Fixed points and approximate fixed points in product spaces

TAIWANESE JOURNAL OF MATHEMATICS Vol. 5, No. 2, pp. 405-416, June 2001 This paper is available online at http://www.math.nthu.edu.tw/tjm/ FIXED POINTS AND APPROXIMATE FIXED POINTS IN PRODUCT SPACES R. Espínola and W. A. Kirk Abstract. The paper deals with the general theme of what is known about the existence of fixed points and approximate fixed points for mappings which satisfy geometric conditions in product spaces. In particular it is shown that if Xand Yare metric spaces each of which has the fixed point property for nonexpansive mappings, then the product space (X×Y)∞has the fixed point property for nonexpansive mappings satisfying various contractive conditions. It is also shown that the product space H=(M×K)∞has the approximate fixed point property for nonexpansive mappings whenever Mis a metric space which has the approximate fixed point property for such mappings and Kis a bounded convex subset of a Banach space. 1. INTRODUCTION The study of fixed point theory for nonexpansive mappings in product spaces is an outgrowth of its analog for continuous mappings. A topological space is said to have the fixed point property if every continuous self-map of the space has a fixed point. It has been known for some time that if both Xand Yhave the fixed point property for continouus mappings, then it need not be the case that X×Yhas the fixed point property for mappings f:X×Y→X×Ywhich are continuous relative to the product topology. Indeed, an example is given in [4] of a metric space Xwhich has the fixed point property, yet the space X×Xfails to have the fixed point property. See, for example, [6] (specifically, Theorem 4.9) for a more extensive discussion. Received February 9, 2000; revised December 15, 2000. Communicated by M.-H. Shih. 2001 Mathematics Subject Classification: 54H25, 47H09; Secondary 47H10. Key words and phrases: Nonexpansive mapping, product space, fixed point, approximate fixed point property. This research was conducted while the first author was visiting the University of Iowa. He acknowledges the kind hospitality of the University of Iowa and also the support of DGICYT research project PB96-1338-C02-01. 405 406 R. Espíinola and W. A. Kirk In 1968, Nadler [17] initiated a study of fixed point properties of mappings T:X×Y→X×Y, where Xis a topological space with the fixed point property, Yis a metric space, and Tis a continuous mapping which is also a local contraction in its second coordinate. Fora continues this approach in [7]. The above results lead naturally to the question of what happens if both Xand Yare metric spaces with the contractive conditions placed directly on T. For this discussion we need to fix some terminology. A mapping fof a metric space (M,d) into a metric space (N,r)is said to be nonexpansive if r(f(x),f(y)) ≤d(x, y)for all x, y ∈M. If r(f(x),f(y)) <d(x, y)for all x, y ∈Mwith x6=y, then fis said to be strictly contractive. A mapping fis said to be a generalized contraction if for each x∈Mthere exists α(x)∈(0,1) such that for each y∈M, r(f(x), f(y)) ≤α(x)d(x, y).Ifαis a constant map, then of course fis a contraction mapping in the sense of Banach. We shall use fix (f)to denote the set of fixed points of a mapping f:M→M. If (X, ρ)and (Y,d)are metric spaces, then the metric d∞on