Abstract-valued Orlicz spaces of range-varying type

1 Introduction

In this paper we study a new type of Orlicz space, whose members are abstract-valued functions taking values in a varying space. Orlicz space, which was introduced firstly by Orlicz [1] in 1931, is a type of semimodular space commonly generated by a Φ or generalized Φ function. Typical examples of this type are Lebesgue and Sobolev spaces with variable exponents, i.e. L^p(x)(Ω) and W^k,p(x)(Ω) (see [2] for references). Due to the wide applications in many fields of applied mathematics, Orlicz space received a growing interest of scholars in the latest decades. Using the anisotropic function spaces, Antontsev-Shmarev in [3, 4, 5] studied the parabolic equations of variable nonlinearity, including a model porus medium problem. By means of time discretization and subdifferential calculus, Akagi etc in [6, 7] dealt with the doubly nonlinear parabolic equations involving variable exponents. The work [8] considered the application of Orlicz space in Navier-Stokes equation, and [9] investigated an obstacle problem with variable growth and low regularity of the data.

To deal with the evolution equations with variable exponents, a new type of functions, called X_θ(⋅)–valued functions, are needed. As the valued space varies upon the time, it is difficult to give a suitable definition of “measurability” for these functions. By introducing the concepts of bounded topological lattice 𝓐, regular Banach space net {X_α : α ∈ 𝓐} and order-continuous exponent θ : I → 𝓐, Zhang- Li in [10] firstly gave definition of the space L⁰(I, X_θ(⋅)), which contains all the X_θ(⋅)–valued functions measurable in a special manner. Like the measurable functions of range-fixed type, members of L⁰(I, X_θ(⋅)) are all norm-measurable. Based on the useful character, from L⁰(I, X_θ(⋅)) the authors extracted two types of function spaces: continuous type C^–(I, X_θ(⋅)) and integral type L^p(⋅)(I, X_θ(⋅)). After showing their completeness and connections between them together with some concrete examples, the authors paid attention to a semilinear evolution equation with the nonlinearity having a time-dependent domain to illustrate the application of the X_θ(⋅)–valued functions.

It is worth remarking that some Banach space net can be produced by a continuous modular net {ϱ_α : α ∈ 𝓐}. According to whether or not being built on the continuous modular nets, Zhang-Li in [11] divided the X_θ(⋅)–valued function spaces of integral type into two subclasses: norm-modular ones and modular-modular ones. A norm-modular space, like L^p(⋅)(I, X_θ(⋅)), is commonly produced by the semimodular

with a generalized Φ function ϕ, while a modular-modular space is derived from a continuous modular net {ϱ_α : α ∈ 𝓐} with the semimodular

Here, M is a continuous operator from a topological linear space X to a closed cone V of another topological linear space W, called a V–modular (refer to [11]).

Here we will drop the extra map M, and use merely {ϱ_α : α ∈ 𝓐} and θ to reconstruct the semimodular, namely

This change brings much convenience to us to study the duality and reflexivity of the abstract-valued Orlicz spaces of modular-modular type.

The main part of this paper is organized as follows: As preparations, in Section 2, we study the abstract-valued Orlicz space generated by a single modular. Section 3 is devoted to the abstract-valued Orlicz space generated by a series of modular. Using different measurability of the X_θ(⋅)–valued functions, we introduce two different spaces: L^ϱ_θ(⋅)(I, X_θ(⋅)) and L+ϱθ(⋅)(I, X_θ(⋅)), both of them are complete according to the same norm. We show that, under some suitable situations, L+ϱθ(⋅)(I, X_θ(⋅)) is separable, and its dual space can be represented by

We also make some evaluations on the proper conditions for the equality

as well as the reflexivity of L^ϱ_θ(⋅)(I, X_θ(⋅)).

For the sake of applications, in Section 4, we make some discussions on the functionals and operators on the modular-modular space L^ϱ_θ(⋅)(I, X_θ(⋅)), including functional Φ͠_φϑ(⋅) defined by another continuous modular system {φ_β : β ∈ 𝓑} and another order-continuous map ϑ : I → 𝓑, and operators Z_θ(⋅) and ∂φ_ϑ(⋅), which are subdifferentials of Φ_ϱθ(⋅) and Φ͠_φϑ(⋅) respectively. Here Z_θ(⋅) plays the role of an extended dual map, and ∂φ_ϑ(⋅) usually arises in a differential equation as the driving operator. Under some extra assumptions, such as the weak lower-continuity of {ϱ_α} and {ϱα∗}, and the strong coercivity of {ϱ_α} and {φ_β}, demicontinuity and coercivity of Z_θ(⋅) and the representation ∂Φ͠_φϑ(⋅)(u)(t) = ∂φ_ϑ(t)(u(t)) are obtained. This is an attempt to extend the convex functional and its subdifferential generated by a single function to that generated by a series of functions (compare to [12, Ch. 2], and [7]).

