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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

Issues

Volume 13 (2015)

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 L+ϱθ()(I, Xθ(⋅)), and discuss their geometrical properties as well as the representation of L+ϱθ()(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 Wper1,ϱθ()(I, Xθ(⋅)), and the intersection space Wper1,ϱθ()(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.

Keywords: Continuous modular net; Xθ(⋅)–valued function space; Geometrical property; Doubly nonlinear differential inclusion

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 L+ϱθ()(I, Xθ(⋅)), both of them are complete according to the same norm. We show that, under some suitable situations, L+ϱθ()(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 {ϱα}, 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 Wper1,ϱθ()(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 uW1,ϱθ()(I,Xθ())C(uLϱθ()(I,Xθ())+uLφϑ()(I,Vϑ())),

in Section 5, we study a type of second order nonlinear differential inclusion ddtϱθ(t)(u(t))+φϑ(t)(u(t))f(t,u(t))for a.e.tI,(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),uXθ(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 Dθ()2 defined by Dθ()2u,vθ()=Iϱθ()(u(t)),v(t)θ(t)dt,u,vWper1,ϱθ()(I,Xθ()),

we obtain a continuous and compact operator (Dθ()2+Φφϑ())1F: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 R+N+1 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={uX:ϱ(λu)<for someλ>0}

endowed with the norm uϱ=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,fL0(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{fp()p,fp()p+}ϱp()(f)max{fp()p,fp()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)={fL0(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

uϱϱ(u)+1,uXϱ,

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 alluX.

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={tI:ϱ(skχE0(t)f(t))ϱ(f(t))},

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

{tI:ϱ(φk(t)f(t))0}=ε>0K1kK{tI:ϱ(φk(t)f(t))ε}ε>0K1kK{tI:ϱ(skχE0(t)f(t))min{ε,ϱ(f(t))}}={tI: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 = n=1 In, where In = I ∩ [0, n], n = 1, 2, ⋯, we can also prove that 𝓢c(I, Xϱ) and Lc (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 Xϱ , 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 ξXϱ , Young’s inequality

ξ,uϱ(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 Xϱ . 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 × XϱandXϱ×Xϱ respectively. As for the multivalued inverse map (∂ϱ)−1, we know that (∂ϱ)−1Xϱ × Xϱ has the same properties as ∂ϱ has, together with the inclusion (∂ϱ)−1∂ϱ* holding. Furthermore, if in addition 𝓡(∂ϱ) = Xϱ , then for all ξXϱ , the image (∂ϱ)−1(ξ) is a nonempty convex and closed subset of Xϱ. Therefore for all ξL0(I, Xϱ ), 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) = Xϱ . Suppose also ϱ* is a modular, and Xϱ 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, Xϱ ).

Proof

For each ξLϱ*(I, Xϱ ), define a functional Λξ through

Λξ,f=Iξ(t),f(t)dt,fLϱ(I,Xϱ).(2)

By Hölder’s inequality

|Iξ(t),f(t)dt|2ξLϱ(I,Xϱ)fLϱ(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, Xϱ ) 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 Xϱ −valued function τn on the set of all measurable subsets of In = I ∩ [0, n] as follows:

τn(E),u=Λ,χEu,uXϱ.

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(λk=1χEku)=Ik=1χEkϱ(λu)dt=ϱ(λu)k=1|Ek|nϱ(λu)<,

we can conclude that

k=1χEku=χk=1EkuinLϱ(I,Xϱ),andk=1χEkuLϱ(I,Xϱ)nuϱ.

Consequently,

τn(k=1Ek),u=k=1Λ,χEku=k=1τn(Ek),u,

and

|τn(k=1Ek),u|n|||Λ|||uϱ.

Therefore τn is an Xϱ −valued measure on In with a bounded total variation no more than n∥|Λ∥|*. Now since Xϱ has the Radon-Nikodym’s property w.r.t. every finite subinterval of I, there is a unique ξnL1(In, Xϱ ) such that

Λ,χEu=τn(E),u=Inξn(t),χEudt

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 Xϱ −valued function satisfying

Λ,f=Iξ(t),fdt(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 fLc (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 fLc (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={tIn:u(t)Xϱn},andun(t)=u(t)χJn.

Then unLc (I, Xϱ) and

Jnϱ(ξ(t))dt+Iϱ(un(t))dt=Iξ(t),un(t)dt=Λ,ununLϱ(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)<unLϱ(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 |In=1 Jn| = 0 into account, we have ∫Iϱ*(ξ(t))dt ≤ 1, which leads to the conclusion ξLϱ*(I, Xϱ ) with the estimate

ξLϱ(I,Xϱ)dt1.

