spaces satisfy the following inequality:(1)d2z,m+d2x,y4≤12d2z,x+d2z,yfor each of the given points x,y,z and some point m, where m is the unique midpoint of x and y, that is, dx,y=2dx,m=2dy,m, since Hadamard spaces are uniquely geodesic.The study of geodesic paths is very relevant in spherical geometry, for instance, in the composition of trajectories through the Earth surface or in common planetary studies of distances. A metric spaceX,d is a geodesic metric space if any two points x,y∈X can be joined by an arc length parameterized continuous curve (geodesic segment) γ:a,b→X,γa=x,γb=y whose length Lγ=dx,y. It is well-known that a geodesic metric space X,d is a CAT(0) space if every geodesic triangle Δ in X satisfies the CAT0 inequality; namely, the distance between any two points of such a triangle is less than or equal to the distance between the corresponding points of the model triangle in the Euclidean space, that is, a triangle Δ- with sides of the same length as the sides of Δ and then of the same perimeter. Therefore, the study of the metric properties of CAT0 spaces has a major importance. In general, a geodesic metric space X,d is a CATk space if every geodesic triangle in X with perimeter less than 2Dk satisfies the CATk inequality. The so-called CAT0 spaces are of curvature k=0 and they are particular spaces of the most general CATk spaces of curvature k∈R. To fix some general basic ideas, let us denote by Mk the unique 2-dimensional Riemannian manifold with constant curvature k of diameter Dk being +∞ if k≤0 and π/k for k>0. The CATk spaces are those whose geodesic triangles Δ, that is, those having geodesic segments as its sides, satisfy the so-called CATk inequality; namely, the distances between points of Δ are less than or equal to the distances between the corresponding points in their comparison triangles Δ′ in the model space Mk (i.e., those triangles whose sides have the same lengths as their counterparts as their corresponding Δ-triangles). Note that if the metric space X,d is a CATk space, then there is a unique geodesic segment with joins x and y (with dx,y<Dk if k>0) [3, 11].Furthermore, it is well-known thatCATk spaces are also CATl spaces for l>k and that n- dimensional hyperbolic spaces with their usual metric are CAT-1 spaces and then CAT0 spaces, whose simpler example is the n-dimensional Euclidean space with its usual metric, so also CAT1 spaces as well, whose simpler example is the unit sphere. Other relevant CAT0 spaces are the so-called Euclidean buildings, which are abstract simplicial complexes, and the so-called CAT0 cube complexes, [12–15]. The first ones give a systematic procedure for geometric interpretation of semisimple Lie groups and the study of semisimple groups over general fields, while the second ones are important, for instance, in the modelling process of robot trajectories on eventually irregular surfaces with eventual obstacles and predesigned admissible corridors for trajectory tracking.This paper has two subsequent body sections. Section2 gives some relating results for some of the various existing concepts of convexity in metric spaces X,d such as 1-convexity, p-convexity, midpoint convexity, convex structure, uniform convexity and near-uniform convexity, and Busemann curvature and its relation to convexity. Section 3 gives and proves some relevant properties related to uniform convexity and near-uniform convexity of geodesic metric CAT(0) spaces. It also studies mappings of the form Qt:X×X→X parameterized by a scalar t∈0,1, defined by QtSx,Ty=tSx⊕1-tTy in a metric space X,d, where S,T:X→X are Lipschitz-continuous while not necessarily contractive mappings; that is, the Lipschitz constants are not necessarily less than unity. In particular, the convergence properties of sequences built from such mappings are formally studied and some conditions of existence of fixed points are given. Some illustrative examples are also presented and discussed. The main aim and motivation of the formal study of the sequences generated by such parameterized mappings, which are constructed with two generator mappings on a metric space X,d, on CAT0 spaces. The obtained and proved results rely on the boundedness and nonexpansive and contractive properties of the sequences generated by such mappings depending on the contractiveness/nonexpansiveness of both generator mappings. A potential application is the generation of admissible corridors whose extreme obstacle-free trajectories are defined by sequences generated by the two mentioned generator mappings. The generators of the corridor extreme trajectories are two primary mappings which define the studied parameterized mapping on the CAT0 space. Those defined extreme trajectories can be bounded and/or convergent for each parameter value of the parameterized mapping of interest and could define the admissible trajectories of a robotic device or a movable body. One of the examples is concerned with this view on some potential applications.Notation
Z and R denote the sets of integer and real numbers.
Z
0
+
=
z
:
z
≥
0.
Z
+
=
z
:
z
>
0.
R
0
+
=
z
:
z
≥
0.
R
+
=
z
:
z
>
0.
c
l
A is the closure of the set A.
c
o
n
x
i
i
∈
I is the closure of the convex hull of the family xii∈I.
F
i
x
T denotes the set of fixed points of a mapping T:X→X.


## 2. Some Preliminary Definitions and Results on Convexity, Uniform Convexity, and Curvature

LetX,d be a complete metric space. It is said that it admits (nonnecessarily unique) midpoints if for any x,y∈X, there is a z=mx,y∈X such that dx,y=2dx,z=2dy,z. Such a point z∈X is said to be a midpoint of x and y and X,d is a geodesic space, [16–18].Definition 1 (p-convexity [16, 19]).
Suppose a metric spaceX,d which admits midpoints (or which has midpoints or which is midpoint convex). Then, X,d is said to be p-convex for some p∈1,∞ if, for each x,y,z∈X and each midpoint mx,y∈X of x and y, (2)dmx,y,z≤121/pdpx,z+dpy,z1/p.For the case p=∞, the right-hand side of (2) is defined as a limit leading to dmx,y,z≤max⁡mx,y,z. If X,d is ∞-convex, it is equivalently said to be ball convex, while if it is 1-convex it is equivalently said to be distance convex [19]. X,d is said to be strictly p-convex for p∈1,∞, if the inequality is strict for x≠y and strictly 1-convex if the inequality is strict for p=1 if dx,y>dx,z-dy,z [16].Definition1 leads to the direct conclusion below.Assertion 2.
If a metric spaceX,d is midpoint convex, then it is 1-convex.Assertion 3.
IfX,d be p-convex for some p∈1,∞ then, for any x,y,z∈X and each midpoint mx,y∈X of x and y, (3)dmx,y,z≤121/pdx,z+dy,z≤12dx,z+dy,z.Proof.
It is direct from (2) and Minkowski inequality or by direct rearrangement of the power 1/p to its inverse in the left-hand side of (2).Inequality (2) leads to the following direct result.Assertion 4.