X×Yis defined in the usual way: d∞((x, u),(y,v)) = max{ρ(x, y),d(u, v)} for (x, u),(y,v)∈X×Y. We shall confine ourselves to the metric d∞in this paper, although all of the results, indeed in some instances even stronger ones, seem to hold for the metrics dp,p∈[1,∞), dp((x, u),(y,v)) = [(ρ(x, y))p+(d(u, v))p]1/p. A basic question now becomes: If (X, ρ)and (Y,d)have the fixed point property for nonexpansive mappings and if T:X×Y→X×Yis nonexpansive relative to the metric d∞,then does Tnecessarily have a fixed point? Although sharp results have been obtained, the full answer to this question remains open. In the next section we summarize what is known about metric fixed point theory in product spaces. In Section 3 we prove some new results for mappings satisfying `contractive' conditions. In Section 4 we prove a new result about the existence of `approximate fixed points' for nonexpansive mappings in product spaces by applying a well-known result about asymptotic regularity of `averaged' nonexpansive mappings. 2. OVERVIEW We begin by summarizing the results of Nadler and Fora. Here and throughout we use P1(resp., P2) to denote the natural coordinate projection of X×Yonto X Fixed Points and Approximate Fixed Points in Product Space 407 (resp., onto Y). Version (C) of this result is due to Nadler; version (C0) to Fora. Alternate proofs of these results are given in [11]. Theorem 2.1. Suppose Xis a topological space which has the fixed point property with respect to continuous mappings,suppose Yis a complete metric space,and suppose T:X×Y→X×Yis a continuous mapping which satisfies (C) for each x∈X, there exists a number λ(x)∈(0,1) such that for all u, v ∈Y, d(P2◦T(x, u),P 2◦T(x, v)) ≤λ(x)d(u, v). Then Thas a fixed point if either (a) Tis uniformly continuous;or (b) Yis locally compact. Assumptions (a) and (b) can be dropped if condition (C) is strengthened to (C0)for each x∈X, there exists a number λ(x)∈(0,1) and a neighborhood Vx such that for each w∈Vxand all u, v ∈Y, d(P2◦T(w, u),P 2◦T(w, v)) ≤λ(x)d(u, v). We now turn to nonexpansive mappings in product spaces. In [15], it was shown that if a bounded closed convex subset Hof a Banach space has the fixed point property for nonexpansive mappings, and if Kis a bounded closed convex subset of either a uniformly convex or uniformly smooth Banach space, then every nonexpansive T:H×K→H×Khas a fixed point. This result led to a sequence of generalizations, culminating in a remarkable result of T. Kuczumow [16]. In order to describe Kuczumow's result, we need some additional facts. It is known that, in general, a weakly compact convex subset of a Banach space need not have the fixed point property for nonexpansive mappings (Alspach [1]), but at the same time weak compactness (or reflexivity of the underlying space) in conjunction with a variety of other geometric conditions (e.g., see [9]) does in fact assure that any closed convex set has the fixed point property for nonexpansive mappings. In view of this, the following definition is quite natural. Definition 2.1. A closed convex subset Kis said to have the generic fixed point property (for nonexpansive mappings)if for every nonexpansive T:K→K and every T-invariant nonempty closed convex H⊆K, fix (T)∩H6=∅. Kuczumow used a retraction approach based on a method of Bruck [3] to prove the following. Theorem 