Finally, using the density of Lc (I, Xϱ) in Lϱ(I, Xϱ), we can conclude that (3) holds for all fLϱ(I, Xϱ). Therefore Λ = Λξ as in (2) and ξLϱ(I,Xϱ) = ∥|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(uv2)εϱ(u)+ϱ(v)2orϱ(uv2)(1δ)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 { Xα : 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 ξXα , xXβ and αβ. It is easy to see that, if {Xα} is uniformly bounded, then { Xα } is also uniformly bounded with the same bounds. However, whether or not { Xα } 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

    limkϱαk(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

limkθJk=limkθJk+=θ(t)

always holds, where θJandθJ+ 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

L0(I,Xθ())={fL0(I,X):f|JL0(J,XθJ)for all JΠ(I)},

and

L0(I,Xθ())={fL0(I,Xθ()):f(t)Xθ(t)for a.e.tI}.

Obviously, both of them are linear spaces according to the sum and scalar multiplication of abstract-valued functions, and L0(I, Xθ(⋅)) ⊆ L0 (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 θn,k±=θJn,k± , for k = 1, 2, ⋯, 2n if I = [0, T] or k = 1, 2, ⋯ if I = [0, ∞). Define a step function θn through θn±(t)=θn,k± for tJn,k. Obviously, { θn± } 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, Xθn±() ) is well defined, on which Φθn±(f)=Iϱθn±(t)(f(t))dt is a semimodular. It induces a Banach space, denoted by Lϱθn±()(I,Xθn±()).

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

L0(I,Xθ())L0(I,Xθ())L0(I,Xθn()).

Thus for each fL0 (I, Xθ(⋅)), the function tϱθn(t)(f(t)) is measurable, n = 1, 2, ⋯. Note that

ϱθ(t)(f(t))=limnϱθn(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,fL0(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,lΦϱθ()(λ(fkfl)=limk,lIϱθ(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,jki,(7)

Especially we have

Iϱθ(t)(2i(fki+1(t)fki(t)))dt<12i,

which in turn yields

|Ei||{tI: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 E=j=1i=jEi , then we have m(E) = 0, and for each tIE, there exists j ∈ ℕ such that tIi=j Ei, or equivalently ∥fki+1(t) − fki(t)∥θ(t) ≤ 1/2i for all ij. Consequently, for the integer lj, we have

i=lfki+1(t)fki(t)θ(t)12l1.

This infers that series fk1(t)+i=l(fki+1(t)fki(t)) 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)))dtlim infjIϱθ(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

C1fLϱθn()(I,Xθn())fLϱθ()(I,Xθ())CfLϱθn+()(I,Xθn+()).

Proposition 3.9

A function fL0 (I, Xθ(⋅)) with the property

K=lim supnfLϱθn()(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

{fL0(I,Xθ()):supJΠ(I)fLϱθJ(I,XθJ)<}

and

fLϱθ()(I,Xθ())supJΠ(I)fLϱθJ(I,XθJ)CfLϱθ()(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 LϱθJ+(J,XθJ+).

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

L+0(I,Xθ())={fL0(I,X):f(t)Xθ(t)for a.e.tI,and there exists a sequence{sn}ofS(I,Xα+)s.t.sn(t)f(t)θ(t)0asnfor a.e.tI}.

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 L+0 (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 L+ϱθ() (I, Xθ(⋅)) through

L+ϱθ()(I,Xθ())={fL+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 L+ϱθ() (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 L+ϱθ() (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, L+ϱθ() (I, Xθ(⋅)) is a Banach space.

Proof

Taken any Cauchy sequence {fn} in L+p() (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 fL+ϱθ() (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(⋅)) = L+0 (I, Xθ(⋅)) is sufficient for Lϱθ(⋅)(I, Xθ(⋅)) = L+ϱθ() (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 L+ϱθ() (I, Xθ(⋅)) is also separable.

This theorem is a straight consequence of Proposition 3.12.

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

ϱα(ξ)=supuXα{ξ,uαϱα(u)},ξXα.

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

ϱ~α(ξ)=ϱα(ξ),ifξXα,,ifξXα+Xα,

then we obtain another family of semimodulars defined on Xα+ , 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 ϱ~αandϱα , and prefer to use { ϱα } instead of { ϱ~α } to denote the 𝓓𝓜𝓝 of {ϱα}.

