ON ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN HYPERCONVEX METRIC SPACES M.A. Khamsi Since bounded hyperconvex metric spaces have the fixed point property for nonexpansive mappings, it is natural to extend such powerful result to asymptotically nonexpansive mappings. Our main result states that the approximate fixed point property holds in this case. The proof is based on the use, for the first time, of the ultrapower of a metric space. Abstract. Introduction The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [AP] who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a nonexpansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [La]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [Sn1] and Soardi [So] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces. For more on the metric fixed point property, the interested reader may consult [AK] and [GK] as well as the most recent book [KK]. Recall also that Jawhari, Misane and Pouzet [JMP] were able to show that Sine and Soardi’s fixed point theorem is equivalent to the classical Tarski’s fixed point theorem in complete ordered sets. This happens via the notion of generalized metric spaces. Therefore, the notion of hyperconvexity should be understood and appreciated in a more abstract formulation. In opposition to the lack of linearity hyperconvexity provides us with a really rich metric structure that leads to a collection of surprising and beautiful results related to different branches of mathematics as, for instance, topology, graph theory, multivalued analysis, fixed point theory. 1991 Mathematics Subject Classification. Primary 47H10, 47E10. Key words and phrases. Nonexpansive mappings, asymptotically nonexpansive mappings, fixed point, hyperconvex. Typeset by AMS-TEX 1 2 M.A. KHAMSI In this work, we investigate some open questions related to the fixed point property (fpp) in hyperconvex metric spaces. Historically nonexpansive mappings have enjoyed most of the interest and were at the core of the fpp in hyperconvex metric spaces. The main motivation of this work was a question by Kirk [Ki2] whether asymptotically nonexpansive mappings have the fpp in bounded hyperconvex metric spaces. This question is still open. But we were able to show that asymptotic fixed point property holds in this case. The proof is nonstandard in nature and uses the notion of ultrapower of a metric space. To the best of our knowledge this is the first time that such notion is considered in the metric setting which lead to some positive new results. Basic Definitions A metric space M is said to be hyperconvex if given any family {xα } of points of M and any family {rα } of real numbers satisfying d(xα , xβ ) ≤ rα + rβ it is the case that ∩α B(xα ; rα ) 6= ∅. The fundamental result of [AP] asserts that a metric space M is hyperconvex if and only if it is injective. Thus M is hyperconvex if given any two metric spaces X and Y with Y a subspace of X, and any nonexpansive mapping f : Y → M, then f has a nonexpansive extension f˜ : X → M. Basic results about injective metric spaces can be found in [Is]. An admissible subset of M is a set of the form \ B(xi ; ri ) i where {B(xi ; ri )} is a family of closed balls centered at points xi ∈ M with respective radii ri . It is quite easy to see that an admissible subset of a hyperconvex metric space is hyperconvex. In what follows we use A(M ) to denote the family of all nonempty admissible subsets of M . The recent interest into hyperconvexity goes back to the results of Sine [Sn1] and Soardi [So] who proved that if H is a bounded hyperconvex metric space and T : H → H is nonexpansive, i.e. d(T (x), T (y)) ≤ d(x, y) for any x, y ∈ H, then there exists a fixed point x ∈ H, i.e. T (x) = x. Moreover the fixed point set F ix(T ) is hyperconvex and consequently is a nonexpansive retract of H. May be the most elegant result in this direction goes to Baillon [Ba] who proved that the conclusion of Sine and Soardi results is still valid when dealing with any family of commutative nonexpansive