2.2. Let Xbe a Banach space. Suppose K⊆Xis weakly compact convex and has the generic fixed point property,and suppose (Y,d)is a metric 408 R. Espíinola and W. A. Kirk space which has the fixed point property for nonexpansive mappings. Then every nonexpansive T:(K×Y)∞→(K×Y)∞has a fixed point. Kuczumow observed that if Xis a conjugate space, then the weak topology in the above result can be replaced by the weak∗topology. Bruck's paper [3] is remarkably rich in ideas, and in fact a different approach found in the same paper can be modified to prove the following result. The details are found in [12]. Theorem 2.3. Let Ebe a Banach space. Suppose X⊆Eis a separable closed convex subset of Ewhich has the generic fixed point property,and suppose (Y,d) is a separable metric space which has the fixed point property for nonexpansive mappings. Then every nonexpansive T:(X×Y)∞→(X×Y)∞has a fixed point. While its method of proof is different, it is not clear to what extent, if any, Theorem 2.3 is actually qualitatively more general than Theorem 2.2. This is because there is no known example of a closed convex subset of a Banach space which has the generic fixed point property yet fails to be weakly compact. There are perhaps two additional results which should be mentioned. While we are basically interested here in the case p=∞,it is quite easy to prove the following for 1≤p<∞. Theorem 2.4. Let Eand Fbe Banach spaces. Suppose X⊆Eand Y⊆ Fboth have the fixed point property for nonexpansive mappings. Then every nonexpansive T:(X×Y)p→(X×Y)phas a fixed point for 1≤p<∞. A proof of the above result is given in [14], based on an argument given for the following result in [11]. Theorem 2.5. Let Eand Fbe Banach spaces. Suppose X⊆Eand Y⊆ Fboth have the fixed point property for generalized contractions. Then every generalized contraction T:(X×Y)p→(X×Y)phas a fixed point for 1≤p≤∞. This completes an overview of what appear to be the most important known results. We now turn to some new observations. 3. CONTRACTIVE MAPPINGS IN PRODUCT SPACES If the assumption of nonexpansiveness is strengthened, then it is possible to prove additional results in a fairly direct manner. Fixed Points and Approximate Fixed Points in Product Space 409 Theorem 3.1. Let (X, ρ)and (Y,d)be metric spaces. Suppose Yhas the fixed point property for nonexpansive mappings and suppose Xhas the fixed point property for strictly contractive mappings,and suppose T:(X×Y)∞→(X×Y)∞ is a nonexpansive mapping which satisfies the additional condition ρ(P1◦T(x, u),P 1◦T(y,v)) <d ∞((x, u),(y,v)) for all (x, u),(y,v)∈X×Ysatisfying ρ(x, y)6=d(u, v).Then Thas a fixed point. Theorem 3.2. Let (X, ρ)and (Y,d)be metric spaces,each of which has the fixed point property for strictly contractive mappings. Then every strictly contractive mapping T:(X×Y)∞→(X×Y)∞has a fixed point. Proof of Theorem 3.1. Fix u∈Yand define Tu:X→Xby setting Tu(x)=P1◦T(x, u),x∈X. Then if x6=y, it follows that ρ(x, y)6=d(u, u)=0,and we have ρ(Tu(x),T u(y))= ρ(P1◦T(x, u)),ρ(P1◦T(y,u)) <d ∞((x, u),(y,u)) =ρ(x, y). Thus Tuis strictly contractive and by assumption has a unique fixed g(u)∈X. Now define ϕ(u)=P2◦T(g(u),u). We show that ϕis nonexpansive. Note that since Tu(g(u)) = g(u)and Tv(g(v)) = g(v),we have g(u)=P1◦T(g(u),u); g(v)=P1◦T(g(v),v), and, moreover, if ρ(g(u),g(v)) 6=d(u, v),then ρ(g(u),g(v))= ρ(P1◦T(g(u),u),P 1◦T(g(v),v)) <d ∞((g(u),u),(g(v),v)) = max{ρ(g(u),g(v)),d(u, v)} =d(u, v). Therefore, ρ(g(u),g(v)) ≤d(u, v)for all u, v ∈Y. It follows that d(ϕ(u),ϕ(v)) = d(P2◦T(g(u),u),P 2◦T(g(v),v)) ≤max{ρ(P1◦T(g(u),u),P 1◦T(g(v),v)),d(P2◦T(g(u),u),P 2◦T(g(v),v))} =d∞(T(g(u),u),T(g(v),v)) ≤d∞((g(u),u),(g(v),v)) = max{ρ(g(u),g(v)),d(u, v)}=d(u, v). 