Linear functional equations and their solutions in generalized Orlicz spaces

Assume that Ω⊂Rk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^k$$\end{document} is an open set, V is a real separable Banach space and f1,…,fN:Ω→Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1,\ldots ,f_N :\Omega \rightarrow \Omega $$\end{document}, g1,…,gN:Ω→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_1,\ldots , g_N:\Omega \rightarrow \mathbb {R}$$\end{document}, h0:Ω→V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_0:\Omega \rightarrow V$$\end{document} are given functions. We are interested in the existence and uniqueness of solutions φ:Ω→V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :\Omega \rightarrow V$$\end{document} of the linear equation φ=∑k=1Ngk·(φ∘fk)+h0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0$$\end{document} in generalized Orlicz spaces.


Introduction
Throughout this paper we fix k, N ∈ N, an open set Ω ⊂ R k , a real separable Banach space (V, · V ) and functions f 1 , . . . , f N : Ω → Ω, g 1 , . . . , g N : Ω → R and h 0 : Ω → V . We are interested in solutions ϕ : Ω → V of the linear equation of the form Solutions of Eq. (1) have been studied by many authors in different classes of functions (for more details see e.g. [12,Chapter XIII], [13,Chapter 6], [2,Chapter 5], [1,Section 4] and the references therein). In this paper we are interested in the existence and uniqueness of solutions of Eq. (1) in generalized Orlicz spaces. This paper is a continuation of investigations started by the authors in [17], where solutions of Eq. (1) were studied in the space L 1 ([0, 1], R). The motivation to study Eq. (1) in the space L 1 ([0, 1], R) came from [20]. However, the interest to consider it in much more general spaces is inspired by [15].
We denote by F the linear space of all functions ψ : Ω → V and fix a subspace F 0 of F. Then we define the operator P : F 0 → F by P ψ = N n=1 g n · (ψ • f n ), (2) and we observe that it is linear and Eq. (1) can be written in the form Note also that if Eq. (1) has a solution ϕ ∈ F 0 such that P ϕ ∈ F 0 , then h 0 ∈ F 0 . Conversely, if h 0 ∈ F 0 , then for every solution ϕ ∈ F 0 of Eq. (1) we have P ϕ ∈ F 0 . Therefore, if we want to look for solutions of Eq. (1) in F 0 , then it is quite natural to assume that h 0 ∈ F 0 and We begin with the following counterpart of [17,Remark 1.2].
Remark 1.1. Assume that F 0 is equipped with a norm, h 0 ∈ F 0 and the operator P given by (2) satisfies (4) and is continuous. If the series ∞ n=0 P n h 0 converges, in the norm, to a function ϕ ∈ F 0 , then (3) holds.
From now on, the series (5) will be called the elementary solution of Eq. (1) in F 0 , provided that it is a well-defined solution of Eq. (1) belonging to F 0 . Let us note that it can happen that Eq. (1) has a solution in F 0 , however its elementary solution in F 0 can fail to exist (see [17,Example 1.4]). Following [17] we are interested in assumptions guaranteeing that the elementary solution of Eq. (1) in F 0 exists, and moreover, that Eq. (1) has no other solutions in F 0 . As it was mentioned at the beginning, in this paper we will focus on F 0 when it is a generalized Orlicz space.
Our first result is a simple generalization of [17,Theorem 3.2], essentially with the same proof. Theorem 1.2. Assume that · is a complete norm in F 0 and let h 0 ∈ F 0 . If the operator P given by (2) satisfies (4) and is a contraction with contraction factor α, then the elementary solution of Eq.

Preliminaries
Let (X, M, μ) and (Y, N , ν) be measure spaces. We say that G : X → Y satisfies Luzin's condition N if for every set N ⊂ Y of measure zero the set G(N ) is also of measure zero. When we integrate a function Φ : X → V , we will use the Bochner integral (for details see e.g. [8, Sections 3.1 and 3.2]).
Recall that a function Φ : X → V is Bochner-measurable if it is equal almost everywhere to the limit of a sequence of measurable simple functions, i.e., Φ(x) = lim n→∞ Φ n (x) for almost all x ∈ X, where each of the functions Φ n : X → V has a finite range and Φ −1 n ({v}) is measurable for every v ∈ V . As we will work with Bochner-integrable solutions of Eq. (1), we need the following observation.
Assume now that (G n ) n∈N is a sequence of measurable simple functions converging to G except on a set M of measure zero. Then (G n · (H n • F )) n∈N is a sequence of measurable simple functions converging to G · (H • F ) except on the set F −1 (N ) ∪ M .
The next result we want to apply is a change of variable formula from [6]. To formulate this theorem, we need to introduce some definitions and notions.
Let F : Ω → R k be measurable. We say that a linear mapping L : R k → R k is an approximate differential of F at x 0 ∈ Ω if for every ε > 0 the set has x 0 as a density point (see [24,Section 2], cf. [23, Chapter IX.12]). We say that F is approximately differentiable at x 0 if the approximate differential of F at x 0 exists. To simplify notation, we will denote the approximate differential of a function F : is called the Banach indicatrix of F . We omit the proof of the next lemma, as it is the same as the version included in [17] in the case where k = 1.
, then so is the other and (6) holds.
Now we are ready to formulate the main assumption about the functions that were fixed at the beginning of this paper. The assumption reads as follows.

