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

Abstract

This paper investigates some fixed point-related questions including the sequence boundedness and convergence properties of mappings defined in spaces, which are parameterized by a scalar , where :   are nonexpansive Lipschitz-continuous mappings and is a metric space which is a space.

1. Introduction

A space, where 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 space is bounded from above by . A particular case of spaces [1–3] is that arising when the curvature is bounded from above by 0 ( spaces. See, for instance, [4–10]). Complete 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:for each of the given points and some point , where is the unique midpoint of and , that is, , 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 space is a geodesic metric space if any two points can be joined by an arc length parameterized continuous curve (geodesic segment) whose length . It is well-known that a geodesic metric space is a space if every geodesic triangle in satisfies the 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 with sides of the same length as the sides of and then of the same perimeter. Therefore, the study of the metric properties of spaces has a major importance. In general, a geodesic metric space is a space if every geodesic triangle in with perimeter less than satisfies the inequality. The so-called spaces are of curvature and they are particular spaces of the most general spaces of curvature . To fix some general basic ideas, let us denote by the unique 2-dimensional Riemannian manifold with constant curvature of diameter being if and for . The spaces are those whose geodesic triangles , that is, those having geodesic segments as its sides, satisfy the so-called inequality; namely, the distances between points of are less than or equal to the distances between the corresponding points in their comparison triangles in the model space (i.e., those triangles whose sides have the same lengths as their counterparts as their corresponding -triangles). Note that if the metric space is a space, then there is a unique geodesic segment with joins and (with if ) [3, 11].

Furthermore, it is well-known that spaces are also spaces for and that - dimensional hyperbolic spaces with their usual metric are spaces and then spaces, whose simpler example is the -dimensional Euclidean space with its usual metric, so also spaces as well, whose simpler example is the unit sphere. Other relevant spaces are the so-called Euclidean buildings, which are abstract simplicial complexes, and the so-called 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. Section 2 gives some relating results for some of the various existing concepts of convexity in metric spaces such as -convexity, -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 spaces. It also studies mappings of the form parameterized by a scalar , defined by in a metric space , where 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 , on 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 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  and denote the sets of integer and real numbers. . . . .  is the closure of the set .  is the closure of the convex hull of the family .  denotes the set of fixed points of a mapping .

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

Let be a complete metric space. It is said that it admits (nonnecessarily unique) midpoints if for any , there is a such that . Such a point is said to be a midpoint of and and is a geodesic space, [16–18].

Definition 1 (-convexity [16, 19]). Suppose a metric space which admits midpoints (or which has midpoints or which is midpoint convex). Then, is said to be -convex for some if, for each and each midpoint of and , For the case , the right-hand side of (2) is defined as a limit leading to . If is -convex, it is equivalently said to be ball convex, while if it is -convex it is equivalently said to be distance convex [19]. is said to be strictly -convex for , if the inequality is strict for and strictly -convex if the inequality is strict for if [16].

Definition 1 leads to the direct conclusion below.

Assertion 2. If a metric space is midpoint convex, then it is -convex.

Assertion 3. If be -convex for some then, for any and each midpoint of and ,

Proof. It is direct from (2) and Minkowski inequality or by direct rearrangement of the power to its inverse in the left-hand side of (2).

Inequality (2) leads to the following direct result.

Assertion 4. If be -convex for some then, for any , each midpoint of and and each midpoint of and ,

Proof. Note from (2) thatand, one gets from (2) and (5) thatleading to the result.

Definition 5 (-Busemann curvature [16, 20]). Suppose a metric space which admits midpoints. Then, is said to satisfy the -Busemann curvature condition for some if, for each , each midpoint of and and each midpoint of and , one has

Assertion 6. Suppose a metric space which admits midpoints, with the midpoint map (or midset) being unique, and which satisfies the -Busemann curvature condition for some . Then, one hasfor any , where and are, respectively, the unique midpoints of and and and .

Proof. Since admits midpoints if , then the midpoint is unique since is unique [19], then . Thus, we can replace leading to an alternative right-hand side in (7) under the replacements and 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 space is midpoint convex (then it is 1-convex from Assertion 2) with unique midpoint map and that it satisfies the -Busemann curvature condition for some . Then, is -convex.