410 R. Espíinola and W. A. Kirk Therefore, ϕ:Y→Yis nonexpansive. Since Yhas the fixed point property for nonexpansive mappings, there exists u∈Ysuch that ϕ(u)=u;whence u=ϕ(u)=P2◦T(g(u),u). Since by assumption g(u)∈fix (Tu),we have Tu(g(u)) = P1◦T(g(u),u). Proof of Theorem 3.2. The argument follows the previous one, except in this case we must show that ϕis strictly contractive. The fact that Tis strictly contractive assures that max{ρ(P1◦T(x, u),P 1◦T(y,v)),d(P2◦T(x, u),P 2◦T(y,v))} <max{ρ(x, y),d(u, v)} if x6=yor u6=v. Following the previous argument step-by-step, we conclude that for u∈Yand x6=y, the mapping Tuis strictly contractive and has a unique fixed point g(u). Also, if u6=vwe have d(ϕ(u),ϕ(v))= d(P2◦T(g(u),u),P 2◦T(g(v),v)) ≤d∞(T(g(u),u),T(g(v),v)) <d ∞((g(u),u),(g(v),v)) =d(u, v). The conclusion now follows as in Theorem 3.1. The following is a variant of Theorem 3.1. The assumptions on the mapping T do not seem to be comparable. Theorem 3.3. Let (X, ρ)and (Y,d)be metric spaces,each of which has the fixed point property for nonexpansive mappings,and suppose T:(X×Y)∞→ (X×Y)∞is a nonexpansive mapping which satisfies the additional condition ρ(P1◦T(x, u, P1◦T(y,v)) <d ∞((x, u),(y,v)) for all (x, u),(y,v)∈X×Ysatisfying u6=vand x6=y. Then Thas a fixed point. Theorem 3.3 has the following immediate corollary. Corollary 3.1. Let (X, ρ)and (Y,d)be metric spaces,each of which has the fixed point property for nonexpansive mappings,and suppose T:(X×Y)∞→ (X×Y)∞is a nonexpansive mapping which is quasi-contractive in the sense that d∞(T(x, u),T(y,v)) <d ∞((x, u),(y,v)) Fixed Points and Approximate Fixed Points in Product Space 411 for all (x, u),(y,v)∈X×Ysatisfying u6=vand x6=y. Then Thas a (unique) fixed point. Note that the condition of the corollary is weaker than the contractive condition of Theorem 3.2. In exchange, a little more is assumed about the spaces; specifically that Xhas the fixed point property for nonexpansive mappings. Proof of Theorem 3.3. Fix u∈Yand as before define Tu:X→Xby setting Tu(x)=P1◦T(x, u),x∈X. Then ρ(Tu(x),T u(y)) ≤max{ρ(P1◦T(x, u),P 1◦T(y,u)),d(P2◦T(x, u),P 2◦T(y,u))} =d∞(T(x, u),T(y,u)) ≤d∞((x, u),(y,u)) =ρ(x, y). Thus Tuis nonexpansive and by assumption has a nonempty fixed point set fix (Tu)⊆X. Let gbe any selection of the mapping u7→ fix (Tu) and define ϕas in Theorem 3.1. We show that ϕis nonexpansive. Since g(u)∈ fix (Tu)and g(v)∈fix (Tv), we have g(u)=P1◦T(g(u),u); g(v)=P1◦T(g(v),v). Now let u, v ∈Y. There are two cases. 1. If g(u)=g(v),then obviously ρ(g(u),g(v)) ≤d(u, v)and we have d(ϕ(u),ϕ(v)) = d(P2◦T(g(u),u),P 2◦T(g(v),v)) ≤max{ρ(P1◦T(g(u),u),P 1◦T(g(v),v)),d(P2◦T(g(u),u),P 2◦T(g(v),v))} =d∞(T(g(u),u),T(g(v),v)) ≤d∞((g(u),u),(g(v),v)) ≤d(u, v). 