mappings. In fact his proof is based on the following structural result: 3 Theorem.. Let H be a bounded hyperconvex metric space. If {Hi }i∈I is a decreasing family of nonempty hyperconvex subsets of H, then we have \ Hi 6= ∅ i∈I and is hyperconvex. The proof is non-intuitive and very complicated. When H is not bounded, then a nonexpansive mappings may not have a fixed point. But it is not hard to see that the nonexpansive mapping T always has an approximate fixed point, i.e. inf{d(x, T (x)); x ∈ H} = 0 . When a map satisfies the above, we say T satisfies the approximate fixed point property. In this case, the set Hε = {x ∈ H; d(x, T (x)) ≤ ε} is not empty for any ε > 0. In fact Sine [Sn2] proved that Hε is hyperconvex. Next we discuss convexity in hyperconvex metric spaces. Historically there are two approaches to this. One is based on Penot’s ideas [Pe] based on the notion of convexity structures who gave the first interesting generalization of the classical Kirk’s fixed point theorem [Ki1] in metric spaces. The other one mimics the linear convexity. Here we will use that one. In order to better understand it, we will use a natural embedding of any metric space M into the Banach space l∞ (M ) (see [EK] for more on this). So if H is hyperconvex, then there exists a nonexpansive retract R : l∞ (H) → H. For any x, y ∈ H, we write tx ⊕ (1 − t)y = R(tx + (1 − t)y) for ant t ∈ [0, 1]. Here we are using the linear convexity of l∞ (H). It is not hard to check that for any z ∈ H we have d(z, tx ⊕ (1 − t)y) ≤ td(z, x) + (1 − t)d(z, y) for ant t ∈ [0, 1]. Ultrapower of Metric Spaces Let (M, d) be a bounded metric space and U Q be a nontrivial ultrafilter on the natural M . Define the equivalence relation ∼ numbers. Consider the cartesian product M = n≥1 on M by (xn ) ∼ (yn ) if and only if lim d(xn , yn ) = 0. U 4 M.A. KHAMSI f. An The limit over U exists since M is bounded. Then we consider the quotient set M f is a subset of M. If (xn ) ∈ x̃, then (yn ) ∈ x̃ if and only if lim d(xn , yn ) = 0. element x̃ ∈ M U f define the metric d˜ by On M ˜ ỹ) = lim d(xn , yn ) d(x̃, U where (xn ) (resp. (yn )) is any element in x̃ (resp. ỹ). It is easy to see that d˜ defines a f which has many nice properties similar to the linear ultrapower of a Banach metric on M f. Indeed, let space. For example, it is obvious that M is isometric to a subset of M g Ṁ = {(x n ); xn = x for any n ≥ 1} . Then it is easy to show that M and Ṁ are isometric. In the sequel we will use the notation f as well. Also it is worth to mention that if M M = Ṁ and see x ∈ M as an element of M f is complete, then M is complete. The proof is similar to the linear one. In the linear case it e is also a finite dimensional is known that if X is a finite dimensional Banach space, then X Banach space with the same dimension as X. The analogue of this is the following result. f is also compact and is isometric to M . Proposition 1. If M is compact then M Proof. Since M is compact, then for any sequence (xn ) ∈ M the limit lim xn = x exists U (in M ) since U is an ultrafilter. So we have lim d(xn , x) = 0 U g f f f or equivalently (x n ) = x. Hence M is a subset of Ṁ , i.e. M = Ṁ . Therefore M is isometric to M and must be compact. ¤ Clearly one may then ask what if M is not compact. In this case it is natural to use measures of noncompactness. The most commonly used were introduced by Hausdorff and Kuratowski (see [ADL] for more). Definition 1. Let (M, d) be a metric space and let B(M ) denote the collection of nonempty, bounded subsets of M . (1) The Kuratowski measure of noncompactness α : B(M ) → [0, ∞) is defined by α(A) = inf{ε > 0; A ⊂ i=n [ Ai with Ai ∈ B(M ) and diam(Ai ) ≤ ε} . i=1 (2) The Hausdorff (or ball) measure of noncompactness χ : B(M ) → [0, ∞) is defined by i=N [ χ(A) = inf{r > 0; A ⊂ B(xi , r) with xi ∈ M } , i=1 where B(x, r) denote the closed ball centered at x with radius r. We have the following more general result: 5 g e = {(x Proposition 2. Let A be a bounded subset of M . Set A n ); xn ∈ A}. Then we have