The following technical definitions are of interest to characterize near-uniform convexity.

Definition 8 (-separated family of points [16]). A family of points is -separated if .

Definition 9 (nearly uniformly convex space [16]). A -convex metric space is said to be nearly uniformly convex if, for any and for any -separated infinite family , with , and any such that , , there is some such that , where is the closure of the convex hull of the family .

It turns out that -convexity implies near-uniform convex but the converse is not true.

Definition 10 (see [21, 22]). Let be a metric space. A mapping is said to be a convex structure on if, for each and ,

Definition 11 (see [21]). A convex metric space is said to be uniformly convex if, for any , there exists such that, for any and with and ,A uniformly convex metric space is also referred to commonly as uniformly 1-convex [16]. This concept may be generalized as follows.

Definition 12. A convex metric space is said to be uniformly -convex if, for any , there exists such that, for any and any with and ,

Proposition 13. If a convex metric space is uniformly convex then it is uniformly -convex for any .

Proof. Since is uniformly convex then, for any , there exists such that (10) holds with any and any subject to and . Thus, . Choosing for any given and any given arbitrary , it follows for that so that is uniformly -convex.

Proposition 14. A convex metric space is nearly uniformly convex if, for any , there exists a strictly increasing such that, for any , any , and any -separated infinite family satisfying andfor some , where denotes an open ball of radius centred at .

Proof. Set for any given . Thus, if , one has from (12) thatsince since so that Since is trivially nonempty and then . Note also that, for any fixed , exists fulfilling (12). From Definition 9, is nearly uniformly convex.

Note that is nearly uniformly convex; then it is not necessarily uniformly convex since it can happen that, for some , any -separated infinite family satisfying and all , there exists some such that However, the converse is true as reflected in the next result.

Proposition 15. If a convex metric space is uniformly convex, then it is nearly uniformly convex.

Proof. Note that if (10) is satisfied with , for any given , then (12) is satisfied for , the ball with center at such and radius .

Proposition 16. If is nearly uniformly convex and strictly -convex then, for any and for any -separated infinite family , with , and any such that , there is such that  ,  and contains at most two points of such that with if and if with a choice .

Proof. Since is nearly uniformly convex and strictly -convex then any nonempty closed convex subset of is a Chebyshev set [16]. Thus, for any , there are balls whose closures are Chebyshev sets consisting of at least two distinct points if their radius [16, 23] for any with . Since Chebyshev sets have a unique nearest neighbor in for each , [24], one has for that is a Chebyshev set of two unique elements one of them being its center . Then, one has since is nearly uniformly convex that    for any ,  for any ,  with if with ,  if with .

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

A metric space is (uniquely) geodesic if every two points of are joined by a unique geodesic segment which is the image of the geodesic path from to , that is, the isometry such that , and . A geodesic triangle consists of three vertices and three geodesic segments joining each pair of vertices. A model comparison triangle of the geodesic triangle is a triangle in the Euclidean space such that for . A geodesic metric space is a CAT space [4, 16, 21, 22] if ; (CAT inequality). See, for instance, [1, 4–8].

In the paper, we write for the unique such that and . Note that the midpoint of is .

Note the following from the basic results in Section 2.

(1) A geodesic space is a CAT space if and only if for any and all the following inequality is satisfied:(Proposition  1.1 [4]).

(2) A CAT space is uniformly -convex for any [16].

(3) A CAT space satisfies inequalities (4) and (5) for any since it is midpoint -convex for any .

(4) A CAT space satisfies the -Busemann curvature condition for any from (7).

A general technical result involving constructions with two self-mappings in a space as follows.

Lemma 17. Let a metric space be a CAT(0) space and let the mapping be defined by for any and let be two self-mappings which satisfy the following conditions:for any given, some positive real constants and . Then, for any given and for any , the following properties hold:(i)(ii)

Proof. From (16) to (18), one gets:On the other hand, one has by completion of squares thatand from the triangle inequality for distances in (22a) and (22b) and the use of (18), one getsThe substitution of (23) into (22a) and (22b) with the use of (18) yieldsNow, the replacement of (18) and (24) into (21) leads towhich implies (19a)–(19c) and the proof of Property (i) is complete. On the other hand, one has directly from (19a)–(19c) after replacing ,  ,  ,   for that (20a)–(20c) holds for any as counterpart of (19a)–(19c) and Property (ii) is proved.