Lebesgue spaces
We begin our considerations on the existence and uniqueness of the elementary solution of Eq. (1) in Lebesgue spaces L p (Ω, V ) of vector-valued functions. As the Lebesgue spaces of vector-valued functions are well known as natural generalizations of the classical Lebesgue spaces of real-valued functions (see e.g. [3] or [8, Chapter 3]), we will not define and describe them in details here.
Since we want to apply Theorem 1.2, we must know that the operator P given by (2) has the properties assumed in this theorem. The next two lemmas serve this purpose.

Lemma 3.1. Assume that (H) holds and let
. . , N} and almost all x ∈ Ω, (7) then the operator P given by (2) satisfies P (L p (Ω, V )) ⊂ L p (Ω, V ) and is continuous with Proof. The proof is similar to that of [17, Lemma 3.1].
Fix h ∈ L p (Ω, V ). First of all observe that applying assertion (i) of Theorem 2.3 with F = f n and H = 1 we conclude that the function N fn (·, Ω) is measurable for every n ∈ {1, . . . , N}. Hence the function h(·) V N fn (·, Ω) is also measurable for every n ∈ {1, . . . , N}. Next by Lemma 2.1 we see that the function g n · (h • f n ) is measurable for every n ∈ {1, . . . , N}, which implies that the function P h is measurable as well. Then, using (7) and Theorem 2.3, we obtain This yields and completes the proof.
Proof. If is enough to observe that It is well known that Lebesgue spaces of real-valued functions are Banach spaces (see e.g. [5,Theorem 6.6]). It turns out that the same is true in the case of vector-valued functions with Banach target spaces, basically with the same proof as in the real-valued case (see e.g. [8, Section 3.2]). Therefore, applying Theorem 1.2 jointly with Lemmas 3.1 and 3.2 we obtain the following result.
The next results concerns the space C(F, V ) of all continuous functions from a compact set F ⊂ R k to V equipped with the supremum norm · sup . Although it is not a Lebesgue space, we will formulate a counterpart of Theorem 3.3 for it. The reason is that its proof is based on the following lemma, the proof of which is the same as the proof of Lemma 3.2. Again, since C(F, V ) with the supremum norm is a Banach space (see e.g. [3,Introduction]), it follows that Theorem 1.2 jointly with Lemma 3.4 gives the following result.

Generalized Orlicz spaces
In this section we will focus on generalized Orlicz spaces (called also Musielak-Orlicz spaces) with values in Banach spaces. Such spaces are a known generalization of the classical Orlicz spaces, and hence they are more general than Lebesgue spaces. Generalized Orlicz spaces were introduced in the case of real valued function in [18] and then generalized also to functions taking values in vector spaces in [10]. There are many results obtained on generalized Orlicz spaces (see e.g. [7,16,19,25] and the references therein). For the convenience of the readers, following [10,11], we recall some basic definitions and facts for our needs. Denote by B(V ) the σ-algebra of all Borel subsets of V and by L k (Ω) the σ-algebra of all Lebesgue measurable subsets of Ω.  ) is even, convex, continuous at zero and lower semicontinuous for almost all x ∈ Ω, (iii) Φ(0, x) = 0 for almost all x ∈ Ω, (iv) there exist functions α, β : From now on the symbol Φ is reserved for N -functions only. Denote by M V the set of all measurable functions h : Ω → V ; as usual, two functions from M V that differ only on a set of measure zero will be considered as equal. Assume that M is a given non-empty subset of M V such that It turns out that L Φ M (Ω, V ) is a linear space; indeed it is enough to note that Typical generalized Orlicz spaces are variable exponent Lebesgue spaces ∞) and real numbers p, q ∈ [1, ∞). Additional interesting examples of generalized Orlicz spaces can be found in [7].

Theorem 4.2.
[see [10,Theorem 2.4]] The formula . Let us note that in the above theorem the assumption about the continuity at zero of Φ(·, x) in condition (ii) of Definition 4.1 can be omitted. However, we will need this assumption to simplify our results. For the same reason we will work only with the generalized Orlicz spaces L Φ MV (Ω, V ), denoted throughout this paper by L Φ (Ω, V ). The norm introduced in Theorem 4.2 is called the Luxemburg norm.  After the short introduction to the generalized Orlicz space, we pass to our investigations.
We are now ready to generalize Lemma 3.1 to generalized Orlicz spaces.
Note that in condition (12) all functions in the space L Φ (Ω, V ) are involved, which makes it a bit difficult to check. However, we can formulate a simple condition that involves no function of the space L Φ (Ω, V ) and implies (12).
Fix x ∈ Ω such that Φ(0, x) = 0 and Φ(·, x) is convex and even. Then for all v 1 , v 2 ∈ V and a, b ∈ R with |a| + |b| ≤ 1 we have This jointly with (14), the non-negativity of Φ, and (13) gives Combining Lemma 4.4 and Theorems 4.2 and 1.2, we obtain the following result.
We now introduce an interesting and widely studied class of N -functions.  Proof. It is enough to apply Theorem 4.6 and Remark 4.5 noting that Eq. 13 holds as Ψ does not depend on x.