e . χ(A) = χ(A) Proof. Let ε > χ(A) and δ > 0. Then by definition of χ, there exists a finite set D = {x1 , x2 , · · · , xN } such that i=N [ A⊂ B(xi , ε) . i=1 e Our previous result implies that D e is compact. So there exists a finite set Consider D. {x̃1 , x̃2 , · · · , x̃K } such that i=K [ e D⊂ B(x̃i , δ) . i=1 From here it is easy to see that which implies e⊂ A i=K [ B(x̃i , ε + δ) i=1 e ≤ε+δ . χ(A) e ≤ ε, which implies Since δ was chosen arbitrarily then we have χ(A) e ≤ χ(A) . χ(A) e Let ε > 0. Set In order to complete our proof, we need to show that χ(A) ≤ χ(A). e + ε. Then there exist x̃i (i = 1, · · · , K) in à such that r = χ(A) e⊂ A i=K [ B(x̃i , r) . i=1 ^ Set x̃i = (x i (n)), for i = 1, · · · , K, with xi (n) ∈ A. We claim that for any δ > 0 there exists n0 ≥ 1 such that i=K [ A⊂ B(xi (n0 ), r + δ) . i=1 Assume not. Then there exists δ0 > 0 such that for any n ≥ 1, there exists xn ∈ A which satisfies i=K [ xn 6∈ B(xi (n), r + δ0 ) . i=1 6 M.A. KHAMSI ^ ∈ Ã. Then Set x̃ = (x(n)) ˜ x̃i ) = lim d(xn , xi (n)) ≥ r + δ0 d(x̃, U for i = 1, · · · , K. Clearly we have x̃ 6∈ i=K [ B(x̃i , r) i=1 which is our desired contradiction. So let δ > 0, we know that there exists n0 ≥ 1 such that i=K [ A⊂ B(xi (n0 ), r + δ) . i=1 This clearly implies e +ε+δ . χ(A) ≤ r + δ = χ(A) Since ǫ and δ were chosen arbitrarly positive, we conclude that e , χ(A) ≤ χ(A) which completes the proof of our proposition. ¤ When M is not compact, more can be said about the ultrapower. f is not separable Proposition 3. Assume M is not compact. Then M Proof. Assume M is not compact. Then there exists a bounded sequence (xn ) with no convergent subsequence. In particular lim xφ(n) does not exists for any subsequence (xφ(n) ) U of (xn ). Moreover we can assume that there exists ε > 0 such that sep(xn ) = inf{d(xn , xm ); n 6= m} ≥ ε . ^ For any subsequence (xφ(n) ) of (xn ) set x̃φ = (x φ(n) ). Clearly we have ˜ φ(n) , xα(n) ) ≥ ε . ˜ φ , x̃α ) = lim d(x d(x̃ U Since any sequence has uncountably many subsequences, the above result implies that f f is not separable. ¤ M has an uncountable ε-separated set. Therefore M It is quite an amazing result since a linear version of it is also known. Next we discuss how Lipschitzian mappings extend naturally to the ultrapower. Indeed, let T : M → M be a Lipschitzian mapping with L as a constant of Lipschitz, i.e. d(T (x), T (y)) ≤ L d(x, y) for x, y ∈ M . 7 Then lim d(xn , yn ) = 0 implies lim d(T (xn ), T (yn )) = 0 . U U f→M f defined by This obviously implies that T̃ : M g ^ T̃ ((x n )) = (T (xn )) is well defined. It is easy to check that T̃ is Lipschitzian with L as a constant of Lipschitz. We also have T̃ (x) = T (x) for any x ∈ M . Before we jump to the next section where the main result of this work will be stated, it is worth to mention that hyperconvexity is not a super-property, i.e. the ultrapower of a hyperconvex metric space is not necessarily hyperconvex. For more on ultrapowers and nonstandard techniques, the interested reader is advised to consult [AK] and [Sm]. Main Result Before we state the main result of this work, we will need some definitions. Let M be a metric space. A map T : M → M is said to be asymptotically nonexpansive if there exists a sequence of positive numbers {kn }, with lim kn = 1, such that n→∞ d(T n (x), T n (y)) ≤ kn d(x, y) for any x, y ∈ M and n = 1, 2, · · · . The main result of our work goes as follows: Theorem. Let H be a bounded hyperconvex metric space and T : H → H be asymptotically nonexpansive mapping. Then T has approximate fixed points, i.e. inf{d(x, T (x)); x ∈ H} = 0 . Proof. In order to prove the above conclusion, we need to show that for any ε > 0, there exists x ∈ H such that d(x, T (x)) ≤ ε. Using the metric convexity of H, we define the map ¶ µ 1 1 n x0 T ⊕ 1− Tn = kn kn where x0 is a fixed point in H and kn is the Lipschitz constant of T n . The maps {Tn } are e of H, over a nontrivial ultrafilter U. Define the nonexpansive. Consider the ultrapower H operators T̂ and T̃ by g g n ^ ^ T̂ (x̃) = T̂ ((x n )) = (T (xn )) and T̃ (x̃) = T̃ ((xn )) = (T (xn )) . 