Continuation methods in certain metric and geodesic spaces

In this paper it is shown that a classical continuation principle due to Granas for contractions holds under weaker contractive assumptions. This leads to a Leray–Schauder principle for such contractions in hyperbolic spaces. Some applications to nonexpansive mappings in hyperbolic geodesic spaces are also discussed.

The original idea may be formulated as follows. Suppose X is a Banach space with D ⊂ X. A mapping T : D → X is said to satisfy the Leray-Schauder condition if there exists z ∈ int (D) such that T (x) − z = λ (x − z) for all x ∈ ∂ D and λ > 1.
To place this condition in a more historical context we refer, for example, to a recent paper by Morales [23] and the references cited therein. We also refer to the recent paper by Garcia-Falset et al. [6] for additional historical comments.
In [23] Morales answered a long-standing open question posed by the first author in 1975 [15]. Specifically, it was proved in [15] that if K is a bounded closed convex subset of a Banach space which has the fixed point property for nonexpansive mappings, and if T : K → X satisfies the Leray-Schauder condition relative to some point z ∈ int (K ), then T has a fixed point under the additional assumption that inf { x − T (x) : x ∈ ∂ K } > 0. Morales proved, among other things, that even for a wider class of mappings this second assumption may be dropped.
The purpose of the present paper is to study the Leray-Schauder condition and related continuation methods in the context of certain metric and geodesic spaces. In particular it is shown that a classical continuation principle due to Granas for contractions holds under slightly weaker assumptions. It is also shown that if G is a bounded open set in a complete hyperbolic space X, and if f : G → X is nonexpansive, then inf d (x, f (x)) : x ∈ G = 0 if there exists p ∈ G such that x / ∈ ( p, f (x)) for all x ∈ ∂G. This condition, in the present context, is equivalent to the Leray-Schauder boundary condition.
We begin with the terminology of Papadopoulos [24]. Let (X, d) be a metric space. Recall that a geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0, l] ⊂ R to X such that c (0) = x, c (l) = y, and d c(t), c t = t − t for all t, t ∈ [0, l]. In particular, c is an isometry and d (x, y) = l. The image α of c is called a geodesic (or metric) segment joining x and y. In this case for t ∈ [0, 1] we use (1 − t) x ⊕ t y to denote the point of α which has distance td (x, y) from x. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X.
Many results in this paper hold in a special class of spaces called hyperbolic spaces. We turn now to the terminology of Kohlenbach [21].
If only condition (i) is satisfied, then (X, d, W ) is a convex metric space in the sense of Takahashi [26]. Conditions (i)-(iii) together are equivalent to (X, d, W ) being a space of hyperbolic type in the sense of Goebel-Kirk [9]. Condition (iii) ensures that the set is a geodesic in the usual sense. In this case we use (1 − λ) x ⊕λy to denote W (x, y, λ). The geodesic segment joining x and y is denoted by [x, y] (with the usual convention for (x, y] and [x, y)).
The relevant observation at this point is that it is not essential that geodesic segments joining each two points of X be unique. It suffices to assume only that some family of geodesic segments satisfy the relevant axioms; in this instance (i)-(iii). Thus the class of spaces of hyperbolic type includes all normed linear spaces (not merely those with strictly convex norm) as well as all convex subsets thereof.
For a more detailed discussion of these concepts we refer, e.g., to Chapters 6 and 9 of Kirk-Shahzad [20].

Continuation methods for contractions
In this section we take two facts as our points of departure. The first is a following fundamental continuation principle due to Andrzej Granas. The second is an extension of Banach's contraction mapping theorem due to Felix Browder. Here, and throughout the remainder of the section, we adopt the terminology of Jachymski and Jóźwik [14].  In what follows we refer to mappings that satisfy the above condition Browder contractions with contractive function ψ. In his survey [13], Jacek Jachymski shows that Browder's contractive condition is equivalent to or, in fact subsumes, many contractive conditions that have subsequently appeared in the literature. By making some modifications in Marlène Frigon's proof of Theorem 2.1 (see [5]) we obtain the following extension of Granas's theorem. We point out that in her original paper, Frigon mentions that condition (d) of Theorem 2.1 may be weakened to condition (d ) below.