IfX,d be p-convex for some p∈1,∞ then, for any x,y,z,w∈X, each midpoint mx,y∈X of x and y and each midpoint mz,w∈X of z and w, (4)dmx,y,mz,w≤14dpx,z+dpx,w+dpy,z+dpy,w1/p.Proof.
Note from (2) that(5)dpy,mx,z≤12dpy,x+dpy,zand, one gets from (2) and (5) that(6)dmx,y,mz,w≤121/pdpx,mz,w+dpy,mz,w1/p≤121/p12dpx,z+dpx,w+dpy,z+dpy,w1/p,leading to the result.Definition 5 (p-Busemann curvature [16, 20]).
Suppose a metric spaceX,d which admits midpoints. Then, X,d is said to satisfy the p-Busemann curvature condition for some p∈1,∞ if, for each x,y,z,w∈X, each midpoint mx,y∈X of x and y and each midpoint mz,w∈X of z and w, one has (7)dmx,y,mz,w≤121/pdpx,z+dpy,w1/p.Assertion 6.
Suppose a metric spaceX,d which admits midpoints, with the midpoint map (or midset) m:X×X→X being unique, and which satisfies the p-Busemann curvature condition for some p∈1,∞. Then, one has(8)dmx,y,mz,w≤121/pmin⁡dpx,z+dpy,w1/p,dpx,w+dpy,z1/pfor any x,y,z,w∈X, where mx,y∈X and mz,w∈X are, respectively, the unique midpoints of x and y and w and z.Proof.
SinceX,d admits midpoints if c=mz,w, then the midpoint is unique since m:X×X→X is unique [19], then c=mw,z. Thus, we can replace mz,w→mw,z leading to an alternative right-hand side in (7) under the replacements z→w and w→z which when combined with (7) leads to (8).Since the right-hand side of (4) is an upper-bound of the right-hand side of (8), we get directly the following important result.Assertion 7.
Assume that a metric spaceX,d is midpoint convex (then it is 1-convex from Assertion 2) with unique midpoint map and that it satisfies the p-Busemann curvature condition for some p∈1,∞. Then, X,d is p-convex.The following technical definitions are of interest to characterize near-uniform convexity.Definition 8 (ε-separated family of points [16]).
A family of pointsxii∈I is ε-separated if infi∈I⁡dxi,xj≥ε.Definition 9 (nearly uniformly convex space [16]).
A∞-convex metric space X,d is said to be nearly uniformly convex if, for any R>0 and for any ε-separated infinite family xii∈I, with xi∈X, and any y∈X such that dxi,y≤r≤R, ∀i∈I, there is some ρ=ρε,R>0 such that B1-ρry∩clconxii∈I≠⌀, where clconxii∈I is the closure of the convex hull of the family xii∈I.It turns out that∞-convexity implies near-uniform convex but the converse is not true.Definition 10 (see [21, 22]).
LetX,d be a metric space. A mapping W:X×X×0,1→X is said to be a convex structure on X if, for each x,y,t∈X×X×0,1 and z∈X, (9)dz,Wx,y,t≤tdz,x+1-tdz,y.Definition 11 (see [21]).
A convex metric spaceX,d,W is said to be uniformly convex if, for any ε>0, there exists δ=δε∈0,1 such that, for any r>0 and x,y,z∈X with max⁡dx,z,dy,z≤r and dx,y≥rε,(10)dz,Wx,y,12≤r1-δ.A uniformly convex metric space X,d,W is also referred to commonly as uniformly 1-convex [16]. This concept may be generalized as follows.Definition 12.
A convex metric spaceX,d,W is said to be uniformly p-convex if, for any ε>0, there exists δ=δε∈0,1 such that, for any r>0 and any x,y,z∈X with max⁡dx,z,dy,z≤r and dx,y≥rε,(11)dz,Wx,y,12≤r1-δ1/p.Proposition 13.
If a convex metric spaceX,d,W is uniformly convex then it is uniformly p-convex for any p≥1.Proof.
SinceX,d,W is uniformly convex then, for any ε1>0, there exists δ1=δε1∈0,1 such that (10) holds with δ=δ1 any r>0 and any x,y,z∈X subject to max⁡dx,z,dy,z≤r and dx,y≥rε1. Thus, dx,y≥rε1=rpε1/rp-1. Choosing ε1=εprp-1 for any given p≥1 and any given arbitrary εp>0, it follows for δp∈max⁡1-r1-p1-δ1,0,1 that dpz,Wx,y,1/2≤rp1-δp≤r1-δ1 so that X,d,W is uniformly p(≥1)-convex.Proposition 14.
A convex metric spaceX,d,W is nearly uniformly convex if, for any ε>0, there exists a strictly increasing δ=δε∈0,1 such that, for any r>0, any z∈X, and any μ=εr-separated infinite family xii∈I⊂X satisfying supi∈I⁡dxi,z≤r and(12)dy,Wxi,xj,12≤r1-δ;∀xi,xj≠xi∈xii∈I⊂Xfor some y∈B1-δrz, where Bαz denotes an open ball of radius αcentred at z.Proof.
Setzij=Wxi,xj,1/2∈X for any given xi,xj≠xi∈xii∈I. Thus, if K⊂X, one has from (12) that(13)dy,clcozij≤dz,zij≤r1-δ,since z∈⋂i,j≠iclcoziji,j≠i∈I since supi∈I⁡dxi,z≤r so that (14)clB1-δrz∩clcoziji,j≠i∈I≠⌀.Since B1-δrz⊃y is trivially nonempty and coziji,j≠i∈I=coxii∈I then B1-δrz∩clcoxii∈I≠⌀. Note also that, for any fixed R≥r, δ=δε,R=sup0<r≤R⁡δε,r exists fulfilling (12). From Definition 9, X,d,W is nearly uniformly convex.Note thatX,d,W is nearly uniformly convex; then it is not necessarily uniformly convex since it can happen that, for some y≠z∈B1-δrz, any εr-separated infinite family xii∈I⊂X satisfying supi∈I⁡dxi,y≤r and all δ=δε∈0,1, there exists some δ0=δ0ε∈0,1 such that (15)dy,Wxi,xj,12≤r1-δ0;dz,Wxi,xj,12>r1-δ.However, the converse is true as reflected in the next result.Proposition 15.
If a convex metric spaceX,d,W is uniformly convex, then it is nearly uniformly convex.Proof.
