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As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the unit ball, retractions of the unit ball onto its boundary, contractions of the unit sphere, and nonzero maps without positive eigenvalues and normalized eigenvectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps. 1. A “folklore” theorem of nonlinear analysis Given a Banach space X, we denote by Br (X) := {x ∈ X : x ≤ r } the closed ball and by Sr (X) := {x ∈ X : x = r } the sphere of radius r > 0 in X; in particular, we use the shortcut B(X) := B1 (X) and S(X) := S1 (X) for the unit ball and sphere. All maps considered in what follows are assumed to be continuous. By ν(x) := x/ x we denote the radial retraction of X \ {0} onto S(X). One of the most important results in nonlinear analysis is Brouwer’s fixed point principle which states that every map f : B(RN ) → B(RN ) has a fixed point. Interestingly, this characterizes finite-dimensional Banach spaces, inasmuch as in each infinite-dimensional Banach space X one may find a fixed point free self-map of B(X). The existence of fixed point free self-maps is closely related to the existence of other “pathological” maps in infinite-dimensional Banach spaces, namely, retractions on balls and contractions on spheres. Recall that a set S ⊂ X is a retract of a larger set B ⊃ S if there exists a map ρ : B → S with ρ(x) = x for x ∈ S; this means that one may extend the identity from S by continuity to B. Likewise, a set S ⊂ X is called contractible if there exists a homotopy h : [0,1] × S → S joining the identity with a constant map, that is, such that h(0,x) = x and h(1,x) ≡ x0 ∈ S. We summarize with the following Theorem 1.1; although this theorem seems to be known in topological nonlinear analysis, we sketch a brief proof which we will use in the sequel. Theorem 1.1. The following four statements are equivalent in a Banach space X: (a) each map f : B(X) → B(X) has a fixed point, (b) S(X) is not a retract of B(X), Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:4 (2004) 317–336 2000 Mathematics Subject Classification: 47H10, 47H09, 47J10 URL: http://dx.doi.org/10.1155/S1687182004406068

318

Banach space constants in fixed point theory

(c) S(X) is not contractible, (d) for each map g : B(X) → X \ {0}, one may find λ > 0 and e ∈ S(X) such that g(e) = λe. Sketch of the proof. (a)⇒(b). If ρ : B(X) → S(X) is a retraction, the map f : B(X) → B(X) defined by f (x) := −ρ(x)

(1.1)

is fixed point free. (b)⇒(c). Given a homotopy h : [0,1] × S(X) → S(X) with h(0,x) = x and h(1,x) ≡ x0 ∈ S(X), for 0 < r < 1 we set    x0  ρ(x) :=   1 − x  ,ν(x) h 1−r

for x ≤ r, for x > r.

(1.2)

Then, ρ : B(X) → S(X) is a retraction. (c)⇒(d). Given g : B(X) → X \ {0}, for 0 < r < 1 we set    x    −g

for x ≤ r,

r

σ(x) := 

 x  − r 1 − x      x− g ν(x) 1−r 1−r

(1.3) for x > r.

Then, there exists z ∈ B(X