Linear Functional Equations and their Solutions in Lorentz Spaces
aa r X i v : . [ m a t h . C A ] J a n LINEAR FUNCTIONAL EQUATIONS AND THEIR SOLUTIONSIN LORENTZ SPACES
JANUSZ MORAWIEC AND THOMAS ZÜRCHER
Abstract.
Assume that Ω ⊂ R k is an open set, V is a separable Banach spaceover a field K ∈ { R , C } and f , . . . , f N : Ω → Ω, g , . . . , g N : Ω → K , h : Ω → V are given functions. We are interested in the existence and uniqueness ofsolutions ϕ : Ω → V of the linear functional equation ϕ = P Nk =1 g k · ( ϕ ◦ f k )+ h in Lorentz spaces. Introduction
Throughout this paper we fix k, N ∈ N , an open set Ω ⊂ R k , a separableBanach space ( V, k·k V ) over a field K ∈ { R , C } and functions f , . . . , f N : Ω → Ω, g , . . . , g N : Ω → K and h : Ω → V . We are interested in the existence anduniqueness of a special solution ϕ : Ω → V of the following linear equation(1) ϕ ( x ) = N X n =1 g n ( x ) ϕ ( f n ( x )) + h ( x ) . Different solutions of equation (1) have been studied by many authors (e.g. [12, Chap-ter XIII], [13, Chapter 6], [3, Chapter 5], [2, Section 4] and the references therein).In order to specify what special solution we are talking about, we need to introducesome notations. But before, let us note that this paper is a continuation of theauthors’ paper [19] and a reader who is already familiar with the content of thatpaper may want to jump to Section 3.Denote by F the linear space of all functions ψ : Ω → V and fix a subspace F of F . Then define the operator P : F → F by(2) P ψ = N X n =1 g n · ( ψ ◦ f n ) , and observe that it is linear and equation (1) can be written in the form(3) ϕ = P ϕ + h . If equation (1) has a solution ϕ ∈ F such that P ϕ ∈ F , then h ∈ F . Con-versely, if h ∈ F , then for every solution ϕ ∈ F of equation (1) we have P ϕ ∈ F .Therefore, if we want to look for solutions of equation (1) in F , then it is quitenatural to assume that h ∈ F and(4) P ( F ) ⊂ F . Mathematics Subject Classification.
Primary 47A50; Secondary 26A24, 39B12, 47B38.
Key words and phrases. linear operators, approximate differentiability, Luzin’s condition N,functional equations, Lorentz spaces.
Remark 1.1 (see [19, Remark 1.1], cf. [18, Remark 1.2]) . Assume that F isequipped with a norm, h ∈ F , and the operator P given by (2) satisfies (4) andis continuous. If the series(5) ∞ X n =0 P n h converges, in the norm, to a function ϕ ∈ F , then (3) holds.From now on, the series (5) will be called the elementary solution of equation (1)in F , provided that it is a well-defined solution of equation (1) belonging to F .Let us note that it can happen that equation (1) has a solution in F , however itselementary solution in F can fail to exist (see [18, Example 1.4]).The investigation of the existence of the elementary solution of equation (1) inthe case F = L ([0 , , R ) was motivated by [21] and studied in [18]. Next, inspiredby [17], the existence of the elementary solution of equation (1) in the case where F is a generalized Orlicz space was examined in [19].The basic result on the existence and uniqueness of the elementary solution ofequation (1) in F reads as follows. Theorem 1.2 (see [19, Theorem 1.2], cf. [18, Theorem 3.2]) . Assume that k·k isa complete norm in F and let h ∈ F . If the operator P given by (2) satis-fies (4) and is a contraction with contraction factor α , then the elementary solutionof equation (1) in F exists, it is the unique solution of equation (1) in F and k P ∞ k = m P k h k ≤ α m − α k h k . 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 alsoof measure zero. When we will integrate a function Φ: X → V , we will use theBochner integral (for details see e.g. [7, Sections 3.1 and 3.2]). Recall that a functionΦ: X → V is Bochner–measurable if it is equal almost everywhere to the limit of asequence 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 − n ( { v } ) ismeasurable for every v ∈ V . As we will work with Bochner–integrable solutions ofequation (1), we need the following observation. Lemma 2.1 (see [19, Lemma 2.1]) . Assume that ( X, M , µ ) is a complete σ –finitemeasure space. Let F : X → X , H : X → V and G : X → K be measurable functions.If for all sets N ⊂ X of measure zero the set F − ( N ) is also of measure zero, thenthe functions H ◦ F and G · ( H ◦ F ) are measurable. The next result we want to apply