Note that if (10) is satisfied with supi∈I⁡dxi,z≤r, infi∈I⁡dxi,y≥εr for any given z∈X, then (12) is satisfied for y=z∈B1-δrz, the ball with center at such z and radius 1-δr.Proposition 16.
IfX,d is nearly uniformly convex and strictly ∞-convex then, for any R>0 and for any ε-separated infinite family xii∈I, with xi∈X, and any y∈X such that dxi,y≤r≤R;∀i∈I, there is ρr=ρrε,ε1,r∈0,1-ε1/r such that dxi,y≥ε1=infx≠y∈X⁡dy,x,clBε1y∩clcon⁡xii∈I≠⌀ and contains at most two points of X such that clBε1y∩clconxii∈I=w with w≠y if y∉xi:i∈I and clBε1y∩clconxii∈I=y,w if y∈clconxii∈I with a choice ρr=1-ε1/r.Proof.
SinceX,d is nearly uniformly convex and strictly ∞-convex then any nonempty closed convex subset of X is a Chebyshev set [16]. Thus, for any y∈X, there are balls Bε1y whose closures clBε1+δy⊇y,w are Chebyshev sets consisting of at least two distinct points if their radius [16, 23] 1-ρrr=ε1+δ≥ε1=infx≠y∈X⁡dy,x for any δ≥0 with ρr=ρrε,ε1,r≤ρ=ρε,R. Since Chebyshev sets have a unique nearest neighbor in X for each y∈X, [24], one has for δ=0 that clBε1y=y,w is a Chebyshev set of two unique elements one of them being its center z. Then, one has since X,d is nearly uniformly convex that clB1-ρrry∩clconxii∈I⊃B1-ρrry∩clconxii∈I≠⌀ for any ρr∈0,1-ε1/r,Bε1-σy∩clconxii∈I=⌀ for any σ>0,clBε1y∩clconxii∈I=w with w≠y if y∉clconxii∈I with ρr=1-ε1/r,clBε1y∩clconxii∈I=y,w if y∈clconxii∈I with ρr=1-ε1/r.


## 3. Some Results on Contractiveness and Nonexpansiveness in CAT(0) Spaces

A metric spaceX,d is (uniquely) geodesic if every two points x,y of X are joined by a unique geodesic segment x,y which is the image of the geodesic path from x to y, that is, the isometry c:0,l→X such that c0=x, cl=y and l=dx,y. A geodesic triangle Δx1,x2,x3 consists of three vertices x1,x2,x3∈X and three geodesic segments joining each pair of vertices. A model comparison triangle Δ-x-1,x-2,x-3 of the geodesic triangle Δx1,x2,x3 is a triangle in the Euclidean space R2 such that dxi,xj=dR2x-i,x-j for i,j=1,2,3. A geodesic metric space is a CAT0 space [4, 16, 21, 22] if dx,y≤dR2x-,y-; ∀x,y∈Δ,∀x-,y-∈Δ- (CAT0 inequality). See, for instance, [1, 4–8].In the paper, we write1-tx⊕ty for the unique zt∈X such that dz,x=tdx,y and dz,y=1-tdx,y. Note that the midpoint of x,y is z1/2=mx,y=1/2x⊕1/2y.Note the following from the basic results in Section2.(1) A geodesic space is a CAT0 space if and only if for any x,y,z∈X and all t∈0,1 the following inequality is satisfied:(16)d21-tx⊕ty,z≤1-td2z,x+td2z,y-t1-td2x,y.(Proposition  1.1 [4]).(2) A CAT0 space is uniformly p-convex for any p≥2 [16].(3) A CAT0 space satisfies inequalities (4) and (5) for any p≥2 since it is midpoint p≥2-convex for any p≥1.(4) A CAT0 space satisfies the p-Busemann curvature condition for any p≥1 from (7).A general technical result involving constructions with two self-mappings in aCAT0 space as follows.Lemma 17.
Let a metric spaceX,d be a CAT(0) space and let the mapping Qt:X×X→X be defined by (17)Qtx,y=tx⊕1-ty;∀t∈0,1for any x,y∈X and let T,S:X→X be two self-mappings which satisfy the following conditions:(18)dTx,Ty≤KTdx,y;dSx,Sy≤KSdx,yfor any given, some positive real constants KT and KS. Then, for any given x,y,p,q∈X and for any t∈0,1, the following properties hold:(i)
(19a)d2QtSp,Tx,QtSq,Ty≤t2KS2d2p,q+1-t2KT2d2x,y+t1-tdSp,Ty+dSq,Ty+dTx,Sq+dSp,Txmin⁡KSdp,q,KTdx,y(19b)d2QtSp,Tx,QtSq,Ty≤t2KS2+1-t2KT2max⁡d2x,y,d2p,q+t1-tdSp,Ty+dSq,Ty+dTx,Sq+dSp,Txmin⁡KSdp,q,KTdx,y(19c)d2QtSp,Tx,QtSq,Ty≤t2KS2+1-t2KT2+4t1-tmin⁡KS,KT×max⁡d2x,y,d2p,q,dSp,Ty,dSq,Ty,dTx,Sq,dSp,Txmin⁡dp,q,dx,y.(ii)
(20a)d2QtSnp,Tnx,QtSnq,Tny≤t2KS2nd2x,y+1-t2KT2nd2p,q+t1-tdSn-1p,Tn-1y+dSn-1q,Tn-1y+dTn-1x,Sn-1q+dSn-1p,Tn-1x×min⁡KSdSn-1p,Sn-1q,KTdTn-1x,Tn-1y.(20b)d2QtSnp,Tnx,QtSnq,Tny≤t2KS2n+1-t2KT2nmax⁡d2x,y,d2p,q+t1-tdSn-1p,Tn-1y+dSn-1q,Tn-1y+dTn-1x,Sn-1q+dSn-1p,Tn-1xdp,q×min⁡KSn,KTnmax⁡dp,q,dTx,Ty(20c)d2QtSnp,Tnx,QtSnq,Tny≤t2KS2n+1-t2KT2n+4t1-tmin⁡KSn,KTn×max⁡d2x,y,d2p,q,dSn-1p,Tn-1y,dSn-1q,Tn-1y,dTn-1x,Sn-1q,dSn-1p,Tn-1x×max⁡dp,q,dTx,Ty;∀n∈Z+Proof.
