A fixed point property of 𝑙₁-product spaces

Kuczumow, Tadeusz; Reich, Simeon; Schmidt, Malgorzata

doi:10.1090/S0002-9939-1993-1155601-9

A fixed point property of $l_ 1$-product spaces
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by Tadeusz Kuczumow, Simeon Reich and Malgorzata Schmidt PDF

Proc. Amer. Math. Soc. 119 (1993), 457-463 Request permission

Abstract:

Let ${X_1}$ and ${X_2}$ be Banach spaces, and let ${X_1} \times {X_2}$ be equipped with the ${l_1}$-norm. If the first space ${X_1}$ is uniformly convex in every direction, then ${X_1} \times {X_2}$ has the fixed point property for nonexpansive mappings (FPP) if and only if $\mathbb {R} \times {X_2}$ (with the ${l_1}$-norm) does. If ${X_1}$ is merely strictly convex, $(\mathbb {R} \times {X_2})$ has the FPP, and ${C_i} \subset {X_i}$ are weakly compact and convex with the FPP (for $i = 1,2$), then the fixed point set of every nonexpansive mapping $T:{C_1} \times {C_2} \to {C_1} \times {C_2}$ is a nonexpansive retract of ${C_1} \times {C_2}$.

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