8 M.A. KHAMSI Since T is asymptotically nonexpansive mapping, the map T̂ is nonexpansive. Moreover we have g ^ T̂ ((x n )) = (Tn (xn )) . Since Tn is nonexpansive, Sine and Soardi’s fixed point theorem implies the existence of a g fixed point xn (of Tn ). The point x̃ = (x n ) is a fixed point of T̂ . Hence the fixed point set e F ix(T̂ ) is a nonempty subset of H. Since the two operators T̂ and T̃ commute, then T̃ leaves invariant the set F ix(T̂ ). It is easy to show that T̃ restricted to F ix(T̂ ) is in fact an isometry (in particular it is nonexpansive). Fix ε > 0. Let x̃i ∈ F ix(T̂ ), i=1,..,N. If ³^´ x̃i = xn (i) , for i = 1, · · · , N , set ³ ´ εn = max d xn (i), Tn (xn (i)) . 1≤i≤N Then we have lim εn = 0 . U Set Hn = {x ∈ H; d(x, Tn (x)) ≤ εn } . Then Hn 6= ∅ because xn (i) ∈ Hn , for i=1,..,N. Since Tn is nonexpansive, Sine [Sn2] proved that Hn is hyperconvex. Therefore, there exists zn (i) = εxn (1) ⊕ (1 − ε)xn (i) ∈ Hn for i = 1,..,N. Consider, the point ³^´ z̃i = zn (i) which we will denote εx̃1 ⊕ (1 − ε)x̃i . Then we have, z̃i ∈ F ix(T̂ ), and d(z̃i , z̃j ) ≤ (1 − ε)d(x̃i , x̃j ) for i,j=2,..,N. Back to our maps T̂ and T̃ . Let x̃ ∈ F ix(T̂ ). Write x̃ = x̃1 . Then from the above ideas, there exists x̃2 ∈ F ix(T̂ ) such that ³ ´ x̃2 = εx̃1 ⊕ (1 − ε)T̃ x̃1 . ³ ´ By induction, we will construct a sequence x̃n of points in F ix(T̂ ) defined by x̃n+1 ³ ´ = εx̃1 ⊕ (1 − ε)T̃ x̃n . 9 We have for any n < m, à ³ ´ ³ d(x̃n , x̃m ) ≤ (1 − ε)d T̃ x̃n−1 , T̃ x̃m−1 ´ ! and since T̃ is nonexpansive when restricted to F ix(T̂ ), we get ´ ³ d(x̃n , x̃m ) ≤ (1 − ε)d x̃n−1 , x̃m−1 . ³ ´ This clearly implies that the sequence x̃n is a Cauchy sequence. Hence it converges to ω̃ ∈ F ix(T̂ ). Moreover we have ³ ´ ³ ´ ³ ´ d ω̃, T̃ (ω̃) = lim d x̃n+1 , T̃ (x̃n ) ≤ ε lim d x̃1 , T̃ (x̃n ) . n→∞ n→∞ If we set δ = diameter(H), we get ³ ´ d ω̃, T̃ (ω̃) ≤ εδ . Therefore, we have proved that for any ε > 0, there exists ω̃ε ∈ F ix(T̂ ), such that ³ ´ d ω̃ε , T̃ (ω̃ε ) ≤ ε . From this it is easy to extract xε ∈ H, such that ³ ´ d xε , T (xε ) ≤ ε for any ε > 0. ¤ Remark. Recall that Kirk’s original question was about the existence of a fixed point for such mappings. This problem is still open. But one may use a simple embedding of any metric space M into l∞ (M ) (see [EK] for more on this), to show that in fact the main problem described here is equivalent to the same problem for the unit ball of l∞ . It is worth to mention that the existence of fixed point for asymptotically nonexpansive mappings is closely related to the existence of fixed point for k-uniformly Lipschitzian mappings in the linear case (see [KX] for more on this). The author wishes to thank the referee for valuable comments on the final version of this work. 10 M.A. KHAMSI References [AK] A.G. Aksoy, M.A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, New York, Berlin (1990). [AP] N. Aronszajn, P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439. [ADL] J.M. Ayerbe, T. Dominguez and G. Lopez, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory, Advances and Applications 99, Birkhuser Verlag, Basilea 1997. [Ba] J.B. Baillon, Nonexpansive mappings and hyperconvex spaces, Contemp. Math. 72 (1988), 11–19. [EK] R. Espinola Garcia and M.A. Khamsi, Introduction to hyperconvex spaces, Handbook of Metric Fixed Point Theory, W.A. Kirk and B. Sims, Editors, Kluwer Academic Publishers (2001). [GK] K. Goebel, and W. A. Kirk,, Topics on Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, (1990). [Is] J.R. 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Math.,no. 160, Queen’s University, Kingston, Ontario (1982). [Sn1] R.C. Sine, On linear contraction semigroups in sup norm spaces, Nonlinear Anal. 3 (1979), 885– 890. [Sn2] R.C. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal. 13 (1989), 863-869. [So] P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25–29. Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968-0514, USA E-mail address: mohamed@math.utep.edu