CHOICE FREE FIXED POINT PROPERTY IN SEPARABLE BANACH SPACES 7 1 VASSILIOSGREGORIADES 0 2 n a J Abstract. We show that the standard approach of minimal invariant sets, 3 whichappliesZorn’sLemmaandisusedtoprovefixedpointtheoremsfornon- 1 expansivemappingsinBanachspacescanbeappliedwithoutanyreferenceto the full Axiom of Choice when the given Banach space is separable. Our ] methodappliesresultsfromclassicalandeffective descriptivesettheory. O L Introduction. A mapping T : X → Y between Banach spaces is non-expansive . h if kTx−Tyk ≤ kx−yk , for all x,y ∈ X. A Banach space X has the fixed t Y X a point property if for all non-empty convex and weakly compact F ⊆ X and for all m non-expansive mappings T :F →F there exists an x∈F with T(x)=x, i.e., x is [ a fixed point of T. A set A is T-invariant if T[A]⊆A . It is well knownthat Banachspaceswhich are uniformly convexor have a “nor- 1 mal” structure have the fixed point property. In fact in these cases one can give v 2 a constructive proof of the existence of a fixed point, which in particular does not 5 use Zorn’s Lemma, see for example [10] and [14]. 7 Nevertheless the standardtechnique for provingthat a Banach space X has the 3 fixed point property is to show -given F and T as above- that there exists a non- 0 empty, T-invariant, convex, weakly compact L⊆F which is minimal with respect . 1 totheseproperties,cf.[9]. ThenoneshowsthatLhaszerodiameterandsoLmust 0 be a singleton, say{x}. Since L is T-invariantwe havethat x is a fixedpoint ofT. 7 1 The typical way of verifying the existence of a minimal set L as above is by : applying Zorn’s Lemma. Our main aim is to show that, in the setting of separable v Banach spaces, it is possible to obtain such a minimal set without the Axiom of i X Choice(AC),cf.Theorem8. Wewilldothisbyapplyingresultsfromclassicaland r effectivedescriptivesettheory. AttheendofthisarticlewepresentaBanachspace a theoretic approachfor proving Theorem 8, which does not use effective theory. Let us see first exactly where the Axiom of Choice is used in order to prove the existence of a minimal set as above. This is an instantiation of the derivation of Zorn’s Lemma from AC. 2010 Mathematics Subject Classification. 47H10,03E15,54H05,54H25. Keywordsandphrases. minimalinvariantsets,non-expansivemappings,fixedpointproperty, AxiomofChoice,effective descriptivesettheory. This a pre-publication draft of the work published in the Proceedings of the American MathematicalSociety 143 (2015), no. 5, 2143-2157. TheauthorwouldliketothankU.Kohlenbach,A.Kreuzer,H.MildenbergerandC.Poulios forhelpfuldiscussions,remarksandsuggestions. Theauthorwouldalsoliketogivespecialthanks totherefereeofthisarticle,fortheirvaluablesuggestionsandforprovidingsomeveryinteresting materialwhichcanbefoundatthelastpagesofthearticle. 1 2 V.GREGORIADES A partially ordered space (P,≤) is inductive if every linearly ordered subset of P has a least upper bound and a mapping f : P → P is expansive if x ≤ f(x) for all x∈P. Theorem 1 (Zermelo’s Fixed Point Theorem). For every inductive space (P,≤) and for every expansive function f :P→P there exists some x∗ ∈P with f(x∗)=x∗. Now consider a Banach space X, a non-empty convex weakly compact F ⊆ X and a non-expansive mapping T :F →F. We define the set P={K ⊆F | K is non-empty, convex, weakly compact and T-invariant} and we consider the relation ≤ of the inverse inclusion, i.e., K ≤L⇐⇒L⊆K for all K,L∈P. Define also the strict part < of ≤ by K r and so the the sequence (τ ) is in fact Σ0(ε,(x ))- xn s xn n∈ω 1 n recursive. Now we go back to the equivalence (1) and we show that the L and x which appear there can be chosen to be in ∆1(ε,(x )) i.e., we claim that 1 n (2) K is weakly compact ⇐⇒ (∀(x ) ⊆K)(∃L∈∆1(ε,(x )))(∃x∈K∩∆1(ε,(x ))) n n∈ω 1 n 1 n (∀x∗ ∈BX∗)[limx∗(xn)=x∗(x)] n∈L for all K ⊆ X. The right-to-left-hand implication is clear from (1) so let us prove the left-to-right-handimplication. Suppose that (x ) is a sequence in K. Since n n∈ω the latter set is weakly compact, it is in particular bounded and so the sequence of functions (τ ) is pointwise bounded. Moreover from the preceding claim xn n∈ω the sequence (τ ) is in ∆1(ε,(x )). Finally since K is weakly compact every xn n∈ω 1 n cluster point of (τxn)n∈ω in RBX∗ is a function of the form τx for some x ∈ K. Hence every cluster point of (τ ) is in fact a continuous function. Therefore xn n∈ω all conditions of Debs’ Theorem are satisfied and so there exists an infinite L ⊆ ω in ∆1(ε,(x )) such that the subsequence (τ ) is pointwise convergent. Using 1 n xn n∈L again the weak compactness of K it follows that the sequence (x ) is weakly n n∈L convergentto some x∈K. From Theorem12 we have that x is in ∆1(ε,(x ) ). 1 n n∈L It is clear that the sequence (x ) is recursive in the pair (L,(x ) ). Since L n n∈L n n∈ω is in ∆1(ε,(x )) we have that x is in ∆1(ε,(x )) as well. This completes the proof 1 n 1 n of the equivalence (2). (The application of Debs’ Theorem found here is similar to the argument for proving Corollary 1.10 in [11], p. 161-162.) Finallyweverifythattheconditionoftherightsideof(2)definesaΠ1 set. The relation R⊆Xω ×X ×BX∗ ×ω defined by e 1 R((y ),x,x∗,s)⇐⇒|x∗(y )−x∗(x)|0 and ε>0 there exists some n∈ω such that ∞ kfk −ε

Choice free Fixed Point Property in separable Banach spaces

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