Suppose α1α2, then Xα1Xα2 , and for all ξXα2 , by (6) we have

ϱα2(ξ)=supuXα2{ξ,uα2ϱα2(u)}supuXα2{ξ,uα21C1ϱα1(u)}+C2C1supuXα1{ξ,uα11C1ϱα1(u)}+C2C1=1C1ϱα1(C1ξ)+C2C1.(8)

Similar to Proposition 3.4, from this property we can show that the dual space family { Xα : 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

limkϱαk(u)=ϱα(u)

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

ϱα(ξ)lim supkϱα~k(ξ)

holds for all ξXα+ , which in turn leads to the successive property of { Xα }. Unfortunately, the inverse inequality, hence continuity of { Xα } can not be guaranteed under present situations. For the sake of convenience, hereinafter, we always assume that Xα+X, and the 𝓓𝓜𝓝{ ϱα } is assumed to be a 𝓒𝓜𝓝 defined on X. We also assume that the 𝓑𝓢𝓝{Xα} and its dual net { Xα } are compatible, i.e. 〈ξ, uα2 = 〈ξ, uα1 provided uXα2, ξXα1 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) = Xα for all α ∈ 𝓐,

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

  • for every α ∈ 𝓐, ϱα is a modular, and

  • Xα has the Radon-Nikodym’s property w.r.t. every JΠ(I).

Then the dual space L+ϱθ() (I, Xθ(⋅))* is equivalent to Lϱθ()(I,Xθ()) in the sense of isomorphism.

Proof

Firstly for each ξLϱθ()(I,Xθ()) , define the linear functional Λξ as follows:

Λξ,fθ()=Iξ(t),f(t)θ(t)dt,fL+ϱθ()(I,Xθ()).(9)

Suppose that ξLϱθ()(I,Xθ())=fL+ϱθ()(I,Xθ())=1 , then by Young’s inequality we have

ξ,fθ()Iϱθ(t)(ξ(t))dt+Iϱθ(t)(f(t))dt2.

Therefore ΛξL+ϱθ() (I, Xθ(⋅))*, and ΛξL+ϱθ()(I,Xθ())2ξLϱθ()(I,Xθ()) . This claim also holds for arbitrary ξLϱθ()(I,Xθ()) by scaling arguments.

Conversely, given a functional ΛL+ϱθ() (I, Xθ(⋅))*, we will find a function ξLϱθ()(I,Xθ()) 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 ΛL+ϱθ()(I,Xθ())=1 . Taking any JΠ(I) and any fLϱθJ+(J,XθJ+) , consider the zero extension of f out of J and denote it by . Obviously, L+ϱθ() (I, Xθ(⋅)) and

f~L+ϱθ()(I,Xθ())CfLϱθJ+(J,XθJ+)(10)

for some constant C > 0 depending on |J| but independent of f, which means that the restriction of Λ to LθJ+(J,XθJ+) , denoted by Λ|J,θJ+ , lies in LθJ+(J,XθJ+) . So by invoking Theorem 2.7, there is a unique function ξJLϱθJ+(I,XθJ+) such that

Λ|J,θJ+,fθJ+=JξJ(t),f(t)θJ+dt

for all fLθJ+(J,XθJ+) , and

ξJLϱθJ+(I,XθJ+)CΛ|J,θJ+LϱθJ+(J,XθJ+)CΛLϱθ()(I,Xθ())(11)

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

Suppose that J1, J2Π(I) satisfy J2J1, then LϱθJ2+(J,XθJ2+) is densely imbedded in LϱθJ1+(J,XθJ1+) , 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 ξL0(I,Xθ()).

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 ξLϱθ()(I,Xθ())andξLϱθ()(I,Xθ())CΛLϱθ()(I,Xθ()) for some C > 0 independent of Λ.

For each fL+ϱθ() (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,φkθ()=Λ|I,α+,φkα+=Iξ(t),φk(t)α+dt=Iξ(t),φk(t)θ(t)dt,

letting k → ∞, we obtain

Λ,fθ()=Iξ(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 ξLϱθ()(J,Xθ()) for all JΠ(I). Consequently ξL0(I, Xθ() ), 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,fnθ()=Λ|[0,n],fnθ()=[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,fθ()=Iξ(t),f(t)θ(t)dt.

