λ Property Of Orlicz-Lorentz Spaces And On A Property Of φ-variational Modular Spaces

This paper contains two parts.The first part has three sections:In the first section, the basical notions of Orlicz-Lorentz spaces and the A property are introduced; In the second and third sections we discuss theλproperty of Orlicz-Lorentz sequence spaces and Orlicz-Lorentz function spaces with the Luxemburg norm, and we aslo discuss the A property of Orlicz-Lorentz function spaces with the Orlicz norm. And the following results are obtained:Theorem 1.1 Letωbe positive and strictly decreasing. Thenλφ,ωhas theλproperty.Theorem 1.2 Letωbe positive, continuous and strictly decreasing on [0,γ]. ThenΛφ,ωhas theλproperty.Theorem 1.3Letωbe positive, continuous and strictly decreasing on [0,γ].ThenΛφ,ωhas uniformlyλproperty if and only ifφis strictly convex.Theorem 1.4 Letωbe positive, continuous and strictly decreasing on [0,γ]. ThenΛφ,ω°has theλproperty.In the second part, L. Maligranda pointed out whether condition (B.1) is satisfied in the variational modular spaceχρφ* is an open problem. We will answer this open problem inχρφ*', a subspace ofχρφ*. And the following result is obtained: Theorem 2.1 Letχbe the space of real-valued functions in the interval[a,b]such that x(a)=0 and let where supremum is taken over all partitionsπ:a=t0<t1<…<tn=b.Defineχρφ*'={x:x∈χρ* and all the discontinuity points of x are isolated} Then condition(B.1)is satisfied in the spaceχρφ*'.According to this theorem,we can draw a conclusion thatχρφ*' can always be F-normed.