From (16) to (18), one gets:(21)d2QtSp,Tx,QtSq,Ty=d2tSp⊕1-tTx,tSq⊕1-tTy≤td2Sp,tSq⊕1-tTy+1-td2Tx,tSq⊕1-tTy-t1-td2Sp,Tx≤t2d2Sp,Sq+t1-td2Sp,Ty-t21-td2Sq,Ty+t1-td2Tx,Sq+1-t2d2Tx,Ty-t1-t2d2Sq,Ty-t1-td2Sp,Tx=t2d2Sp,Sq+1-t2d2Tx,Ty+t1-td2Sp,Ty+d2Tx,Sq-t1-t2+t21-td2Sq,Ty-t1-td2Sp,Tx=t2d2Sp,Sq+1-t2d2Tx,Ty+t1-td2Sp,Ty+d2Tx,Sq-d2Sq,Ty-d2Sp,TxOn the other hand, one has by completion of squares that(22a)d2Sp,Ty+d2Tx,Sq-d2Sq,Ty-d2Sp,Tx=dSp,Ty+dSq,TydSp,Ty-dSq,Ty+dTx,Sq+dSp,TxdTx,Sq-dSp,Tx(22b)d2Sp,Ty+d2Tx,Sq-d2Sq,Ty-d2Sp,Tx=dSp,Ty+dSp,TxdSp,Ty-dSp,Tx+dTx,Sq+dSq,TydTx,Sq-dSq,Tyand from the triangle inequality for distances in (22a) and (22b) and the use of (18), one gets(23)max⁡dSp,Ty-dSq,Ty,dTx,Sq-dSp,Tx≤min⁡dSp,Sq,dTx,Ty≤min⁡KSdp,q,KTdx,y.The substitution of (23) into (22a) and (22b) with the use of (18) yields(24)d2Sp,Ty+d2Tx,Sq-d2Sq,Ty-d2Sp,Tx≤dSp,Ty+dTx,Sq+dSq,Ty+dSp,Tx×min⁡KSdp,q,KTdx,y.Now, the replacement of (18) and (24) into (21) leads to(25)d2QtSp,Tx,QtSq,Ty≤t2KS2d2p,q+1-t2KT2d2x,y+t1-tdSp,Ty+dSq,Ty+dTx,Sq+dSp,Txmin⁡KSdp,q,KTdx,y≤t2KS2+1-t2KT2max⁡d2p,q,d2x,y+t1-tdSp,Ty+dSq,Ty+dTx,Sq+dSp,Txmin⁡KSdp,q,KTdx,y,which implies (19a)–(19c) and the proof of Property (i) is complete. On the other hand, one has directly from (19a)–(19c) after replacing p→Sn-1p,  q→Sn-1q,  x→Sn-1x,  y→Sn-1y for n∈Z+ that (20a)–(20c) holds for any n∈Z+ as counterpart of (19a)–(19c) and Property (ii) is proved.From Lemma17(ii), we get the following result.Lemma 18.
Let a metric spaceX,d be a CAT(0) space and consider the mapping defined in (17) via two self-mappings T,S:X→X satisfying (18). Then, the following properties hold:
(i) Assume thatmax⁡KS,KT<1 (i.e., S,T:X→X are both strictly contractive). Then, limn→∞⁡dQtSnp,Tnx,QtSnq,Tny=0 for any x,y,p,q∈X and t∈0,1.
(ii) Assume that(a)
eitherKS∈0,1 (i.e., S:X→X is strictly contractive), KT=1 (i.e., T:X→X is nonexpansive but noncontractive), and T:X→X has a fixed point(b)
KT∈0,1 (i.e., T:X→X is strictly contractive), KS=1 (i.e., S:X→X is nonexpansive but noncontractive), and S:X→X has a fixed point.
Then,(26)lim⁡supn→∞⁡d2QtSnp,Tnx,QtSnq,Tny-min⁡t2,1-t2max⁡d2x,y,d2p,q≤0for any x,y,p,q∈X and t∈0,1.
(iii) IfT,S:C→C are both nonexpansive, where C is a nonempty closed convex subset of X, then (27)lim⁡supn→∞⁡d2QtSnp,Tnx,QtSnq,Tny-t2+1-t2max⁡d2x,y,d2p,q+t1-tMp,q,x,ymax⁡dp,q,dx,y≤0for any x,y,p,q∈X and t∈0,1.Proof.
It follows for constantsKS∈0,1 and KT∈0,1 that both T:X→X and T:X→X are strict contractions on X so that one gets from (20a)–(20c) that(28)lim⁡supn→∞⁡d2QtSnp,Tnx,QtSnq,Tny-t2KS2nd2x,y-1-t2KT2nd2p,q+t1-tmin⁡KSn,KTnmax⁡dp,q,dx,y×dSn-1p,Tn-1y+dSn-1q,Tn-1y+dTn-1x,Sn-1q+dSn-1p,Tn-1x≤0.Since T:X→X and S:X→X are strict contractions on X, KSn→0 and KSTn→0. Thus, ∃limn→∞⁡dQtSnp,Tnx,QtSnq,Tny=0 for any x,y,p,q∈X and t∈0,1. Property (i) has been proved.
Now, assume that eitherKS∈0,1 and KT=1 (then T:X→X has a fixed point since X,d is a CAT0 space, [5, 17, 18]), or KT∈0,1 and KS=1 (then S:X→X has a fixed point) and t∈0,1. Then, one has from (20a)–(20c) that min⁡KSn,KTn→0:(29)lim⁡supn→∞⁡d2QtSnp,Tnx,QtSnq,Tny-min⁡t2,1-t2max⁡KS2nd2x,y,KT2nd2p,q+t1-tmin⁡KSn,KTnmax⁡dp,q,dx,y×dSn-1p,Tn-1y+dSn-1q,Tn-1y+dTn-1x,Sn-1q+dSn-1p,Tn-1x≤0.Assume with no loss in generality that T:X→X is strictly contractive KT<1 with a unique fixed point y∗∈X as a result, since X,d is complete, and S:X→X is nonexpansive KS=1 and has a fixed point p∗∈X by hypothesis. Then,(30)dSnp,Tny≤dSnp,Snp∗+dp∗,y∗+dTny∗,Tny≤dp,p∗+dp∗,y∗+KTndy,y∗≤dp,p∗+dp∗,y∗+dy,y∗<∞;∀n∈Z0+dSnq,Tny≤dq,p∗+dp∗,y∗+dy,y∗<∞;∀n∈Z0+dSnp,Tnx≤dp,p∗+dp∗,y∗+dx,y∗<∞;∀n∈Z0+dSnq,Tnx≤dq,p∗+dp∗,y∗+dx,y∗<∞;∀n∈Z0+.Thus, the sequence dSnp,Tny+dSnq,Tny+dTnx,Snq+dSnp,Tnx is bounded for any given p,q,x,y∈X for any distance d:X×X→R0+, and (31)t1-tmin⁡KSn,KTnmax⁡dp,q,dx,y⟶0since min⁡KSn,KTn→0. Furthermore, since KS,KT∈0,1 with either KS=1 or KT=1, then one has from (28) that for t∈0,1:(32)lim⁡supn→∞⁡KS2nt2+KT2n1-t2max⁡d2x,y,d2p,q≤min⁡t2,1-t2max⁡d2x,y,d2p,q≤max⁡d2x,y,d2p,q,the one which is contractive has a unique fixed point since X,d is complete; then Snz, Tnw, dSn-1p,Tn-1y and then dSn-1q,Tn-1y, dTn-1x,Sn-1q and dSn-1p,Tn-1x are bounded sequences for any given z,w,x,y,p,q∈X. Thus, one gets from (29)–(32) that (26) holds. Property (ii) has been proved.
