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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Abstract-valued Orlicz spaces of range-varying type

Qinghua Zhang
• Corresponding author
• Department of Mathematics, Nantong University, 9 Seyuan Road, Nantong City, 226019, Jiangsu Province, China
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Published Online: 2018-08-20 | DOI: https://doi.org/10.1515/math-2018-0080

## Abstract

This paper mainly deals with the abstract-valued Orlicz spaces of range-varying type. Using notions of Banach space net and continuous modular net etc., we give definitions of Lϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)), and discuss their geometrical properties as well as the representation of $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅))*. We also investigate some functionals and operators on Lϱθ(⋅)(I, Xθ(⋅)), giving expression for the subdifferential of the convex functional generated by another continuous modular net. After making some investigations on the Bochner-Sobolev spaces W1, ϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)), and the intersection space $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)) ∩ Lφϑ(⋅)(I, Vϑ(⋅)), a second order differential inclusion together with an anisotropic nonlinear elliptic equation with nonstandard growth are also taken into account.

MSC 2010: 46B10; 46E30; 46E40

## 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. Lp(x)(Ω) and Wk,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 Xθ(⋅)–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 {Xα : α ∈ 𝓐} and order-continuous exponent θ : I → 𝓐, Zhang- Li in [10] firstly gave definition of the space L0(I, Xθ(⋅)), which contains all the Xθ(⋅)–valued functions measurable in a special manner. Like the measurable functions of range-fixed type, members of L0(I, Xθ(⋅)) are all norm-measurable. Based on the useful character, from L0(I, Xθ(⋅)) the authors extracted two types of function spaces: continuous type C(I, Xθ(⋅)) and integral type Lp(⋅)(I, Xθ(⋅)). 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 Xθ(⋅)–valued functions.

It is worth remarking that some Banach space net can be produced by a continuous modular net {ϱα : α ∈ 𝓐}. According to whether or not being built on the continuous modular nets, Zhang-Li in [11] divided the Xθ(⋅)–valued function spaces of integral type into two subclasses: norm-modular ones and modular-modular ones. A norm-modular space, like Lp(⋅)(I, Xθ(⋅)), is commonly produced by the semimodular $ϱϕ(f)=∫Aϕ(t,∥f(t)∥θ(t))dt$

with a generalized Φ function ϕ, while a modular-modular space is derived from a continuous modular net {ϱα : α ∈ 𝓐} with the semimodular $Φϱ(f)=∫Iϱ(Mf(t))dt.$

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 {ϱα : α ∈ 𝓐} and θ to reconstruct the semimodular, namely $Φϱθ(⋅)(f)=∫Iϱθ(t)(f(t))dt.$

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 Xθ(⋅)–valued functions, we introduce two different spaces: Lϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)), both of them are complete according to the same norm. We show that, under some suitable situations, $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)) is separable, and its dual space can be represented by $L+ϱθ(⋅)(I,Xθ(⋅))∗=Lϱθ(⋅)∗(I,Xθ(⋅)∗).$

We also make some evaluations on the proper conditions for the equality $L+ϱθ(⋅)(I,Xθ(⋅))=Lϱθ(⋅)(I,Xθ(⋅))$

as well as the reflexivity of Lϱθ(⋅)(I, Xθ(⋅)).

For the sake of applications, in Section 4, we make some discussions on the functionals and operators on the modular-modular space Lϱθ(⋅)(I, Xθ(⋅)), including functional Φ͠φϑ(⋅) defined by another continuous modular system {φβ : β ∈ 𝓑} and another order-continuous map ϑ : I → 𝓑, and operators Zθ(⋅) and ∂φϑ(⋅), which are subdifferentials of Φϱθ(⋅) and Φ͠φϑ(⋅) respectively. Here Zθ(⋅) plays the role of an extended dual map, and ∂φϑ(⋅) usually arises in a differential equation as the driving operator. Under some extra assumptions, such as the weak lower-continuity of {ϱα} and $\begin{array}{}\left\{{\varrho }_{\alpha }^{\ast }\right\}\end{array}$, and the strong coercivity of {ϱα} and {φβ}, demicontinuity and coercivity of Zθ(⋅) and the representation ∂Φ͠φϑ(⋅)(u)(t) = ∂φϑ(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 W1, ϱθ(⋅)(I, Xθ(⋅)) and $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)), and the intersection $W=Wper1,ϱθ(⋅)(I,Xθ(⋅))∩Lφϑ(⋅)(I,Vϑ(⋅)),$

including the continuous and compact embedding of W1, ϱθ(⋅)(I, Xθ(⋅)) into the space Lϱθ(⋅)(I, Xθ(⋅)) along with the estimate $∥u∥W1,ϱθ(⋅)(I,Xθ(⋅))≤C(∥u′∥Lϱθ(⋅)(I,Xθ(⋅))+∥u∥Lφϑ(⋅)(I,Vϑ(⋅))),$

in Section 5, we study a type of second order nonlinear differential inclusion $−ddt∂ϱθ(t)(u′(t))+∂φϑ(t)(u(t))∋f(t,u(t))for a.e.t∈I,$(1)

with the periodic boundary condition, where the operator f : I × XX owns a nonstandard growth $ϱθ(t)∗(f(t,u))≤μϱθ(t)(u)+h(t),u∈Xθ(t)$

for a small number μ > 0 and a nonnegative function hL1(I). By introducing the Nemytskij operator F(u) = f(⋅, u), and the second order differential operator $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ defined by $〈〈Dθ(⋅)2u,v〉〉θ(⋅)=∫I〈∂ϱθ(⋅)(u′(t)),v′(t)〉θ(t)dt,u,v∈Wper1,ϱθ(⋅)(I,Xθ(⋅)),$

we obtain a continuous and compact operator $(Dθ(⋅)2+∂Φφϑ(⋅))−1∘F:Lϱθ(⋅)(I,Xθ(⋅))→Lϱθ(⋅)(I,Xθ(⋅)),$

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 $\begin{array}{}{\mathbb{R}}_{+}^{N+1}\end{array}$ with a Caratheodory type nonlinearity μ g(t, x, u) are investigated. Because of the nonstandard growth $|g(t,x,u)|≤C(1+|u|p(t,x)−1)$

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:

• ϱ(0) = 0,

• for every uX, the function λ ↦ ϱu) is left continuous on [0, ∞), and

• ϱu) = ϱ(u) provided |λ| = 1,

• ϱu) = 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 $Xϱ={u∈X:ϱ(λu)<∞for someλ>0}$

endowed with the norm $∥u∥ϱ=inf{λ>0:ϱ(uλ)≤1}$

becomes a normed linear space. Xϱ is called the semimodular space, while ∥⋅∥ϱ is called the Luxemburg norm. Both of them are generated by ϱ. Recall that in Xϱ the unit ball property is holding, that is ∥uϱ ≤ 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 limt → 0ϕ(t) = 0, limt → ∞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 $ϱϕ(f)=∫Ωϕ(|f(x)|)dμ$

defines a semimodular on L0(A, μ). The corresponding semimodular space, denoted by Lϕ(A, μ), is called an Orlicz space. According to the Luxemburg norm Lϕ(A, μ) is a Banach space. Moreover, ϱϕ 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. xA, ϕ(x, ⋅) is a Φ functions, and for all t ∈ [0, ∞), the function xϕ(x, t) is measurable on A, then for all fL0(A, μ), integration Ωϕ(x, |f(x)|) 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 ⊆ ℝN, and a measurable exponent p : Ω → [1, ∞), define A = Ω with Lebesgue measure, and ϕ(x, t) = tp(x). Then we obtain a generalized Φ function, from which we can construct an integral modular ϱp(⋅) through $ϱp(⋅)(f)=∫Ω|f(x)|p(x)dx,f∈L0(Ω),$

and induce an important Musielak-Orlicz space, denoted by Lp(⋅)(Ω), and called the Lebesgue space with variable exponent. One knows that if p+ = esssupxΩp(x) < ∞, then Lp(⋅)(Ω) is separable, and the unit ball property turns to be $min{∥f∥p(⋅)p−,∥f∥p(⋅)p+}≤ϱp(⋅)(f)≤max{∥f∥p(⋅)p−,∥f∥p(⋅)p+},$

where ∥⋅∥p(⋅) := ∥⋅∥ϱp(⋅) is the Luxemburg norm. Furthermore, if additionally p = essinfxΩp(x) > 1, then Lp(⋅)(Ω) is uniformly convex (of course reflexive) with the dual space Lp′(⋅)(Ω), where 1/p′(x)+1/p(x) = 1 for a.e. xΩ.

## 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 Xϱ 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 : IXϱ is said to be measurable, if for every open set GX, the preimage {tI : f(t) ∈ G} is a measurable subset of I. Moreover, f is called strongly measurable, if there is a sequence of Xϱ–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, Xϱ) the set of all strongly measurable Xϱ–valued functions defined on I. Recall that a semimodular ϱ is lower-continuous on the induced space Xϱ, thus for all a > 0, the set {uXϱ : ϱ(u) > a} is open in Xϱ. Consequently, for each fL0(I, Xϱ), the multifunction tϱ(f(t)) is also measurable. Hence integration $Φϱ(f)=∫Iϱ(f(t))dt$

makes sense. One can easily verify that Φϱ is also a semimodular on L0(I, Xϱ) with the semimodular space $Lϱ(I,Xϱ)={f∈L0(I,Xϱ):Φϱ(λf)<∞for someλ>0}$

and the Luxemburg norm denoted by ∥⋅∥Lϱ(I, Xϱ).

#### Theorem 2.1

Lϱ(I, Xϱ) is a Banach space in case that Xϱ is complete.

This theorem is a special case of Theorem 3.7, which is given in § 3 with a proof.

#### Remark 2.2

Suppose that fL(I, Xϱ), and the one-dimension Lebesgue measure of the set E0 = {tI : f(t) ≠ 0} is finite. Then we have

$Φϱ(fM+1)≤|E0|<∞,$

where M ≥ 0 is the essential supremum off(t)∥ϱ. Thus fLϱ(I, Xϱ) andfLϱ(I,Xϱ) ≤ (M + 1)max{1, |E0|}. Furthermore, by the estimate

$∥u∥ϱ≤ϱ(u)+1,∀u∈Xϱ,$

we also have

$L∞(I,Xϱ)↪Lϱ(I,Xϱ)↪L1(I,Xϱ)$

in case that I = [0, T] is bounded.

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

$ϱ(2u)≤d2ϱ(u)for allu∈X.$

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

• uXϱ if and only if ϱu) < ∞ for all λ > 0, and

• unu in Xϱ if and only if ϱ(unu) → 0.

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

#### Proposition 2.3

If ϱ satisfies the Δ2condition, then the set of all simple Xϱ-valued functions, say 𝓢(I, Xϱ), is dense in Lϱ(I, Xϱ).

#### Proof

For each fLϱ(I, Xϱ), there is correspondingly a sequence of simple Xϱ− valued functions {sk} such that sk(t) → f(t) and consequently skχE0(t) → f(t) in Xϱ for a.e. tI for the set E0 defined in Remark 2.2. Let

$Ek={t∈I:ϱ(skχE0(t)−f(t))≤ϱ(f(t))},$

and let φk = skχE0χEk, which is also a simple Xϱ-valued function. Notice that

${t∈I:ϱ(φk(t)−f(t))→0}=⋂ε>0⋃K≥1⋂k≥K{t∈I:ϱ(φk(t)−f(t))≤ε}⊇⋂ε>0⋃K≥1⋂k≥K{t∈I:ϱ(skχE0(t)−f(t))≤min{ε,ϱ(f(t))}}={t∈I:ϱ(skχE0(t)−f(t))→0},$

we have ϱ(φk(t) − f(t)) → 0 a.e. on I as k → ∞, which combined with ϱ(φk(t) − f(t)) ≤ ϱ(f(t)) for a.e. tI, and Lebesgue’s convergence theorem, yields Φϱ(φkf) → 0 or equivalently φkf in Lϱ(I, Xϱ) as k → ∞. Thus density of 𝓢(I, Xϱ) in Lϱ(I, Xϱ) has been proved. □

#### Remark 2.4

Since I = $\begin{array}{}\bigcup _{n=1}^{\mathrm{\infty }}\end{array}$ In, where In = I ∩ [0, n], n = 1, 2, ⋯, we can also prove that 𝓢c(I, Xϱ) and $\begin{array}{}{L}_{c}^{\mathrm{\infty }}\end{array}$ (I, Xϱ), subsets of 𝓢(I, Xϱ) and L(I, Xϱ) respectively, containing the functions with compact supports, are both dense in Lϱ(I, Xϱ).

#### Corollary 2.5

Under the Δ2condition of ϱ and the separability assumption of Xϱ, Lϱ(I, Xϱ) is a separable space.

#### Remark 2.6

Given a semimodular ϱ, recall that the dual functional ϱ* is also a semimodular on $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ , and the double dual ϱ** is equal to ϱ on the space Xϱ (cf. [2, § 2.2] or [14, § 3.2]). Moreover, for all uXϱ and ξ$\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ , Young’s inequality

$〈ξ,u〉≤ϱ(u)+ϱ∗(ξ)$

holds. The equality also holds if and only if ξ∂ϱ(u) or equivalently u∂ϱ*(ξ) if we regard Xϱ as a closed subspace of $\begin{array}{}{X}_{\varrho }^{\ast \ast }\end{array}$ . Here ∂ϱ is the the subdifferential operator of ϱ and ∂ϱ* is that of ϱ*. Recall that as the subdifferential operators of lower-semicontinuous and convex proper functionals, ∂ϱ and ∂ϱ* can be viewed as two maximal monotone and semiclosed subsets of the product spaces Xϱ × $\begin{array}{}{X}_{\varrho }^{\ast }\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{X}_{\varrho }^{\ast }×{X}_{\varrho }^{\ast \ast }\end{array}$ respectively. As for the multivalued inverse map (∂ϱ)−1, we know that (∂ϱ)−1$\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ × Xϱ has the same properties as ∂ϱ has, together with the inclusion (∂ϱ)−1∂ϱ* holding. Furthermore, if in addition 𝓡(∂ϱ) = $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ , then for all ξ$\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ , the image (∂ϱ)−1(ξ) is a nonempty convex and closed subset of Xϱ. Therefore for all ξL0(I, $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ ), the multifunction t ↦ (∂ϱ)−1(ξ(t)) is graph measurable with nonempty closed image everywhere, consequently under the additional separability assumption of Xϱ, we can assert that (∂ϱ)−1(ξ(⋅)) has a measurable selection, or in other words, there is a function uL0(I, Xϱ) such that u(t) ∈ (∂ϱ)−1(ξ(t)) for a.e. tI (see [15, § 8.3] for references).

#### Theorem 2.7

Suppose that ϱ satisfies the Δ2condition, Xϱ is a separable Banach space, and 𝓡(∂ϱ) = $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ . Suppose also ϱ* is a modular, and $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ has the Radon-Nikodym’s property w.r.t. every finite subinterval of I. Then the dual space Lϱ(I, Xϱ)* is isomorphic to Lϱ*(I, $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ ).