is a change of variable formula from [6]. Toformulate 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 ∈ Ω if for every ε > (cid:26) x ∈ Ω \ { x } : k F ( x ) − F ( x ) − L ( x − x ) kk x − x k < ε (cid:27) has x as a density point (see [24, Section 2], cf. [22, Chapter IX.12]). We saythat F is approximately differentiable at x if the approximate differential of F at x exists. To simplify notation, we will denote the approximate differential of INEAR EQUATIONS IN LORENTZ SPACES 3 a function F : Ω → R k at x by F ′ ( x ). Moreover, if a function F : Ω → R k isalmost everywhere approximately differentiable, then as usual we denote by F ′ thefunction Ω ∋ x F ′ ( x ), adopting the convention that F ′ ( x ) = 0 for every point x ∈ Ω at which F is not approximately differentiable. If E ⊂ Ω, then the function N F ( · , E ): R k → N ∪ {∞} defined by N F ( y, E ) = card( F − ( y ) ∩ E )is called the Banach indicatrix of F .As we are working with functions equal almost everywhere, we need the followingobservation. Lemma 2.2 (see [18, Lemma 2.1], cf. [19, Lemma 2.1]) . Let F , F : Ω → R k be func-tions such that F = F almost everywhere. If F is approximately differentiablealmost everywhere, then F is as well. Moreover, whenever F or F is approxi-mately differentiable at a point, the other function is as well, and the approximatederivatives agree at this point. Now we are in a position to formulate the change of variable formula in which J F denotes the determinant of the Jacobi matrix of F . Theorem 2.3 (see [6, Theorem 2]) . Assume that F : Ω → R k is a measurablefunction satisfying Luzin’s condition N and being almost everywhere approximatelydifferentiable. If H : R k → R is a measurable function, then for every measurableset E ⊂ Ω the following statements are true: (i) The functions ( H ◦ F ) | J F | and HN F ( · , E ) are measurable; (ii) If H ≥ , then (6) Z E ( H ◦ F )( x ) | J F ( x ) | dx = Z R k H ( y ) N F ( y, E ) dy ;(iii) If one of the functions ( H ◦ F ) | J F | and HN F ( · , E ) is integrable ( for ( H ◦ F ) | J F | integrability is considered with respect to E ) , then so is the otherand (6) holds. Now we are ready to formulate the main assumption about the functions thatwere fixed at the beginning of this paper. The assumption reads as follows.(H)
The functions f , . . . , f N are measurable and almost everywhere approxi-mately differentiable and satisfy Luzin’s condition N. For all n ∈ { , . . . , N } and sets M ⊂ R k of measure zero the set f − n ( M ) is of measure zero.There exists K ∈ N such that for every n ∈ { , . . . , N } the set { x ∈ Ω :card f − n ( x ) > K } is of measure zero. The functions g , . . . , g N and h aremeasurable. Moreover, L = max { l ∈ { , . . . , N } : l n ,...,n l > for some n < n < · · · < n l } , where l n ,...,n l is the k -dimensional Lebesgue measure of the intersection T li =1 f n i (Ω) for all l ∈ { , . . . , N } and distinct n , . . . , n l ∈ { , . . . , N } . Lorentz spaces of complex valued functions
In this paper we are interested in the existence of the elementary solution ofequation (1) in F being a Lorentz space. In fact, we are interested in assumptionsguaranteeing that the elementary solution of equation (1) in a given Lorentz spaceexists, and moreover, that equation (1) has no other solutions in this Lorentz space. J. MORAWIEC AND T. ZÜRCHER
Lorentz spaces were introduced originally in [14]. They play an important rolewhen studying generalizations of Sobolev maps and are used to obtain sharp con-ditions for Sobolev maps to be differentiable almost everywhere or to send sets ofmeasure zero to sets of measure zero (see [10] for more details). Lorentz spacesare also important in interpolation theory (see e.g. [5, Theorem 3.5.15] or [4, The-orem 4.4.13]).We begin our investigations with Lorentz spaces that consist of complex or realvalued functions, i.e. with the case where V = K . For the convenience of the reader,following [16] we recall some basic definitions and facts for our need. More detailson Lorentz spaces can be found e.g. in [4, 5, 23]. Definition 3.1 (see [16, Definition 2.2]) . A non-decreasing left-continuous andconvex function Ψ: [0 , ∞ ) → [0 , ∞ ] is said to be a Young function , iflim t → Ψ( t ) = Ψ(0) = 0 and lim t →∞ Ψ( t ) = ∞ . From now on the symbol Ψ is reserved for Young functions only.