Theorem 2.3
Let U be a domain in a complete metric space X, let ψ : [0, ∞) → [0, ∞) be monotone nondecreasing, continuous from the right, and such that ψ (t) < t for all t > 0 and let let f, g : U → X be two Browder contractions with common contractive function ψ. Suppose also that there exists H :

Proof of Theorem 2.3 Let
where ψ and φ are as in (c ) and (d )). Then the function H (·, λ) : B (x; r ) → B (x; r ) . Indeed, for every y ∈ B (x; r ) we have (using (c ) and (d) ) To see that Q is closed in [0, 1] , suppose λ n ∈ Q, n = 1, 2, . . . , and suppose λ n → λ as n → ∞. For each n ∈ N there exists x n ∈ U such that Then if m, n ∈ N, If {x n } is not a Cauchy sequence then by passing to a subsequence we may suppose d (x m , x n ) → t > 0 as m, n → ∞. Moreover, by passing to a subsequence again we may suppose that either d ( In the first case, since ψ is continuous from the right, ψ (d (x m , x n )) →ψ (t) as m, n→∞. In the second case, because ψ is nondecreasing, ψ (d (x m , x n )) → r ≤ ψ (t) as m, n → ∞. In either case, this leads to the contradiction It follows that {x n } is a Cauchy sequence in U , so there exists x ∈ U such that x n → x as n → ∞. Moreover, since ψ (0) = 0 and ψ is continuous from the right, This proves that Remark 1 A routine argument shows that it is possible to replace condition (d ) in Remark 2 Theorem 2.3 extends Corollary 3.2 of Agarwal, et al. [1] in that ψ is merely assumed to be continuous from the right rather than continuous.
Now let δ = inf δ r : r * ≤ r ≤ r . We now show that δ > 0. If δ = 0, then there exists a sequence r n ⊂ [r * , r ] such that r n −α r n r n → 0 as n → ∞. If α r n 1 we may pass to a subsequence r n k of r n such that α r n k → t > 0 as k → ∞. Then lim inf contradicting r n k − α r n k r n k → 0 as k → ∞. Therefore α r n → 1. However α r n → 1 implies r n → 0, and this contradicts the fact that r n ≥ r * . Therefore it must be the case that δ > 0, and for each y H (y, λ)) ≤ r. Hence for such λ, H (·, λ) : B (x; r ) → B (x; r ). The remainder of the proof is straightforward, following the method of Theorem 2.3.

Remark 3
In a subsequent paper [8], Geraghty weakened his contractive condition, requiring only that α (t n ) → 1 ⇒ t n → 0 for monotone decreasing sequences {t n } ⊂ R + . Mappings satisfying this condition are called Geraghty (II) contractions in [14]. It is shown in [11] that the class of Geraghty (II) contractions coincides with a class of contractions introduced by Boyd and Wong in [2]. The Boyd-Wong contractions are similar to the Browder contractions except that no monotonicity condition is required on the Browder contractive function ψ, and it is only assumed that ψ is upper semicontinuous from the right. It is noted in [14] that Boyd-Wong contractions properly contain the Browder contractions. For a comprehensive comparison of these and numerous related contractive conditions, we refer to the survey [14] by Jachymski and Jóźwik.

Question.
We leave open the question of whether Theorem 2.3 holds for the Geraghty (II) (= Boyd-Wong) contractions.

The Leray-Schauder condition for contractions in hyperbolic spaces
The following is an application of Theorem 2.3.

Theorem 3.1 Let G be a bounded domain in a complete hyperbolic space (X, d) and suppose f : G → X is either a Browder or a Geraghty (III) contraction. Suppose also that there exists p ∈ G such that x / ∈ ( p, f (x)) for all x ∈ ∂G. Then f has a unique fixed point in G.
Proof We assume that f is a Browder contraction. Thus f : G → X satisfies (d (x, y)) for all x, y ∈ G If f has a fixed point in ∂G there is nothing to prove, so we may assume that x / ∈ ( p, f (x)) for all x ∈ ∂G. This assures that condition (b) of Theorem 2.3 holds. Also, using condition (iv) of the definition of a hyperbolic space (taking x = y) we have: y) for each x, y ∈ G, and because G is bounded it follows that f (G) is bounded. Therefore there exists M > 0 such that d ( p, f (x)) ≤ M for all x ∈ G. Hence by condition (ii),  f d (x, f (x)) : x ∈ G = 0.
Proof As above, for x ∈ G and t ∈ [0, 1), let f t (x) be the point of the segment [ p, f (x)] with distance td ( p, f (x)) from p. Then f t : G → X is a contraction mapping and x / ∈ ( p, f t (x)) for each x ∈ ∂G, so by Theorem 3.1 for each such t there exists Our next observation involves the CAT(0) spaces of Gromov (see [3]). It is known ( [17], Theorem 21) that if K is a bounded closed convex subset of a complete CAT(0) space X and if T : K → X is a nonexpansive mapping for which inf {d (x, f (x)) : x ∈ K } = 0, then T has a fixed point. In conjunction with Theorem 4.1 this shows that the Leray-Schauder boundary condition implies the existence of a fixed point for such mappings if int (K ) = ∅. However in this case it is known that the convexity assumption on K is not even needed. The following is Theorem 3.3 of [18]. (X, d), and suppose f : G → X is nonexpansive. Suppose also that there exists p ∈ G such that x / ∈ ( p, f (x)) for all x ∈ ∂G. Then f has a fixed point in G.