2. On the other hand, if g(u)6=g(v), then it must also be the case that u6=v. Therefore, ρ(g(u),g(v)) = ρ(P1◦T(g(u),u),P 1◦T(g(v),v)) <d ∞((g(u),u),(g(v),v)) = max{ρ(g(u),g(v)),d(u, v)} =d(u, v) 412 R. Espíinola and W. A. Kirk and it follows that d(ϕ(u),ϕ(v)) = d(P2◦T(g(u),u),P 2◦T(g(v),v)) ≤max{ρ(P1◦T(g(u),u),P 1◦T(g(v),v)),d(P2◦T(g(u),u),P 2◦T(g(v),v))} =d∞(T(g(u),u),T(g(v),v)) ≤d∞((g(u),u),(g(v),v)) = max{ρ(g(u),g(v)),d(u, v)}=d(u, v). Therefore, in either case, d(ϕ(u),ϕ(v)) ≤d(u, v),and ϕ:Y→Yis nonexpansive. The conclusion again follows as in Theorem 3.1. In the preceding proof, the question might arise as to whether fix (Tu)is a singleton. Suppose otherwise, and let g1(u)and g2(u)be distinct choices for the selection u7→ fix (Tu).Then according to case 2, for any v∈Y, d(g1(u),g 2(u)) ≤d(g1(u),v)+d(g2(u),v)<2d(u, v). Obviously, this can happen only if uis an isolated point of Y. To facilitate comparison, we summarize the foregoing results as follows: Theorem 3.4. Let (X, ρ)and (Y,d)be metric spaces,and suppose T:(X× Y)∞→(X×Y)∞is a nonexpansive mapping. Then Thas a fixed point if any one of the following conditions holds. (a) Xand Yhave the fixed point property for nonexpansive mappings and T satisfies d∞(T(x, u),T(y,v)) <d ∞((x, u),(y,v)) for all (x, u),(y,v)∈X×Ysatisfying u6=vand x6=y. (b) Yhas the fixed point property for nonexpansive mappings, Xhas the fixed point property for strictly contractive mappings, and T:(X×Y)∞→ (X×Y)∞satisfies ρ(P1◦T(x, u),P 1◦T(y,v)) <d ∞((x, u),(y,v)) for all (x, u),(y,v)∈X×Ysatisfying ρ(x, y)6=d(u, v). (c) Xand Yhave the fixed point property for strictly contractive mappings and Tis strictly contractive. Fixed Points and Approximate Fixed Points in Product Space 413 4. APPROXIMATE FIXED POINTS IN PRODUCT SPACES In this section we prove an approximate fixed point theorem for nonexpansive mappings in certain product spaces. A metric space (M,d)is said to have the approximate fixed point property if any nonexpansive mapping T:M→Mhas an approximate fixed point sequence, that is, a sequence {un}in Mfor which limnd(un,T(un)) = 0.This of course is equivalent to saying inf{d(x, T(x)) : x∈M}=0. Our theorem is based on the following result, which was proved for a single mapping (and for a more general convergence process) by Ishikawa [10]. Edelstein and O'Brien [5] showed that the convergence is uniform over K, and subsequently Goebel and Kirk [8] showed that in fact the convergence is uniform over x0in K and over the class of all nonexpansive mappings T:K→K. Another proof of this fact is given in [13]. For a technical study of the rate of uniform convergence and a comprehensive review of the literature, see [2]. Theorem 4.1. Let Kbe a bounded convex subset of a Banach space and let ε>0.Then there exists N∈Nsuch that if n≥N, if x0∈K, and if T:K→K is nonexpansive,then kfn(x0)−fn+1(x0)k≤ε, where f=(1/2)(I+T). An interesting feature of the proof given below is the fact that the penultimate step of the proof requires the uniformity of the convergence of {fn(x0)}in the above result over the class of all nonexpansive T:K→K. Theorem 4.2. Suppose Mis a metric space which has the approximate fixed point property for nonexpansive mappings and suppose Kis a bounded closed convex subset of a Banach space X. Let H=(K×M)∞. Then Hhas the approximate fixed point property for nonexpansive mappings. Proof. Let T:H→Hbe nonexpansive and let P1and P2denote the respective coordinate projections of Honto Kand M. Fix y∈Mand define Ty:K→K by setting Ty(x)=P1◦T(x, y),x∈K. Now fix x0∈Kand set fy=(I+Ty)/2.