Definition 3.2 (see [16, Definition 2.4]) . A Young function ψ : [0 , ∞ ) → [0 , ∞ ) issaid to satisfy condition ∆ globally , if there exists a real number d ∈ (1 , ∞ ) suchthat ψ (2 t ) ≤ dψ ( t ) for every t ∈ (0 , ∞ ) . To the end of this paper we fix a strictly increasing Young function ψ satisfyingthe condition ∆ globally and such that(7) lim t → ψ ( t ) t = lim t →∞ tψ ( t ) = 0 . According to [16, page 35] the fixed function ψ has left-sided and right-sidedderivatives, which coincide except on a possibly countable set. When we write ψ ′ r ,we refer henceforth to the right-sided derivative. Note that ψ satisfies (7) if andonly if 0 < ψ ′ r ( t ) < ∞ for every t ∈ (0 , ∞ ), ψ ′ r (0) = 0 and lim t →∞ ψ ′ r ( t ) = ∞ .We now define a function τ : [0 , ∞ ) → [0 , ∞ ) by putting τ ( t ) = ( , if t = 0 , ψ ( t ) , if t ∈ (0 , ∞ ) . Note that the just defined function τ is a strictly increasing Young function satis-fying condition ∆ globally andlim t → τ ( t ) t = lim t →∞ tτ ( t ) = 0(see [16, page 68]).Denote by µ the k -dimensional Lebesgue measure on R k and by M K the spaceof all Lebesgue measurable functions from Ω to K . Definition 3.3 (see [16, Definition 9.1]) . If f ∈ M K , then the function µ f : [0 , ∞ ) → [0 , ∞ ] defined by µ f ( s ) = µ ( { x ∈ Ω : | f ( x ) | > s } )is said to be the distribution function of f . INEAR EQUATIONS IN LORENTZ SPACES 5
The linear space L ψ, (Ω , K ) consisting of all functions f ∈ L (Ω , K ) (i.e. f ∈ L ( K, K ) for every compact set K ⊂ Ω) satisfying the following condition Z ∞ τ − ( µ f ( s )) ds < ∞ is called the Lorentz space (see [16, Definition 9.2]). The Lorentz space equippedwith the norm(8) k f k L ψ, (Ω , K ) = Z ∞ τ − ( µ f ( s )) ds is a Banach space (see [16, page 68]; we will give a sketch of a proof of this fact atthe beginning of Section 4).The next observation relates Lorentz and Orlicz spaces. But before, let us recallthe definition of Orlicz spaces. Definition 3.4 (see [16, Definition 2.7]) . Let Ω ⊂ R k and let Ψ: [0 , ∞ ) → [0 , ∞ ]be a Young function. The Orlicz spaces L Ψ (Ω , K ) is the set of all measurablefunctions u : Ω → K such that Z Ω Ψ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) u ( x ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) dx < ∞ for some t > L Ψ (Ω , K ) with the Luxemburg norm k u k L Ψ (Ω , K ) = inf (cid:26) t > Z Ω Ψ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) u ( x ) t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) dx ≤ (cid:27) it becomes a Banach space (see e.g. [11, Theorem 2.4]). Lemma 3.5 (see [16, Lemmas 9.3 and 9.4]) . Assume that h ∈ M K . (i) If h ∈ L ψ, (Ω , K ) with k h k L ψ, (Ω , K ) = 1 , then there exists a Young func-tion Ψ such that (9) Z Ω Ψ( | h ( x ) | ) dx ≤ Z ∞ ( τ ′ r ) − (cid:16) ψ ′ ( t ) (cid:17) dt. (ii) If Ψ is a Young function satisfying (9) , then h ∈ L ψ, (Ω , K ) and k h k L ψ, (Ω , K ) ≤ k h k L Ψ (Ω , K ) . The next result is a counterpart of [19, Theorem 4.6] for Lorentz spaces.
Theorem 3.6.
