Abstract-valued Orlicz spaces of range-varying type

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 de nitions of Lθ(⋅)(I, Xθ(⋅)) and Lθ(⋅) + (I, Xθ(⋅)), and discuss their geometrical properties as well as the representation of L %θ(⋅) + (I, Xθ(⋅)). We also investigate some functionals and operators on Lθ(⋅)(I, Xθ(⋅)), giving expression for the subdi erential of the convex functional generated by another continuousmodular net. Aftermaking some investigations on the Bochner-Sobolev spacesW1,%θ(⋅)(I, Xθ(⋅)) andW 1,%θ(⋅) per (I, Xθ(⋅)), and the intersection spaceW 1,%θ(⋅) per (I, Xθ(⋅))∩ Lθ(⋅)(I, Vθ(⋅)), a second order di erential inclusion together with an anisotropic nonlinear elliptic equation with nonstandard growth are also taken into account.


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 rstly by Orlicz [1] in 1931, is a type of semimodular space commonly generated by a Φ or generalized Φ function. Typical examples of this type are Lebesgue and Sobolev spaces with variable exponents, i.e. L p(x) (Ω) and W k,p(x) (Ω) (see [2] for references). Due to the wide applications in many elds 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 subdi erential calculus, Akagi etc in [6,7] dealt with the doubly nonlinear parabolic equations involving variable exponents. The work [8] considered the application of Orlicz space in Navier-Stokes equation, and [9] investigated an obstacle problem with variable growth and low regularity of the data.
To deal with the evolution equations with variable exponents, a new type of functions, called X θ(⋅) −valued functions, are needed. As the valued space varies upon the time, it is di cult to give a suitable de nition of "measurability" for these functions. By introducing the concepts of bounded topological lattice A, regular Banach space net {X α ∶ α ∈ A} and order-continuous exponent θ ∶ I → A, Zhang-Li in [10] rstly gave de nition of the space L (I, X θ(⋅) ), which contains all the X θ(⋅) −valued functions measurable in a special manner. Like the measurable functions of range-xed type, members of L (I, X θ(⋅) ) are all normmeasurable. Based on the useful character, from L (I, X θ(⋅) ) the authors extracted two types of function spaces: continuous type C − (I, X θ(⋅) ) and integral type L p(⋅) (I, X θ(⋅) ). After showing their completeness and connections between them together with some concrete examples, the authors paid attention to a semilinear evolution equation with the nonlinearity having a time-dependent domain to illustrate the application of the X θ(⋅) −valued functions.
It is worth remarking that some Banach space net can be produced by a continuous modular net { α ∶ α ∈ A}. According to whether or not being built on the continuous modular nets, Zhang-Li in [11] divided the X θ(⋅) −valued function spaces of integral type into two subclasses: norm-modular ones and modular-modular ones. A norm-modular space, like L p(⋅) (I, X θ(⋅) ), is commonly produced by the semimodular with a generalized Φ function φ, while a modular-modular space is derived from a continuous modular net { α ∶ α ∈ A} with the semimodular Here, M is a continuous operator from a topological linear space X to a closed cone V of another topological linear space W, called a V−modular (refer to [11]).
Here we will drop the extra map M, and use merely { α ∶ α ∈ A} and θ to reconstruct the semimodular, namely This change brings much convenience to us to study the duality and re exivity 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 abstractvalued Orlicz space generated by a single modular. Section 3 is devoted to the abstract-valued Orlicz space generated by a series of modular. Using di erent measurability of the X θ(⋅) −valued functions, we introduce two di erent spaces: L θ(⋅) (I, X θ(⋅) ) and L θ(⋅) + (I, X θ(⋅) ), both of them are complete according to the same norm. We show that, under some suitable situations, L θ(⋅) + (I, X θ(⋅) ) is separable, and its dual space can be represented by L θ(⋅) + (I, X θ(⋅) ) * = L * θ(⋅) (I, X * θ(⋅) ).
After making some investigations on the Bochner-Sobolev spaces W , θ(⋅) (I, X θ(⋅) ) and W in Section 5, we study a type of second order nonlinear di erential inclusion − d dt ∂ θ(t) (u ′ (t)) + ∂ϕ ϑ(t) (u(t)) ∋ f (t, u(t)) for a.e. t ∈ I, with the periodic boundary condition, where the operator f ∶ I × X → X owns a nonstandard growth * θ(t) (f (t, u)) ≤ µ θ(t) (u) + h(t), u ∈ X θ(t) for a small number µ > and a nonnegative function h ∈ L (I). By introducing the Nemytskij operator F(u) = f (⋅, u), and the second order di erential operator D θ(⋅) de ned by we obtain a continuous and compact operator (D θ(⋅) + ∂Φ ϕ ϑ(⋅) ) − ○ F ∶ L θ(⋅) (I, X θ(⋅) ) → L θ(⋅) (I, X θ(⋅) ), which by Leray-Schauder's alternative theorem contains a xed point solving the di erential inclusion (1) in the weak sense. To illustrate these results, at the end of the paper, an anisotropic elliptic equation de ned on a cylinder I × Ω of R N+ + with a Caratheodory type nonlinearity µg(t, x, u) are investigated. Because of the nonstandard growth for a.e. (t, x) ∈ I × Ω and all u ∈ R ful lled by the nonlinearity, and the periodic boundary condition u( , 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 rstly 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.
If in addition, (u) = means u = , then is called a modular. Given a semimodular on X, the corresponding subspace becomes a normed linear space. X is called the semimodular space, while ⋅ is called the Luxemburg norm. Both of them are generated by . Recall that in X the unit ball property is holding, that is u ≤ if and only if (u) ≤ . Scalar Orlicz space is a common semimodular space produced by the integral semimodular. Suppose that φ ∶ [ , ∞) → [ , ∞] is a Φ function, i.e., φ is convex, left continuous, φ( ) = and lim t→ φ(t) = , lim t→∞ φ(t) = ∞. Suppose also (A, µ) is a σ− nite and complete measure space, and L (A, µ) is the linear space containing all the measurable scalar function de ned on A. Then integration de nes a semimodular on L (A, µ). The corresponding semimodular space, denoted by L φ (A, µ), is called an Orlicz space. According to the Luxemburg norm L φ (A, µ) is a Banach space. Moreover, φ is a modular in case that φ is positive, i.e., φ(t) > whenever λ > . Suppose further φ ∶ A × [ , ∞) → [ , ∞] is a generalized Φ function, that is, for a.e. x ∈ A, φ(x, ⋅) is a Φ functions, and for all t ∈ [ , ∞), the function x ↦ φ(x, t) is measurable on A, then for all f ∈ L (A, µ), integration ∫ Ω φ(x, f (x) )dµ makes sense. This de nes another semimodular and induces another semimodular space, which is the generalization of Orlicz space, called a Musielak-Orlicz space.
Taking a measurable subset Ω ⊆ R N , and a measurable exponent p ∶ Ω → [ , ∞), de ne A = Ω with Lebesgue measure, and φ(x, t) = t p(x) . Then we obtain a generalized Φ function, from which we can construct an integral modular p(⋅) through and induce an important Musielak-Orlicz space, denoted by L p(⋅) (Ω), and called the Lebesgue space with variable exponent. One knows that if p + = esssup x∈Ω p(x) < ∞, then L p(⋅) (Ω) is separable, and the unit ball property turns to be min{