#### Proof

For each ξLϱ*(I, $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ ), define a functional Λξ through

$〈〈Λξ,f〉〉=∫I〈ξ(t),f(t)〉dt,∀f∈Lϱ(I,Xϱ).$(2)

By Hölder’s inequality

$|∫I〈ξ(t),f(t)〉dt|≤2∥ξ∥Lϱ∗(I,Xϱ∗)∥f∥Lϱ(I,Xϱ),$

it is easy to check that ΛξLϱ(I, Xϱ)*, and ∥Λξ∥ ≤ 2∥ξLϱ*(I, Xϱ*).

Conversely, for each ΛLϱ(I, Xϱ)*, we will prove the existence of a unique ξLϱ*(I, $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ ) such that Λ = Λξ as in (2). If Λ = 0, then take ξ = 0 and there is nothing to do. If Λ ≠ 0, then ∥|Λ∥|* > 0, where ∥|Λ∥|* denotes the Lϱ(I, Xϱ)*-norm of Λ. Without loss of generality, in the following discussions we may assume that ∥|Λ∥|* = 1. For each positive integer n, define an $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ −valued function τn on the set of all measurable subsets of In = I ∩ [0, n] as follows:

$〈τn(E),u〉=〈〈Λ,χEu〉〉,∀u∈Xϱ.$

Here χE is the characteristic function of E.

Suppose that {Ek} is a sequence of mutually disjoint measurable subsets of In and uXϱ. Since for all λ > 0,

$Φϱ(λ∑k=1∞χEku)=∫I∑k=1∞χEkϱ(λu)dt=ϱ(λu)∑k=1∞|Ek|≤nϱ(λu)<∞,$

we can conclude that

$∑k=1∞χEku=χ∪k=1∞EkuinLϱ(I,Xϱ),and∥∑k=1∞χEku∥Lϱ(I,Xϱ)≤n∥u∥ϱ.$

Consequently,

$〈τn(⋃k=1∞Ek),u〉=∑k=1∞〈〈Λ,χEku〉〉=∑k=1∞〈τn(Ek),u〉,$

and

$|〈τn(⋃k=1∞Ek),u〉|≤n|||Λ|||∗∥u∥ϱ.$

Therefore τn is an $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ −valued measure on In with a bounded total variation no more than n∥|Λ∥|*. Now since $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ has the Radon-Nikodym’s property w.r.t. every finite subinterval of I, there is a unique ξnL1(In, $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ ) such that

$〈〈Λ,χEu〉〉=〈τn(E),u〉=∫In〈ξn(t),χEu〉dt$

for every measurable subset E of In. By the uniqueness of ξn in the above representation, we have that ξn+1(t) = ξn(t) a.e. on In. Let ξ(t) = ξn(t) for tIn, then we obtain a globally defined and strongly measurable $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ −valued function satisfying

$〈〈Λ,f〉〉=∫I〈ξ(t),f〉dt$(3)

for the function f = E with a bounded measurable subset E of I and a point uXϱ and consequently for all f ∈ 𝓢(I, Xϱ) with compact supports.

Given a function f$\begin{array}{}{L}_{c}^{\mathrm{\infty }}\end{array}$ (I, Xϱ), from Corollary 2.3, we can find a sequence of Xϱ−valued simple functions with compact supports, say {sk}, such that skf in both L(I, Xϱ) and Lϱ(I, Xϱ), and (3) is satisfied by sk for all k ∈ ℕ. Taking limits as k → ∞ in both sides of (3), we can deduce that (3) is also satisfied by f$\begin{array}{}{L}_{c}^{\mathrm{\infty }}\end{array}$ (I, Xϱ).

Remark 2.6 shows that the multivalued function t ↦ (∂ϱ)−1(ξ(t)) is measurable, and it has a strongly measurable selection since Xϱ is separable. Denote the selection by u, then we have u(t) ∈ (∂ϱ)−1(ξ(t)) ⊆ ∂ϱ*(ξ(t)) a.e. on I. For each n ∈ ℕ+, let

$Jn={t∈In:∥u(t)∥Xϱ≤n},andun(t)=u(t)χJn.$

Then un$\begin{array}{}{L}_{c}^{\mathrm{\infty }}\end{array}$ (I, Xϱ) and

$∫Jnϱ∗(ξ(t))dt+∫Iϱ(un(t))dt=∫I〈ξ(t),un(t)〉dt=〈〈Λ,un〉〉≤∥un∥Lϱ(I,Xϱ).$(4)

As Λ ≠ 0 and ϱ* is a modular, neither ξ(t) nor ϱ*(ξ(t)) is equal to 0 a.e. on I. Thus ∫Jnϱ*(ξ(t))dt > 0, consequently from (4) we get

$Φϱ(un)<∥un∥Lϱ(I,Xϱ)$

for n large enough. Hence by the unit ball property, we assert that ∥unLϱ(I,Xϱ) ≤ 1, which in turn yields ∫Jnϱ*(ξ(t))dt ≤ 1. Let n → ∞, and take the fact |I$\begin{array}{}\bigcup _{n=1}^{\mathrm{\infty }}\end{array}$ Jn| = 0 into account, we have ∫Iϱ*(ξ(t))dt ≤ 1, which leads to the conclusion ξLϱ*(I, $\begin{array}{}{X}_{\varrho }^{\ast }\end{array}$ ) with the estimate

$∥ξ∥Lϱ∗(I,Xϱ∗)dt≤1.$

Finally, using the density of $\begin{array}{}{L}_{c}^{\mathrm{\infty }}\end{array}$ (I, Xϱ) in Lϱ(I, Xϱ), we can conclude that (3) holds for all fLϱ(I, Xϱ). Therefore Λ = Λξ as in (2) and $\begin{array}{}\parallel \xi {\parallel }_{{L}^{{\varrho }^{\ast }}\left(I,{X}_{\varrho }^{\ast }\right)}\end{array}$ = ∥|Λξ∥|* = 1. Thus the proof has been completed in the case ∥|Λξ∥|* = 1, and the general case can be dealt with by the scaling arguments. □

#### Remark 2.8

Here the separability assumption of Xϱ can be replaced by the strict convexity assumption of ϱ. As a matter of fact, if ϱ is strictly convex, then ∂ϱ is injective, or equivalently (∂ϱ)−1 is single-valued.

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

#### Corollary 2.9

Suppose that both ϱ and its dual ϱ* satisfy the Δ2condition, the semimodular space Xϱ is reflexive and separable. Then

$Lϱ(I,Xϱ)∗≅Lϱ∗(I,Xϱ∗),Lϱ∗(I,Xϱ∗)∗≅Lϱ(I,Xϱ),$

and the function space Lϱ(I, Xϱ) is also reflexive.

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

$ϱ(u−v2)≤εϱ(u)+ϱ(v)2orϱ(u−v2)≤(1−δ)ϱ(u)+ϱ(v)2$(5)

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 Φϱ. Summing up, we have

#### Theorem 2.10

Under the uniform convexity assumption and the Δ2condition of ϱ, Lϱ(I, Xϱ) is a uniformly convex space.

## 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 {αk} is said to be approaching a point β, if the two conditions αkβ, ∀ k ∈ ℕ and limk→∞ αk = β are both fulfilled.

#### Definition 3.1

Given a family of Banach spaces {Xα : α ∈ 𝓐}, we say it is a Banach space net, or 𝓑𝓢𝓝 for short, provided

• αβ implies XβXα.

We say {Xα} is norm-continuous, if

• for every sequence {αk} approaching β, the limit of norms limk→∞xαk = ∥xβ holds at all xXβ.

{Xα} is called uniformly bounded, whenever

• there is a constant C ≥ 1 such that for all α, β ∈ 𝓐 with αβ and all xXβ, inequalityxαCxβ always holds.

And {Xα} is said to be successive, if

• for any sequence {αk} approaching β and any point xXα with the constraints: xXαk for all k ∈ ℕ and C = supk→∞xαk < ∞, we have xXβ andxβC.

Finally, a 𝓑𝓢𝓝{Xα} is called regular provided it is norm-continuous, uniformly bounded and successive at the same time.

#### Remark 3.2

Given a 𝓑𝓢𝓝{Xα : α ∈ 𝓐}, the family of dual spaces { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ : α ∈ 𝓐}, where 𝓐 takes the inverse orderinstead of ≺, is also a 𝓑𝓢𝓝, called the dual space net or 𝓓𝓢𝓝 in symbol. Here we use the convention: 〈ξ, xα = 〈ξ, xβ provided ξ$\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ , xXβ and αβ. It is easy to see that, if {Xα} is uniformly bounded, then { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ } is also uniformly bounded with the same bounds. However, whether or not { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ } inherits the norm-continuity and successive property from {Xα} is not clear.

#### Definition 3.3

Suppose that X is a linear space, and {ϱα : α ∈ 𝓐} is a family of semimodulars defined on X. We say {ϱα} is a continuous modular net, or 𝓒𝓜𝓝 in abbreviation, i.e. we mean that the following hypotheses are satisfied:

1. every ϱα generates a Banach space Xϱα =: Xα,

2. there exist two positive constants Ci, i = 1, 2 for which inequality

$ϱα1(u)≤C1ϱα2(u)+C2,$(6)

holds for all uX and all αi ∈ 𝓐, i = 1, 2 with α1α2, and

3. if {αk} approaches α in 𝓐, then

$limk→∞ϱαk(u)=ϱα(u).$

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

#### Proposition 3.4

Given a 𝓒𝓜𝓝{ϱα : α ∈ 𝓐}, the family of semimodular spaces {Xα : α ∈ 𝓐} is a regular 𝓑𝓢𝓝.

#### Remark 3.5

Similar to the scalar ones, for two indexes αi ∈ 𝓐, i = 1, 2 with α1α2, we have Lϱα2(I, Xα2) ↪ Lϱα1 (I, Xα1) with the imbedding constant C = max{1, C1 + C2T} in the case I = [0, T].

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 {JkΠ(I) : k = 1, 2, ⋯} shrinking to t, the limit

$limk→∞θJk−=limk→∞θJk+=θ(t)$

always holds, where $\begin{array}{}{\theta }_{J}^{-}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{\theta }_{J}^{+}\end{array}$ and θJ+ denote the order infimum and supremum of θ on J respectively.

#### Remark 3.6

Here we give up the extra assumption that θ is continuous according to the topology of 𝓐, which was stated but not used in [11].

Define

$L−0(I,Xθ(⋅))={f∈L0(I,X):f|J∈L0(J,XθJ−)for all J∈Π(I)},$

and

$L0(I,Xθ(⋅))={f∈L−0(I,Xθ(⋅)):f(t)∈Xθ(t)for a.e.t∈I}.$

Obviously, both of them are linear spaces according to the sum and scalar multiplication of abstract-valued functions, and L0(I, Xθ(⋅)) ⊆ $\begin{array}{}{L}_{-}^{0}\end{array}$ (I, Xθ(⋅)).

For each positive integer n, let tn,k = kT/2n or tn,k = k/2n, Jn,1 = [0, tn,1], Jn,k+1 = (tn,k, tn,k+1] and $\begin{array}{}{\theta }_{n,k}^{±}={\theta }_{{J}_{n,k}}^{±}\end{array}$ , for k = 1, 2, ⋯, 2n if I = [0, T] or k = 1, 2, ⋯ if I = [0, ∞). Define a step function θn through $\begin{array}{}{\theta }_{n}^{±}\left(t\right)={\theta }_{n,k}^{±}\end{array}$ for tJn,k. Obviously, { $\begin{array}{}{\theta }_{n}^{±}\end{array}$ } is decreasing (increasing) in n and converging to θ(t) as n → ∞ for all tI. Similar to the constant ones, for every n ∈ ℕ, function space L0(I, $\begin{array}{}{X}_{{\theta }_{n}^{±}\left(\cdot \right)}\end{array}$ ) is well defined, on which $\begin{array}{}{\mathit{\Phi }}_{{\theta }_{n}^{±}}\left(f\right)={\int }_{I}{\varrho }_{{\theta }_{n}^{±}\left(t\right)}\left(f\left(t\right)\right)dt\end{array}$ is a semimodular. It induces a Banach space, denoted by $\begin{array}{}{L}^{{\varrho }_{{\theta }_{n}^{±}\left(\cdot \right)}}\left(I,{X}_{{\theta }_{n}^{±}\left(\cdot \right)}\right)\end{array}$.

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

$L0(I,Xθ(⋅))⊆L−0(I,Xθ(⋅))⊆L0(I,Xθn−(⋅)).$

Thus for each f$\begin{array}{}{L}_{-}^{0}\end{array}$ (I, Xθ(⋅)), the function tϱθn(t)(f(t)) is measurable, n = 1, 2, ⋯. Note that

$ϱθ(t)(f(t))=limn→∞ϱθn(t)(f(t))$

by the continuity of {ϱα}, so the composite function tϱθ(t)(f(t)) is also measurable. Let

$Φϱθ(⋅)(f)=∫Iϱθ(t)(f(t))dt,f∈L−0(I,Xθ(⋅)).$

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

#### Theorem 3.7

For every 𝓒𝓜𝓝{ϱα : α ∈ 𝓐} and every order-continuous map θ : I → 𝓐, Lϱθ(⋅)(I, Xθ(⋅)) is a Banach space.

#### Proof

Suppose that {fk} is a cauchy sequence in Lϱθ(⋅)(I, Xθ(⋅)). Then for every λ > 0, we have

$limk,l→∞Φϱθ(⋅)(λ(fk−fl)=limk,l→∞∫Iϱθ(t)(λ(fk(t)−fl(t))dt=0.$

Thus there is sequence of positive integers, say {ki}, satisfying ki < ki+1 for all i ∈ ℕ, limi→∞ ki = ∞, and

$∫Iϱθ(t)(2i(fk(t)−fj(t)))dt<12iwheneverk,j≥ki,$(7)

Especially we have

$∫Iϱθ(t)(2i(fki+1(t)−fki(t)))dt<12i,$

which in turn yields

$|Ei|≤|{t∈I:ϱθ(t)(2i(fki+1(t)−fki(t)))>1}|<12i,$

where Ei = {tI : ∥fki+1(t) − fki(t)∥θ(t) > 1/2i}, i = 1, 2, ⋯.

Let $\begin{array}{}E=\bigcap _{j=1}^{\mathrm{\infty }}\bigcup _{i=j}^{\mathrm{\infty }}{E}_{i}\end{array}$ , then we have m(E) = 0, and for each tIE, there exists j ∈ ℕ such that tI$\begin{array}{}\bigcup _{i=j}^{\mathrm{\infty }}\end{array}$ Ei, or equivalently ∥fki+1(t) − fki(t)∥θ(t) ≤ 1/2i for all ij. Consequently, for the integer lj, we have

$∑i=l∞∥fki+1(t)−fki(t)∥θ(t)≤12l−1.$

This infers that series $\begin{array}{}{f}_{{k}_{1}}\left(t\right)+\sum _{i=l}^{\mathrm{\infty }}\left({f}_{{k}_{i+1}}\left(t\right)-{f}_{{k}_{i}}\left(t\right)\right)\end{array}$ is absolutely continuous in Xθ(t) on the set IE. Then by the completeness of Xθ(t), we conclude that {fki(t)} is convergent in Xθ(t) a.e. on I, and the limit function f belongs to L0(I, Xθ(⋅)).