Thus Λ = Λξ by the arbitrariness of f. The remaining task for us is to show ξLϱθ()(I,Xθ()) . For this purpose, notice that the effective domain 𝓓(ϱα) is equal to Xα and the latter is separable, so the dual modular ϱα can be represented by

ϱα(η)=supk1{η,vkαϱα(vk)},ηXα,(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 ηXθ(t) for all tI.

For each n ∈ ℕ, define

rn(t)=χ[0,n](t)max1kn{ξ(t),vkθ(t)ϱθ(t)(vk)}.

Obviously, {rn} is a nondecreasing sequence of nonnegative (v1 = 0) measurable functions converging to ϱθ(t) (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

1Φϱθ()(Λ)Λ,snΦϱθ()(sn)=I{ξ(t),sn(t)θ(t)ϱθ(t)(sn(t))}dt,

where Φϱθ() is the dual modular of Φϱθ(⋅). Taking limit of the second line as n → ∞, we obtain

Φϱθ()(ξ)=Iϱθ(t)(ξ(t))dt1.

Therefore ξLϱθ()(I,Xθ())andξLϱθ()(I,Xθ())1.

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

ξLϱθ()(I,Xθ())ΛLϱθ()(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 Φϱθ()(ξ)=Φϱθ()(Λξ) for all ξLϱθ()(I,Xθ()) . 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(⋅)) = L+0 (I, Xθ(⋅)), then

Lϱθ()(I,Xθ())Lϱθ()(I,Xθ()).(13)

Theorem 3.19

Suppose the following conditions are all satisfied.

  • both {ϱα} and { ϱα } satisfy the Δ2condition,

  • {Xα} and { Xα } are two dense 𝓑𝓢𝓝s,

  • Xα+ and Xα are both separable, and

  • for every α ∈ 𝓐, Xα is reflexive,

  • L0(I, Xθ(⋅)) = L+0(I,Xθ()),andL0(I,Xθ())=L+0(I,Xθ()).

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)={pP(Ω):1pp+<},

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 ϱp 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={pPb(Ω):p(x)[p_,p¯]fora.e.xΩ}.

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 p_=pQ,p¯=pQ+ , and define θ(t) = p(t, ⋅) =: p(t), then we obtain an order-continuous exponent θ : I → 𝓐b with

θJ+=sup{p(t,x):tJ},θJ=inf{p(t,x):tJ}

for all JΠ(I), and Xθ(t) = Lp(t,x)(Ω), Xθ(t) = 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)) = L+ϱp() (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, ϱα , as well as Φϱθ(⋅) and Φϱθ() . Coercivity, which says

ϱ(u)uXasuX,

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)unXKfor 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 ϱα satisfies Dom( ϱα ) = Xα . As a matter of fact, taking any ξXα , by the coercivity of ϱα, there is a constant M > 0 for which ϱα(u) ≥ ( ξα + 1)∥uα provided ∥uαM. Consequently,

ϱα(ξ)=supuαM{ξ,uαϱα(u)}Mξα<.

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(uγ1(s))ϱα(u)sfor alluX,s>0andαA,(14)

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

limsγ(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)uα12ϱα(uγ1(uα/2))12uα/2γ1(uα/2)ϱα(uuα/2)12uα/2γ1(uα/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)γ1(Φϱθ()(u)))dt1Φϱθ()(u)Iϱθ(t)(u(t))dt=1,

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

limuLϱθ()(I,Xθ())Φϱθ()(u)uLϱθ()(I,Xθ())limΦϱθ()(u)Φϱθ()(u)γ1(Φϱθ()(u))=,

which shows the coercivity of Φϱθ(⋅).

In the following arguments, we also need the strict convexity of ϱα for all α ∈ 𝓐, and the weak lower-semicontinuity of {ϱα} and { ϱα }, 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 Xβ , we have

    ϱβ(u)lim infkϱαk(uk)andϱβ(ξ)lim infkϱαk(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 ϱα , and weak lower-semicontinuity of {ϱα} and { ϱα } are put together, denoted by H(B), which jointly with the strict convexity assumption of ϱα and ϱα 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 { ϱα }.

Consider the subdifferential operator ∂ϱα. Similar to Remark 2.8, we can check that ∂ϱα is single-valued provided ϱα is strict convex. Furthermore, by the coercivity of ϱα, we can also find that ∂ϱα : XαXα is a demicontinuous, monotone and coercive operator, whose range is the whole space Xα . Since

ϱα(u),uα=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 Xα are involved.

Lemma 4.1

For all uL(I, Xα), the compound operator t∂ϱα(u(t)) lies in L(I, Xα ). Moreover, if uLϱα(I, Xα), then ∂ϱα(u(⋅)) ∈ Lϱα(I,Xα).