Property (iii) follows since (27) holds from (28) with a finite real M depending on the points p,q,x,y of X.(33)M=Mp,q,x,y≥dSnp,Tny+dSnq,Tny+dTnx,Snq+dSnp,Tnx.Related to Lemma18(ii), note that if T,S:C→C are both nonexpansive, where C is a nonempty closed convex subset of X, and at least one of them is strictly contractive and the other one has a fixed point then (26) holds. This occurs despite that although a CAT0 space is uniformly p≥2 convex [16], it can be nonuniformly convex, in general. However, the existence of (at least) a fixed point for the (noncontractive) expansive mapping T or S on C is guaranteed [5, 17, 18]. The results below are concerned with sufficient conditions for the uniform convexity and near-uniform convexity of CAT(0) spaces.Proposition 19.
Let a geodesic metric spaceX,d,W be a CAT(0) space. Then, it is uniformly convex if, for any ε∈0,2, there exists a strictly increasing δ=δε∈0,1-2-ε2/4, with δε>0 for ε>0, such that dz,Wx,y,t≤r1-δ for t∈0,1, any r>0 and any x,y,z∈X with max⁡dx,z,dy,z≤r and 2r≥dx,y≥rε.Proof.
A geodesic space is a uniformly convex CAT(0) space if, for anyt∈0,1, (34)d21-tx⊕ty,z≤1-td2z,x+td2z,y-t1-td2x,y≤2r2-t1-td2x,y≤2-t1-tε2r2≤r21-δ2;t∈0,1[4]. Note that (34) holds for any x,y,z∈X fulfilling max⁡dx,z,dy,z≤r and any t∈0,1 if (35)d2x,y≤mint∈0,1t-1+1-t-1r2=r2maxt∈0,1⁡tt-1=4r2.For coherency of the above constraints with the distance properties for any given x,y,z∈X with max⁡dx,z,dy,z≤r and εr≤dx,y≤2r for all ε∈0,2 subject to(36)0≤δ≤1-maxs∈0,12-s1-sε2=1-2-ε24≤1-2-t1-tε;t∈0,1so that δ:0,2→0,1 satisfying δ=δε∈0,1-2-ε2/4 is strictly increasing with δ0=0 for any ε∈0,2.Proposition 20.
Let a geodesic metric spaceX,d,W be a uniformly convex CAT(0) space satisfying the conditions of Proposition 19. Then X,d,W is nearly uniformly convex.Proof.
Since the CAT(0) space satisfies Proposition19 then it is uniformly convex and uniformly p(≥1) convex and then ∞-uniformly convex so it is nearly uniformly convex (Definition 9).There are some particular results of Lemma17 of interest concerning the role of just one of the two involved mappings. For instance, Lemma 17(i) yields directly the following result.Theorem 21.
Let the metric spaceX,d be a CAT(0) space and consider the mapping defined in (17) via two self-mappings T,S:C→C satisfying (18), where C is a nonempty closed convex subset of X. The following properties hold:
(i)(37)dQtSmp,Tx,QtSmp,Ty≤1-tKTdx,y;∀m∈Z0+,d2QtSmp,Tnx,QtSmp,Tny≤1-tKTndx,y;∀n,m∈Z0+,dQtSp,Tmx,QtSq,Tmx≤tKSdp,q;∀m∈Z0+,d2QtSnp,Tmx,QtSnq,Tmx≤tKSndp,q;∀n,m∈Z0+.
(ii) IfX,d is complete then Qt:Smp×TC→C is a strict contraction for each t∈0,1 and each m∈Z0+, irrespective of the mapping S:C→C, for any given p,q=p∈X provided that T:C→C is strictly contractive. Thus, dQtSmp,Tnx,QtSmp,Tny→0, ∀m∈Z0+ as n→∞, and has a unique fixed point:(38)z=ztm,S,p,y∗=tSmp⊗1-ty∗for each t∈0,1 and each given p∈C and m∈Z0+ and the unique fixed point y∗=Ty∗∈C of T:C→C and Tny→y∗. In particular, if t=0, z0y∗=y∗ and if t=1, z1m,S,p,y∗=Smp.
If bothS,T:C→C are strictly contractive then Smp→p∗=Sp∗ with FixS=p∗ and then z=limm→∞⁡ztm,S,p∗,y∗=tp∗⊗1-ty∗ for each t∈0,1 is the unique fixed point of Qt:Smp×TC→C for each t∈0,1; ∀m∈Z0+.
IfT:C→C is strictly contractive and S:C→C is nonexpansive with p∗∈FixS(FixS=p∗ if S:C→C is strictly contractive) then Qt:Smp∗×TC→C has a unique fixed point zt=tp∗⊗1-ty∗  for each t∈0,1; ∀m∈Z0+.
(iii) IfX,d is complete then Qt:X×X→X is a strict contraction, irrespective of the mapping T:C→C, for any given x,y=x∈C provided that S:C→C is strictly contractive so that it has a unique fixed point and then dQtSnp,Tmx,QtSnp,Tmy→0; ∀m∈Z0+ as n→∞ and has a unique fixed point: (39)z=ztm,T,p∗,x=tp∗⊗1-tTmxfor each t∈0,1 and each given x∈C and m∈Z0+ and the unique existing fixed point p∗=Sp∗∈C of the nonexpansive mapping S:C→C.Proof.