Taking any λ > 0 and ε > 0, there exists i ∈ ℕ such that 2i > λ and 1/2i < ε, thus using inequality (7), Fatou’s lemma together with the lower semicontinuity of ϱθ(t), we obtain

$∫Iϱθ(t)(λ(uk(t)−u(t)))dt≤lim infj→∞∫Iϱθ(t)(2i(uk(t)−ukj(t)))dt<ε,∀k>ki,$

which means that uLϱθ(⋅)(I, Xθ(⋅)) and uku in Lϱθ(⋅)(I, Xθ(⋅)) as k → ∞. This shows the completeness of Lϱθ(⋅)(I, Xθ(⋅)). □

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

#### Proposition 3.8

Suppose that I is a bounded interval, then

$Lϱθn+(⋅)(I,Xθn+(⋅))↪Lϱθ(⋅)(I,Xθ(⋅))↪Lϱθn−(⋅)(I,Xθn−(⋅)),$

and there exists a constant C > 0 such that

$C−1∥f∥Lϱθn−(⋅)(I,Xθn−(⋅))≤∥f∥Lϱθ(⋅)(I,Xθ(⋅))≤C∥f∥Lϱθn+(⋅)(I,Xθn+(⋅)).$

#### Proposition 3.9

A function f$\begin{array}{}{L}_{-}^{0}\end{array}$ (I, Xθ(⋅)) with the property

$K=lim supn→∞∥f∥Lϱθn−(⋅)(I,Xθn−(⋅))<∞$

lies in fLϱθ(⋅)(I, Xθ(⋅)) definitely with the estimatefLϱθ(⋅)(I,Xθ(⋅))K.

#### Corollary 3.10

Under the bounded assumption of I, function space Lϱθ(⋅)(I, Xθ(⋅)) is equivalent to

${f∈L−0(I,Xθ(⋅)):supJ∈Π(I)∥f∥LϱθJ−(I,XθJ−)<∞}$

and

$∥f∥Lϱθ(⋅)(I,Xθ(⋅))≤supJ∈Π(I)∥f∥LϱθJ−(I,XθJ−)≤C∥f∥Lϱθ(⋅)(I,Xθ(⋅))$

for some constant C > 0.

Assume that {Xα} generated by {ϱα} is a dense 𝓑𝓢𝓝, i.e. Xα2 is a dense subspace of Xα1 whenever α1α2. It is easy to see that, under this situation 𝓢(I, Xα+) is contained in L0(I, Xθ(⋅)), consequently for every f ∈ 𝓢(I, Xα+), the multifunction tϱθ(t)(f(t)) is measurable. Moreover, by invoking Proposition 2.3 we can prove that, if every modular ϱα satisfies the Δ2−condition, then for each JΠ(I), 𝓢(J, Xα+) hence Lϱα+(J, Xα+) is dense in $\begin{array}{}{L}^{{\varrho }_{{\theta }_{J}^{+}}}\left(J,{X}_{{\theta }_{J}^{+}}\right)\end{array}$.

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

$L+0(I,Xθ(⋅))={f∈L0(I,X):f(t)∈Xθ(t)for a.e.t∈I,and there exists a sequence{sn}ofS(I,Xα+)s.t.∥sn(t)−f(t)∥θ(t)→0asn→∞for a.e.t∈I}.$

#### Remark 3.11

In this definition, the set 𝓢(I, Xα+) can be replaced by L0(I, Xα+), both of which are contained in L0(I, Xθ(⋅)). As a result, one can easily check that $\begin{array}{}{L}_{+}^{0}\end{array}$ (I, Xθ(⋅)) is a subspace of L0(I, Xθ(⋅)).

Suppose that {ϱα : α ∈ 𝓐} is a 𝓒𝓜𝓝 generating a dense 𝓑𝓢𝓝{Xα}. Similar to Lϱθ(⋅)(I, Xθ(⋅)), we can define $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)) through

$L+ϱθ(⋅)(I,Xθ(⋅))={f∈L+0(I,Xθ(⋅)):Φϱθ(⋅)(λf)<∞for someλ>0}$

with the same Luxemburg norm. Note that in such situation, 𝓢(I, Xα+) is a linear subspace of $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)).

We say that the 𝓒𝓜𝓝{ϱα} satisfies the Δ2−condition, if every ϱα 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, Φϱθ(⋅) is also a Δ2−type modular. Hence following the same process as in Proposition 2.3, we can prove that

#### Proposition 3.12

Under the Δ2condition of {ϱα} and the density assumption of {Xα}, each function of $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)) can be approximated by a sequence of 𝓢(I, Xα+) according to the norm ∥⋅∥Lϱθ(⋅)(I,Xθ(⋅)).

#### Theorem 3.13

Under the same assumptions as above, $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)) is a Banach space.

#### Proof

Taken any Cauchy sequence {fn} in $\begin{array}{}{L}_{+}^{p\left(\cdot \right)}\end{array}$ (I, Xθ(⋅)), by virtue of Theorem 3.7, there is a function fLϱθ(⋅)(I, Xθ(⋅)) for which ∥fnfLϱθ(⋅)(I,Xθ(⋅)) → 0 as n → ∞. For each n ∈ ℕ, on account of Proposition 3.12, there is a φn ∈ 𝓢(I, Xα+) such that ∥φnfnLϱθ(⋅)(I,Xθ(⋅)) < 1/n, which in turn yields ∥φnfLϱθ(⋅)(I,Xθ(⋅)) → 0 as n → ∞. Hence there is a subsequence, say {φn} itself, satisfying φn(t) → f(t) in Xθ(t) as n → ∞ for a.e. tI. Therefore f$\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)) and the proof is completed. □

#### Remark 3.14

By reviewing the above proof, one can easily find that, under present situations, condition L0(I, Xθ(⋅)) = $\begin{array}{}{L}_{+}^{0}\end{array}$ (I, Xθ(⋅)) is sufficient for Lϱθ(⋅)(I, Xθ(⋅)) = $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)).

#### Theorem 3.15

Besides the Δ2condition of {ϱα} and the density assumption of {Xα}, assume that Xα+ is separable. Then the function space $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)) is also separable.

This theorem is a straight consequence of Proposition 3.12.

For each α ∈ 𝓐, denote by $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ the Fenchel duality of ϱα, i.e.

$ϱα∗(ξ)=supu∈Xα{〈ξ,u〉α−ϱα(u)},ξ∈Xα∗.$

Since ϱα is a semimodular, $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ is also a semimodular on $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ , and the semimodular space derived by $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ is exactly $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ itself (see [2, § 2.2]). Define

$ϱ~α∗(ξ)=ϱα∗(ξ),if ξ∈Xα∗,∞,if ξ∈Xα+∗∖Xα∗,$

then we obtain another family of semimodulars defined on $\begin{array}{}{X}_{{\alpha }^{+}}^{\ast }\end{array}$ , called the dual modular net or in symbol 𝓓𝓜𝓝 of {ϱα}. Since for each α ∈ 𝓐, the effective domains and the induced semimodular spaces are equal, in the coming arguments, we will not distinguish $\begin{array}{}{\stackrel{~}{\varrho }}_{\alpha }^{\ast }\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{\varrho }_{\alpha }^{\ast }\end{array}$ , and prefer to use { $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ } instead of { $\begin{array}{}{\stackrel{~}{\varrho }}_{\alpha }^{\ast }\end{array}$ } to denote the 𝓓𝓜𝓝 of {ϱα}.

Suppose α1α2, then $\begin{array}{}{X}_{{\alpha }_{1}}^{\ast }↪{X}_{{\alpha }_{2}}^{\ast }\end{array}$ , and for all ξ$\begin{array}{}{X}_{{\alpha }_{2}}^{\ast }\end{array}$ , by (6) we have

$ϱα2∗(ξ)=supu∈Xα2{〈ξ,u〉α2−ϱα2(u)}≤supu∈Xα2{〈ξ,u〉α2−1C1ϱα1(u)}+C2C1≤supu∈Xα1{〈ξ,u〉α1−1C1ϱα1(u)}+C2C1=1C1ϱα1∗(C1ξ)+C2C1.$(8)

Similar to Proposition 3.4, from this property we can show that the dual space family { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ : α ∈ 𝓐}, where 𝓐 takes the inverse order, is a uniformly bounded net. Moreover, assume that the function αϱα is sequently continuous, in other words, if {αk} converges to α in 𝓐, then for all uX, the limit

$limk→∞ϱαk(u)=ϱα(u)$

holds. Under this assumption, we can deduce that, for all sequences {α̃k} satisfying α̃kα and α̃kα, inequality

$ϱα∗(ξ)≤lim supk→∞ϱα~k∗(ξ)$

holds for all ξ$\begin{array}{}{X}_{{\alpha }^{+}}^{\ast }\end{array}$ , which in turn leads to the successive property of { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ }. Unfortunately, the inverse inequality, hence continuity of { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ } can not be guaranteed under present situations. For the sake of convenience, hereinafter, we always assume that $\begin{array}{}{X}_{{\alpha }^{+}}^{\ast }\end{array}$X, and the 𝓓𝓜𝓝{ $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ } is assumed to be a 𝓒𝓜𝓝 defined on X. We also assume that the 𝓑𝓢𝓝{Xα} and its dual net { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ } are compatible, i.e. 〈ξ, uα2 = 〈ξ, uα1 provided uXα2, ξ$\begin{array}{}{X}_{{\alpha }_{1}}^{\ast }\end{array}$ and α1α2. This convention has been already used in (8). All the assumptions mentioned above will be used later without any other comments.

#### Theorem 3.16

Suppose that the following hypotheses are all satisfied:

• {ϱα} satisfies the Δ2condition, and 𝓡(∂ϱα) = $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ for all α ∈ 𝓐,

• {Xα} is a dense 𝓑𝓢𝓝, and Xα+ is separable,

• for every α ∈ 𝓐, $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ is a modular, and

• $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ has the Radon-Nikodym’s property w.r.t. every JΠ(I).

Then the dual space $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅))* is equivalent to $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ in the sense of isomorphism.

#### Proof

Firstly for each ξ$\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ , define the linear functional Λξ as follows:

$〈〈Λξ,f〉〉θ(⋅)=∫I〈ξ(t),f(t)〉θ(t)dt,∀f∈L+ϱθ(⋅)(I,Xθ(⋅)).$(9)

Suppose that $\begin{array}{}\parallel \xi {\parallel }_{{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)}=\parallel f{\parallel }_{{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\left(I,{X}_{\theta \left(\cdot \right)}\right)}=1\end{array}$ , then by Young’s inequality we have

$〈〈ξ,f〉〉θ(⋅)≤∫Iϱθ(t)∗(ξ(t))dt+∫Iϱθ(t)(f(t))dt≤2.$

Therefore Λξ$\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅))*, and $\begin{array}{}\parallel {\mathit{\Lambda }}_{\xi }{\parallel }_{{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\left(I,{X}_{\theta \left(\cdot \right)}{\right)}^{\ast }}\le 2\parallel \xi {\parallel }_{{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)}\end{array}$ . This claim also holds for arbitrary ξ$\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ by scaling arguments.

Conversely, given a functional Λ$\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅))*, we will find a function ξ$\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ such that Λ = Λξ in the sense of (9) with the norm equivalent to that of ξ. If Λ = 0, then we take ξ = 0 and there is nothing to do. If Λ ≠ 0, then without loss of generality, assume that $\begin{array}{}\parallel \mathit{\Lambda }{\parallel }_{{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\left(I,{X}_{\theta \left(\cdot \right)}{\right)}^{\ast }}=1\end{array}$ . Taking any JΠ(I) and any $\begin{array}{}f\in {L}^{{\varrho }_{{\theta }_{J}^{+}}}\left(J,{X}_{{\theta }_{J}^{+}}\right)\end{array}$ , consider the zero extension of f out of J and denote it by . Obviously, $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)) and

$∥f~∥L+ϱθ(⋅)(I,Xθ(⋅))≤C∥f∥LϱθJ+(J,XθJ+)$(10)

for some constant C > 0 depending on |J| but independent of f, which means that the restriction of Λ to $\begin{array}{}{L}^{{\theta }_{J}^{+}}\left(J,{X}_{{\theta }_{J}^{+}}\right)\end{array}$ , denoted by $\begin{array}{}\mathit{\Lambda }{|}_{J,{\theta }_{J}^{+}}\end{array}$ , lies in $\begin{array}{}{L}^{{\theta }_{J}^{+}}\left(J,{X}_{{\theta }_{J}^{+}}{\right)}^{\ast }\end{array}$ . So by invoking Theorem 2.7, there is a unique function $\begin{array}{}{\xi }_{J}\in {L}^{{\varrho }_{{\theta }_{J}^{+}}^{\ast }}\left(I,{X}_{{\theta }_{J}^{+}}^{\ast }\right)\end{array}$ such that

$〈〈Λ|J,θJ+,f〉〉θJ+=∫J〈ξJ(t),f(t)〉θJ+dt$

for all $\begin{array}{}f\in {L}^{{\theta }_{J}^{+}}\left(J,{X}_{{\theta }_{J}^{+}}\right)\end{array}$ , and

$∥ξJ∥LϱθJ+∗(I,XθJ+∗)≤C∥Λ|J,θJ+∥LϱθJ+(J,XθJ+)∗≤C∥Λ∥Lϱθ(⋅)(I,Xθ(⋅))∗$(11)

for some constant C > 0 depending only on the length of J.

Suppose that J1, J2Π(I) satisfy J2J1, then $\begin{array}{}{L}^{{\varrho }_{{\theta }_{{J}_{2}}^{+}}}\left(J,{X}_{{\theta }_{{J}_{2}}^{+}}\right)\end{array}$ is densely imbedded in $\begin{array}{}{L}^{{\varrho }_{{\theta }_{{J}_{1}}^{+}}}\left(J,{X}_{{\theta }_{{J}_{1}}^{+}}\right)\end{array}$ , hence by the uniqueness of the representation ξJ1, we can assert that ξJ2(t) = ξJ1(t) for a.e. tJ1. Define ξ(t) = ξJ(t) if tJ for arbitrary JΠ(I), then we obtain a well defined function $\begin{array}{}\xi \in {L}_{-}^{0}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$.

In case that I = [0, T] is bounded, all the constants C in (10) and (11) can be selected independent of JΠ(I), thus via Corollary 3.10, we can derive that $\begin{array}{}\xi \in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\parallel \xi {\parallel }_{{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)}\le C\parallel \mathit{\Lambda }{\parallel }_{{L}^{{\varrho }_{\theta \left(\cdot \right)}}\left(I,{X}_{\theta \left(\cdot \right)}{\right)}^{\ast }}\end{array}$ for some C > 0 independent of Λ.