Orlicz space generated by a single modular
Let X be a linear space and ∶ X → [ , ∞] be a semimodular, which induces a semimodular space X with the Luxemburg norm ⋅ . Let I be a nite or in nite interval, namely I = [ , T] for some < T < ∞ or I = [ , ∞). A function f ∶ I → X is said to be measurable, if for every open set G ⊆ X, the preimage {t ∈ I ∶ f (t) ∈ G} is a measurable subset of I. Moreover, f is called strongly measurable, if there is a sequence of X −valued simple functions convergent to f almost everywhere. Of course, a strongly measurable function is measurable de nitely, and vice versa provided X is separable (cf. [13, §1.2]). Denote by L (I, X ) the set of all strongly measurable X −valued functions de ned on I. Recall that a semimodular is lower-continuous on the induced space X , thus for all a > , the set {u ∈ X ∶ (u) > a} is open in X . Consequently, for each f ∈ L (I, X ), the multifunction t ↦ (f (t)) is also measurable. Hence integration makes sense. One can easily verify that Φ is also a semimodular on L (I, X ) with the semimodular space and the Luxemburg norm denoted by ⋅ L (I,X ) .
This theorem is a special case of Theorem 3.7, which is given in §3 with a proof.

Remark 2.2.
Suppose that f ∈ L ∞ (I, X ), and the one-dimension Lebesgue measure of the set E = {t ∈ I ∶ f (t) ≠ } is nite. Then we have where M ≥ is the essential supremum of f (t) . Thus f ∈ L (I, X ) and f L (I,X ) ≤ (M + ) max{ , E }. Furthermore, by the estimate A semimodular is said to be satisfying the ∆ −condition, if there exists a constant d ≥ such that Recall that, under the ∆ −condition, turns to be a continuous modular satisfying u ∈ X if and only if (λu) < ∞ for all λ > , and u n → u in X if and only if (u n − u) → .
Moreover, Φ also satis es the ∆ −condition with the same constant d . Proof. For each f ∈ L (I, X ), there is correspondingly a sequence of simple and let ϕ k = s k χ E χ E k , which is also a simple X −valued function. Notice that we have (ϕ k (t) − f (t)) → a.e. on I as k → ∞, which combined with (ϕ k (t) − f (t)) ≤ (f (t)) for a.e. t ∈ I, and Lebesgue's convergence theorem, yields Φ (ϕ k − f ) → or equivalently ϕ k → f in L (I, X ) as k → ∞. Thus density of S(I, X ) in L (I, X ) has been proved.  Remark 2.6. Given a semimodular , recall that the dual functional * is also a semimodular on X * , and the double dual * * is equal to on the space X (cf. [2, §2.2] or [14, §3.2]). Moreover, for all u ∈ X and ξ ∈ X * , Young's inequality ⟨ξ, u⟩ ≤ (u) + * (ξ) holds. The equality also holds if and only if ξ ∈ ∂ (u) or equivalently u ∈ ∂ * (ξ) if we regard X as a closed subspace of X * * . Here ∂ is the the subdi erential operator of and ∂ * is that of * . Recall that as the subdi erential operators of lower-semicontinuous and convex proper functionals, ∂ and ∂ * can be viewed as two maximal monotone and semiclosed subsets of the product spaces X × X * and X * × X * * respectively. As for the multivalued inverse map (∂ ) − , we know that (∂ ) − ⊆ X * × X has the same properties as ∂ has, together with the inclusion (∂ ) − ⊆ ∂ * holding. Furthermore, if in addition R(∂ ) = X * , then for all ξ ∈ X * , the image (∂ ) − (ξ) is a nonempty convex and closed subset of X . Therefore for all ξ ∈ L (I, X * ), the multifunction t ↦ (∂ ) − (ξ(t)) is graph measurable with nonempty closed image everywhere, consequently under the additional separability assumption of X , we can assert that (∂ ) − (ξ(⋅)) has a measurable selection, or in other words, there is a function u ∈ L (I, X ) such that u(t) ∈ (∂ ) − (ξ(t)) for a.e. t ∈ I (see [15, §8.3] for references).