Equations (37) follow directly from ((19a)–(19c) and (20a)–(20c)) and Property (i) is proved directly. If, in addition, the CAT0 space is a complete; then if T:C→C is strictly contractive, then and C is nonempty, closed, and convex; T:C→C has a unique fixed point y∗∈C, Tnx→y∗ for any x∈C and Snp and Tnx are bounded for any x,p∈C since C is bounded. From the second inequality of (37), QtSmp,Tnx-QtSmp,Tny→0 as n→∞, ∀m∈Z0+ for all t∈0,1. Then, QtSmp,Tnx is a Cauchy sequence with a unique limit z=ztm,S,p,y∗=tSmp⊗1-ty∗ for each t∈0,1 and each m∈Z0+. Note that, for each p∈C, z=ztm,S,p,y∗ is a unique fixed point of Qt:Smp×TC→C since(40)QtSmp,Ty∗=tSmp⊕1-tTy∗=tSmp⊕1-ty∗=QtSmp,y∗.The corresponding results for the case that S is also contractive or nonexpansive are direct counterparts of the above reasoning. Property (ii) has been proved. The proof of Property (iii) is similar and then it is omitted.Different classes of iterative schemes and their stability properties related to fixed point theory as Halpern, Jungck, Ishikawa, and many of its variants and extensions have been studied in a number of papers. See, for instance, [1, 2, 4, 5, 25, 25–28] and some references therein. The following result links an iterative scheme based on two maps to the convergence properties in CAT(0) spaces.Theorem 22.
Let the metric spaceX,d be a CAT(0) space and consider the iterative scheme:(41)xn+2=QtnSnxn+1,Tnxn=tnSnxn+1⊕1-tnTnxn;∀n∈Z0+subject to any initial conditions x0,x1∈X and define(42)θn+1=dSnxn+2,Tnxn+dSnxn+1,Tnxn+dTnxn+1,Snxn+1+dSnxn+2,Tnxn+1max⁡dxn+1,xn+2,dxn,xn+1;∀n∈Z0+with θn being dependent, in general, on x0 and x1.Then, the following properties hold:(i) Assume thatsupn∈Z0+⁡θn≤θ<+∞ with θ being dependent, in general, on x0 and x1. Then(a)
ifTn is a sequence of nonexpansive (resp., strictly contractive) sequence of mappings then there is a real sequence tn⊂0,1 of sufficiently small elements such that ρn⊂0,1 (resp., ρn⊂0,1),(b)
ifSn is a sequence of nonexpansive (resp., strictly contractive) sequence of mappings then there is a real sequence tn⊂0,1 of elements being sufficiently close to unity such that ρn⊂0,1 (resp., ρn⊂0,1).(ii)(43)d2xn+2,xn+3≤ρn+1max⁡d2xn+1,xn+2,d2xn,xn+1;∀n∈Z0+,where(44)ρn+1=tn+12KSn+12+1-tn+12KTn+12+tn+11-tn+1min⁡KSn+1,KTn+1θn+1;∀n∈Z0+for any tn∈0,1; ∀n∈Z0+, and(45)d2xn+mn,xn+mn+1≤ρn+mn-1max⁡d2xn+mn-1,xn+mn,d2xn+mn-2,xn+mn-1≤∏i=0mn-1ρn+imax⁡d2xn+1,xn+2,d2xn,xn+1;mn≥2,∀n∈Z0+.(iii) Ifx0,x1∈X and x2=Qt0Sx1,Tx0 are such that dx1,x2≤dx0,x1 and ρn≤1, ∀n∈Z0+, then(46)dxn+2,xn+3;∀n∈Z0+.The above inequality is strict if ρn<1, ∀n∈Z0+.(iv) If there is a sequencemn of integers satisfying 2≤ma≤mn≤mb<∞, ∀n∈Z0+, such that ∏i=0mn-1ρn+i≤ρ<1 then limn→∞⁡dxn,xn+1=limn→∞⁡dxn+mn,xn+mn+1=0.(v) Define the following sequences:(47)αn=tn2KSn2+1-tn2KTn2βn=tn1-tnminKSn,KTndSnxn+1,Tnxn-1+dSnxn,Tnxn-1+dTnxn,Snx+dSnxn+1,Tnxn;∀n∈Z+.If αn+βn<1, ∀n∈Z+, then dxn+1,xn→0 as n→∞. If, in addition, X,d is a complete CAT0 space then (1) xn converges to a limit x∗∈X which is unique if tn=t∈0,1, ∀n∈Z0+. (2) If tn⊂0,1→t∈0,1 and the sequence of mappings Qtn converges point-wise to Qt, then Qt is a strict contraction and xn∗→x∗, where Fix⁡Qtn=xn∗, ∀n∈Z0+, and x∗∈Fix⁡T∗. (3) If tn⊂0,1→t∈0,1 and the sequence of mappings Qtn with Fix⁡Qtn=xn∗; ∀n∈Z0+ converges uniformly to a strict contraction Qt with Fix⁡Qt=x∗, then xn∗→x∗.(vi) Property (v) also holds if the constraintαn+βn<1, ∀n∈Z+, is replaced with ρn<1; ∀n∈Z+.Proof.
Note that, for any sequenceKTn⊂0,1, a sequence t-n⊂0,1 exists such that if supn∈Z0+⁡θn≤θ<+∞, tn∈0,t-n; ∀n∈Z0+, ρn⊂0,1 since(48)ρn=ρntn=tn2KSn2+1-tn2KTn2+tn1-tnmin⁡KSn,KTnθn=tn1-tnmin⁡KSn,KTndSnxn+2,Tnxn+dSnxn+1,Txn+dTnxn+1,Snxn+1+dSnxn+2,Tnxn+1max⁡dxn+1,xn+2,dxn,xn+1+tn-12KTn2+tn2KSn2≤1;∀n∈Z0+.Note that ρnt is a continuous real function on R0+with ρn0=KTn2≤1ifKTn2≤1, that is, if Tn is a nonexpansive sequence of mappings, and ρn0<1ifKTn2<1; ∀n∈Z0+ so that there is a sequence t-n⊆0,ε⊆0,1 such that ρntn≤1 if tn∈0,t-n; ∀n∈Z0+, KTn⊂0,1, ρntn=KTn2<1 and ρntn<1if t-n⊂0,1 and KTn⊂0,1, that is, if Tn is a strictly contractive sequence. The same conclusions arise for Sn being a nonexpansive sequence of mappings, respectively, a strictly contractive sequence (i.e., KSn⊂0,1, resp., KSn⊂0,1) if t-n⊆1-ε,1⊆0,1, respectively, t-n⊂0,1 since ρn0=KSn2≤1ifKSn2≤1 and ρn0=KSn2<1ifKSn2<1. Property (i) has been proved.