For each f$\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$ (I, Xθ(⋅)), select a sequence {φk} ⊆ 𝓢(I, Xα+) converging to f according to the Lϱθ(⋅)(I, Xθ(⋅))-norm. Since for every k ∈ ℕ,

$〈〈Λ,φk〉〉θ(⋅)=〈〈Λ|I,α+,φk〉〉α+=∫I〈ξ(t),φk(t)〉α+dt=∫I〈ξ(t),φk(t)〉θ(t)dt,$

letting k → ∞, we obtain

$〈〈Λ,f〉〉θ(⋅)=∫I〈ξ(t),f(t)〉θ(t)dt,$

which shows that Λ = Λξ.

It remains to prove (9) in the case I = [0, ∞). Firstly the above discussions tell us that $\begin{array}{}\xi \in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(J,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ for all JΠ(I). Consequently ξL0(I, $\begin{array}{}{X}_{\theta \left(\cdot \right)}^{\ast }\end{array}$ ), and the scalar function t ↦ 〈ξ(t), f(t)〉θ(t) is measurable on I whenever fLϱθ(⋅)(I, Xθ(⋅)).

Given a function fLϱθ(⋅)(I, Xθ(⋅)), for each n ∈ ℕ, let fn = [0,n], then we obtain an approximate sequence of f in Lϱθ(⋅)(I, Xθ(⋅)) satisfying

$〈〈Λ,fn〉〉θ(⋅)=〈〈Λ|[0,n],fn〉〉θ(⋅)=∫[0,n]〈ξ(t),f(t)〉θ(t)dt.$

Let n → ∞, using the fact limn→∞ 〈〈Λ, fn〉〉θ(⋅) = 〈〈Λ, f〉〉θ(⋅), we can deduce that t ↦ 〈ξ(t), f(t)〉θ(t) is integrable on I, and

$〈〈Λ,f〉〉θ(⋅)=∫I〈ξ(t),f(t)〉θ(t)dt.$

Thus Λ = Λξ by the arbitrariness of f. The remaining task for us is to show $\begin{array}{}\xi \in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ . For this purpose, notice that the effective domain 𝓓(ϱα) is equal to Xα and the latter is separable, so the dual modular $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ can be represented by

$ϱα∗(η)=supk≥1{〈η,vk〉α−ϱα(vk)},∀η∈Xα∗,$(12)

where {vk} is a countable dense subset of Xα. By the density of {Xα}, if we take {vk} as the dense sequence of Xα+ with v1 = 0, then (12) holds with α = θ(t) and η$\begin{array}{}{X}_{\theta \left(t\right)}^{\ast }\end{array}$ for all tI.

For each n ∈ ℕ, define

$rn(t)=χ[0,n](t)max1≤k≤n{〈ξ(t),vk〉θ(t)−ϱθ(t)(vk)}.$

Obviously, {rn} is a nondecreasing sequence of nonnegative (v1 = 0) measurable functions converging to $\begin{array}{}{\varrho }_{\theta \left(t\right)}^{\ast }\end{array}$ (ξ(t)) almost everywhere. Moreover, there is a sequence of simple functions {sn} ⊆ 𝓢(I, Xα+) such that

$rn(t)=〈ξ(t),sn(t)〉θ(t)−ϱθ(t)(sn(t)).$

Due to the facts 𝓢(I, Xα+) ⊆ Lϱθ(⋅)(I, Xθ(⋅)) and ∥ΛLϱθ(⋅)(I,Xθ(⋅))* = 1, we have

$1≥Φϱθ(⋅)∗(Λ)≥〈〈Λ,sn〉〉−Φϱθ(⋅)(sn)=∫I{〈ξ(t),sn(t)〉θ(t)−ϱθ(t)(sn(t))}dt,$

where $\begin{array}{}{\mathit{\Phi }}_{{\varrho }_{\theta \left(\cdot \right)}}^{\ast }\end{array}$ is the dual modular of Φϱθ(⋅). Taking limit of the second line as n → ∞, we obtain

$Φϱθ(⋅)∗(ξ)=∫Iϱθ(t)∗(ξ(t))dt≤1.$

Therefore $\begin{array}{}\xi \in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\parallel \xi {\parallel }_{{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)}\le 1.\end{array}$

Finally by means of scaling transformation, we can obtain the desired estimate

$∥ξ∥Lϱθ(⋅)∗(I,Xθ(⋅)∗)≤∥Λ∥Lϱθ(⋅)(I,Xθ(⋅))∗.$

Thus we have completed the proof. □

#### Remark 3.17

There is a by-product produced from the above proof, that is under all the hypotheses of Theorem 3.16, we have $\begin{array}{}{\mathit{\Phi }}_{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(\xi \right)={\mathit{\Phi }}_{{\varrho }_{\theta \left(\cdot \right)}}^{\ast }\left({\mathit{\Lambda }}_{\xi }\right)\end{array}$ for all $\begin{array}{}\xi \in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ . This is a natural extension of that of the scalar case.

#### Corollary 3.18

In addition to the assumptions of the above theorem, assume that L0(I, Xθ(⋅)) = $\begin{array}{}{L}_{+}^{0}\end{array}$ (I, Xθ(⋅)), then

$Lϱθ(⋅)(I,Xθ(⋅))∗≅Lϱθ(⋅)∗(I,Xθ(⋅)∗).$(13)

#### Theorem 3.19

Suppose the following conditions are all satisfied.

• both {ϱα} and { $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ } satisfy the Δ2condition,

• {Xα} and { $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ } are two dense 𝓑𝓢𝓝s,

• Xα+ and $\begin{array}{}{X}_{{\alpha }^{-}}^{\ast }\end{array}$ are both separable, and

• for every α ∈ 𝓐, Xα is reflexive,

• L0(I, Xθ(⋅)) = $\begin{array}{}{L}_{+}^{0}\left(I,{X}_{\theta \left(\cdot \right)}\right),\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{L}^{0}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)={L}_{+}^{0}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$.

Then Lϱθ(⋅)(I, Xθ(⋅)) is a reflexive space.

Given a 𝓒𝓜𝓝{ϱα}, assume that it is uniformly convex, in other words, every ϱα is uniformly convex, and for each ε ∈ (0, 1), the corresponding number δ ∈ (0, 1) appearing in (5) is independent of α.

#### Theorem 3.20

Under the uniform convexity assumption and the Δ2condition of {ϱα}, the function space Lϱθ(⋅)(I, Xθ(⋅)) is uniformly convex.

#### Remark 3.21

Putting all the hypotheses in Theorem 3.19, 3.20 together, we obtain not only the uniform convexity of Lϱθ(⋅)(I, Xθ(⋅)), but the representation (13) as well.

#### Example 3.22

Let Ω ⊆ ℝN be a bounded domain, and let 𝓟(Ω) be the set of all measurable functions taking values in [1, ∞], and

$Pb(Ω)={p∈P(Ω):1≤p−≤p+<∞},$

where notations p+ and p denote the essential supremum and infimum of p on Ω respectively. For any p ∈ 𝓟b(Ω), functional

$ϱp(f)=∫Ω1p(x)|f(x)|p(x)dx$

is a continuous modular on the linear space X = L0(Ω), which induces a separable Banach space Xp : = Lp(x)(Ω). Evidently ϱp satisfies the Δ2condition with the function ω(t) = tp+. If in addition p > 1, then ϱp is uniformly convex, its dual modular $\begin{array}{}{\varrho }_{p}^{\ast }\end{array}$ equals ϱp′, and the dual space Lp(x)(Ω)* is equivalent to Lp′(x)(Ω). Here p′(x) is the conjugate exponent of p(x), that is 1/p(x) + 1/p′(x) = 1 for a.e. xΩ. It is also easy to see that, Lp2(x)(Ω) is a dense subspace of Lp1(x)(Ω) provided p1(x) ≤ p2(x) a.e. on Ω.

Fix two numbers p and in [1, ∞) with p, let

$Ab={p∈Pb(Ω):p(x)∈[p_,p¯]fora.e.x∈Ω}.$

Then equipped with the order: pq by p(x) ≤ q(x) a.e. on Ω, and the topology determined by: pnp in 𝓐b if and only if pn(x) → p(x) a.e. on Ω, 𝓐b becomes a 𝓑𝓣𝓛. Meanwhile, {ϱp : p ∈ 𝓐b} is a 𝓒𝓜𝓝 defined on X satisfying the Δ2condition with the common function ω(t) = t, and {Xp : p ∈ 𝓐b} is a dense regular 𝓑𝓢𝓝 generated by {ϱp} (cf. [11]).

Assume that I = [0, T], and Q = I × Ω is a cylinder. Recall that, each uL0(Q) has an Xrealization Pu in L0(I, X) satisfying Pu(t)(x) = u(t, x) for a.e. xΩ and a.e. tI, and conversely, each uL0(I, X) has a scalar realization ũ in L0(Ω) satisfying ũ(t, x) = u(t)(x) for a.e. (t, x) ∈ Q. Moreover, for all q ∈ (1, ∞), the projection P : Lq(Q) → Lq(I, Lq(Ω)) is a linear isometrical isomorphism with the inverse P−1u = ũ. If q ∈ 𝓟b(Ω) is a variable exponent, then P : Lq(⋅)(Q) → Lq(I, Lq(⋅)(Ω)) is also continuous (refer to [7]). In the following discussion we will omit the notation P and simply use a single letter u to represent a scalar function and its Xrealization, or an Xvalued function and its scalar realization without any other remarks, if there is no confusion arising.

Suppose that p ∈ 𝓟b(Q) is a Caratheodory type function satisfying

1. p(t, ⋅) is measurable on Ω for every tI, and

2. p(⋅, x) is continuous on I for a.e. xΩ.

Let $\begin{array}{}\underset{_}{p}={p}_{Q}^{-},\overline{p}={p}_{Q}^{+}\end{array}$ , and define θ(t) = p(t, ⋅) =: p(t), then we obtain an order-continuous exponent θ : I → 𝓐b with

$θJ+=sup{p(t,x):t∈J},θJ−=inf{p(t,x):t∈J}$

for all JΠ(I), and Xθ(t) = Lp(t,x)(Ω), $\begin{array}{}{X}_{\theta \left(t\right)}^{\ast }\end{array}$ = Lp′(t,x)(Ω) for all tI. [11] reveals that

$L0(I;Lp(⋅,x)(Ω))=L+0(I;Lp(⋅,x)(Ω)).$

Thus, Lϱp(⋅)(I; Lp(⋅,x)(Ω)) = $\begin{array}{}{L}_{+}^{{\varrho }_{p\left(\cdot \right)}}\end{array}$ (I; Lp(⋅,x)(Ω)) is a separable Banach space. Furthermore, if p > 1, then

$L0(I;Lp′(⋅,x)(Ω))=L+0(I;Lp′(⋅,x)(Ω)),$

and all the assumptions arising in Theorem 3.19, 3.20 are fulfilled, thus by Remark 3.21, we get the uniform convexity, hence the reflexivity of Lϱp(⋅)(I; Lp(⋅,x)(Ω)), together with the expression

$Lϱp(⋅)(I;Lp(⋅,x)(Ω))∗≅Lϱp′(⋅)(I;Lp′(⋅,x)(Ω)).$

It is worth mentioning that for the same exponent p(⋅, ⋅), projection P is also an isometrical isomorphism from Lp(t,x)(Q) onto Lϱp(⋅)(I; Lp(⋅,x)(Ω)) (see [11] for references). This is a natural extension of the property of P from the case of constant exponents to the case of variable ones.

## 4 Functionals and Operators On Lϱθ(⋅)(I, Xθ(⋅))

In this section we will study some functionals and operators on the function space Lϱθ(⋅)(I, Xθ(⋅)), including the subdifferential of Φϱθ(⋅), whose representation will be taken into account. For this purpose, we need the coercive assumption on ϱα, $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ , as well as Φϱθ(⋅) and $\begin{array}{}{\mathit{\Phi }}_{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\end{array}$ . Coercivity, which says

$ϱ(u)∥u∥X→∞as∥u∥X→∞,$

is an important property of a lower semicontinuous (or lsc for short) and proper convex function ϱ defined on a Banach space X. Using the coercive property of ϱ, we can obtain the boundedness of a sequence in X under some situations. For example, if there is a sequence {un} ⊆ X satisfying

$ϱ(un)∥un∥X≤Kfor someK>0,$

then there is a constant C > 0 depending only on K such that ∥unXC for all n ∈ ℕ.

It is easy to check that if ϱα is a coercive modular, then its dual $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ satisfies Dom( $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ ) = $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ . As a matter of fact, taking any ξ$\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ , by the coercivity of ϱα, there is a constant M > 0 for which ϱα(u) ≥ ( $\begin{array}{}\parallel \xi {\parallel }_{\alpha }^{\ast }\end{array}$ + 1)∥uα provided ∥uαM. Consequently,

$ϱα∗(ξ)=sup∥u∥α≤M{〈ξ,u〉α−ϱα(u)}≤M∥ξ∥α∗<∞.$

In general, coercivity of Φϱθ(⋅) could not be derived from the coercive assumption of all the ϱαs naturally. Under some special conditions, however, all of ϱα, α ∈ 𝓐 and Φϱθ(⋅) are coercive simultaneously. The following assumption, which is called strong coercivity of {ϱα}, is a desired one.

$ϱα(uγ−1(s))≤ϱα(u)sfor allu∈X,s>0andα∈A,$(14)

where γ : [1, ∞) → [1, ∞) is strictly increasing function satisfying

$lims→∞γ(s)s=∞.$(15)

By (15), there is a constant K ≥ 1 such that γ−1(s) ≤ s whenever sK. Now taking any α ∈ 𝓐 and uXα with ∥uα ≥ 2K, and using (14), we can deduce that

$ϱα(u)∥u∥α≥12ϱα(uγ−1(∥u∥α/2))≥12∥u∥α/2γ−1(∥u∥α/2)ϱα(u∥u∥α/2)≥12∥u∥α/2γ−1(∥u∥α/2),$

which combined with (15) yields the coercivity of ϱα.

Furthermore, for any uLϱθ(⋅)(I, Xθ(⋅)) with ∥uLϱθ(⋅)(I,Xθ(⋅)) ≥ 1, we have that

$∫Iϱθ(t)(u(t)γ−1(Φϱθ(⋅)(u)))dt≤1Φϱθ(⋅)(u)∫Iϱθ(t)(u(t))dt=1,$

which means ∥uLϱθ(⋅)(I,Xθ(⋅))γ−1(Φϱθ(⋅)(u)). Consequently,

$lim∥u∥Lϱθ(⋅)(I,Xθ(⋅))→∞Φϱθ(⋅)(u)∥u∥Lϱθ(⋅)(I,Xθ(⋅))≥limΦϱθ(⋅)(u)→∞Φϱθ(⋅)(u)γ−1(Φϱθ(⋅)(u))=∞,$

which shows the coercivity of Φϱθ(⋅).