Theorem 2.7. Suppose that satis es the ∆ −condition, X is a separable Banach space, and R(∂ ) = X * . Suppose also * is a modular, and X * has the Radon-Nikodym's property w.r.t. every nite subinterval of I. Then the dual space L (I, X ) * is isomorphic to L * (I, X * ).
Conversely, for each Λ ∈ L (I, X ) * , we will prove the existence of a unique ξ ∈ L * (I, X * ) such that Λ = Λ ξ as in (2). If Λ = , then take ξ = and there is nothing to do. If Λ ≠ , then Λ * > , where Λ * denotes the L (I, X ) * −norm of Λ. Without loss of generality, in the following discussions we may assume that Λ * = . For each positive integer n, de ne an X * −valued function τ n on the set of all measurable subsets of I n = I ∩ [ , n] as follows: Here χ E is the characteristic function of E.
Suppose that {E k } is a sequence of mutually disjoint measurable subsets of I n and u ∈ X . Since for all λ > , Consequently, Therefore τ n is an X * −valued measure on I n with a bounded total variation no more than n Λ * . Now since X * has the Radon-Nikodym's property w.r.t. every nite subinterval of I, there is a unique ξ n ∈ L (I n , X * ) such that ⟨⟨Λ, χ E u⟩⟩ = ⟨τ n (E), u⟩ = I n ⟨ξ n (t), χ E u⟩dt for every measurable subset E of I n . By the uniqueness of ξ n in the above representation, we have that ξ n+ (t) = ξ n (t) a.e. on I n . Let ξ(t) = ξ n (t) for t ∈ I n , then we obtain a globally de ned and strongly measurable X * −valued function satisfying for the function f = uχ E with a bounded measurable subset E of I and a point u ∈ X and consequently for all f ∈ S(I, X ) with compact supports.
Given a function f ∈ L ∞ c (I, X ), from Corollary 2.3, we can nd a sequence of X −valued simple functions with compact supports, say {s k }, such that s k → f in both L ∞ (I, X ) and L (I, X ), and (3) is satis ed by s k for all k ∈ N. Taking limits as k → ∞ in both sides of (3), we can deduce that (3) is also satis ed by f ∈ L ∞ c (I, X ). Remark 2.6 shows that the multivalued function t ↦ (∂ ) − (ξ(t)) is measurable, and it has a strongly measurable selection since X is separable. Denote the selection by u, then we have u(t) ∈ (∂ ) − (ξ(t)) ⊆ ∂ * (ξ(t)) a.e. on I. For each n ∈ N + , let J n = {t ∈ I n ∶ u(t) X ≤ n}, and u n (t) = u(t)χ J n .
As Λ ≠ and * is a modular, neither ξ(t) nor * (ξ(t)) is equal to a.e. on I. Thus ∫ J n * (ξ(t))dt > , consequently from (4) we get Φ (u n ) < u n L (I,X ) for n large enough. Hence by the unit ball property, we assert that u n L (I,X ) ≤ , which in turn yields ∫ J n * (ξ(t))dt ≤ . Let n → ∞, and take the fact I ∖ ⋃ ∞ n= J n = into account, we have ∫ I * (ξ(t))dt ≤ , which leads to the conclusion ξ ∈ L * (I, X * ) with the estimate ξ L * (I,X * ) dt ≤ .
Finally, using the density of L ∞ c (I, X ) in L (I, X ), we can conclude that (3) holds for all f ∈ L (I, X ). Therefore Λ = Λ ξ as in (2) and ξ L * (I,X * ) = Λ ξ * = . Thus the proof has been completed in the case Λ ξ * = , and the general case can be dealt with by the scaling arguments.