Property (ii) follows by direct calculation from (19a)–(19c). On the other hand, if dx1,x2≤dx0,x1 proceed by complete induction and contradiction by assuming that dxj+2,xj+3≤dxj,xj+1;  ∀j<n∈Z0+ and some given n>2∈Z+ and that dxn+2,xn+3>dxn,xn+1. Thus,(49)d2xn,xn+1<d2xn+2,xn+3≤max⁡d2xn+1,xn+2,d2xn,xn+1≤d2xn,xn+1∀n∈Z0+.Then, dxn+2,xn+3≤dxn,xn+1;  ∀n∈Z0+ if ρn≤1 and dxj+2,xj+3≤dxj,xj+1;  ∀j<n∈Z0+;  ∀n∈Z0+ and Property (iii) is proved. If ρn<1;  ∀n∈Z0+, then the relevant inequalities by reasoning by contradiction become(50)d2xn,xn+1≤d2xn+2,xn+3<max⁡d2xn+1,xn+2,d2xn,xn+1≤d2xn,xn+1;∀n∈Z0+,again a contradiction. Then, dxn+2,xn+3<dxn,xn+1 if ρn<1;  ∀n∈Z0+.
From the fact that∏i=0mn-2ρn+i≤ρ<1, one gets that limn→∞⁡dxn,xn+1=limn→∞⁡dxn+mn,xn+mn+1=0 so that it follows by combining the above result with those of Properties (ii)-(iii) that Property (iv) is proved.
To prove Property (v), first note that(51)d2xn+2,xn+3≤αnmax⁡d2xn+1,xn+2,d2xn,xn+1+βn+1max⁡d2xn+1,xn+2,d2xn,xn+1.
Case 1.
m
a
x
d
2
x
n
+
1
,
x
n
+
2
,
d
2
x
n
,
x
n
+
1
≥
1 so that(52)d2xn+2,xn+3≤αn+1+βn+1max⁡d2xn+1,xn+2,d2xn,xn+1.
Case 2.
m
a
x
d
2
x
n
+
1
,
x
n
+
2
,
d
2
x
n
,
x
n
+
1
≤
1 so that(53)d2xn+2,xn+3≤αn+1+βn+1max⁡dxn+1,xn+2,dxn,xn+1.Assume that Case  1 holds. Then since αn+βn<1 and max⁡d2xn+1,xn+2,d2xn,xn+1≥1;  ∀n∈Z+ then lim⁡supn→∞⁡d2xn+2,xn+3≤1. Otherwise, if lim⁡supn→∞⁡d2xn+2,xn+3>1 then there is a subsequence d2xnk+2,xnk+3 such that d2xnk+2,xnk+3>1 for the subsequence nk⊂Z0+, k∈Z0+ and, since αn+βn<1;  ∀n∈Z+, we get the contradiction 0≥lim⁡supk→∞⁡d2xnk+2,xnk+3>1. Thus, lim⁡supn→∞⁡d2xn+2,xn+3≤1.
Now, iflim⁡supn→∞⁡d2xn+2,xn+3=1 which implies limn→∞⁡d2xn+2,xn+3=1 this contradicts the contractive property associated with the condition αn+βn<1;  ∀n∈Z+. So, lim⁡supn→∞⁡d2xn+2,xn+3<1. Now, Case 2 applies and two situations can arise for each n∈Z+, namely, either(54)d2xn+2,xn+3≤dxn+2,xn+3≤αn+1+βn+1max⁡dxn+1,xn+2,dxn,xn+1or(55)dxn+2,xn+3≥max⁡αn+1+βn+1max⁡dxn+1,xn+2,dxn,xn+1,d2xn+2,xn+3=αn+1+βn+1max⁡dxn+1,xn+2,dxn,xn+1≥d2xn+2,xn+3.Assume that there is a subsequence dxnk+2,xnk+3 which does not converge to zero. Then, the whole sequence dxn+2,xn+3 does not converge to zero. If dxnk+2,xnk+3 is bounded, we get from (55) the following contradiction:(56)0<lim⁡infm→∞⁡dxn+m+2,xn+m+3≥limm→∞⁡∏j=1mαn+j+βn+jmax⁡dxn+1,xn+2,dxn,xn+1=0≥lim⁡infm→∞⁡d2xn+m+2,xn+m+3.Thus, dxnk+2,xnk+3 cannot be bounded if it does not converge to zero so that it is unbounded. But, in Case 2, the distance is bounded since it is less than unity. As a result, dxn+1,xn→0 as n→∞ and xn converges to a unique limit if the CAT(0) metric space is complete if tn=t∈0,1 since then Qt is a strict contraction on X of constant supn∈Z0+⁡αn+bn<1. If tn⊂0,1→t∈0,1 and the sequence of (strictly) contractive mappings Qtn (i.e., supn∈Z0+⁡αn+bn<1) converges point-wise to Qt then Qt is a strict contraction and the sequence xn∗→x∗, where xn∗∈Fix⁡Qtn is unique; ∀n∈Z0+ and x∗∈Fix⁡T∗. If tn∈0,1→t, xn∗∈Fix⁡Qtn, ∀n∈Z0+, Qtn→Qt uniformly, where Qt is a strict contraction with FixQt=x∗, then there is a subsequence Qtnk of Qtn of mappings on XQtnk→Qt (being convergent strict contractions to Qt since Qt is a strict contraction) such that Fix⁡Qtnk=xnk∗, ∀k∈Z0+ and xnk∗→x∗ [27, 28]. Property (v) has been proved. It is evident from Properties (ii) and (v) that Property (vi) also holds if the constraint αn+βn<1; ∀n∈Z+ is replaced with ρn<1; ∀n∈Z+.Example 23.