In the following arguments, we also need the strict convexity of $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ for all α ∈ 𝓐, and the weak lower-semicontinuity of {ϱα} and { $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ }, which says

• For any sequence {αk} approaching β in 𝓐 and any sequence {uk} ⊆ Xβ converging weakly to u in Xβ, {ξk} converging star-weakly to ξ in $\begin{array}{}{X}_{\beta }^{\ast }\end{array}$ , we have

$ϱβ(u)≤lim infk→∞ϱαk(uk)andϱβ∗(ξ)≤lim infk→∞ϱαk∗(ξk).$(16)

For the sake of convenience, all the hypotheses listed in Theorem 3.19, 3.20 are denoted by H(A) as a whole. Furthermore, assumptions of strong coercivity of ϱα and $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ , and weak lower-semicontinuity of {ϱα} and { $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ } are put together, denoted by H(B), which jointly with the strict convexity assumption of ϱα and $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ for all α ∈ 𝓐 are denoted by H(B). Without any other specific comments, in further discussion we always assume that H(A) and H(B) are both verified for the given 𝓒𝓜𝓝{ϱα} and its 𝓓𝓜𝓝 { $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ }.

Consider the subdifferential operator ∂ϱα. Similar to Remark 2.8, we can check that ∂ϱα is single-valued provided $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ is strict convex. Furthermore, by the coercivity of ϱα, we can also find that ∂ϱα : Xα$\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ is a demicontinuous, monotone and coercive operator, whose range is the whole space $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ . Since

$〈∂ϱα(u),u〉α=ϱα(u)+ϱα∗(∂ϱα(u))$

for all uXα, ϱα(u) can be regarded as the extension of the traditional dual map where not modulars but norms of Xϱ and $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ are involved.

#### Lemma 4.1

For all uL(I, Xα), the compound operator t∂ϱα(u(t)) lies in L(I, $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ ). Moreover, if uLϱα(I, Xα), then ∂ϱα(u(⋅)) ∈ $\begin{array}{}{L}^{{\varrho }_{\alpha }^{\ast }}\left(I,{X}_{\alpha }^{\ast }\right)\end{array}$.

#### Proof

By the demicontinuity of ∂ϱα, one can easily see that the function ξ(t) = ∂ϱα(u(t)) is strongly measurable. Furthermore by Remark 2.6, we have

$ϱα∗(ξ(t))+ϱα(u(t))=〈ξ(t),u(t)〉α≤∥ξ(t)∥α∗∥u(t)∥α,$

which together with the coercivity of $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ , yields the boundedness of $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ (ξ(t)) uniformly for a.e. tI, hence the inclusion ξL(I, $\begin{array}{}{X}_{{\varrho }_{\alpha }}^{\ast }\end{array}$).

Suppose that uLϱα(I, Xϱα), and {sk} ⊆ 𝓢(I, Xϱα) converges to u in Lϱα(I, Xα). From above arguments, we know that for every k ∈ ℕ, the $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ −valued simple function ξk(t) = ∂ϱα(sk(t)) lies in L(I, $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ ), consequently it lies $\begin{array}{}{L}^{{\varrho }_{\alpha }^{\ast }}\left(I,{X}_{{\varrho }_{\alpha }}^{\ast }\right)\end{array}$ , and

$ϱα∗(ξk(t))+ϱα(sk(t))=〈ξk(t),sk(t)〉αa.e. onI.$(17)

Taking integrations on both sides and using generalized Hölder’s inequality, we have

$Φϱα∗(ξk)+Φϱα(sk)=∫I〈ξk),sk(t)〉αdt≤2∥ξk∥Lϱα∗(I,Xα∗)∥sk∥Lϱα(I,Xα).$

Then by the coercivity of $\begin{array}{}{\mathit{\Phi }}_{{\varrho }_{\alpha }^{\ast }}\end{array}$ and the boundedness of {ξk} in Lϱα(I, Xα), we get the boundedness of {ξk} in $\begin{array}{}{L}^{{\varrho }_{\alpha }^{\ast }}\end{array}$(I, $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$). Therefore there is a subsequence, say {ξk} itself, convergent to some ξ̃ weakly in $\begin{array}{}{L}^{{\varrho }_{\alpha }^{\ast }}\end{array}$(I, $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$). Suppose that {vi} is a countable dense subset of Xα, then for every two positive integers i, n, we have

$limk→∞∫In〈ξk(t)−ξ~(t),vi〉αdt=0.$

It follows that the scalar function hk(t) = 〈ξk(t) – ξ̃(t), viα is convergent to 0 in measure on In. As a result, {hk} has a subsequence convergent to 0 a.e. on In. Then by means of the diagonalizing method, we can find another subsequence, denoted still by {hk} such that limk→∞ hk(t) = 0 for a.e. tI, which combined with the boundedness of {ξk(t)} derived from (17) and the density of {vi} in Xα, results in the weak convergence of ξk(t) to ξ̃(t) in $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ as k → ∞ for a.e. tI. Now taking limits in (17), and using continuity of ϱα and weak lower-continuity of $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$, we obtain

$ϱα∗(ξ~(t))+ϱα(u(t))≤lim infk→∞ϱα∗(ξk(t))+limk→∞ϱα(sk(t)) ≤lim infn→∞(ϱα∗(ξk(t))+ϱα(u(t))) ≤limk→∞〈ξk(t),sk(t)〉α=〈ξ~(t),u(t)〉α,$

which in turn tields ξ̃(t) = ∂ϱα(u(t)) for a.e. tI. Thus the lemma has been proved since ξ̃$\begin{array}{}{L}^{{\varrho }_{\alpha }^{\ast }}\end{array}$(I, $\begin{array}{}{X}_{{\varrho }_{\alpha }}^{\ast }\end{array}$).□

In studying the subdifferential of the Nemytzkij functional of a series of modulars, we always assume that I = [0, T].

#### Proposition 4.2

For each uLϱθ(⋅)(I, Xθ(⋅)), the $\begin{array}{}{X}_{\theta \left(\cdot \right)}^{\ast }\end{array}$-valued function ∂ϱθ(t)(u(t)) belongs to $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\end{array}$(I, $\begin{array}{}{X}_{\theta \left(\cdot \right)}^{\ast }\end{array}$).

#### Proof

For every 2n-mean partition of I, define the step function $\begin{array}{}{\theta }_{n}^{-}\end{array}$(t) as in the preceding section, and let ξn(t) = $\begin{array}{}\mathrm{\partial }{\varrho }_{{\theta }_{n}^{-}\left(t\right)}\end{array}$(u(t)) for a.e. tI. From Lemma 4.1, we can derive that ξn lies in $\begin{array}{}{L}^{{\varrho }_{{\theta }_{n}^{-}\left(\cdot \right)}}\left(I,{X}_{{\theta }_{n}^{-}\left(\cdot \right)}^{\ast }\right),\end{array}$ hence it lies in $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right).\end{array}$ Much similar to the proof of Lemma 4.1, and using the strong coercivity of $\begin{array}{}\left\{{\varrho }_{\alpha }^{\ast }\right\}\end{array}$, density of {ϱα}, separability of Xα+ together with the relations Lϱθ(⋅)(I, Xθ(⋅)) = $\begin{array}{}{L}_{+}^{{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)) and $\begin{array}{}{L}^{{\varrho }_{{\theta }_{n}^{-}\left(\cdot \right)}^{\ast }}\left(I,{X}_{{\theta }_{n}^{-}\left(\cdot \right)}^{\ast }\right)\end{array}$$\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$, we can deduce that

$ϱθn−(t)∗(ξn(t))+ϱθn−(t)(u(t))=〈ξn(t),u(t)〉θn−(t)a.e. onI,$(18)

{ξn} is bounded in $\begin{array}{}{L}^{{\varrho }_{{\theta }_{n}^{-}\left(\cdot \right)}^{\ast }}\left(I,{X}_{{\theta }_{n}^{-}\left(\cdot \right)}^{\ast }\right)\end{array}$ and consequently in $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$. Thus by passing to a subsequence, we can assume that {ξn} converges weakly to some ξ in $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$, and ξn(t) ⇀ ξ(t) in $\begin{array}{}{X}_{\theta \left(t\right)}^{\ast }\end{array}$ as n → ∞ a.e. on I. Now taking limits in (18), and using (13), we obtain

$ϱθ(t)∗(ξ(t))+ϱθ(t)(u(t))≤lim infn→∞ϱθn−(t)∗(ξn(t))+limn→∞ϱθn−(t)(u(t)) ≤limn→∞〈ξn(t),u(t)〉θn−(t)=〈ξ(t),u(t)〉θ(t)$

for a.e. tT. Therefore ξ(t) = ∂ϱθ(t)(u(t)) almost everywhere and the proposition has been proved.□

#### Remark 4.3

In terms of Lemma 4.1 and Remark 3.17, we can claim that for every uLϱα(I, Xϱα), the subdifferential ∂Φϱα(u) equals ∂ϱα(u(⋅)). Moreover, if we drop the strict convexity assumption of $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$, then we have the classical representation:

$∂Φϱα(u)={ξ∈Lϱα∗(I,Xα∗):ξ(t)∈∂ϱα(u(t))for a.e.t∈I}.$

Similarly, under the conditions of Proposition 4.2, we have ∂Φϱθ(⋅)(u) = ∂ϱθ(⋅)(u(⋅)) for all uLϱθ(⋅)(I, Xθ(⋅)), and

$∂Φϱθ(⋅)(u)={ξ∈Lϱθ(⋅)∗(I,Xθ(⋅)∗):ξ(t)∈∂ϱθ(t)(u(t))for a.e.t∈I},$

if the strict convexity assumptions of $\begin{array}{}{\varrho }_{\alpha }^{\ast }\end{array}$ for all α ∈ 𝓐 are moved.

The following theorem is a natural corollary of Proposition 4.2 and Remark 4.3.

#### Theorem 4.4

Under hypotheses H(A) and H(B), the operator Zθ(⋅) defined through Zθ(⋅)(u) = ∂ϱθ(⋅)(u(⋅)) is demicontinuous, coercive and bounded, together with

$〈〈Zθ(⋅)(u),u〉〉θ(⋅)=Φϱθ(⋅)(u)+Φϱθ(⋅)∗(u)$

for all uLϱθ(⋅)(I, Xθ(⋅). In a word, Zθ(⋅) : Lϱθ(⋅)(I, Xθ(⋅)) → $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ is a generalized dual map.

#### Remark 4.5

Due to the facts that 𝓓(Φθ(⋅)) = Lϱθ(⋅)(I, Xθ(⋅)) and ∂Φθ(⋅) = Zθ(⋅) is single-valued, functional Φθ(⋅) is Gâteaux differential, and its Gâteaux differential $\begin{array}{}{\mathit{\Phi }}_{\theta \left(\cdot \right)}^{\prime }\end{array}$ equals Zθ(⋅).

Suppose that 𝓑 is another 𝓑𝓣 𝓛, {φβ : β ∈ 𝓑} is another 𝓒𝓜𝓝 on YX and {Vβ : β ∈ 𝓑} is the corresponding 𝓑𝓢𝓝 generated by {φβ}. We say {φβ} is stronger than {ϱα}, we mean that

• Vβ is imbedded continuously and densely in Xα, and the dual product 〈ξ, u〉 has the same value in both $\begin{array}{}{V}_{\beta }^{\ast }\end{array}$ × Vβ and $\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ × Xα for all uVβ and ξ$\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$, and all α ∈ 𝓐, β ∈ 𝓑.

Under this assumption, for all C > 0,

$EC={u∈Xα:φβ(u)≤C}$

is a bounded and weakly closed subset of Vβ. Hence by the inclusion VβXα, we have

#### Lemma 4.6

If Vβ is reflexive, then φβ is also a lower semicontinuous and convex function on Xα.

#### Corollary 4.7

Under the reflexivity assumption of Vβ, for every uL0(I, Xα), the multifunction tφβ(u(t)) is measurable.

Suppose that ϑ : I → 𝓑 is also an order-continuous map, then based on the above results and the continuity of {φβ}, we can check that

#### Proposition 4.8

Assume that for every β ∈ 𝓑, the modular space Vβ is reflexive. Then for each uL0(I, Xθ(⋅)), functions $\begin{array}{}\left\{{\phi }_{{\vartheta }_{n}^{-}\left(\cdot \right)}\left(u\left(\cdot \right)\right){\right\}}_{n=1}^{\mathrm{\infty }}\end{array}$ are all measurable, hence as the limit function, φϑ(⋅)(u(⋅)) is also measurable.

Now we can define the Nemytzkij functional of {φϑ(t) : tI} through

$Φφϑ(⋅)(u)=∫Iφϑ(t)(u(t))dt,u∈L0(I,Xθ(⋅)).$

#### Lemma 4.9

Suppose that φβ satisfies the Δ2-condition, and Vβ is a reflexive and separable space. Then a function uL0(I, Xα) is also a member of Lφβ(I, Vβ) provided Φφβ(u) < ∞.

#### Proof

By the condition Φφβ(u) < ∞, it suffices to show the inclusion uL0(I, Vβ). Taking any r > 0 and u0Vβ, denote by

$BVβ(u0,r)={u∈Vβ:∥u−u0∥Vβ

and

$Bφβ(u0,r)={u∈Vβ:φβ(u−u0)

By the unit ball property, we know that BVβ(u0, r) ⊆ Bφβ(u0, r) provided 0 < r < 1. Moreover, for each r > 0, by the Δ2-condtion of φβ, there is a δ > 0 such that Bφβ(u0, δ) ⊆ BVβ(u0, r) (cf. [2, P. 43]). Thus for any subset E of Vβ, one can check that

$E¯=⋂n≥1⋃u∈EBφβ(u,1n).$

Take an arbitrary nonempty closed subset F of Vβ. By the separability of Vβ, F has a countable dense subset {vk} making

$F=⋂n≥1⋃k≥1Bφβ(vk,1n),$

which results in

${t∈I:u(t)∈F}=⋂n≥1⋃k≥1{t∈I:φβ(u(t)−vk)<1n}=:⋂n≥1⋃k≥1Ek,n.$

Evidently, for each k ∈ ℕ, function tu(t) – vk belongs to L0(I, Xα), so the set Ek,n is measurable for all n ∈ ℕ. Consequently as the intersection and union of countable measurable sets, {tI : u(t) ∈ F} is also measurable, which leads to the measurability of u as a Vβ-valued function. Since Vβ is separable, we have that uL0(I, Vβ), and the proof has been completed.□

#### Proposition 4.10

Suppose that the every φβ satisfies the Δ2-condition, and every Vβ is reflexive and separable. Then for all functions uL0(I, Xθ(⋅)) fulfilling Φφϑ(⋅)(u) < ∞, we have uL0(I, Vϑ(⋅)).

This proposition can be proved with the aid of the interim spaces L0 $\begin{array}{}\left(I,{V}_{{\vartheta }_{n}^{-}\left(\cdot \right)}\right)\end{array}$ (n = 1, 2, ⋯).

