Some geometrical properties and fixed point theorems in Orlicz spaces

AbstractLet (G, ∑, μ) be a finite, atomless measure space and let Lφ be an Orlicz space of measurable functions on G. We consider some geometrical properties of the functional ρ(ƒ) = ∝Gφ(ƒ(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 ⊂ Lφ, that are nonexpansive with respect to the Orlicz modular ρ, i.e., ρ(Tƒ − Tg) ⩽ ρ(ƒ − g) for all ƒ and g in B. We prove also existence and uniqueness in Lφ of the best approximant with respect to ρ and some convex subsets of Lφ. 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