Assume that (H) holds and let h ∈ L ψ, (Ω , K ) . If there exists areal constant α ∈ [0 , ) such that | g n ( x ) |≤ α min (cid:26) | J f n ( x ) | KL , N (cid:27) for all n ∈ { , . . . , N } andalmost all x ∈ Ω , (10) then the elementary solution of equation (1) in L ψ, (Ω , K ) exists, it is the uniquesolution of equation (1) in L ψ, (Ω , K ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = m P k h (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ψ, (Ω , K ) ≤ (2 α ) m − α k h k L ψ, (Ω , K ) . J. MORAWIEC AND T. ZÜRCHER
Proof.
We want to apply Theorem 1.2. For this purpose it suffices to check that
P h ∈ L ψ, (Ω , K ) and k P h k L ψ, (Ω , K ) < α for every h ∈ L ψ, (Ω , K ) with k h k L ψ, (Ω , K ) =1. Fix h ∈ L ψ, (Ω , K ) with k h k L ψ, (Ω , K ) = 1. By assertion (i) of Lemma 3.5 thereexists a Young function Ψ such that (9) holds. Thus h ∈ L Ψ (Ω , K ) and k h k L Ψ (Ω , K ) ≤
1. Applying [19, Remark 4.5 and Lemma 4.4] (note that we may apply theseresults as we are actually dealing with the norms of h and P h , respectively), weconclude that
P h ∈ L Ψ (Ω , K ) and k P h k L Ψ (Ω , K ) ≤ α . This jointly with assertion (ii)of Lemma 3.5 implies that P h ∈ L ψ, (Ω , K ) and k P h k L ψ, (Ω , K ) ≤ k P h k L Ψ (Ω , K ) ≤ α . (cid:3) Now, we fix and integer m > L ψ m , (Ω , K ) thatis generated by the Young function of the form ψ m ( t ) = mt m for every t ∈ [0 , ∞ )(see [5, Section 3.4.1]; cf. [10] where mappings with derivatives in those Lorentzspaces are considered). The following result follows from Theorem 3.6. Corollary 3.7.
Assume that (H) holds. Let m > and let h ∈ L ψ m , (Ω , K ) . If (10) holds with a real constant α ∈ [0 , ) , then the elementary solution of equation (1) in L ψ m , (Ω , K ) exists, it is the unique solution of equation (1) in L ψ m , (Ω , K ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = m P k h (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ψm, (Ω , K ) ≤ (2 α ) m − α k h k L ψm, (Ω , K ) . Lorentz spaces of vector valued functions
We begin with a definition, which collects the nice properties of Lebesgue spaces.To formulate it, we denote by M + K the subclass of M K of functions that are almosteverywhere nonnegative and accept that the symbol a k ↑ a means that ( a k ) k ∈ N ∈ [0 , ∞ ] N is increasing and converges to a ∈ [0 , ∞ ]. Definition 4.1 (see [5, Definition 3.1.1]) . A Banach function norm on (Ω , µ ) isa map ρ : M + K → [0 , ∞ ] such that for all functions f, g ∈ M + K , sequences ( f k ) k ∈ N of functions from M + K , scalars λ ≥ µ -measurable sets E ⊂ Ω, the followingconditions hold:(P1) ρ ( f ) = 0 ⇐⇒ f = 0 µ -a.e., ρ ( λf ) = λρ ( f ) and ρ ( f + g ) ≤ ρ ( f ) + ρ ( g );(P2) g ≤ f µ -a.e. = ⇒ ρ ( g ) ≤ ρ ( f );(P3) f k ↑ f µ -a.e. = ⇒ ρ ( f k ) ↑ ρ ( f );(P4) µ ( E ) < ∞ = ⇒ ρ ( χ E ) < ∞ ;(P5) µ ( E ) < ∞ = ⇒ R E f dµ ≤ C ( E ) ρ ( f ) with some C ( E ) < ∞ .If ρ is a Banach function norm on (Ω , µ ), then the set X K = { f ∈ M K : ρ ( | f | ) < ∞} is called a Banach function space ; as usual, we identify functions that are equal a.e.The basic result on Banach functions spaces reads as follows.