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)αξ(t)αu(t)α,

which together with the coercivity of ϱα , yields the boundedness of ϱα (ξ(t)) uniformly for a.e. tI, hence the inclusion ξL(I, Xϱα).

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 Xα −valued simple function ξk(t) = ∂ϱα(sk(t)) lies in L(I, Xα ), consequently it lies Lϱα(I,Xϱα) , 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)=Iξk),sk(t)αdt2ξkLϱα(I,Xα)skLϱα(I,Xα).

Then by the coercivity of Φϱα and the boundedness of {ξk} in Lϱα(I, Xα), we get the boundedness of {ξk} in Lϱα(I, Xα). Therefore there is a subsequence, say {ξk} itself, convergent to some ξ̃ weakly in Lϱα(I, Xα). Suppose that {vi} is a countable dense subset of Xα, then for every two positive integers i, n, we have

limkInξk(t)ξ~(t),viαdt=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 Xα as k → ∞ for a.e. tI. Now taking limits in (17), and using continuity of ϱα and weak lower-continuity of ϱα, we obtain

ϱα(ξ~(t))+ϱα(u(t))lim infkϱα(ξk(t))+limkϱα(sk(t))lim infn(ϱα(ξk(t))+ϱα(u(t)))limkξk(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 ξ̃Lϱα(I, Xϱα).□

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 Xθ()-valued function ∂ϱθ(t)(u(t)) belongs to Lϱθ()(I, Xθ()).

Proof

For every 2n-mean partition of I, define the step function θn(t) as in the preceding section, and let ξn(t) = ϱθn(t)(u(t)) for a.e. tI. From Lemma 4.1, we can derive that ξn lies in Lϱθn()(I,Xθn()), hence it lies in Lϱθ()(I,Xθ()). Much similar to the proof of Lemma 4.1, and using the strong coercivity of {ϱα}, density of {ϱα}, separability of Xα+ together with the relations Lϱθ(⋅)(I, Xθ(⋅)) = L+ϱθ()(I, Xθ(⋅)) and Lϱθn()(I,Xθn())Lϱθ()(I,Xθ()), 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 Lϱθn()(I,Xθn()) and consequently in Lϱθ()(I,Xθ()). Thus by passing to a subsequence, we can assume that {ξn} converges weakly to some ξ in Lϱθ()(I,Xθ()), and ξn(t) ⇀ ξ(t) in Xθ(t) 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 infnϱθn(t)(ξn(t))+limnϱθn(t)(u(t))limnξn(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 ϱα, then we have the classical representation:

Φϱα(u)={ξLϱα(I,Xα):ξ(t)ϱα(u(t))fora.e.tI}.

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

Φϱθ()(u)={ξLϱθ()(I,Xθ()):ξ(t)ϱθ(t)(u(t))fora.e.tI},

if the strict convexity assumptions of ϱα 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),uθ()=Φϱθ()(u)+Φϱθ()(u)

for all uLϱθ(⋅)(I, Xθ(⋅). In a word, Zθ(⋅) : Lϱθ(⋅)(I, Xθ(⋅)) → Lϱθ()(I,Xθ()) 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 Φθ() 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 Vβ × Vβ and Xα × Xα for all uVβ and ξXα, and all α ∈ 𝓐, β ∈ 𝓑.

Under this assumption, for all C > 0,

EC={uXα: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 {φϑn()(u())}n=1 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,uL0(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)={uVβ:uu0Vβ<r},

and

Bφβ(u0,r)={uVβ:φβ(uu0)<r}.

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¯=n1uEBφβ(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=n1k1Bφβ(vk,1n),

which results in

{tI:u(t)F}=n1k1{tI:φβ(u(t)vk)<1n}=:n1k1Ek,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 (I,Vϑn()) (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)={ξLφϑ()(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)={uW1,1(Ω):u,DiuLq(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

uW1,q(x)(Ω)=inf{λ>0:ϱq(uλ)+i=1Nϱq(Diuλ)},

or equivalently

uLq(x)(Ω)+uLq(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 qPlogω(Ω) in symbol, which means that

|q(x)q(y)|ω(|xy|),for|xy|<1,(19)

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

Cω=supr>0ω(r)log1r<.(20)

Under this situation, C(Ω) is dense in W1,q(⋅)(Ω), and W01,q(x)(Ω) can be defined as the complement of C0(Ω) in W1,q(x)(Ω). By this definition, W01,q()(Ω)=W01,1(Ω)W1,q(x)(Ω), and Poincaré’s inequality

uLq(x)(Ω)CuLq(x)(Ω),uW01,q()(Ω)

remains true. Therefore W01,q()(Ω) is topologically equivalent to the homogeneous Sobolev space D01,q()(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),ifuW01,1(Ω),,ifuL0(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 = 𝓐bPlogω(Ω) is also a 𝓑𝓣𝓛, on which {φq : q ∈ 𝓑ω} is a 𝓒𝓜𝓝, and { W01,q(x)(Ω) : q ∈ 𝓑ω} is the corresponding regular 𝓑𝓢𝓝.