Proposition 4.10 shows that Φφϑ(⋅) is also a modular defined on the linear space L0(I, Xθ(⋅)), and the semimodular space derived by Φφϑ(⋅) is exactly Lφϑ(⋅)(I, Vϑ(⋅)), a separable Banach space. Moreover, suppose that {φβ} satisfies H(A) and H(B) with the supremum β+ of 𝓑 instead of α+, then Lφϑ(⋅)(I, Vϑ(⋅)) is uniformly convex with the dual

$Lφϑ(⋅)(I,Vϑ(⋅))∗≅Lφϑ(⋅)∗(I,Vϑ(⋅)∗).$

Moreover, similar to Remark 4.3, the restriction of Φφϑ(⋅) on Lφϑ(⋅)(I, Vϑ(⋅)), denoted by Φ͠φϑ(⋅), is convex and locally Lipschitz everywhere. Its subdifferential operator has the form

$∂Φ~φϑ(⋅)(u)={ξ∈Lφϑ(⋅)∗(I,Vϑ(⋅)∗):ξ(t)∈∂φϑ(t)(u(t))a.e. onI}$

for all uLφϑ(⋅)(I, Vϑ(⋅)).

#### Example 4.11

Let us pay attention to the Sobolev space of variable exponent type

$W1,q(x)(Ω)={u∈W1,1(Ω):u,Diu∈Lq(x)(Ω),i=1,2,⋯,N},$

where q ∈ 𝓟b(Ω) with the notation 𝓟b(Ω) introduced in Example 3.22, and Diu = ∂u/∂xi denotes the i-th weak derivative of u. Recall that endowed with the norm

$∥u∥W1,q(x)(Ω)=inf{λ>0:ϱq(uλ)+∑i=1Nϱq(Diuλ)},$

or equivalently

$∥u∥Lq(x)(Ω)+∥∇u∥Lq(x)(Ω),$

W1,q(x)(Ω) turns to be a separable Banach space. It is uniformly convex, and of course, reflexive provided q > 1.

Assume that ∂ΩC1 and qC(Ω) is a log-Hölder continuous exponent, or q$\begin{array}{}{\mathcal{P}}_{log}^{\omega }\end{array}$(Ω) in symbol, which means that

$|q(x)−q(y)|≤ω(|x−y|),for|x−y|<1,$(19)

where ω : [0, ∞) → [0, ∞) is a nondecreasing function fulfilling ω(0) = 0 and

$Cω=supr>0ω(r)log⁡1r<∞.$(20)

Under this situation, C(Ω) is dense in W1,q(⋅)(Ω), and $\begin{array}{}{W}_{0}^{1,q\left(x\right)}\end{array}$(Ω) can be defined as the complement of $\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$(Ω) in W1,q(x)(Ω). By this definition, $\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\left(\mathit{\Omega }\right)={W}_{0}^{1,1}\left(\mathit{\Omega }\right)\cap {W}^{1,q\left(x\right)}\left(\mathit{\Omega }\right),\end{array}$ and Poincaré’s inequality

$∥u∥Lq(x)(Ω)≤C∥∇u∥Lq(x)(Ω),∀u∈W01,q(⋅)(Ω)$

remains true. Therefore $\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) is topologically equivalent to the homogeneous Sobolev space $\begin{array}{}{D}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) (see [2, Ch. 8, 9] for relative discussions), and it can be regarded as a semimodular space derived from X = L0(Ω) by the modular

$φq(u)=∑i=1Nϱq(Diu),if u∈W01,1(Ω),∞,if u∈L0(Ω)∖W01,1(Ω).$

Take 𝓐b as in Example 3.22 and fix ω fulfilling (20). Then as an ordered topological subspace of 𝓐b, the intersection 𝓑ω = 𝓐b$\begin{array}{}{\mathcal{P}}_{log}^{\omega }\end{array}$(Ω) is also a 𝓑𝓣𝓛, on which {φq : q ∈ 𝓑ω} is a 𝓒𝓜𝓝, and { $\begin{array}{}{W}_{0}^{1,q\left(x\right)}\end{array}$(Ω) : q ∈ 𝓑ω} is the corresponding regular 𝓑𝓢𝓝.

For any q$\begin{array}{}{\mathcal{P}}_{log}^{\omega }\end{array}$(Ω) with q > 1, if we take $\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) and $\begin{array}{}{D}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) as the same space, then every member of the dual space $\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω)* =: W–1,q′(⋅)(Ω) has a representation in Lq′(⋅)(Ω)N, i.e. for each ξW–1,q′(⋅)(Ω), there is an F = (f1, ⋯, fN) ∈ Lq′(⋅)(Ω)N such that ξ = $\begin{array}{}-\sum _{i=1}^{N}{D}_{i}{f}_{i}\end{array}$ in 𝓓′(Ω), or equivalently,

$〈ξ,u〉=∑i=1N∫Ωfi(x)Diu(x)dx$

for all u$\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) (see [2], §12.3). Thus for the 𝓑𝓣𝓛 𝓑ω with q > 1, we can fix a basis {εk} of W–1,q(Ω), whose representations {Gk} are also fixed in Lq(Ω)N. Based on this selection, for any q ∈ 𝓐0, due to the density of W–1,q(Ω) in W–1,q′(⋅)(Ω), every element ξW–1,q′(⋅)(Ω) has a unique representation in Lq′(⋅)(Ω)N. Conversely, every vector function of Lq′(⋅)(Ω)N represents a unique member of W–1,q′(⋅)(Ω). Naturally, W–1,q′(⋅)(Ω) can be viewed as a semimodular space deduced by the modular

$ψq′(ξ)=∑i=1Nϱq′(fi),$

where F = (f1, ⋯, fN) ∈ Lq′(⋅)(Ω)N is the representation of ξ determined by {Gk}. Notice that, for all u$\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) and ξ = $\begin{array}{}-\sum _{i=1}^{N}{D}_{i}{f}_{i}\end{array}$W–1,q′(⋅)(Ω), we have

$〈ξ,u〉=∑i=1N∫Ωfi(x)Diu(x)dx≤φq(u)+ψq′(ξ),$

and equality holds if and only if fi∂ϱq(Dui) or equivalently Dui∂ϱq(fi) for all i ∈ {1, 2, ⋯, N}. In this sense, ψq can be viewed as the dual modular of φq, and {ψq : q ∈ 𝓑ω} can be regarded as the 𝓓𝓜𝓝 of {φq : q ∈ 𝓑ω}. By the continuity of {ϱq}, one can easily check that {ψq} is also a 𝓒𝓜𝓝, and correspondingly { $\begin{array}{}{V}_{q}^{\ast }\end{array}$ = W–1,q′(⋅)(Ω) : q ∈ 𝓑ω} is a regular 𝓑𝓢𝓝, which is the 𝓓𝓢𝓝 of {Vq = $\begin{array}{}{W}_{0}^{1,q\left(\cdot \right)}\end{array}$(Ω) : q ∈ 𝓑ω}.

If we set ϑ(t) = q(t, ⋅), we then obtain another order-continuous exponent ϑ : I → 𝓑ω. The authors in [11] revealed that for a continuous exponent q(t, x) satisfying (19) uniformly for all tI with a fixed ω verifying (20) and 1 < q < ∞,

$L0(I;W01,q(x,⋅)(Ω))=L+0(I;W01,q(x,⋅)(Ω))$

and

$L0(I;W−1,q′(x,⋅)(Ω))=L+0(I;W−1,q′(x,⋅)(Ω)).$

Thus by invoking Corollary 3.18 and Theorem 3.19 again, we get the reflexivity of Lφq(⋅)(I; $\begin{array}{}{W}_{0}^{1,q\left(x\right)}\end{array}$(Ω)), and the representation

$Lφq(⋅)(I;W01,q(x,⋅)(Ω))∗≅Lψq′(⋅)(I;W−1,q′(x,⋅)(Ω)).$

#### Remark 4.12

Given an exponent p ∈ 𝓟(Ω) with 1 < p < p+ < ∞, it is easy to check that

$ϱp(ut1/p−)≤ϱp(u)t.$

Hence, ϱp satisfies the strongly coercive property (14) with γ(t) = tp verifying (15). Furthermore, under the assumptions 1 < p < < ∞ and 1 < q < < N together with

$p(t,x)≤Nq(t,x)N−q(t,x)=:q∗(t,x),$

{φq : q ∈ 𝓑ω} is stronger than {ϱp : p ∈ 𝓐b}, and all the modular nets {ϱp}, {ϱp′}, {φq} and {ψq′} satisfy the strong coercivity with γ(t) = tp, tp̄′, tq and tq̄′ respectively. Thus taking the strict convexity of Lp′(x)(Ω) and W–1,q′(x)(Ω) into account, we can assert that hypotheses H(A) and H(B) are fulfilled by both {ϱp : p ∈ 𝓐b} and {φq : q ∈ 𝓑ω}. Therefore in our setting,

$Zθ(⋅)(u)(t)=|u|p(t,x)−2u=∂ϱp(t)(u)$

defines a bounded, coercive and demicontinuous operator

$Zθ(⋅):Lϱp(⋅)(I;Lq(⋅,x)(Ω))→Lϱp′(⋅)(I;Lp′(⋅,x)(Ω))$

as in Theorem 4.4, and

$∂Φ~φϑ(⋅)(u)(t)=∂φϑ(t)(u(t))=−div(|∇u|q(t,x)−2∇u)$

defines the subdifferential of Φ͠φϑ(⋅) at uLψq(⋅)(I; $\begin{array}{}{W}_{0}^{1,q\left(\cdot ,x\right)}\end{array}$(Ω)).

#### Remark 4.13

Replace q by a vector exponent q = (q1, q2, ⋯, qN) in Example 4.11, and define the anisotropic space

$W01,q(x)(Ω)={u∈W01,1(Ω):Diu∈Lqi(x)(Ω)}.$

Here qi$\begin{array}{}{\mathcal{P}}_{log}^{\omega }\end{array}$ with a common ω, and qi(x) ∈ [q, ] ⊆ (1, ∞) for all xΩ, i = 1, 2, ⋯, N. Similar to the isotropic ones, $\begin{array}{}{W}_{0}^{1,q\left(x\right)}\end{array}$(Ω) can be viewed as a semimodular space derived by

$φq(u)=∑i=1Nϱqi(Diu),if u∈W01,1(Ω),∞,if u∈L0(Ω)∖W01,1(Ω),$

and its dual space, denoted by W–1,q′(x)(Ω), can also be represented by the product space $\begin{array}{}\prod _{i=1}^{N}{L}^{{q}_{i}^{\prime }\left(x\right)}\end{array}$(Ω). Therefore following the same discussions as in Example 4.11, we can conclude that {φq : q$\begin{array}{}{\mathcal{B}}_{\omega }^{N}\end{array}$}, where $\begin{array}{}{\mathcal{B}}_{\omega }^{N}\end{array}$ is equipped with the product topology and the order : qr if and only if qiri for all i = 1, 2, ⋯, N, is a 𝓒𝓜𝓝 fulfilling hypotheses H(A) and H(B), and stronger than {ϱp : p ∈ 𝓐b}. { $\begin{array}{}{W}_{0}^{1,q\left(x\right)}\end{array}$(Ω) : q$\begin{array}{}{\mathcal{B}}_{\omega }^{N}\end{array}$} is a 𝓑𝓢𝓝 derived by {φq}, its 𝓓𝓢𝓝 {W–1,q′(x)(Ω)} is generated by {ϱq} with a basis fixed in W–1,q′(x)(Ω) and its representation fixed in $\begin{array}{}\prod _{i=1}^{N}{L}^{{\underset{_}{{q}_{i}}}^{\prime }}\end{array}$(Ω), where q = (q1, q2, ⋯, qN), and

$ϱq′(F)=∑i=1Nϱqi′(fi),F=(f1,f2,⋯,fN)∈L0(Ω)N.$

Therefore, for a continuous vector exponent q : I × Ω → [q, ]N with every component qi satisfying (19) uniformly for tI, the modular-modular space Lφq(⋅)(I, $\begin{array}{}{W}_{0}^{1,\mathbf{q}\left(\cdot ,x\right)}\end{array}$(Ω)) deduced from L0(I, Lp(⋅,x)(Ω)) by the modular

$Φφq(⋅)(u)=∫Iφq(t)(u(t))dt$

is a separable and uniformly convex space with the dual

$Lφq(⋅)(I,W01,q(⋅,x)(Ω))∗≅Lφq′(⋅)(I,W−1,q′(⋅,x)(Ω)).$

Moreover, the restriction functional Φ͠φq(⋅) is convex and locally Lipschitz everywhere on Lφq(⋅)(I, $\begin{array}{}{W}_{0}^{1,\mathbf{q}\left(\cdot ,x\right)}\end{array}$(Ω)), its subdifferential ∂Φ͠φq(⋅) at uLφq(⋅)(I, $\begin{array}{}{W}_{0}^{1,\mathbf{q}\left(\cdot ,x\right)}\end{array}$(Ω)) has the expression

$∂Φ~φq(⋅)(u)=∂φq(t)(u(t))=−∑i=1NDi(|Diu|qi(t,x)−2Diu),$

or equivalently,

$⟨⟨∂Φ~φq(⋅)(u),v⟩⟩ϑ(⋅)=∑i=1N∫I∫Ω|Diu(t,x)|qi(t,x)−2Diu(t,x)Div(t,x)dxdt$

for all vLφq(⋅)(I, $\begin{array}{}{W}_{0}^{1,\mathbf{q}\left(\cdot ,x\right)}\end{array}$(Ω)).

## 5 Bochner-Sobolev spaces of modular-modular type and applications in doubly nonlinear differential equations

We begin with the Bochner-Sobolev space of range-fixed type, that is

$W1,ϱα(I,Xα)={u∈Lϱα(I,Xα):u′∈Lϱα(I,Xα)},$

where u′ denotes the derivative of u in the sense of distribution, i.e. for all ξ$\begin{array}{}{X}_{\alpha }^{\ast }\end{array}$ and all γ$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$(I), equality

$∫I⟨u′(t),γ(t)ξ⟩αdt=−∫I⟨u(t),γ′(t)ξ⟩αdt$

holds. It is easy to check that, endowed with the norm

$∥u∥W1,ϱα(I,Xα)=∥u∥Lϱα(I,Xα)+∥u′∥Lϱα(I,Xα),$

which is equivalent to

$inf{λ>0:Φϱα(u/λ)+Φϱα(u′/λ)≤1},$

W1,ϱα(I, Xα) turns to be a Banach space.

#### Theorem 5.1

Function space W1,ϱα(I, Xα) can be embedded into the space of continuous functions C(I, Xα). If in addition Vβ is embedded into Xα compactly, then W1,ϱα(I, Xα) ∩ Lφβ(I, Vβ) is embedded compactly into Lϱα(I, Xα).