Remark 2.8.
Here the separability assumption of X can be replaced by the strict convexity assumption of . As a matter of fact, if is strictly convex, then ∂ is injective, or equivalently (∂ ) − is single-valued.
Recall that every re exive space satis es the Radon-Nikodym's property with respect to every complete and nite measure space. Furthermore if X is re exive, then ∂ * = (∂ ) − and ∂ = (∂ * ) − . Putting these facts into Theorem 2.7, we have Corollary 2.9. Suppose that both and its dual * satisfy the ∆ −condition, the semimodular space X is re exive and separable. Then and the function space L (I, X ) is also re exive.
Given a semimodular ∶ X → [ , ∞], we say is uniformly convex, i.e. we mean that for every ε ∈ ( , ), there is a δ ∈ ( , ), for which either holds. According to [2, §2.4], we know that every uniformly convex semimodular satisfying the ∆ −condition generates a uniformly convex space. Similarly, for a semimodular , its uniform convexity can be inherited by the Nemytskij functional Φ . Summing up, we have Theorem 2.10. Under the uniform convexity assumption and the ∆ −condition of , L (I, X ) is a uniformly convex space.

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 in mum exist in A simultaneously. In this paper, A is always assumed to be a totally order-bounded topological lattice, or BT L in abbreviation. Its order supremum and in mum are denoted by α + and α − respectively. In a BT L A, a sequence {α k } is said to be approaching a point β, if the two conditions α k ≺ β, ∀ k ∈ N and lim k→∞ α k = β are both ful lled.

De nition 3.1. Given a family of Banach spaces
for any sequence {α k } approaching β and any point x ∈ X α − with the constraints: x ∈ X α k for all k ∈ N and C = sup k→∞ x α k < ∞, we have x ∈ X β and x β ≤ C. Finally, a BSN {X α } is called regular provided it is norm-continuous, uniformly bounded and successive at the same time.

Remark 3.2.
Given a BSN {X α ∶ α ∈ A}, the family of dual spaces {X * α ∶ α ∈ A}, where A takes the inverse order ≻ instead of ≺, is also a BSN , called the dual space net or DSN in symbol. Here we use the convention: ⟨ξ, x⟩ α = ⟨ξ, x⟩ β provided ξ ∈ X * α , x ∈ X β and α ≺ β. It is easy to see that, if {X α } is uniformly bounded, then {X * α } is also uniformly bounded with the same bounds. However, whether or not {X * α } inherits the normcontinuity and successive property from {X α } is not clear.

De nition 3.3.
Suppose that X is a linear space, and { α ∶ α ∈ A} is a family of semimodulars de ned on X. We say { α } is a continuous modular net, or CMN in abbreviation, i.e. we mean that the following hypotheses are satis ed: holds for all u ∈ X and all α i ∈ A, i = , with α ≺ α , and The following proposition reveals the relationship between CMN and BSN . For its proof, please refer to [10].

Remark 3.5. Similar to the scalar ones, for two indexes
Let I be an interval as in Section 2, and Π(I) be the collection of all bounded subintervals of I. Consider the map θ ∶ I → A. When we say θ is order-continuous, we mean that for any nest of intervals {J k ∈ Π(I) ∶ k = , , ⋯} shrinking to t, the limit lim always holds, where θ − J and θ + J denote the order in mum 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].
Obviously, both of them are linear spaces according to the sum and scalar multiplication of abstract-valued functions, and L (I, X θ(⋅) ) ⊆ L − (I, X θ(⋅) ).
For each positive integer n, let t n,k = kT n or t n,k = k n , J n, = [ , t n, ], J n,k+ = (t n,k , t n,k+ ] and Obviously, {θ ± n } is decreasing (increasing) in n and converging to θ(t) as n → ∞ for all t ∈ I. Similar to the constant ones, for every n ∈ N, function space ). There is a natural relation among the three types of function spaces mentioned above, that is .
. Then for every λ > , we have Thus there is sequence of positive integers, say {k i }, satisfying k i < k i+ for all i ∈ N, lim i→∞ k i = ∞, and Especially we have This infers that series a.e. on I, and the limit function f belongs to L (I, X θ(⋅) ).
Taking any λ > and ε > , there exists i ∈ N such that i > λ and i < ε, thus using inequality (7), Fatou's lemma together with the lower semicontinuity of θ(t) , we obtain The following propositions can be proved by inequality (6), continuity of { α }, Fatou's lemma, together with the unit ball property.