For any qPlogω(Ω) with q > 1, if we take W01,q()(Ω) and D01,q()(Ω) as the same space, then every member of the dual space W01,q()(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 = i=1NDifi in 𝓓′(Ω), or equivalently,

ξ,u=i=1NΩfi(x)Diu(x)dx

for all uW01,q()(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 uW01,q()(Ω) and ξ = i=1NDifiW–1,q′(⋅)(Ω), we have

ξ,u=i=1NΩfi(x)Diu(x)dxφq(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 { Vq = W–1,q′(⋅)(Ω) : q ∈ 𝓑ω} is a regular 𝓑𝓢𝓝, which is the 𝓓𝓢𝓝 of {Vq = W01,q()(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;W1,q(x,)(Ω))=L+0(I;W1,q(x,)(Ω)).

Thus by invoking Corollary 3.18 and Theorem 3.19 again, we get the reflexivity of Lφq(⋅)(I; W01,q(x)(Ω)), and the representation

Lφq()(I;W01,q(x,)(Ω))Lψq()(I;W1,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)Nq(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)2u)

defines the subdifferential of Φ͠φϑ(⋅) at uLψq(⋅)(I; W01,q(,x)(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)={uW01,1(Ω):DiuLqi(x)(Ω)}.

Here qiPlogω with a common ω, and qi(x) ∈ [q, ] ⊆ (1, ∞) for all xΩ, i = 1, 2, ⋯, N. Similar to the isotropic ones, W01,q(x)(Ω) can be viewed as a semimodular space derived by

φq(u)=i=1Nϱqi(Diu),ifuW01,1(Ω),,ifuL0(Ω)W01,1(Ω),

and its dual space, denoted by W–1,q′(x)(Ω), can also be represented by the product space i=1NLqi(x)(Ω). Therefore following the same discussions as in Example 4.11, we can conclude that {φq : qBωN}, where BωN 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}. { W01,q(x)(Ω) : qBωN} 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 i=1NLqi_(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, W01,q(,x)(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,W1,q(,x)(Ω)).

Moreover, the restriction functional Φ͠φq(⋅) is convex and locally Lipschitz everywhere on Lφq(⋅)(I, W01,q(,x)(Ω)), its subdifferential ∂Φ͠φq(⋅) at uLφq(⋅)(I, W01,q(,x)(Ω)) has the expression

Φ~φq()(u)=φq(t)(u(t))=i=1NDi(|Diu|qi(t,x)2Diu),

or equivalently,

Φ~φq()(u),vϑ()=i=1NIΩ|Diu(t,x)|qi(t,x)2Diu(t,x)Div(t,x)dxdt

for all vLφq(⋅)(I, W01,q(,x)(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)={uLϱα(I,Xα):uLϱα(I,Xα)},

where u′ denotes the derivative of u in the sense of distribution, i.e. for all ξXα and all γC0(I), equality

Iu(t),γ(t)ξαdt=Iu(t),γ(t)ξαdt

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

uW1,ϱα(I,Xα)=uLϱα(I,Xα)+uLϱα(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

uXαϱα(u)+1,uXα,

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

0Thϱα(u(t+h)u(t))dt0Thtt+hϱα(u(τ))dτdt=(0h0τ+hThτhτ+ThTτhTh)ϱα(u(τ))dtdτ=0hτϱα(u(τ))dτ+hThhϱα(u(τ))dτ+ThT(Tτ)ϱα(u(τ))dτhIϱα(u(τ))dτCh.(21)

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

Mru(t)=1rtt+ru(τ)dτ,t[0,Tr].

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

φβ(Mru(t))1rtt+rφβ(u(τ))dτ,t[0,Tr].