Theorem 4.2 Let G be a bounded open set in a complete CAT(0) space
We now turn to another application. It is easy to see that if B := B ( p; r ) is a closed ball in a complete hyperbolic space and if f : B → B ( p; r + ε) for some ε > 0, then This is because Theorem 4.1 implies inf {d (x, f (x)) : x ∈ B} = 0 if f satisfies the Leray-Schauder condition on ∂ B for some x or, if the Leray-Schauder condition fails there exists a point x ∈ ∂ B such that x ∈ ( p, f (x)) , which implies d (x, f (x)) ≤ ε. We now examine the extent to which this observation extends to arbitrary convex sets.
Let K be a bounded closed convex subset of a complete linear hyperbolic space X in the sense that each two distinct points x, y of X lie on a unique geodesic (metric) line containing the geodesic segment [x, y]. (This is the approach taken in [25]). The following is an immediate extension of the observation about approximate fixed points in closed balls. The ε-neighborhood of K for ε > 0 is the set:

Theorem 4.3 Let K be a bounded closed convex subset of a complete linear hyperbolic space, and suppose int
Proof Since int (K ) = ∅ there exist p ∈ int (K ) and r > 0 such that B ( p; r ) ⊂ K and B p; r K if r > r . Let ε > ε be arbitrary. If the Leray-Schauder condition holds on ∂ K relative to p then by Theorem 4.1 there is nothing to prove. Otherwise there exists y ∈ ∂ K such that y ∈ ( p, f (y)) . Also, since f (y) ∈ N ε (K ) there exists z ∈ ∂ K such that d ( f (y) , z) ≤ ε . If z = y we are finished. Otherwise, there is a point q on the geodesic line passing through p and z such that d (q, p) = r and Since B ( p; r ) is an arbitrary ball in K and ε > ε is arbitrary, the conclusion follows.
We do not know whether the estimate in Theorem 4.3 is optimal. Indeed, as we show in the Appendix a sharper estimate holds in a Banach space.

Further remarks
The so-called R-trees (or metric trees) are a very special case of the CAT(0) spaces (see [3, p.167]). The following is another result of [18]. In this instance boundedness of the domain may be replaced by geodesic boundedness.
Theorem 5.1 ([18]) Let (X, d) be a complete R-tree, suppose K is a closed, geodesically bounded, convex subset of X, and suppose p ∈ int (K ). If f : K → X is continuous and satisfies the Leray-Schauder condition: Then f has a fixed point.
Crucial to the proof of the above result is the fact that a continuous self-mapping of a geodesically bounded closed convex subset of a complete R-tree always has a fixed point. For a detailed discussion, see [19].
Finally, we remark in passing that it is shown in [12] that for a nonexpansive mapping T : B → H, where B is the unit ball in a Hilbert space H, the existence of a fixed point for T and the Leray-Schauder condition are mutually exclusive alternatives, and that this fact characterizes Hilberts space among Banach spaces. It is not clear whether a similar fact characterizes CAT(0) spaces among Busemann spaces.
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Appendix
The following is a Banach space extension of Theorem 4.3. However we do not even know whether this estimate is optimal. Theorem 6.1 ([16]) Let K be a bounded closed convex subset of a Banach space X with int (K ) = ∅, and let f : K → N ε (K ) (ε > 0) be nonexpansive. Then Proof Because [16] may not be readily available we include the details. Since int (K ) = ∅ there exists r > 0 such that B ( p; r ) ⊂ K , and we may further suppose p = 0. If the Leray-Schauder condition holds for f on ∂ K relative to p then by Theorem 4.1 (or earlier Banach space results) there is nothing to prove, so we suppose that for some y ∈ ∂ K and λ > 1, f (y) = λy. Now let ε > ε > 0 be arbitrary. Since f (y) ∈ N ε (K ) there exists z ∈ ∂ K such that d ( f (y), z) ≤ ε . Let w ∈ X satisfy y = 1 − λ −1 w + λ −1 z. (6.1) Then, since y = λ −1 f (y), we have From (6.2), and it follows that 1 − λ −1 w + ε ≤ ε . (6.4) Multiplying both sides by f (y) we have However f (y) ≤ diam (K ) − r + ε, so Also, since both y and z lie on ∂ K , it follows that w / ∈ int (K ). In particular w ≥ r . Therefore Since ε > ε is arbitrary and B ( p, r ) is an arbitrary ball in K , the conclusion follows.
It is easy to check that diam (K ) −r + ε r + ε ε ≤ diam (K ) − 2r + ε. This is because diam (K ) ≥ 2r if B ( p; r ) ⊂ K , with equality holding only if diam (K ) = 2r . Thus in general the estimate in Theorem 6.1 is better that the one given by Theorem 4.3. At the same time, if K is a closed ball each estimate reduces to inf {d (x, f (x)) : x ∈ K } ≤ ε.