Proposition 3.8. Suppose that I is a bounded interval, then
and there exists a constant C > such that for some constant C > .
Assume that {X α } generated by { α } is a dense BSN , i.e. X α is a dense subspace of X α whenever α ≺ α . It is easy to see that, under this situation S(I, X α + ) is contained in L (I, X θ(⋅) ), consequently for every is measurable. Moreover, by invoking Proposition 2.3 we can prove that, if every modular α satis es the ∆ −condition, then for each J ∈ Π(I), S(J, . Analogous to the range-invariant ones, we can de ne the space of strongly measurable functions with varying ranges, that is Remark 3.11. In this de nition, the set S(I, X α + ) can be replaced by L (I, X α + ), both of which are contained in L (I, X θ(⋅) ). As a result, one can easily check that L + (I, X θ(⋅) ) is a subspace of L (I, X θ(⋅) ).

Theorem 3.15.
Besides the ∆ −condition of { α } and the density assumption of {X α }, assume that X α + is separable. Then the function space L θ(⋅) This theorem is a straight consequence of Proposition 3.12.
For each α ∈ A, denote by * α the Fenchel duality of α , i.e. * Since α is a semimodular, * α is also a semimodular on X * α , and the semimodular space derived by * α is exactly X * α itself (see [2, §2.2]). De ne˜ * then we obtain another family of semimodulars de ned on X * α + , called the dual modular net or in symbol DMN of { α }. Since for each α ∈ A, the e ective domains and the induced semimodular spaces are equal, in the coming arguments, we will not distinguish˜ * α and * α , and prefer to use Suppose α ≺ α , then X * α ↪ X * α , and for all ξ ∈ X * α , by (6) we have * Similar to Proposition 3.4, from this property we can show that the dual space family {X * α ∶ α ∈ A}, where A takes the inverse order, is a uniformly bounded net. Moreover, assume that the function α ↦ α is sequently continuous, in other words, if {α k } converges to α in A, then for all u ∈ X, the limit holds. Under this assumption, we can deduce that, for all sequences holds for all ξ ∈ X * α + , which in turn leads to the successive property of {X * α }. Unfortunately, the inverse inequality, hence continuity of {X * α } can not be guaranteed under present situations. For the sake of convenience, hereinafter, we always assume that X * α + ⊆ X, and the DMN { * α } is assumed to be a CMN de ned on X. We also assume that the BSN {X α } and its dual net {X * α } are compatible, i.e. ⟨ξ, u⟩ α = ⟨ξ, u⟩ α provided u ∈ X α , ξ ∈ X * α and α ≺ α . 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 satis ed: -{ α } satis es the ∆ −condition, and R(∂ α ) = X * α for all α ∈ A, -{X α } is a dense BSN , and X α + is separable, for every α ∈ A, * α is a modular, and -X * α has the Radon-Nikodym's property w.r.t. every J ∈ Π(I). Then the dual space L θ(⋅) + (I, X θ(⋅) ) * is equivalent to L * θ(⋅) (I, X * θ(⋅) ) in the sense of isomorphism.
Given a function f ∈ L θ(⋅) (I, X θ(⋅) ), for each n ∈ N, let f n = f χ [ ,n] , then we obtain an approximate sequence of f in L θ(⋅) (I, X θ(⋅) ) satisfying Let n → ∞, using the fact lim n→∞ ⟨⟨Λ, Thus Λ = Λ ξ by the arbitrariness of f . The remaining task for us is to show ξ ∈ L * θ(⋅) (I, X * θ(⋅) ). For this purpose, notice that the e ective domain D( α ) is equal to X α and the latter is separable, so the dual modular * α can be represented by * where {v k } is a countable dense subset of X α . By the density of {X α }, if we take {v k } as the dense sequence of X α + with v = , then (12) holds with α = θ(t) and η ∈ X * θ(t) for all t ∈ I. For each n ∈ N, de ne Obviously, {r n } is a nondecreasing sequence of nonnegative (v = ) measurable functions converging to * θ(t) (ξ(t)) almost everywhere. Moreover, there is a sequence of simple functions {s n } ⊆ S(I, X α + ) such that r n (t) = ⟨ξ(t), s n (t)⟩ θ(t) − θ(t) (s n (t)).
Due to the facts S(I, X α + ) ⊆ L θ(⋅) (I, X θ(⋅) ) and Λ L θ(⋅) (I,X θ(⋅) ) * = , we have is the dual modular of Φ θ(⋅) . Taking limit of the second line as n → ∞, we obtain Finally by means of scaling transformation, we can obtain the desired estimate 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
). This is a natural extension of that of the scalar case.
Theorem 3.19. Suppose the following conditions are all satis ed.
Given a CMN { α }, assume that it is uniformly convex, in other words, every α is uniformly convex, and for each ε ∈ ( , ), the corresponding number δ ∈ ( , ) appearing in (5) is independent of α.   where notations p + and p − denote the essential supremum and in mum of p on Ω respectively. For any p ∈ P b (Ω), functional is a continuous modular on the linear space X = L (Ω), which induces a separable Banach space X p ∶= L p(x) (Ω). Evidently p satis es the ∆ −condition with the function ω(t) = t p + . If in addition p − > , then p is uniformly convex, its dual modular * p equals p ′ , and the dual space L p(x) (Ω) * is equivalent to L p ′ (x) (Ω).