After making some investigations on the Bochner-Sobolev spaces W^{1, ϱ_θ(⋅)}(I, X_θ(⋅)) and Wper1,ϱθ(⋅)(I, X_θ(⋅)), and the intersection

including the continuous and compact embedding of W^{1, ϱ_θ(⋅)}(I, X_θ(⋅)) into the space L^ϱ_θ(⋅)(I, X_θ(⋅)) along with the estimate

in Section 5, we study a type of second order nonlinear differential inclusion

with the periodic boundary condition, where the operator f : I × X → X owns a nonstandard growth

for a small number μ > 0 and a nonnegative function h ∈ L¹(I). By introducing the Nemytskij operator F(u) = f(⋅, u), and the second order differential operator Dθ(⋅)2 defined by

we obtain a continuous and compact operator

which by Leray-Schauder’s alternative theorem contains a fixed point solving the differential inclusion (1) in the weak sense. To illustrate these results, at the end of the paper, an anisotropic elliptic equation defined on a cylinder I × Ω of R+N+1 with a Caratheodory type nonlinearity μ g(t, x, u) are investigated. Because of the nonstandard growth

for a.e. (t, x) ∈ I × Ω and all u ∈ ℝ fulfilled by the nonlinearity, and the periodic boundary condition u(0, x) = u(T, x), study of the anisotropic elliptic equation seems somewhat meaningful.

Framework of our study can be incorporated in the theory of convex analysis and function spaces with variable exponents. Results obtained here have their meaning in the study of nonlinear evolution equations with nonstandard growth.

As preliminaries, let us firstly make a brief review on the Orlicz space of scalar type. For the detailed discussions please refer to [2, Ch. 2] with the references therein.

Let 𝕂 be a scalar (real or complex) field, and X be a 𝕂–linear space. A convex function ϱ : X → [0, ∞] is called a semimodular on X, if the following hypotheses are all satisfied:

1. ϱ(0) = 0,

2. for every u ∈ X, the function λ ↦ ϱ(λ u) is left continuous on [0, ∞), and

3. ϱ(λ u) = ϱ(u) provided |λ| = 1,

4. ϱ(λ u) = 0 for all λ > 0 implies that u = 0.

If in addition, ϱ(u) = 0 means u = 0, then ϱ is called a modular. Given a semimodular ϱ on X, the corresponding subspace

endowed with the norm

becomes a normed linear space. X_ϱ is called the semimodular space, while ∥⋅∥_ϱ is called the Luxemburg norm. Both of them are generated by ϱ. Recall that in X_ϱ the unit ball property is holding, that is ∥u∥_ϱ ≤ 1 if and only if ϱ(u) ≤ 1.

Scalar Orlicz space is a common semimodular space produced by the integral semimodular. Suppose that ϕ : [0, ∞) → [0, ∞] is a Φ function, i.e., ϕ is convex, left continuous, ϕ(0) = 0 and lim_{t → 0}ϕ(t) = 0, lim_{t → ∞}ϕ(t) = ∞. Suppose also (A, μ) is a σ–finite and complete measure space, and L⁰(A, μ) is the linear space containing all the measurable scalar function defined on A. Then integration

defines a semimodular on L⁰(A, μ). The corresponding semimodular space, denoted by L^ϕ(A, μ), is called an Orlicz space. According to the Luxemburg norm L^ϕ(A, μ) is a Banach space. Moreover, ϱ_ϕ is a modular in case that ϕ is positive, i.e., ϕ(t) > 0 whenever λ > 0. Suppose further ϕ : A × [0, ∞) → [0, ∞] is a generalized Φ function, that is, for a.e. x ∈ A, ϕ(x, ⋅) is a Φ functions, and for all t ∈ [0, ∞), the function x ↦ ϕ(x, t) is measurable on A, then for all f ∈ L⁰(A, μ), integration ∫_Ωϕ(x, |f(x)|)dμ makes sense. This defines another semimodular and induces another semimodular space, which is the generalization of Orlicz space, called a Musielak-Orlicz space.