From Lemma 17(ii), we get the following result.

Lemma 18. Let a metric space be a CAT(0) space and consider the mapping defined in (17) via two self-mappings satisfying (18). Then, the following properties hold:
(i) Assume that (i.e., are both strictly contractive). Then, for any and .
(ii) Assume that(a)either (i.e., is strictly contractive), (i.e., is nonexpansive but noncontractive), and has a fixed point(b) (i.e., is strictly contractive), (i.e., is nonexpansive but noncontractive), and has a fixed point.Then,for any and .
(iii) If are both nonexpansive, where is a nonempty closed convex subset of , then for any and .

Proof. It follows for constants and that both and are strict contractions on so that one gets from (20a)–(20c) thatSince and are strict contractions on , and . Thus, for any and . Property (i) has been proved.
Now, assume that either and (then has a fixed point since is a space, [5, 17, 18]), or and (then has a fixed point) and . Then, one has from (20a)–(20c) that :Assume with no loss in generality that is strictly contractive with a unique fixed point as a result, since is complete, and is nonexpansive and has a fixed point by hypothesis. Then,Thus, the sequence is bounded for any given for any distance , and since . Furthermore, since with either or , then one has from (28) that for :the one which is contractive has a unique fixed point since is complete; then , , and then , and are bounded sequences for any given . 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 depending on the points of .

Related to Lemma 18(ii), note that if are both nonexpansive, where is a nonempty closed convex subset of , 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 space is uniformly convex [16], it can be nonuniformly convex, in general. However, the existence of (at least) a fixed point for the (noncontractive) expansive mapping or on 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 space be a CAT(0) space. Then, it is uniformly convex if, for any , there exists a strictly increasing , with for , such that for , any and any with and .

Proof. A geodesic space is a uniformly convex CAT(0) space if, for any , [4]. Note that (34) holds for any fulfilling and any if For coherency of the above constraints with the distance properties for any given with and for all subject toso that satisfying is strictly increasing with for any .

Proposition 20. Let a geodesic metric space be a uniformly convex CAT(0) space satisfying the conditions of Proposition 19. Then is nearly uniformly convex.

Proof. Since the CAT(0) space satisfies Proposition 19 then it is uniformly convex and uniformly convex and then -uniformly convex so it is nearly uniformly convex (Definition 9).

There are some particular results of Lemma 17 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 space be a CAT(0) space and consider the mapping defined in (17) via two self-mappings satisfying (18), where is a nonempty closed convex subset of . The following properties hold:
(i)(ii) If is complete then is a strict contraction for each and each , irrespective of the mapping , for any given provided that is strictly contractive. Thus, , as , and has a unique fixed point:for each and each given and and the unique fixed point of and . In particular, if , and if , .
If both are strictly contractive then with and then for each is the unique fixed point of for each ; .
If is strictly contractive and is nonexpansive with if is strictly contractive) then has a unique fixed point   for each ; .
(iii) If is complete then is a strict contraction, irrespective of the mapping , for any given provided that is strictly contractive so that it has a unique fixed point and then ; as and has a unique fixed point: for each and each given and and the unique existing fixed point of the nonexpansive mapping .

Proof. Equations (37) follow directly from ((19a)–(19c) and (20a)–(20c)) and Property (i) is proved directly. If, in addition, the space is a complete; then if is strictly contractive, then and is nonempty, closed, and convex; has a unique fixed point , for any and and are bounded for any since is bounded. From the second inequality of (37), as , for all . Then, is a Cauchy sequence with a unique limit for each and each . Note that, for each , is a unique fixed point of sinceThe corresponding results for the case that 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 space be a CAT(0) space and consider the iterative scheme:subject to any initial conditions and definewith being dependent, in general, on and .

Then, the following properties hold:

(i) Assume that with being dependent, in general, on and . Then(a)if is a sequence of nonexpansive (resp., strictly contractive) sequence of mappings then there is a real sequence of sufficiently small elements such that (resp., ),(b)if is a sequence of nonexpansive (resp., strictly contractive) sequence of mappings then there is a real sequence of elements being sufficiently close to unity such that (resp., ).

