On a connection between the fixed point property and L∞(Ω,Σ,μ)L∞(Ω,Σ,μ)

For a finite and positive measure space (Ω,Σ,μ)(Ω,Σ,μ) and any weakly compact convex subset of L∞(Ω,Σ,μ)L∞(Ω,Σ,μ) a fixed point theorem for a class of nonexpansive self-mappings is proved. An analogous result is obtained for the space C(Ω)C(Ω). Also, for a bounded domain with C1C1 boundary Ω⊂RNΩ⊂RN, we construct a family of Banach spaces Hp(Ω)Hp(Ω), p>1p>1, which are continuously embedded in L∞(Ω)L∞(Ω) and possess the fixed point property.