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

0Thϱα(Mru(t)u(t))dt=0Thϱα(1r0r(u(t+τ)u(t))dτ)dt1r0Th0rϱα(u(t+τ)u(t))dτdt1r0r0Tτϱα(u(t+τ)u(t))dtdτCh

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())={uW1,ϱα(I,Xα):u,uLϱθ()(I,Xθ())}.

Similarly, equipped with the norm

uW1,ϱθ()(I,Xθ())=uLϱθ()(I,Xθ())+uLϱθ()(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,

    uθ(t)Cuα1δ(t)uϑ(t)δ(t)foralluVϑ(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{uθ(t)p~1(t),uθ(t)p~2(t)}uXθ(t),(23)

    and

    min{uϑ(t)q~1(t),uϑ(t)q~2(t)}Cφϑ(t)(u),uVϑ(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

Iuk(t)θ(t)p~2(t)dtCIuk(t)αp~2(t)(1δ(t))uk(t)ϑ(t)p~2(t)δ(t)dtC(Iuk(t)αp~2(t)(1δ(t))Kdt)1/K(Iuk(t)ϑ(t)p~2(t)δ(t)Kdt)1/KC(Iuk(t)αp~2(t)(1δ(t))Kdt)1/K[T1/K+({uk(t)ϑ(t)>1}uk(t)ϑ(t)p~2(t)δ(t)Kdt)1/K]C(Iuk(t)αp~2(t)(1δ(t))Kdt)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

uXαCuVβ(25)

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

uL1(I,Xα)CuLϱθ()(I,Xθ()),uL1(I,Xα)CuLφϑ()(I,Vϑ())

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

uLϱθ()(I,Xθ())+uLφϑ()(I,Vϑ())1,

estimates (24) turn to be

Iu(t)θ(t)p~2(t)dtCT1/Kmax{(maxtIu(t)α)p~2+(1δ),(maxtIu(t)α)p~2(1δ+)}(Iu(t)ϑ(t)p~2(t)δ(t)Kdt)1/KCT1/Kmax{uW1,1(I,Xα)p~2+(1δ),uW1,1(I,Xα)p~2(1δ+)}[T1/K+({uk(t)Vϑ(t)>1}uk(t)Vϑ(t)p~2(t)δ(t)Kdt)1/K]CT1/K(uLϱθ()(I,Xθ())+uLφϑ()(I,Vϑ()))p~2(1δ+)[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

uW1,ϱθ()(I,Xθ())C(uLϱθ()(I,Xθ())+uLφϑ()(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(⋅)( vk) + 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(δ1))Φϱθ()(u)+Cσ(δ1)

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)={uW1,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 Dθ()2 : Wper1,ϱθ()(I, Xθ(⋅)) → Wper1,ϱθ()(I, Xθ(⋅))* defined through

Dθ()2(u),v=Iϱθ(t)(u(t)),v(t)θ(t)dt,u,vW1,ϱα(I,Xα).(27)

By the definition, it is easy to check that Dθ()2 is a single-valued, monotone and demicontinuous operator, hence it is maximal monotone. In addition, taking v(t) = γ(t)w in (27) with γC0(I) and wXα+, we have

Dθ()2(u),v=Iϱθ(t)(u(t)),γ(t)wα+dt=Iddtϱθ(t)(u(t)),γ(t)wα+dt=Iddtϱθ(t)(u(t)),γ(t)wθ(t)dt,

which means that Dθ()2 is a second order nonlinear differential operator, i.e.

Dθ()2(u)=ddtϱθ()(u())

in the sense of distribution.

Let Ψ͠(u) = Φϱθ(⋅)(u′) at uWper1,ϱθ()(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) = Dθ()2(u). Thus the subdifferential operator ∂Ψ͠ is single-valued, and ∂Ψ͠(u) = Dθ()2(u) for all uWper1,ϱθ()(I, Xθ(⋅)). Consider the extension of Ψ͠ onto Lϱθ(⋅)(I, Xθ(⋅))

Ψ(u)=,ifuLϱθ()(I,Xθ())Wper1,ϱθ()(I,Xθ())Ψ~(u),ifuWper1,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)={uW1,ϱθ()(I,Xθ()):there is anξLϱθ()(I,Xθ())such thatIξ(t),v(t)θ(t)dt=IZθ()(u)(t),v(t)θ(t)dtfor allvWper1,ϱθ()(I,Xθ())},(28)

together with ∂Ψ(u) = Dθ()2(u) = ξ in Wper1,ϱθ()(I, Xθ(⋅))* for the function ξLϱθ()(I,Xθ()) satisfying (28) and all u ∈ 𝓓(∂Ψ). In this sense, we can say that ∂ΨDθ()2.