#### Proof

Firstly, from the inequality

$∥u∥Xα≤ϱα(u)+1,∀u∈Xα,$

we can deduce that Lϱα(I, Xα) ↪ L1(I, Xα). Similarly, we have W1,ϱα(I, Xα) ↪ W1,1(I, Xα) and Lφβ(I, Vβ) ↪ L1(I, Vβ). The first conclusion comes since the embedding W1,1(I, Xα)) ↪ C(I, Xα) holds.

Given a bounded subset F of W1,ϱα(I, Xα) ∩ Lφβ(I, Vβ), it is also bounded in W1,1(I, Xα)) ∩ L1(I, Vβ)). Assume that

$Φϱα(u′)+Φϱα(u)+Φφβ(u)≤C$

for some C > 0 independent of uF. Then for any uF and 0 < h < min{1, T/2} and t, t + hI, we have

$ϱα(u(t+h)−u(t))≤∫tt+hϱα(u′(τ))dτ,$

consequently

$∫0T−hϱα(u(t+h)−u(t))dt≤∫0T−h∫tt+hϱα(u′(τ))dτdt=(∫0h∫0τ+∫hT−h∫τ−hτ+∫T−hT∫τ−hT−h)ϱα(u′(τ))dtdτ=∫0hτϱα(u′(τ))dτ+∫hT−hhϱα(u′(τ))dτ+∫T−hT(T−τ)ϱα(u′(τ))dτ≤h∫Iϱα(u′(τ))dτ≤Ch.$(21)

Taking any r ∈ (0, T), consider the average operator Mr on Lϱα(I, Xα) defined by

$Mru(t)=1r∫tt+ru(τ)dτ,t∈[0,T−r].$

Obviously, for all uLφβ(I, Vβ), MruC([0, Tr], Vβ) with the estimate

$φβ(Mru(t))≤1r∫tt+rφβ(u(τ))dτ,t∈[0,T−r].$

Moreover, due to the boundedness of F in W1,ϱα(I, Xα) and the estimate (21), precompactness of the set Fr = {Mru : uF} in C([0, Tr], Xα) can be reached (refer to [16, 17]).

In addition, from (21), one can deduce that

$∫0T−hϱα(Mru(t)−u(t))dt=∫0T−hϱα(1r∫0r(u(t+τ)−u(t))dτ)dt ≤1r∫0T−h∫0rϱα(u(t+τ)−u(t))dτdt ≤1r∫0r∫0T−τϱα(u(t+τ)−u(t))dtdτ ≤Ch$

provided 0 < rh, which means that Mruu in Lϱα([0, Th], Xα) as r → 0 uniformly for uF. This fact, combined with the precompactness of Fr in C([0, Tr], Xα) for every fixed r ∈ (0, h], leads to the precompactness of F in Lϱα1([0, Th], Xα). The final conclusion comes if we make the same discussions on the set = {(t) = u(Tt) : uF} (see [16]).□

Using the facts Lϱα(I, Xα) ↪ L1(I, Xα) and W1,ϱα(I, Xα) ↪ C(I, Xα), we can also deduce that

#### Corollary 5.2

Under hypotheses of the above theorem, W1,ϱα(I, Xα) ∩ Lφβ(I, Vβ) can be embedded compactly into Lp(I, Xα) for any 1 ≤ p < ∞, hence Lp(⋅)(I, Xα) for all p ∈ 𝓟b(I, Xα).

Given two 𝓒𝓜𝓝s {ϱα :∈ 𝓐} and {φβ : β ∈ 𝓑} satisfying H(A) + H(B) and H(A) + H(B)′ respectively, and the latter stronger than the former, introduce the Bochner-Sobolev space of range-varying type

$W1,ϱθ(⋅)(I,Xθ(⋅))={u∈W1,ϱα−(I,Xα−):u,u′∈Lϱθ(⋅)(I,Xθ(⋅))}.$

Similarly, equipped with the norm

$∥u∥W1,ϱθ(⋅)(I,Xθ(⋅))=∥u∥Lϱθ(⋅)(I,Xθ(⋅))+∥u′∥Lϱθ(⋅)(I,Xθ(⋅)),$

which is equivalent to

$inf{λ>0:Φϱθ(⋅)(u/λ)+Φϱθ(⋅)(u′/λ)≤1},$

W1,ϱθ(⋅)(I, Xθ(⋅)) becomes a Banach space.

#### Theorem 5.3

Besides the assumptions upon {ϱα} and {φβ} as above, assume that there are scalar functions i, i, δC(I) (i = 1, 2) such that 1 ≤ 1(t) ≤ 2(t) < ∞, 1 ≤ 1(t) ≤ 2(t) < ∞, δ(t) ∈ (0, 1), and 2(t)δ(t) < 2(t) for all tI. Suppose also

• Vϑ ↪↪ Xα+ and there is a constant C > 0 such that

• (Xα, Vϑ(t))δ(t)Xθ(t) uniformly for tI, in other words,

$∥u∥θ(t)≤C∥u∥α−1−δ(t)∥u∥ϑ(t)δ(t)for allu∈Vϑ(t),$(22)

where notation (Xα, Vϑ(t))δ(t) represents the real or complex interpolation space between Xα and Vϑ(t) with the index δ(t);

• for all tI,

$ϱθ(t)(u)≤Cmax{∥u∥θ(t)p~1(t),∥u∥θ(t)p~2(t)}u∈Xθ(t),$(23)

and

$min{∥u∥ϑ(t)q~1(t),∥u∥ϑ(t)q~2(t)}≤Cφϑ(t)(u),u∈Vϑ(t).$

Then space W1,ϱθ(⋅)(I, Xθ(⋅)) ∩ Lφϑ(⋅)(I, Vϑ(⋅)) is embedded into Lϱθ(⋅)(I, Xθ(⋅)) compactly.

#### Proof

Firstly, by (23) and Remark 4.8 in [11], we have that

$Lp~2(⋅)(I,Xθ(⋅))↪Lϱθ(⋅)(I,Xθ(⋅)).$

Thus from Theorem 5.1 and its corollary, it suffices to show that a sequence {uk} bounded in Lφϑ(⋅)(I, Vϑ(⋅)) and convergent in Lp(⋅)(I, Xα) for all p ∈ 𝓟b(I, Xα) is convergent in L2(⋅)(I, Xθ(⋅)) definitely. Without loss of generality, assume that the limit of {uk} is 0. Take K > 1 so close to 1 that 2(t)δ(t)K2(t) and 2(t)(1 – δ(t))K′ ≥ 1 (1/K + 1/K′ = 1) for all tI, then by (22), we have

$∫I∥uk(t)∥θ(t)p~2(t)dt≤C∫I∥uk(t)∥α−p~2(t)(1−δ(t))∥uk(t)∥ϑ(t)p~2(t)δ(t)dt≤C(∫I∥uk(t)∥α−p~2(t)(1−δ(t))K′dt)1/K′(∫I∥uk(t)∥ϑ(t)p~2(t)δ(t)Kdt)1/K≤C(∫I∥uk(t)∥α−p~2(t)(1−δ(t))K′dt)1/K′⋅[T1/K+(∫{∥uk(t)∥ϑ(t)>1}∥uk(t)∥ϑ(t)p~2(t)δ(t)Kdt)1/K]≤C(∫I∥uk(t)∥α−p~2(t)(1−δ(t))K′dt)1/K′[T1/K+(∫Iφϑ(t)(uk(t))dt)1/K],$(24)

which leads to the desired conclusion.□

#### Remark 5.4

Under all the hypotheses of Theorem 5.3 with the compact embedding of Vϑ into Xα+ replaced by the following condition

$∥u∥Xα≤C∥u∥Vβ$(25)

uniformly for α ∈ 𝓐 and β ∈ 𝓑, we have

$∥u′∥L1(I,Xα−)≤C∥u′∥Lϱθ(⋅)(I,Xθ(⋅)),∥u∥L1(I,Xα−)≤C∥u∥Lφϑ(⋅)(I,Vϑ(⋅))$

for all uW1,ϱθ(⋅)(I, Xθ(⋅)) ∩ Lφϑ(⋅)(I, Vϑ(⋅)), thus under the condition

$∥u′∥Lϱθ(⋅)(I,Xθ(⋅))+∥u∥Lφϑ(⋅)(I,Vϑ(⋅))≤1,$

estimates (24) turn to be

$∫I∥u(t)∥θ(t)p~2(t)dt≤CT1/K′max{(maxt∈I∥u(t)∥α−)p~2+(1−δ−),(maxt∈I∥u(t)∥α−)p~2−(1−δ+)}⋅(∫I∥u(t)∥ϑ(t)p~2(t)δ(t)Kdt)1/K≤CT1/K′max{∥u∥W1,1(I,Xα−)p~2+(1−δ−),∥u∥W1,1(I,Xα−)p~2−(1−δ+)}⋅[T1/K+(∫{∥uk(t)∥Vϑ(t)>1}∥uk(t)∥Vϑ(t)p~2(t)δ(t)Kdt)1/K]≤CT1/K′(∥u′∥Lϱθ(⋅)(I,Xθ(⋅))+∥u∥Lφϑ(⋅)(I,Vϑ(⋅)))p~2−(1−δ+)⋅[T1/K+(∫Iφϑ(t)(u(t))dt)1/K]≤C,$

which in turn producesuL2(t)(I, Xθ(t))C, and consequentlyuLϱθ(⋅)(I, Xθ(⋅))C since L2(⋅)(I, Xθ(⋅)) ↪ Lϱθ(⋅)(I, Xθ(⋅)). Then by means of scaling transformation, we get

$∥u∥W1,ϱθ(⋅)(I,Xθ(⋅))≤C(∥u′∥Lϱθ(⋅)(I,Xθ(⋅))+∥u∥Lφϑ(⋅)(I,Vϑ(⋅))).$(26)

This is an important inequality for later use.

In spite of the estimate (26), we do not expect the control of the modular Φϱθ(⋅)(u) by the sum Φϱθ(⋅)(u′) + Φ͠φϑ(⋅)(u). Conversely, we have

#### Proposition 5.5

For every r ≥ 0, there is an εr > 0 such that for all 0 ≤ εεr, the a priori estimate

$Φϱθ(⋅)(u′)+Φ~φϑ(⋅)(u)≤εrΦϱθ(⋅)(u)+r$

defines a bounded subset of W1,ϱθ(⋅)(I, Xθ(⋅)).

#### Proof

We proceed by contradiction. Assume that there is a sequence {uk} in W1,ϱθ(⋅)(I, Xθ(⋅)) verifying

$Φϱθ(⋅)(uk′)+Φ~φϑ(⋅)(uk)≤1kΦϱθ(⋅)(uk)+r,$

but Φϱθ(⋅)(uk) ≥ k. Let vk = uk/Φϱθ(⋅)(uk). Then we have

$Φϱθ(⋅)(vk′)+Φ~φϑ(⋅)(vk)≤Φϱθ(⋅)(uk′)+Φ~φϑ(⋅)(uk)Φϱθ(⋅)(uk)≤1+rk.$

Therefore Φϱθ(⋅)( $\begin{array}{}{v}_{k}^{\prime }\end{array}$) + Φ͠φϑ(⋅)(vk) → 0, and consequently vk → 0 in W1,ϱθ(⋅)(I, Xθ(⋅)) as k → ∞. Thus limk→∞ Φϱθ(⋅)(vk) = 0, which contradicts to fact that Φϱθ(⋅)(vk) ≥ 1/2 for all k ∈ ℕ.□

#### Remark 5.6

As a preparation for later arguments, following the same process as above, we can prove that for every C > 0, there exist two small numbers δ0 > 0 and μ0 > 0 such that all the functions satisfying the a priori estimate

$Φϱθ(⋅)(u′)+Φ~φϑ(⋅)(u)≤(δ+μσ(δ−1))Φϱθ(⋅)(u)+Cσ(δ−1)$

comprise a bounded subset of Lϱθ(⋅)(I, Xθ(⋅)) as long as 0 < δδ0 and 0 < μμ0.

Define two function spaces with periodic boundary condition, one is

$Wper1,ϱα(I,Xα)={u∈W1,ϱα(I,Xα):u(0)=u(T)},$

the other is

$Wper1,ϱθ(⋅)(I,Xθ(⋅))=W1,ϱθ(⋅)(I,Xθ(⋅))∩Wper1,ϱα−(I,Xα−)$

under the condition θ(0) = θ(t). Evidently, the two spaces are closed subspaces of W1,ϱα(I, Xα) and W1,ϱθ(⋅)(I, Xθ(⋅)) respectively.

Let us make some investigations on the operator $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ : $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)) → $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅))* defined through

$〈〈Dθ(⋅)2(u),v〉〉=∫I〈∂ϱθ(t)(u′(t)),v′(t)〉θ(t)dt,u,v∈W1,ϱα(I,Xα).$(27)

By the definition, it is easy to check that $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ is a single-valued, monotone and demicontinuous operator, hence it is maximal monotone. In addition, taking v(t) = γ(t)w in (27) with γ$\begin{array}{}{C}_{0}^{\mathrm{\infty }}\end{array}$(I) and wXα+, we have

$〈〈Dθ(⋅)2(u),v〉〉=∫I〈∂ϱθ(t)(u′(t)),γ′(t)w〉α+dt=−∫I〈ddt∂ϱθ(t)(u′(t)),γ(t)w〉α+dt=−∫I〈ddt∂ϱθ(t)(u′(t)),γ(t)w〉θ(t)dt,$

which means that $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ is a second order nonlinear differential operator, i.e.

$Dθ(⋅)2(u)=−ddt∂ϱθ(⋅)(u′(⋅))$

in the sense of distribution.

Let Ψ͠(u) = Φϱθ(⋅)(u′) at u$\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)), we obtain a continuous and convex functional. A direct calculation shows that Ψ͠ is Gâteaux differentiable everywhere with the Gâteaux differential Ψ͠′(u) = $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$(u). Thus the subdifferential operator ∂Ψ͠ is single-valued, and ∂Ψ͠(u) = $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$(u) for all u$\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)). Consider the extension of Ψ͠ onto Lϱθ(⋅)(I, Xθ(⋅))

$Ψ(u)=∞,if u∈Lϱθ(⋅)(I,Xθ(⋅))∖Wper1,ϱθ(⋅)(I,Xθ(⋅))Ψ~(u),if u∈Wper1,ϱθ(⋅)(I,Xθ(⋅)).$

It is also easy to verify that Ψ is a semicontinuous and convex proper functional, whose subdifferential operator has the domain

$D(∂Ψ)={u∈W1,ϱθ(⋅)(I,Xθ(⋅)):there is anξ∈Lϱθ(⋅)∗(I,Xθ(⋅)∗)such that ∫I〈ξ(t),v(t)〉θ(t)dt=∫I〈Zθ(⋅)(u′)(t),v′(t)〉θ(t)dt for allv∈Wper1,ϱθ(⋅)(I,Xθ(⋅))},$(28)

together with ∂Ψ(u) = $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$(u) = ξ in $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅))* for the function ξ$\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ satisfying (28) and all u ∈ 𝓓(∂Ψ). In this sense, we can say that ∂Ψ$\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$.