Taking a measurable subset Ω ⊆ ℝ^N, and a measurable exponent p : Ω → [1, ∞), define A = Ω with Lebesgue measure, and ϕ(x, t) = t^p(x). Then we obtain a generalized Φ function, from which we can construct an integral modular ϱ_p(⋅) through

and induce an important Musielak-Orlicz space, denoted by L^p(⋅)(Ω), and called the Lebesgue space with variable exponent. One knows that if p⁺ = esssup_{x ∈ Ω}p(x) < ∞, then L^p(⋅)(Ω) is separable, and the unit ball property turns to be

where ∥⋅∥_p(⋅) := ∥⋅∥_ϱp(⋅) is the Luxemburg norm. Furthermore, if additionally p⁻ = essinf_{x ∈ Ω}p(x) > 1, then L^p(⋅)(Ω) is uniformly convex (of course reflexive) with the dual space L^p′(⋅)(Ω), where 1/p′(x)+1/p(x) = 1 for a.e. x ∈ Ω.

2 Orlicz space generated by a single modular

Let X be a linear space and ϱ : X → [0, ∞] be a semimodular, which induces a semimodular space X_ϱ with the Luxemburg norm ∥⋅∥_ϱ. Let I be a finite or infinite interval, namely I = [0, T] for some 0 < T < ∞ or I = [0, ∞). A function f : I → X_ϱ is said to be measurable, if for every open set G ⊆ X, the preimage {t ∈ I : f(t) ∈ G} is a measurable subset of I. Moreover, f is called strongly measurable, if there is a sequence of X_ϱ–valued simple functions convergent to f almost everywhere. Of course, a strongly measurable function is measurable definitely, and vice versa provided X is separable (cf. [13, § 1.2]). Denote by L⁰(I, X_ϱ) the set of all strongly measurable X_ϱ–valued functions defined on I. Recall that a semimodular ϱ is lower-continuous on the induced space X_ϱ, thus for all a > 0, the set {u ∈ X_ϱ : ϱ(u) > a} is open in X_ϱ. Consequently, for each f ∈ L⁰(I, X_ϱ), the multifunction t ↦ ϱ(f(t)) is also measurable. Hence integration

makes sense. One can easily verify that Φ_ϱ is also a semimodular on L⁰(I, X_ϱ) with the semimodular space

and the Luxemburg norm denoted by ∥⋅∥_{L^ϱ(I, Xϱ)}.

A semimodular ϱ is said to be satisfying the Δ₂−condition, if there exists a constant d₂ ≥ 2 such that

Recall that, under the Δ₂−condition, ϱ turns to be a continuous modular satisfying

1. u ∈ X_ϱ if and only if ϱ(λu) < ∞ for all λ > 0, and

2. u_n → u in X_ϱ if and only if ϱ(u_n − u) → 0.

Moreover, Φ_ϱ also satisfies the Δ₂−condition with the same constant d₂.

Recall that every reflexive space satisfies the Radon-Nikodym’s property with respect to every complete and finite measure space. Furthermore if X_ϱ is reflexive, then ∂ϱ^* = (∂ϱ)⁻¹ and ∂ϱ = (∂ϱ^*)⁻¹. Putting these facts into Theorem 2.7, we have

Given a semimodular ϱ : X → [0, ∞], we say ϱ is uniformly convex, i.e. we mean that for every ε ∈ (0, 1), there is a δ ∈ (0, 1), for which either

holds. According to [2, § 2.4], we know that every uniformly convex semimodular satisfying the Δ₂−condition generates a uniformly convex space. Similarly, for a semimodular ϱ, its uniform convexity can be inherited by the Nemytskij functional Φ_ϱ. Summing up, we have

3 Orlicz space generalized by a series of semimodular

Suppose that 𝓐 is a topological lattice, i.e. 𝓐 is an ordered topological space, and for every order-bounded subset of 𝓐 its order supremum and order infimum exist in 𝓐 simultaneously. In this paper, 𝓐 is always assumed to be a totally order-bounded topological lattice, or 𝓑𝓣𝓛 in abbreviation. Its order supremum and infimum are denoted by α⁺ and α⁻ respectively. In a 𝓑𝓣𝓛 𝓐, a sequence {α_k} is said to be approaching a point β, if the two conditions α_k ≺ β, ∀ k ∈ ℕ and lim_k→∞α_k = β are both fulfilled.

The following proposition reveals the relationship between 𝓒𝓜𝓝 and 𝓑𝓢𝓝. For its proof, please refer to [10].

Let I be an interval as in Section 2, and Π(I) be the collection of all bounded subintervals of I. Consider the map θ : I → 𝓐. When we say θ is order-continuous, we mean that for any nest of intervals {J_k ∈ Π(I) : k = 1, 2, ⋯} shrinking to t, the limit

always holds, where θJ− and θJ+ and θ_J⁺ denote the order infimum and supremum of θ on J respectively.

Define

and

Obviously, both of them are linear spaces according to the sum and scalar multiplication of abstract-valued functions, and L⁰(I, X_θ(⋅)) ⊆ L−0 (I, X_θ(⋅)).