Theorem 4.2 (see [5, Theorem 3.1.3]) . Any Banach function space X K equippedwith the norm k f k X K = ρ ( | f | ) is a Banach space. INEAR EQUATIONS IN LORENTZ SPACES 7
Now, we will give a brief sketch of the fact that the Lorentz space L ψ, (Ω , K )equipped with the norm defined by (8) is complete. According to Theorem 4.2 itsuffices to show that the map ρ : M + K → [0 , ∞ ] given by ρ ( f ) = Z ∞ τ − ( µ f ( s )) ds is Banach function norm on (Ω , µ ).We only look at the more involved conditions of Definition 4.1. Let us focusfirst on the triangle inequality in (P1). For this purpose we need the followingdefinition in which we use the convention that inf ∅ = ∞ . Definition 4.3 (see [5, Definition 3.2.3]) . The non-increasing rearrangement ofan almost everywhere finite function f ∈ M K is the function f ∗ : [0 , ∞ ) → [0 , ∞ ]defined by f ∗ ( t ) = inf { λ ∈ [0 , ∞ ) : µ f ( λ ) ≤ t } . Switching from the distribution function µ f to the nonincreasing rearrangement f ∗ we have(11) ρ ( f ) = Z ∞ f ∗ ( τ ( s )) ds. We know that τ is convex, but following the proof of the theorem in [1], we can provethat it is even bi-Lipschitz on compact intervals. This allows us to use Theorem 2.3and obtain(12) ρ ( f ) = Z ∞ f ∗ ( s )( τ − ) ′ ( s ) ds. Since τ is convex, it has a nondecreasing right derivative (see e.g. [20, Theorem 1.3.3]or [9, Theorems 3.7.3 and 3.7.4]). Finally, replacing this derivative with a nonde-creasing function that is right continuous and agree with the original function almosteverywhere, we apply the following result. Proposition 4.4 (see [15, Proposition 2.7]) . If ϕ : (0 , ∞ ) → R is right continuous,nonnegative and nonincreasing, then the operator f Z ∞ ϕ ( t ) f ∗ ( t ) dt is subadditive. Condition (P3) can be proven by using the fact that τ − is continuous on compactintervals, whereas condition (P5) can be seen by applying [5, Proposition 3.2.5] andthe fact that t τ − ( t ) /t is nonincreasing, which follows from [1, Lemma].Banach function spaces are originally defined as spaces of K -valued functions. Wewill show that the corresponding spaces for functions with Banach space targetsare Banach spaces as well. We begin with formal definition of these spaces.Denote by M V the space of all Lebesgue measurable functions from Ω to V . Definition 4.5.
The set XV = { f ∈ M V : k f k V ∈ X K } is called a Banach function space ( of vector valued functions ); as in the K -valuedcase we identify functions that are equal a.e. J. MORAWIEC AND T. ZÜRCHER
It is easy to check that the formula(13) k f k XV = kk f k V k X K defines a norm on the Banach function space XV . We want to prove that the justdefined norm is complete. The next lemma, inspired by [5, Lemma 3.1.2], is thebasic step in the proof. Lemma 4.6.
Assume that ( f n ) n ∈ N ia a sequence of functions from the Banach func-tion space XV such that lim n →∞ f n ( x ) exists for almost all x ∈ Ω . If lim inf n →∞ k f n k XV < ∞ , then lim n →∞ f n ∈ XV and k lim n →∞ f n k XV ≤ lim inf n →∞ k f n k XV . Proof.
It is clear that the formula f ( x ) = ( lim n →∞ f n ( x ) , if lim n →∞ f n ( x ) exists0 , otherwisedefines a function belonging to M V .For every n ∈ N we define the function g n : Ω → [0 , ∞ ] by putting g n ( x ) = inf m ≥ n k f m ( x ) k V . Then for almost all x ∈ Ω we havelim n →∞ g n ( x ) = lim n →∞ inf m ≥ n k f m ( x ) k V = lim inf n →∞ k f n ( x ) k V = k f ( x ) k V . Hence(14) k f k XV = kk f k V k X K = k lim n →∞ g n k X K = lim n →∞ k inf m ≥ n k f m k V k X K . Since for all l, n ∈ N with l ≥ n we have inf m ≥ n k f m k V ≤ k f l k V , it followsby (P2) that k inf m ≥ n k f m k V k X K ≤ kk f l k V k X K . Therefore, k inf m ≥ n k f m k V k X K = inf l ≥ n k inf m ≥ n k f m k V k X K ≤ inf l ≥ n kk f l k V k X K , which jointly with (14) gives k f k XV ≤ lim n →∞ inf l ≥ n kk f l k V k X K = lim inf n →∞ kk f n k V k X K = lim inf n →∞ k f n k XV < ∞ . In consequence, f ∈ XV . (cid:3) Theorem 4.7.