Here p ′ (x) is the conjugate exponent of p(x), that is p(x) + p ′ (x) = for a.e. x ∈ Ω. It is also easy to see that, L p (x) (Ω) is a dense subspace of L p (x) (Ω) provided p (x) ≤ p (x) a.e. on Ω.
Fix two numbers p andp in [ , ∞) with p ≤p, let Then equipped with the order: p ≺ q by p(x) ≤ q(x) a.e. on Ω, and the topology determined by: p n → p in A b if and only if p n (x) → p(x) a.e. on Ω, A b becomes a BT L. Meanwhile, { p ∶ p ∈ A b } is a CMN de ned on X satisfying the ∆ −condition with the common function ω(t) = tp, and {X p ∶ p ∈ A b } is a dense regular BSN generated by { p } (cf. [11]). Assume that I = [ , T], and Q = I × Ω is a cylinder. Recall that, each u ∈ L (Q) has an X−realization Pu in L (I, X) satisfying Pu(t)(x) = u(t, x) for a.e. x ∈ Ω and a.e. t ∈ I, and conversely, each u ∈ L (I, X) has a scalar realizationũ in L (Ω) satisfyingũ(t, x) = u(t)(x) for a.e. (t, x) ∈ Q. Moreover, for all q ∈ ( , ∞), the projection P ∶ L q (Q) → L q (I, L q (Ω)) is a linear isometrical isomorphism with the inverse P − u =ũ. If q ∈ P b (Ω) is a variable exponent, then P ∶ L q(⋅) (Q) → L q − (I, L q(⋅) (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 X−realization, or an X−valued function and its scalar realization without any other remarks, if there is no confusion arising.

Functionals and Operators On L θ(⋅) (I, X θ(⋅) )
In this section we will study some functionals and operators on the function space L θ(⋅) (I, X θ(⋅) ), including the subdi erential of Φ θ(⋅) , whose representation will be taken into account. For this purpose, we need the coercive assumption on α , * α , as well as Φ θ(⋅) and Φ * θ(⋅) . Coercivity, which says is an important property of a lower semicontinuous (or lsc for short) and proper convex function de ned on a Banach space X. Using the coercive property of , we can obtain the boundedness of a sequence in X under some situations. For example, if there is a sequence {u n } ⊆ X satisfying (u n ) u n X ≤ K for some K > , then there is a constant C > depending only on K such that u n X ≤ C for all n ∈ N.
It is easy to check that if α is a coercive modular, then its dual * α satis es Dom( * α ) = X * α . As a matter of fact, taking any ξ ∈ X * α , by the coercivity of α , there is a constant M > for which α (u) In general, coercivity of Φ θ(⋅) could not be derived from the coercive assumption of all the α s naturally. Under some special conditions, however, all of α , α ∈ A and Φ θ(⋅) are coercive simultaneously. The following assumption, which is called strong coercivity of { α }, is a desired one. where By (15), there is a constant K ≥ such that γ − (s) ≤ s whenever s ≥ K. Now taking any α ∈ A and u ∈ X α with u α ≥ K, and using (14), we can deduce that which combined with (15) yields the coercivity of α . Furthermore, for any u ∈ L θ(⋅) (I, X θ(⋅) ) with u L θ(⋅) (I,X θ(⋅) ) ≥ , we have that which shows the coercivity of Φ θ(⋅) . In the following arguments, we also need the strict convexity of * α for all α ∈ A, and the weak lowersemicontinuity of { α } and { * α }, which says -For any sequence {α k } approaching β in A and any sequence {u k } ⊆ X β converging weakly to u in X β , {ξ k } converging star-weakly to ξ in X * β , we have Consider the subdi erential operator ∂ α . Similar to Remark (2.8), we can check that ∂ α is single-valued provided * α is strict convex. Furthermore, by the coercivity of α , we can also nd that ∂ α ∶ X α → X * α is a demicontinuous, monotone and coercive operator, whose range is the whole space X * α . Since for all u ∈ X α , α (u) can be regarded as the extension of the traditional dual map where not modulars but norms of X and X * α are involved.
Taking integrations on both sides and using generalized Hölder's inequality, we have Then by the coercivity of Φ * α and the boundedness of {ξ k } in L α (I, X α ), we get the boundedness of {ξ k } in L * α (I, X * α ). Therefore there is a subsequence, say {ξ k } itself, convergent to someξ weakly in L * α (I, X * α ). Suppose that {v i } is a countable dense subset of X α , then for every two positive integers i, n, we have It follows that the scalar function h k (t) = ⟨ξ k (t) −ξ(t), v i ⟩ α is convergent to in measure on I n . As a result, {h k } has a subsequence convergent to a.e. on I n . Then by means of the diagonalizing method, we can nd another subsequence, denoted still by {h k } such that lim k→∞ h k (t) = for a.e. t ∈ I, which combined with the boundedness of {ξ k (t)} derived from (17) and the density of {v i } in X α , results in the weak convergence of ξ k (t) toξ(t) in X * α as k → ∞ for a.e. t ∈ I. Now taking limits in (17), and using continuity of α and weak lower-continuity of * α , we obtain * which in turn tieldsξ(t) = ∂ α (u(t)) for a.e. t ∈ I. Thus the lemma has been proved sinceξ ∈ L * α (I, X * α ). In studying the subdi erential of the Nemytzkij functional of a series of modulars, we always assume that I = [ , T].