Taking the intersection

W=Wper1,ϱθ()(I,Xθ())Lφϑ()(I,Vϑ())

as the work space, where the norm ∥⋅∥𝓦 takes the value uWper1,ϱθ()(I,Xθ())+uLφϑ()(I,Vϑ()), 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 Wper1,ϱθ()(I, Xθ(⋅)) × Wper1,ϱθ()(I, Xθ(⋅))*. Moreover, by (26) and the coercivity of Φϱθ(⋅), Φφϑ(⋅), we have

Φ~(u),uuWCΨ~(u)+Φ~φϑ()(u)uLϱθ()(I,Xθ())+uLφϑ()(I,Vϑ())asuW,

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 fLϱθ()(I,Xθ()) ⊆ 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 fLϱθ()(I,Xθ()), second order differential inclusion

ddtϱθ(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

Iϱθ()(u(t)),v(t)θ(t)dt+Iξ(t),v(t)ϑ(t)dt=If(t),v(t)θ(t)dt

for all v ∈ 𝓦.

The above theorem shows that the sum operator Dθ()2 + ∂Φφϑ(⋅) is both injective and surjective, its inverse ( Dθ()2 + ∂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 Lϱθ()(I,Xθ()) weakly, the corresponding sequence {( Dθ()2 + ∂Φ͠φϑ(⋅))–1 fk} converges to ( Dθ()2 + ∂Φ͠φϑ(⋅))–1 f in Lϱθ(⋅)(I, Xθ(⋅)) strongly.

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

  • for any JΠ(I) and uXθJ+, tf(t, u) lies in L0(I, XθJ+),

  • for every tI, f(t, ⋅) : Xθ(t)Xθ(t) 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) ∈ L0(I,Xθ()), and by (31), it comes F(u) ∈ Lϱθ()(I,Xθ()). Consequently, from the density of 𝓢(I, Xα+) in Lϱθ(⋅)(I, Xθ(⋅)) and the demicontinuity assumption upon f, we can derive that F(u) ∈ Lϱθ()(I,Xθ()) provided uLϱθ(⋅)(I, Xθ(⋅)). Moreover, we have

Proposition 5.9

F is a bounded and demicontinuous operator from Lϱθ(⋅)(I, Xθ(⋅)) to Lϱθ()(I,Xθ()).

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))dt0asn.

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

limnϱθ(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 Lϱθ()(I,Xθ()). Consequently, there is a subsequence, without loss of generality, assuming also {un} itself, and a function ξLϱθ()(I,Xθ()), such that F(un) → ξ weakly, or equivalently

limnIf(t,un(t))ξ(t),v(t)θ(t)dt=0

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

limnEf(t,un(t))ξ(t),vθ(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

limnf(t,un(t)),vθ(t)=f(t,u(t)),vθ(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)),vθ(t)f(t,u(t)),vθ(t)uniformly onEε

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

limnEf(t,un(t))f(t,u(t)),vθ(t)dt=0.(33)

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

Ef(t,u(t))ξ(t),vθ(t)dt=0,

which implies

f(t,u(t))ξ(t),vθ(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 Xθ(t) on E0. Therefore F(u) = ξ and F(un) ⇀ F(u) in Lϱθ()(I,Xθ()) 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())1F: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

ddtϱθ(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={uW:u=λF(u)for some0<λ<1}.

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

ddtϱθ(t)(λ1u(t))+φϑ(t)(λ1u(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()(λ1u)+Φ~φϑ()(λ1u)F(u),λ1uθ()Φϱθ()(δ1F(u))+Φϱθ()(δλ1u)σ(δ1)[μΦϱθ()(λ1u)+hL1(I)]+δΦϱθ()(λ1u)=(δ+μσ(δ1))Φϱθ()(λ1u)+hL1(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 ( Dθ()2 + ∂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),xΩ,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

infxΩ(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=1NQ(|Diu|qi(x,t)2Diu)Divdxdt=Qf(t,u)v(t,x)dxdt

for all vCper1(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

uLp(t,x)(Q)CuLq(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 W01,p(t,x)(Q) as the work space, instead, an anisotropic space Wper1,ϱp()(I, Lp(⋅, x)(Ω)) ∩ Lϱq(⋅)(I, W01,q(,x)(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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About the article

Received: 2017-12-16

Accepted: 2018-05-08

Published Online: 2018-08-20


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 924–954, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0080.

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© 2018 Zhang, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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