For each positive integer n, let t_n,k = kT/2ⁿ or t_n,k = k/2ⁿ, J_n,1 = [0, t_n,1], J_n,k+1 = (t_n,k, t_n,k+1] and θn,k±=θJn,k± , for k = 1, 2, ⋯, 2ⁿ if I = [0, T] or k = 1, 2, ⋯ if I = [0, ∞). Define a step function θ_n through θn±(t)=θn,k± for t ∈ J_n,k. Obviously, { θn± } is decreasing (increasing) in n and converging to θ(t) as n → ∞ for all t ∈ I. Similar to the constant ones, for every n ∈ ℕ, function space L⁰(I, Xθn±(⋅) ) is well defined, on which Φθn±(f)=∫Iϱθn±(t)(f(t))dt is a semimodular. It induces a Banach space, denoted by Lϱθn±(⋅)(I,Xθn±(⋅)).

There is a natural relation among the three types of function spaces mentioned above, that is

Thus for each f ∈ L−0 (I, X_θ(⋅)), the function t ↦ ϱ_θn(t)(f(t)) is measurable, n = 1, 2, ⋯. Note that

by the continuity of {ϱ_α}, so the composite function t ↦ ϱ_θ(t)(f(t)) is also measurable. Let

we then obtain a semimodular, whose semimodular space is denoted by L^ϱ_θ(⋅)(I, X_θ(⋅)). Obviously, every member of L^ϱ_θ(⋅)(I, X_θ(⋅)) lies in L⁰(I, X_θ(⋅)), hence Φ_ϱθ(⋅) can also be considered as a semimodular on L⁰(I, X_θ(⋅)), correspondingly L^ϱ_θ(⋅)(I, X_θ(⋅)) can be regarded as the semimodular space generated from L⁰(I, X_θ(⋅)).

The following propositions can be proved by inequality (6), continuity of {ϱ_α}, Fatou’s lemma, together with the unit ball property.

Assume that {X_α} generated by {ϱ_α} is a dense 𝓑𝓢𝓝, i.e. X_α2 is a dense subspace of X_α1 whenever α₁ ≺ α₂. It is easy to see that, under this situation 𝓢(I, X_α⁺) is contained in L⁰(I, X_θ(⋅)), consequently for every f ∈ 𝓢(I, X_α⁺), the multifunction t ↦ ϱ_θ(t)(f(t)) is measurable. Moreover, by invoking Proposition 2.3 we can prove that, if every modular ϱ_α satisfies the Δ₂−condition, then for each J ∈ Π(I), 𝓢(J, X_α⁺) hence L^ϱ_α+(J, X_α⁺) is dense in LϱθJ+(J,XθJ+).

Analogous to the range-invariant ones, we can define the space of strongly measurable functions with varying ranges, that is

Suppose that {ϱ_α : α ∈ 𝓐} is a 𝓒𝓜𝓝 generating a dense 𝓑𝓢𝓝{X_α}. Similar to L^ϱ_θ(⋅)(I, X_θ(⋅)), we can define L+ϱθ(⋅) (I, X_θ(⋅)) through

with the same Luxemburg norm. Note that in such situation, 𝓢(I, X_α⁺) is a linear subspace of L+ϱθ(⋅) (I, X_θ(⋅)).

We say that the 𝓒𝓜𝓝{ϱ_α} satisfies the Δ₂−condition, if every ϱ_α satisfies the Δ₂−condition, and the constant C₂ or the related function σ is independent of α. It is easy to see that under this condition, Φ_ϱθ(⋅) is also a Δ₂−type modular. Hence following the same process as in Proposition 2.3, we can prove that

This theorem is a straight consequence of Proposition 3.12.

For each α ∈ 𝓐, denote by ϱα∗ the Fenchel duality of ϱ_α, i.e.

Since ϱ_α is a semimodular, ϱα∗ is also a semimodular on Xα∗ , and the semimodular space derived by ϱα∗ is exactly Xα∗ itself (see [2, § 2.2]). Define

then we obtain another family of semimodulars defined on Xα+∗ , called the dual modular net or in symbol 𝓓𝓜𝓝 of {ϱ_α}. Since for each α ∈ 𝓐, the effective domains and the induced semimodular spaces are equal, in the coming arguments, we will not distinguish ϱ~α∗ and ϱα∗ , and prefer to use { ϱα∗ } instead of { ϱ~α∗ } to denote the 𝓓𝓜𝓝 of {ϱ_α}.

Suppose α₁ ≺ α₂, then Xα1∗↪Xα2∗ , and for all ξ ∈ Xα2∗ , by (6) we have