Remark 4.3.
In terms of Lemma 4.1 and Remark 3.17, we can claim that for every u ∈ L α (I, X α ), the subdi erential ∂Φ α (u) equals ∂ α (u(⋅)). Moreover, if we drop the strict convexity assumption of * α , then we have the classical representation:

Similarly, under the conditions of Proposition 4.2, we have ∂Φ
if the strict convexity assumptions of * α for all α ∈ A are moved.
The following theorem is a natural corollary of Proposition 4.2 and Remark 4.3.

Remark 4.5. Due to the facts that
Suppose that B is another BT L, {ϕ β ∶ β ∈ B} is another CMN on Y ⊆ X and {V β ∶ β ∈ B} is the corresponding BSN generated by {ϕ β }. We say {ϕ β } is stronger than { α }, we mean that -V β is imbedded continuously and densely in X α , and the dual product ⟨ξ, u⟩ has the same value in both V * β × V β and X * α × X α for all u ∈ V β and ξ ∈ X * α , and all α ∈ A, β ∈ B.
Under this assumption, for all C > , is a bounded and weakly closed subset of V β . Hence by the inclusion V β ↪ X α , we have Lemma 4.6. If V β is re exive, then ϕ β is also a lower semicontinuous and convex function on X α .
Suppose that ϑ ∶ I → B is also an order-continuous map, then based on the above results and the continuity of {ϕ β }, we can check that Proposition 4.8. Assume that for every β ∈ B, the modular space V β is re exive. Then for each u ∈ L (I, X θ(⋅) ), functions {ϕ ϑ − n (⋅) (u(⋅))} ∞ n= are all measurable, hence as the limit function, ϕ ϑ(⋅) (u(⋅)) is also measurable.
Lemma 4.9. Suppose that ϕ β satis es the ∆ −condition, and V β is a re exive and separable space. Then a function u ∈ L (I, X α ) is also a member of L ϕ β (I, V β ) provided Φ ϕ β (u) < ∞.
Proof. By the condition Φ ϕ β (u) < ∞, it su ces to show the inclusion u ∈ L (I, V β ). Taking any r > and u ∈ V β , denote by By the unit ball property, we know that . Thus for any subset E of V β , one can check that Take an arbitrary nonempty closed subset F of V β . By the separability of V β , F has a countable dense subset {v k } making Evidently, for each k ∈ N, function t ↦ u(t) − v k belongs to L (I, X α ), so the set E k,n is measurable for all n ∈ N. Consequently as the intersection and union of countable measurable sets, {t ∈ I ∶ u(t) ∈ F} is also measurable, which leads to the measurability of u as a V β −valued function. Since V β is separable, we have that u ∈ L (I, V β ), and the proof has been completed.

Example 4.11. Let us pay attention to the Sobolev space of variable exponent type
where q ∈ P b (Ω) with the notation P b (Ω) introduced in Example 3.22, and D i u = ∂u ∂x i denotes the i−th weak derivative of u. Recall that endowed with the norm W ,q(x) (Ω) turns to be a separable Banach space. It is uniformly convex, and of course, re exive provided q − > . Assume that ∂Ω ∈ C and q ∈ C(Ω) is a log-Hölder continuous exponent, or q ∈ P ω log (Ω) in symbol, which means that where ω ∶ [ , ∞) → [ , ∞) is a nondecreasing function ful lling ω( ) = and Under this situation, C ∞ (Ω) is dense in W ,q(⋅) (Ω), and W ,q(x) (Ω) can be de ned as the complement of remains true. Therefore W ,q(⋅) (Ω) is topologically equivalent to the homogeneous Sobolev space D ,q(⋅) (Ω) (see [2,Ch. 8,9] for relative discussions), and it can be regarded as a semimodular space derived from X = L (Ω) by the modular if u ∈ L (Ω) W , (Ω).