The Banach function space XV equipped with the norm defined by (13) is a Banach space.Proof. We will follow the proof of [5, Theorem 3.1.3].Assume that ( f n ) n ∈ N is a Cauchy sequence of functions from XV and let ( g n ) n ∈ N be one of its subsequences such that k g n +1 − g n k XV ≤ n +1 for every n ∈ N . Setting g = 0, for every n ∈ N we put h n = g n − g n − and note that h n ∈ M V . Next forall x ∈ Ω and N ∈ N we put G N ( x ) = N X n =1 k h n ( x ) k V and G ( x ) = ∞ X n =1 k h n ( x ) k V . INEAR EQUATIONS IN LORENTZ SPACES 9
Then G N , G ∈ M K and(15) k G N k X K ≤ ∞ X n =1 kk h n k V k X K = ∞ X n =1 k h n k XV ≤ k h k XV + 1 < ∞ for every N ∈ N . Since G N ↑ G , it follows by (P3) that G ∈ X K .Fix ε > E ⊂ Ω such that µ ( E ) < ∞ . By (P5), we see thatlim N →∞ µ ( { x ∈ E : | G ( x ) − G N ( x ) | > ε } ) ≤ lim N →∞ ε Z E | G ( x ) − G N ( x ) | dµ ( x ) ≤ C ( E ) ε lim N →∞ k| G − G N |k X K = C ( E ) ε lim N →∞ ∞ X n = N k g n +1 − g n k XV = C ( E ) ε lim N →∞ N = 0 . Thus the sequence ( G N ) N ∈ N converges to G in Lebesgue measure on E . Apply-ing the Riesz theorem (see e.g. [8, Theorem 11.26]) there exists a subsequenceof ( G N ) N ∈ N which converges to G a.e. on E . Since the k -dimensional Lebesguemeasure is σ -finite, we can apply the denationalization method to obtain that thereexists a subsequence of ( G N ) N ∈ N which converges to G a.e. on Ω. As we know that G ∈ X K , it follows that the series P ∞ n =1 k h n ( x ) k V is finite for almost all x ∈ Ω.Thus ∞ X n =1 h n ( x ) ∈ V for almost all x ∈ Ω.Now, we define the function g ∈ XV by putting g ( x ) = ( lim n →∞ g n ( x ) , if P ∞ n =1 h n ( x ) ∈ V , otherwise . Our goal is to show that g is the limit of ( f n ) n ∈ N ; note that to achieve this, it isenough to show that it is the limit of ( g n ) n ∈ N .Fix m ∈ N . Thenlim inf n →∞ k g m − g n k XV = lim inf n →∞ (cid:13)(cid:13)(cid:13) n X k = m +1 h k (cid:13)(cid:13)(cid:13) XV ≤ lim inf n →∞ ∞ X k = m +1 k h k k XV ≤ m . (16)Applying now Lemma 4.6 we conclude that g m − g ∈ XV and k g m − g k XV ≤ lim inf n →∞ k g m − g n k XV . Hence g = g m − ( g m − g ) ∈ XV and by (16) we havelim m →∞ k g m − g k XV ≤ lim m →∞ lim inf n →∞ k g m − g n k XV ≤ lim m →∞ m = 0 . Hence ( g n ) n ∈ N converges to some g . Since ( f n ) n ∈ N is a Cauchy sequence and( g n ) n ∈ N one of its subsequences, it follows that ( f n ) n ∈ N converges to g , which com-pletes the proof. (cid:3) We end this paper with a counterpart of Theorem 3.6 for vector valued functions.We omit its proof as the boundedness of the considered operator can be proven bylooking at the norm of the function instead of at the function itself.
Theorem 4.8.
Assume that (H) holds and let h ∈ L ψ, (Ω , V ) . If there exists areal constant α ∈ [0 , ) such that | g n ( x ) |≤ α min (cid:26) | J f n ( x ) | KL , N (cid:27) for all n ∈ { , . . . , N } andalmost all x ∈ Ω , (17) then the elementary solution of equation (1) in L ψ, (Ω , V ) exists, it is the uniquesolution of equation (1) in L ψ, (Ω , V ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k = m P k h (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ψ, (Ω ,V ) ≤ (2 α ) m − α k h k L ψ, (Ω ,V ) . Acknowledgement.
The research was supported by the University of SilesiaMathematics Department (Iterative Functional Equations and Real Analysis pro-gram).
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