Taking the intersection

$W=Wper1,ϱθ(⋅)(I,Xθ(⋅))∩Lφϑ(⋅)(I,Vϑ(⋅))$

as the work space, where the norm ∥⋅∥𝓦 takes the value $\begin{array}{}\parallel u{\parallel }_{{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\left(I,{X}_{\theta \left(\cdot \right)}\right)}+\parallel u{\parallel }_{{L}^{{\phi }_{\vartheta \left(\cdot \right)}}\left(I,{V}_{\vartheta \left(\cdot \right)}\right)},\end{array}$ and consider the sum

$Φ~(u)=Ψ~(u′)+Φ~φϑ(⋅)(u)$

as the potential functional. Evidently, Φ͠(u) is a continuous modular on 𝓦 with the effective domain 𝓓(Φ͠) = 𝓦, its subdifferential operator

$∂Φ~=Dθ(⋅)2+∂Φ~φϑ(⋅)$

is a maximal monotone subset of $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅)) × $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{\theta \left(\cdot \right)}}\end{array}$(I, Xθ(⋅))*. Moreover, by (26) and the coercivity of Φϱθ(⋅), Φφϑ(⋅), we have

$〈〈∂Φ~(u),u〉〉∥u∥W≥CΨ~(u′)+Φ~φϑ(⋅)(u)∥u′∥Lϱθ(⋅)(I,Xθ(⋅))+∥u∥Lφϑ(⋅)(I,Vϑ(⋅))→∞as∥u∥W→∞,$

which yields the coercivity, hence surjectivity of ∂Φ͠. Furthermore, suppose that for every β ∈ 𝓑, φβ is strictly convex, then Φ͠φϑ(⋅) hence Φ͠ is also strictly convex, which in turn leads to the injectivity of ∂Φ͠. Summing up, we conclude that

#### Theorem 5.7

Suppose all the hypotheses mentioned above are satisfied, then for every $\begin{array}{}f\in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ ⊆ 𝓦*, there is a unique u ∈ 𝓦 and a corresponding selection ξ∂Φ͠φϑ(⋅)(u) such that

$Dθ(⋅)2(u)+ξ=finW∗.$(29)

In other words, for every $\begin{array}{}f\in {L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$, second order differential inclusion

$−ddt∂ϱθ(t)(u′(t))+∂φϑ(t)(u(t))∋f(t)a.e. onI$(30)

with periodic boundary values admits a unique weak solution u in the sense

$∫I〈∂ϱθ(⋅)(u′(t)),v′(t)〉θ(t)dt+∫I〈ξ(t),v(t)〉ϑ(t)dt=∫I〈f(t),v(t)〉θ(t)dt$

for all v ∈ 𝓦.

The above theorem shows that the sum operator $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ + ∂Φφϑ(⋅) is both injective and surjective, its inverse ( $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ + ∂Φφϑ(⋅))–1 is existing and single-valued. Moreover, in light of Proposition 5.5, together with the compact imbedding 𝓦 ↪↪ Lϱθ(⋅)(I, Xθ(⋅)), we can prove that

#### Theorem 5.8

If all the hypotheses of Theorem 5.3 together with (25) are satisfied, then the inverse operator

$(Dθ(⋅)2+∂Φ~φϑ(⋅))−1:Lϱθ(⋅)∗(I,Xθ(⋅)∗)→Lϱθ(⋅)(I,Xθ(⋅))$

is both bounded and strongly continuous in the sense that for any sequence {fk} convergent to f in $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ weakly, the corresponding sequence {( $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ + ∂Φ͠φϑ(⋅))–1 fk} converges to ( $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ + ∂Φ͠φϑ(⋅))–1 f in Lϱθ(⋅)(I, Xθ(⋅)) strongly.

Let us consider the operator f : I × XX. Assume that

• for any JΠ(I) and u$\begin{array}{}{X}_{{\theta }_{J}^{+}}\end{array}$, tf(t, u) lies in L0(I, $\begin{array}{}{X}_{{\theta }_{J}^{+}}^{\ast }\end{array}$),

• for every tI, f(t, ⋅) : Xθ(t)$\begin{array}{}{X}_{\theta \left(t\right)}^{\ast }\end{array}$ is demicontinuous, and

• there is an μ > 0 and a nonnegative integrable function h for which inequality

$ϱθ(t)∗(f(t,u))≤μϱθ(t)(u)+h(t)$(31)

holds for all uXθ(t).

Taken any uL0(I, Xθ(⋅)), denote by F(u) = f(t, u). It is easy to see that under the first assumption upon f, if u ∈ 𝓢(I, Xα+), then F(u) ∈ $\begin{array}{}{L}_{-}^{0}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right),\end{array}$ and by (31), it comes F(u) ∈ $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$. Consequently, from the density of 𝓢(I, Xα+) in Lϱθ(⋅)(I, Xθ(⋅)) and the demicontinuity assumption upon f, we can derive that F(u) ∈ $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ provided uLϱθ(⋅)(I, Xθ(⋅)). Moreover, we have

#### Proposition 5.9

F is a bounded and demicontinuous operator from Lϱθ(⋅)(I, Xθ(⋅)) to $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$.

#### Proof

Boundedness of F comes from (31) immediately. For the weak continuity, suppose that {un} is a sequence of Lϱθ(⋅)(I, Xθ(⋅)) converging to u strongly, that is

$∫Iϱθ(t)(un(t)−u(t))dt→0asn→∞.$

Thus there is a subsequence, say {un} itself satisfying

$limn→∞ϱθ(t)(un(t)−u(t))=0a.e. onI.$

From the boundedness of {un} in Lϱθ(⋅)(I, Xθ(⋅)) and (31), we get the boundedness of {F(un)} in $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$. Consequently, there is a subsequence, without loss of generality, assuming also {un} itself, and a function ξ$\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$, such that F(un) → ξ weakly, or equivalently

$limn→∞∫I〈f(t,un(t))−ξ(t),v(t)〉θ(t)dt=0$

for all vLϱθ(⋅)(I, Xθ(⋅)), which in turn yields

$limn→∞∫E〈f(t,un(t))−ξ(t),v〉θ(t)dt=0$(32)

for all measurable subsets E of I and all elements v of Xα+ since 𝓢(I, Xα+) is dense in Lϱθ(⋅)(I, Xθ(⋅)) by H(A).

On the other hand, since

$limn→∞〈f(t,un(t)),v〉θ(t)=〈f(t,u(t)),v〉θ(t)$

a.e. on I, for each ε > 0, by Egorov’s theorem, there is a measurable set EεI with |IEε| < ε verifying

$〈f(t,un(t)),v〉θ(t)→〈f(t,u(t)),v〉θ(t)uniformly onEε$

as n → ∞, consequently for any subset E of Eε, we have

$limn→∞∫E〈f(t,un(t))−f(t,u(t)),v〉θ(t)dt=0.$(33)

Putting (32) and (33) together with the same subset E, we obtain

$∫E〈f(t,u(t))−ξ(t),v〉θ(t)dt=0,$

which implies

$〈f(t,u(t))−ξ(t),v〉θ(t)=0$(34)

a.e. on Eε and eventually a.e. on I by the arbitrariness of E and ε respectively.

Suppose that {vk} is a dense and countable subset of Xα+, then there is a subset E0 of I with zero complement on which (34) holds with v replaced by vk for all k ∈ ℕ. Finally, using the density of Xα+ in Xθ(t), we deduce that f(t, u(t)) = ξ(t) in $\begin{array}{}{X}_{\theta \left(t\right)}^{\ast }\end{array}$ on E0. Therefore F(u) = ξ and F(un) ⇀ F(u) in $\begin{array}{}{L}^{{\varrho }_{\theta \left(\cdot \right)}^{\ast }}\left(I,{X}_{\theta \left(\cdot \right)}^{\ast }\right)\end{array}$ as n → ∞. Thus the proof has been completed.□

Putting Theorem 5.8 and Proposition 5.9 together, we can easily see that the composite operator

$F:=(Dθ(⋅)2+∂Φ~φϑ(⋅))−1∘F:Lϱθ(⋅)(I,Xθ(⋅))→Lϱθ(⋅)(I,Xθ(⋅))$

is both continuous and compact. Moreover, we have

#### Theorem 5.10

Under all the assumptions mentioned above, there is an μ0 > 0 such that as long as 0 ≤ μμ0, 𝓕 has a fixed point in Lϱθ(⋅)(I, Xθ(⋅)), or in other words, second order differential inclusion

$−ddt∂ϱθ(t)(u′(t))+∂φϑ(t)(u(t))∋f(t,u(t))a.e. onI$(35)

has a weak solution in 𝓦.

#### Proof

Consider the set

$S={u∈W:u=λF(u)for some0<λ<1}.$

Evidently, every member of S is a weak solution of the inclusion

$−ddt∂ϱθ(t)(λ−1u′(t))+∂φϑ(t)(λ−1u(t))∋f(t,u(t))a.e. onI.$

By multiplying both sides of the above inclusion by λ–1 u′(t), and integrate on I, then using the assumption (31), we can deduce that

$Φϱθ(⋅)(λ−1u′)+Φ~φϑ(⋅)(λ−1u)≤〈〈F(u),λ−1u〉〉θ(⋅) ≤Φϱθ(⋅)∗(δ−1F(u))+Φϱθ(⋅)(δλ−1u) ≤σ(δ−1)[μΦϱθ(⋅)(λ−1u)+∥h∥L1(I)]+δΦϱθ(⋅)(λ−1u) =(δ+μσ(δ−1))Φϱθ(⋅)(λ−1u)+∥h∥L1(I).$

By taking δ0 > 0 and μ0 > 0 as in Remark 5.6 and letting 0 < δδ0, 0 ≤ μμ0, we can claim that {λ–1u : uS} is bounded in Lϱθ(⋅)(I, Xθ(⋅)). Therefore, there is a constant C > 0 independent of λ such that ∥uLϱθ(⋅)(I, Xθ(⋅))C for all uS. Finally by invoking Leray-Schauder’s alternative theorem for the compact and strongly continuous operators (refer to [18, Ch. 13] or [19]), we can assert the existence of the fixed point of 𝓕.□

#### Remark 5.11

From the demicontinuity of F and the compactness of the inverse ( $\begin{array}{}{D}_{\theta \left(\cdot \right)}^{2}\end{array}$ + ∂Φφϑ(⋅))–1, we can also check that solution set of (35) is a nonempty and compact subset of 𝓦.

#### Remark 5.12

In our setting, periodic boundary condition can be replaced by the following one:

$u(T)=Ku(0),$

where K : Xθ(0)Xθ(t) is a bounded linear operator with other conditions unchanged.

At the end of this paper, let us choose an anisotropic elliptic partial differential equation of second order to illustrate our results,

$−Dt(|Dtu|p(x,t)−2Dtu)−∑i=1NDi(|Diu|qi(x,t)−2Diu)=μg(t,x,u),(t,x)∈I×Ω,u(0,x)=u(T,x),x∈Ω,u(t,x)=0,(t,x)∈I×∂Ω.$(36)

Here Dt denotes the partial differential derivative with respect to t, p and qi are doubly variable exponents introduced in Example 3.22 and Remark 4.13 respectively.

Suppose that g : I × Ω × ℝ → ℝ is a Caratheodory function with a nonstandard growth, i.e. for a.e. (t, x) ∈ I × Ω, ug(t, x, u) is continuous, and for all u ∈ ℝ, (t, x) ↦ g(t, x, u) is measurable, together with

$|g(t,x,u)|≤C(1+|u|p(t,x)−1)$(37)

holding for a.e. (t, x) ∈ I × Ω and all u ∈ ℝ.

Let f(t, u)(x) = g(t, x, u(t, x)). Since L0(I, L0(Ω)) is equivalent to L0(Q), one can easily check that f(t, u) lies in L0(I, L0(Ω)) provided u does. Moreover, combining (37) with the continuity of g(t, x, u) with respect to u, we can find that f(t, u) ∈ L0(I, L*(Ω)) for all uL0(I, L(Ω)), and f(t, u) is weakly continuous from Lp(t, x)(Ω) to Lp′(t, x)(Ω) for a.e. tI, together with (31) verified.

Suppose that p(t, x) and qi(t, x) (i = 1, 2, ⋯, N) are variable exponents as in Example 3.22 and Remark 4.13 fulfilling

$infx∈Ω(qi∗(t,x)−p(t,x))>0,i=1,2,⋯,N.$(38)

Using the notations and definitions in Examples 3.22, 4.11 and Remark 4.13, from equation (36), we then derive an abstract evolution equation (35) on the space

$W=Wper1,ϱp(⋅)(I,Lp(⋅,x)(Ω))∩Lϱq(⋅)(I,W01,q(⋅,x)(Ω)).$

From all the assumptions of Example 3.22 and Remark 4.13 together with (38), one can easily verify all the conditions listed in Theorem 5.3 as well as (25). Thus in terms of Theorem 5.10, equation (35) has a solution u in 𝓦, whose scalar version, still denoted by u, solves (36) in the sense of distribution, i.e.

$∫Q(|Dtu|p(x,t)−2Dtu)Dtvdxdt+∑i=1N∫Q(|Diu|qi(x,t)−2Diu)Divdxdt=∫Qf(t,u)v(t,x)dxdt$

for all v$\begin{array}{}{C}_{\text{per}}^{1}\end{array}$(I, C(Ω)). In conclusion, under all the conditions upon h, p and qi (i = 1, 2, ⋯, N) including (38), there is a μ0 > 0 such that for all 0 ≤ μμ0, Equation (36) has a weak solution.

#### Remark 5.13

Unlike the traditional one, here we do not need u(t, x) = 0 on the whole boundary ∂Q and do not require the whole log-Hölder continuity of p and qi (i = 1, 2, ⋯, N). Hence Poincaré′s inequality

$∥u∥Lp(t,x)(Q)≤C∥∇u∥Lq(t,x)(Q)$

could not be applied even if p(t, x) ≤ qi(t, x) for all i ∈ {1, 2, ⋯, N}. By these reasons, here we give up $\begin{array}{}{W}_{0}^{1,p\left(t,x\right)}\end{array}$(Q) as the work space, instead, an anisotropic space $\begin{array}{}{W}_{\text{per}}^{1,{\varrho }_{p\left(\cdot \right)}}\end{array}$(I, Lp(⋅, x)(Ω)) ∩ Lϱq(⋅)(I, $\begin{array}{}{W}_{0}^{1,\mathbf{q}\left(\cdot ,x\right)}\end{array}$(Ω)) is taken into account, while the elliptic equation (36) turns to be an abstract second order evolution equation. To our best knowledge, this way to deal with the anisotropic elliptic equations with nonstandard growth is new, and different to that applied in available literature.

## Acknowledgement

This work is supported by Preresearch Project of Nantong University (17ZY01).

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Accepted: 2018-05-08

Published Online: 2018-08-20

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 924–954, ISSN (Online) 2391-5455,

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