Bochner-Sobolev spaces of modular-modular type and applications in doubly nonlinear di erential equations
We begin with the Bochner-Sobolev space of range-xed type, that is where u ′ denotes the derivative of u in the sense of distribution, i.e. for all ξ ∈ X * α and all γ ∈ C ∞ (I), equality holds. It is easy to check that, endowed with the norm which is equivalent to inf λ > ∶ Φ α (u λ) + Φ α (u ′ λ) ≤ , W , α (I, X α ) turns to be a Banach space.
Theorem 5.1. Function space W , α (I, X α ) can be embedded into the space of continuous functions C(I, X α ). If in addition V β is embedded into X α compactly, then W , α (I, X α ) ∩ L ϕ β (I, V β ) is embedded compactly into L α (I, X α ).
for some C > independent of u ∈ F. Then for any u ∈ F and < h < min{ , T } and t, t + h ∈ I, we have Taking any r ∈ ( , T), consider the average operator M r on L α (I, X α ) de ned by Obviously, for all u ∈ L ϕ β (I, V β ), M r u ∈ C([ , T − r], V β ) with the estimate Moreover, due to the boundedness of F in W , α (I, X α ) and the estimate (21), precompactness of the set can be reached (refer to [16,17]). In addition, from (21), one can deduce that , X α ) as r → uniformly for u ∈ F. This fact, combined with the precompactness of F r in C([ , T − r], X α ) for every xed r ∈ ( , h], leads to the precompactness of F in L α ([ , T − h], X α ). The nal conclusion comes if we make the same discussions on the setF = {ũ(t) = u(T − t) ∶ u ∈ F} (see [16]).
Given two CMN s { α ∶∈ A} and {ϕ β ∶ 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 Similarly, equipped with the norm ) becomes a Banach space.
, and p (t)δ(t) <q (t) for all t ∈ I. Suppose also -V ϑ − ↪↪ X α + and there is a constant C > such that -(X α − , V ϑ(t) ) δ(t) ↪ X θ(t) uniformly for t ∈ I, in other words, where notation (X α − , V ϑ(t) ) δ(t) represents the real or complex interpolation space between X α − and V ϑ(t) with the index δ(t); for all t ∈ I, and min u .
Proof. Firstly, by (23) and Remark 4.8 in [11], we have that Thus from Theorem 5.1 and its corollary, it su ces to show that a sequence {u k } bounded in L ϕ ϑ(⋅) (I, V ϑ(⋅) ) and convergent in L p(⋅) (I, X α − ) for all p ∈ P b (I, X α ) is convergent in Lp (⋅) (I, X θ(⋅) ) de nitely. Without loss of generality, assume that the limit of {u k } is . Take K > so close to thatp (t)δ(t)K ≤q (t) andp (t)( − δ(t))K ′ ≥ ( K + K ′ = ) for all t ∈ I, then by (22), we have which leads to the desired conclusion.

Remark 5.4.
Under all the hypotheses of Theorem 5.3 with the compact embedding of V ϑ − into X α + replaced by the following condition uniformly for α ∈ A and β ∈ B, we have for all u ∈ W , θ(⋅) (I, X θ(⋅) ) ∩ L ϕ ϑ(⋅) (I, V ϑ(⋅) ), thus under the condition
This is an important inequality for later use.
Proof. Boundedness of F comes from (31) immediately. For the weak continuity, suppose that {u n } is a sequence of L θ(⋅) (I, X θ(⋅) ) converging to u strongly, that is I θ(t) (u n (t) − u(t))dt → as n → ∞.
Putting (32) and (33) together with the same subset E, we obtain E ⟨f (t, u(t)) − ξ(t), v⟩ θ(t) dt = , which implies ⟨f (t, u(t)) − ξ(t), v⟩ θ(t) = a.e. on E ε and eventually a.e. on I by the arbitrariness of E and ε respectively. Suppose that {v k } is a dense and countable subset of X α + , then there is a subset E of I with zero complement on which (34) holds with v replaced by v k for all k ∈ N. Finally, using the density of X α + in X θ(t) , we deduce that f (t, u(t)) = ξ(t) in X * θ(t) on E . Therefore F(u) = ξ and F(u n ) ⇀ F(u) in L * θ(⋅) (I, X * θ(⋅) ) as n → ∞. Thus the proof has been completed.
Evidently, every member of S is a weak solution of the inclusion u(t)) a.e. on I.
By taking δ > and µ > as in Remark 5.6 and letting < δ ≤ δ , ≤ µ ≤ µ , we can claim that {λ − u ∶ u ∈ S} is bounded in L θ(⋅) (I, X θ(⋅) ). Therefore, there is a constant C > independent of λ such that u L θ(⋅) (I,X θ(⋅) ) ≤ C for all u ∈ S. 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 xed point of F. Remark 5.11. From the demicontinuity of F and the compactness of the inverse (D θ(⋅) + ∂Φ ϕ ϑ(⋅) ) − , 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: where K ∶ X θ( ) → X θ(T) is a bounded linear operator with other conditions unchanged.