(ii)wherefor any ; , and

(iii) If and are such that and , , thenThe above inequality is strict if , .

(iv) If there is a sequence of integers satisfying , , such that then .

(v) Define the following sequences:If , , then as . If, in addition, is a complete space then (1) converges to a limit which is unique if , . (2) If and the sequence of mappings converges point-wise to , then is a strict contraction and , where , , and . (3) If and the sequence of mappings with ; converges uniformly to a strict contraction with , then .

(vi) Property (v) also holds if the constraint , , is replaced with ; .

Proof. Note that, for any sequence , a sequence exists such that if , ; , sinceNote that is a continuous real function on with , that is, if is a nonexpansive sequence of mappings, and ; so that there is a sequence such that if ; , , and if and , that is, if is a strictly contractive sequence. The same conclusions arise for being a nonexpansive sequence of mappings, respectively, a strictly contractive sequence (i.e., , resp., ) if , respectively, since and . Property (i) has been proved.
Property (ii) follows by direct calculation from (19a)–(19c). On the other hand, if proceed by complete induction and contradiction by assuming that ;   and some given and that . Thus,Then, ;   if and ;  ;   and Property (iii) is proved. If ;  , then the relevant inequalities by reasoning by contradiction becomeagain a contradiction. Then, if ;  .
From the fact that , one gets that 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 Case 1. so thatCase 2. so thatAssume that Case  1 holds. Then since and ;   then . Otherwise, if then there is a subsequence such that for the subsequence , and, since ;  , we get the contradiction . Thus, .
Now, if which implies this contradicts the contractive property associated with the condition ;  . So, . Now, Case applies and two situations can arise for each , namely, eitherorAssume that there is a subsequence which does not converge to zero. Then, the whole sequence does not converge to zero. If is bounded, we get from (55) the following contradiction:Thus, cannot be bounded if it does not converge to zero so that it is unbounded. But, in Case , the distance is bounded since it is less than unity. As a result, as and converges to a unique limit if the CAT(0) metric space is complete if since then is a strict contraction on of constant . If and the sequence of (strictly) contractive mappings (i.e., ) converges point-wise to then is a strict contraction and the sequence , where is unique; and . If , , , uniformly, where is a strict contraction with , then there is a subsequence of of mappings on (being convergent strict contractions to since is a strict contraction) such that , and [27, 28]. Property (v) has been proved. It is evident from Properties (ii) and (v) that Property (vi) also holds if the constraint ; is replaced with ; .

Example 23. Assume that ; . Then, Thus, ; if ; . Since if for each given then and if have nonzero imaginary values () or if they are real () and ; . Thus, Theorem 22(vi) is applicable if, for any ,or, equivalently, for any given ; .

Example 24. Assume that , , with , , and ; and assume that the metric is homogeneous and norm-induced, that is, for any and . Assume also that thenNote that (60) holds, in particular, if . Then, one gets from (57) and (48) thatand, and from (60) and (61) thatif and only ifNote that if and (63) holds with , and Theorem 22(vi) is applicable, if for each ,

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. Let be a metric space which is a space. Define and for , which are points of , parameterized by in for any given being points of , with and being nonexpansive mappings on , such thatDefine ; as in (19a)–(19c) for and so that we can characterize the sequences generated via by the Lipschitz-continuous generator mappings and . The interpretation of these mappings and 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 cube complexes. See, for instance, [12–14]. The values of the parameter in allocate the points and , and their successive sequential points generated via , within segments of and and their successive points obtained from and for , 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 given then the corridor generator mappings and on are defined as appropriate contractive mappings which define the obstacle-free corridor which converges asymptotically to a single trajectory dependent on taking values in [Lemma 18(i)],(b)if one of the mappings 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 of then converges (as ) to a unique fixed point for each which depends on the fixed points of the two generator mappings and the parameter . If one of them is contractive (say ) while the other has a finite number of iterations (say has iterations) then converges (as ) to a point which depends on , each value of the parameter and the fixed point of . Such a -parameterized point becomes to be a function of as takes values on [Theorem 21].

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

The author is very grateful to the Spanish Government and European Fund of Regional Development FEDER for Grant DPI2015-64766-R and to UPV/EHU for Grant PGC 17/33.