Some geometrical properties and fixed point theorems in Orlicz spaces

Let (G, ∑, μ) be a finite, atomless measure space and let L^φ be an Orlicz space of measurable functions on G. We consider some geometrical properties of the functional ρ(f{hook}) = ∝_Gφ(f{hook}(t)) dμ(t), called the Orlicz modular. These properties, like strict convexity, uniform convexity or uniform convexity in every direction, can be equivalently expressed in terms of the properties of the corresponding Orlicz function φ. We use these properties in order to prove some fixed point results for mappings T: B → B, B ⊂ L^φ, that are nonexpansive with respect to the Orlicz modular ρ, i.e., ρ(Tf{hook} - Tg) ≤ ρ(f{hook} - g) for all f{hook} and g in B. We prove also existence and uniqueness in L^φ of the best approximant with respect to ρ and some convex subsets of L^φ. Our results are valid also in the case when the Orlicz function φ does not satisfy the Δ₂-condition. This demonstrates the advantage of our method because, in the latter case, both Luxemburg's and Orlicz's norms cannot possess suitable convexity properties.