Assume thatSn=Tn; ∀n∈Z0+. Then, (57)dSnxn+2,Tnxn+dSnxn+1,Tnxn+dTnxn+1,Snxn+1+dSnxn+2,Tnxn+1≤2dTnxn+2,Tnxn+1+dTnxn+1,Tnxn≤2KTdxn+2,xn+1+dxn+1,xn≤4KTnmax⁡dxn+2,xn+1,dxn+1,xn;∀n∈Z0+.Thus, ρn=1+2tn1-tnKTn2<1; ∀n∈Z0+ if qtn,KTn=1-ρn=1-1+2tn1-tnKTn2>0; ∀n∈Z0+. Since qtn,KTn=0 if tn1,2=1±2-KTn-2 for each given n∈Z0+ then qtn,KTn>0 and tn∈0,1 if tn1,2 have nonzero imaginary values (KTn∈0,1/2) or if they are real (KTn∈1/2,1) and tn∈0,tn1∪tn2,1; ∀n∈Z0+. Thus, Theorem 22(vi) is applicable if, for any n∈Z+,(58)KTn∈0,12,tn∈0,1,KTn∈12,1,tn∈0,tn1∪tn2,1or, equivalently, KTn<1/1+2tn1-tn for any given tn∈0,1; ∀n∈Z0+.Example 24.
Assume thatSn=μnI, Tn=νnI, with μn=αnνn, KSn=μn∈R+,KTn=νn∈R+, and αn∈0,1; ∀n∈Z0+ and assume that the metric d:X×X→R0+ is homogeneous and norm-induced, that is, dcx,cy=cdx,y=cx-y for any c∈R and x,y∈X. Assume also that (59)limsupn→∞1-αn-dxn,xn+1+dxn+1,xn+2xn+xn+1≤0;then(60)limsupn→∞dαnxn,xn+dαnxn+1,xn+1-dxn+2,xn+1-dxn+1,xn≤limsupn→∞1-αnxn+xn+1-dxn+2,xn+1-dxn+1,xn≤0.Note that (60) holds, in particular, if αn→1. Then, one gets from (57) and (48) that(61)dαnνnxn+2,νxn+dαnνnxn+1,νnxn+dαnνnxn+1,νnxn+1+dαnνnxn+2,νnxn+1=νndαnxn+2,xn+dαnxn+1,xn+dαnxn+1,xn+1+dαnxn+2,xn+1≤νndαnxn+2,αnxn+1+dαnxn+1,xn+dαnxn+1,xn+νndαnxn+1,xn+1+dαnxn+1,xn+1+dαxn+2,αnxn+1=νn2αndxn+2,xn+1+2dαnxn+1,xn+2dαnxn+1,xn+1≤νn2αndxn+2,xn+1+2dαnxn+1,αnxn+2dαnxn,xn+2dαnxn+1,xn+1=2νnαndxn+2,xn+1+dxn+1,xn+dαnxn,xn+dαnxn+1,xn+1;∀n∈Z0+and, and from (60) and (61) that(62)lim⁡supn→∞⁡dαnνnxn+2,νxn+dαnνnxn+1,νnxn+dαnνnxn+1,νnxn+1+dαnνnxn+2,νnxn+1-8νnαnmax⁡dxn+2,xn+1,dxn+1,xn≤0,limsupn→∞tn2αn2νn2+1-tn2νn2+8tn1-tnαn2νn2<1⟹limsupn→∞ρn<1r1tn,αn,νn=tn2αn2νn2+νn2+tn2νn2-2tnνn2+8tnαn2νn2-8tn2αn2νn2-1=νn21+tn2-2tn+8tnαn2-7tn2αn2-1=νn21+tn21-7αn2-2tn1-4αn2-1<0;∀n∈Z0+if and only if(63)rtn,αn,νn=tn21-7αn2+2tn4αn2-1+1-νn-2<0;∀n∈Z0+.Note that rtn1,2,αn,νn=0 if tn1,2=1-4αn2±1-4αn22+1-7αn2νn-2-1/1-7αn2 and (63) holds with tn⊂0,1, and Theorem 22(vi) is applicable, if for each n∈Z0+, (64)tn∈max⁡0,mintn1,tn2,min⁡1,min⁡tn1,tn2if1-4αn22+1-7αn2νn-2-1≥0,0,1if1-4αn22+1-7αn2νn-2-1<0.Example 25.
A potential application of the given results on motions of robots, or movable bodies in general, on surfaces subject to the definition of admissible obstacle-free corridors is now described. LetX,d be a metric space which is a CAT0 space. Define zt=tSp⊕1-tTx and ωt=tSq⊕1-tTy for t∈0,1, which are points of X, parameterized by t in 0,1 for any given x,y,p,q being points of X, with S and T being nonexpansive mappings on X, such that(65)dzt,Sp=1-tdSp,Tx;dzt,Tx=tdSp,Txdωt,Sq=1-tdSq,Ty;dωt,Ty=tdSq,Ty.Define Qtx,y=tx⊕1-ty; ∀t∈0,1 as in (19a)–(19c) for x,y∈X and t∈0,1 so that we can characterize the sequences generated via Qt by the Lipschitz-continuous generator mappings S and T. The interpretation of these mappings S and T is that they generate sequences of points which define the extreme trajectories for given initial points of obstacle-free corridor to accommodate the trajectories of a movable robot or body, in general. The mappings are defined so that the admissible trajectories are obstacle-free in the admissible motion surface. This obstacle-free motion surface can be modelled, for instance, according to Euclidean buildings, other abstract simplicial complexes or CAT(0) cube complexes. See, for instance, [12–14]. The values of the parameter t in 0,1 allocate the points zt and ωt, and their successive sequential points generated via Qt, within segments of Sq and Ty and their successive points obtained from Snp and Tnx for n∈Z+, respectively. Now, Lemma 18 and Theorem 21 can be invoked in this application according to the specific needs. In particular,(a)
if the convergence of the motion trajectory is suited as objective for any givent then the corridor generator mappings S and T on X are defined as appropriate contractive mappings which define the obstacle-free corridor which converges asymptotically to a single trajectory dependent on t taking values in 0,1 [Lemma 18(i)],(b)
if one of the mappingsS,T is nonexpansive and specifically noncontractive, while the other one is contractive, then the extreme trajectories define a corridor which becomes for a sufficiently large iteration step as narrow as suited accordingly to the initial values of the trajectories definition [Lemma 18(ii)-(iii)],(c)
if both mappings are contractive and defined on a nonempty convex closed convex ofX then QtSmp,Tnx converges (as m,n→∞) to a unique fixed point for each t which depends on the fixed points of the two generator mappings and the parameter t. If one of them is contractive (say T) while the other has a finite number of iterations (say S has m iterations) then QtSmp,Tnx converges (as n→∞) to a point which depends on m,p, each value of the parameter t and the fixed point of T. Such a t-parameterized point becomes to be a function of m as m takes values on Z+ [Theorem 21].



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*Source: 1470582-2017-07-19.xml*