Averaging principle and hyperbolic evolution equations
aa r X i v : . [ m a t h . A P ] O c t Averaging principle and hyperbolic evolutionequations
Aleksander Ćwiszewski Faculty of Mathematics and Computer ScienceNicolaus Copernicus Universityul. Chopina 12/18, 87-100 Toruń, Poland e-mail: [email protected] 12.04.2011
Abstract
An averaging principle is derived for the abstract nonlinear evolution equationwhere the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It gen-eralizes classical Henry’s results for perturbations of sectorial operators on fractionalspaces. It is also proved that the main hypothesis of the nonlinear averaging principleis satisfied for general hyperbolic evolution equations introduced by Kato.
We are concerned with the limit behavior with regard to λ → + of evolution systemsof the form ( P λ ) ˙ u ( t ) = A ( t/λ ) u ( t ) + F ( t/λ, u ( t )) , t > , where { A ( t ) } t ≥ is a family of operators generating C semigroups of bounded linear oper-ators on a Banach space E , F : [0 , + ∞ ) × E → E is a continuous map satisfying the localLipschitz condition with respect to the second variable and λ > is a parameter. Theso-called averaging principle is a well known tool in the theory of ordinary differential equa-tions, i.e., when E is finite dimensional. Roughly speaking, it says that if F is periodic intime, then trajectories of ˙ u ( t ) = F ( t/λ, u ( t )) converge to trajectories of the averaged equa-tion as λ → + (see [2]). This averaging idea is of importance when studying qualitativebehavior of nonautonomous equations. It enables to perceive the dynamics of a nonau-tonomous equation in terms of the related averaged one. For instance, by this approach,one may examine global attractors for dissipative equations, periodic solutions and otherdynamic features such as bounded or recurrent solutions. Therefore extending the methodto infinite dimension and applying it to partial differential equations is a natural and vitalissue attracting much attention. The averaging principle in the infinite dimensional casewas obtained by Henry [9] who assumed that the (independent of time) operator A is asectorial one on a Banach space E and F : [0 , + ∞ ) × E α → E , where E α , ≤ α < ,is the fractional power space determined by A , is bouneded and continuous. Averagingfor time dependent (set-valued) perturbations of a C group generator was considered byKamenskii, Obukhovskii and Zecca in [11], where A was a C semigroup generator and F was an upper semicontinuous k -set conctraction with respect to a measure of noncompact-ness. Averaging principle, in the context of attractors and Conley-Rybakowski index, for : 47J35, 47J15, 37L05 Key words : semigroup, evolution system, evolution equation, averaging.The research supported by the MNiSzW Grant no. N N201 395137. parabolic partial differential equations on R N was used by Antocci and Prizzi [1] and Prizzi[15]. Recently a version of averaging principle has been also obtained by the author in [3]where A was a C semigroup generator and F a time periodic continuous perturbation.In this paper we look for a general averaging scheme in the abstract operator settingwith time dependent A and apply it to hyperbolic evolution equations. We shall prove ageneral principle, a version of which can be stated as follows (cf. Theorem 2.2 and Remark2.10). Theorem 1.1
Let { R ( λ ) ( t, s ) } t ≥ s ≥ , λ > , be linear evolution systems on a separableBanach space E , corresponding to the problems (cid:26) ˙ u ( t ) = A ( t/λ ) u ( t ) , t > s,u ( s ) = ¯ u ∈ E. Suppose that ( A there are M ≥ and ω ∈ R such that k R ( λ ) ( t, s ) k ≤ M e ω ( t − s ) if t ≥ s ≥ ; ( A there exists a C semigroup { b S ( t ) } t ≥ of bounded linear operators on E with theinfinitesimal generator b A such that, for any ¯ u ∈ E and t, s ≥ with t ≥ , lim λ → + , ¯ v → ¯ u R ( λ ) ( t, s )¯ v = b S ( t − s )¯ u uniformly with respect to t , s from bounded intervals; ( A F : [0 , + ∞ ) × E → E is Lipschitz on bounded subsets and has sublinear growthuniformly with respect to the second variable; ( A for each ¯ u ∈ E , the set { F ( t, ¯ u ) | t ≥ } is relatively compact and there is a locallyLipschitz mapping b F : E → E such that, for any ¯ u ∈ E and h > , b F (¯ u ) = lim T → + ∞ , ¯ v → ¯ u T Z T F ( τ + h, ¯ v ) d τ uniformly with respect to h .Then, for any ( λ n ) in (0 , + ∞ ) and (¯ u n ) in E such that λ n → + and ¯ u n → ¯ u for some ¯ u ∈ E , the mild solutions u n : [0 , + ∞ ) → E of ( P λ n ) satisfying u n (0) = ¯ u n , n ≥ ,converge uniformly on bounded intervals to the mild solution of the averaged problem (cid:26) ˙ u ( t ) = b Au ( t ) + b F ( u ( t )) , t > ,u (0) = ¯ u . The assumptions ( A and ( A actually state that the averaging principle holds for thelinear equation. Obviously, it is always the case if A is independent of time and is an in-finitesimal generator of a C semigroup. We will verify (A1) and (A2) for hyperbolic typelinear evolution systems introduced by Kato – see Theorem 3.3. Assumption (A3) andthe separability of E are to assure the existence of unique mild solutions for initial valueproblems associated with ( P λ ) , the boundedness of solutions starting from bounded setsand the relative compactness of semiorbits of relatively compact sets (see ( H ) − ( H ) ).Finally, ( A simply says that F has the average b F . It is worth mentioning that ( A isfulfilled if F is almost periodic with respect to time (see [13]) and it is always the casewhen F is time-periodic. The obtained theorem generalizes those known in the literature– see Remark 2.6 adn besides the proof is rather straightforward.The paper is organized as follows. Section 2 is devoted to the general version of aver-aging principle while in Section 3 we are concerned with its verification for abstract linearhyperbolic evolution systems. Section 4 provides an example of application to first orderhyperbolic partial differential equations. Notation By R we denote the field of real numbers; by [ x ] we mean the integer (or floor) part of x ∈ R .If X is a metric space and B ⊂ X , then ∂B and clB stand for the boundary of B and theclosure of B , respectively. If x ∈ X and r > , then B ( x , r ) := { x ∈ M | d ( x, x ) < r } .If E is a normed space, then by k · k we denote its norm. If V is another normedspace then L ( V, E ) stands for the space of all bounded linear operators with domain V and values in E with the operator norm denoted by k · k L ( V,E ) or simply k · k if no confusionmay appear. Recall that a family of bounded linear operators { R ( t, s ) : E → E } t ≥ s ≥ on a Banachspace E is an evolution system if and only if R ( t, t ) = I , R ( t, s ) R ( s, r ) = R ( t, r ) , whenever t ≥ s ≥ r ≥ , and the mapping ( s, t ) R ( t, s )¯ u is continuous for any ¯ u ∈ E . Inthis section we deal with general evolution systems, i.e. we do not indicate how they aregenerated.Evolution systems come up naturally in equations involving time-dependent families oflinear operators. Namely, if { A ( t ) } t ≥ is a family of linear operators in a Banach space E satisfying suitable assumptions, then for any s ≥ and ¯ u ∈ E , the problem (cid:26) ˙ u ( t ) = A ( t ) u ( t ) , t > su ( s ) = ¯ u admits a unique solution u s, ¯ u : [ s, + ∞ ) → E (understood in an appropriate sense). Forinstance, this is the case if A ( t ) = A , for each t ≥ , with some A being a generatorof a C semigroup of bounded linear operators on E , as well as if { A ( t ) } t ≥ satisfies theso-called parabolic or hyperbolic conditions (see e.g. [16], [14] or [6]). Moreover, the formula R ( t, s )¯ u := u s, ¯ u ( t ) for t ≥ s defines an evolution system { R ( t, s ) } t ≥ s ≥ on E . In what follows we assume that cosideredevolution systems are generated by family time indexed families of operators in the abovemanner. We briefly say that the evolution system { R ( t, s ) } t ≥ s ≥ is determined by or cor-respond to the family { A ( t ) } t ≥ .Let { R ( t, s ) } t ≥ s ≥ be an arbitrary evolution system determined by a family { A ( t ) : D ( A ( t )) → E } t ≥ of linear operators in E . Consider the problem (cid:26) ˙ u ( t ) = A ( t ) u ( t ) + F ( t, u ( t )) , t ∈ [0 , ω ) ,u (0) = ¯ u. where ¯ u ∈ E , ω ∈ (0 , + ∞ ] and F : [0 , + ∞ ) × E → E is a continuous mapping. By a mildsolution of the above problem we understand a continuous function u : [0 , ω ) → E suchthat u ( t ) = R ( t, u + Z t R ( t, τ ) F ( τ, u ( τ )) d τ for any t ∈ [0 , ω ) . We shall say that a family { R ( µ ) } µ ∈ P of evolution systems, where P is a metric spaceof parameters, is continuous if, for any ¯ u ∈ E and ( µ n ) in P with µ n → µ , R ( µ n ) ( t, s )¯ u → R ( µ ) ( t, s )¯ u uniformly with respect to t ≥ s ≥ from bounded intervals.Now we pass to the averaging principle. For the sake of generality and future reference,we shall consider its parameterized version. To this end we take families of operators { A ( µ ) ( t ) } t ≥ , µ ∈ P , where P is a metric space of parameters, determining correspondingevolution systems { R ( µ ) ( t, s ) } t ≥ s ≥ , µ ∈ P , on a Banach space E . Assume that the family { R ( µ ) } µ ∈ P is continuous and that F : [0 , + ∞ ) × E × P → E is a continuous mapping. Letfamilies { A ( µ,λ ) ( t ) } t ≥ , µ ∈ P , λ > , be defined by A ( µ,λ ) ( t ) := A ( µ ) ( t/λ ) , t ≥ and F ( µ,λ ) : [0 , + ∞ ) × E → E , µ ∈ P , λ > , by F ( µ,λ ) ( t, ¯ u ) := F ( t/λ, ¯ u, µ ) , t ≥ , ¯ u ∈ E. The evolution system determined by { A ( µ,λ ) ( t ) } t ≥ , for µ ∈ P , λ > , is denoted by { R ( µ,λ ) ( t, s ) } t ≥ s ≥ .We shall assume that the following conditions hold ( H ) for any ¯ u ∈ E , µ ∈ P and λ > , the problem (cid:26) ˙ u ( t ) = A ( µ,λ ) u ( t ) + F ( µ,λ ) ( t, u ( t )) , t > u (0) = ¯ u admits a unique maximal mild solution u ( · ; ¯ u, µ, λ ) : [0 , ω ¯ u,µ,λ ) → E with ω ¯ u,µ,λ ∈ (0 , + ∞ ] ; ( H ) given a bounded set Q ⊂ E , the sets F ([0 , + ∞ ) × Q × P ) and { u ( t ; ¯ u, µ, λ ) | t ∈ [0 , ¯ t ] , ¯ u ∈ Q, µ ∈ P, λ > } , where ¯ t > is such that ¯ t < ω ¯ u,µ,λ for any ¯ u ∈ E , µ ∈ P and λ > , are bounded; ( H ) if Q ⊂ E and P ⊂ P are relatively compact and < ¯ t < ω ¯ u,µ,λ for any ¯ u ∈ E , µ ∈ P and λ > , then { u (¯ t ; ¯ u, µ, λ ) | ¯ u ∈ Q , µ ∈ P , λ > } is relatively compact. ( H ) there are M ≥ and ω ∈ R such that, for any µ ∈ P and λ > , k R ( µ,λ ) ( t, s ) k ≤ M e ω ( t − s ) whenever ≤ s ≤ t ;( H ) there are C semigroups b S ( µ ) = { b S ( µ ) ( t ) : E → E } t ≥ , µ ∈ P , of bounded linearoperators on E with the infinitesimal generators b A ( µ ) : D ( b A ( µ ) ) → E , µ ∈ P , suchthat, for any t ≥ , s ∈ [0 , t ] , µ ∈ P and ¯ u ∈ E , lim λ → + , ¯ v → ¯ u, ν → µ R ( ν,λ ) ( t, s )¯ v = b S ( µ ) ( t − s )¯ u and the convergence is uniform for t and s from bounded intervals; ( H ) F : [0 , + ∞ ) × E × P → E is continuous uniformly with respect to the first variable,the set { F ( t, ¯ u, µ ) | t ≥ } is relatively compact for any ¯ u ∈ E and µ ∈ P , and thereis a continuous b F : E × P → E such that, for any ¯ u ∈ E , µ ∈ P and h > , b F (¯ u, µ ) = lim T → + ∞ , ¯ v → ¯ u, ν → µ T Z T F ( τ + h, ¯ v, ν ) d τ (1)where the convergence is uniform with respect to h > ; ( H ) for any ¯ u ∈ E and µ ∈ P the averaged problem (cid:26) ˙ u ( t ) = b A ( µ ) u ( t ) + b F ( u ( t ) , µ ) , t > u (0) = ¯ u admits a unique maximal mild solution b u ( · ; ¯ u, µ ) : [0 , b ω ¯ u,µ ) → E with some b ω ¯ u,µ ∈ (0 , + ∞ ] . Remark 2.1 (i) Assumptions ( H ) and ( H ) are standard local existence properties, which hold if F and b F are locally Lipschitz in the state variable. We shall show in Proposition 2.7 that ( H ) and ( H ) hold for a large class of F . Property ( H ) is natural and is satisfied forexample for the class of hyperbolic evolution systems considered in Section 3.(ii) Note that ( H ) is a sort of an almost periodicity assumption (cf. [13]). Moreover,it is always satisfied if F is continuous and time periodic.(iii) Note that in ( H ) we actually require that the averaging principle is true in thelinear case. In Section 3 we shall prove it in the general hyperbolic case – see Theorem 3.3. Theorem 2.2 (Abstract averaging principle)
Let ( H ) − ( H ) be satisfied. If (¯ u n ) in E , ( t n ) in [0 , + ∞ ) , ( µ n ) in P and ( λ n ) in (0 , + ∞ ) are such that ¯ u n → ¯ u , t n → t , µ n → µ , λ n → + as n → + ∞ , for some t ≥ , ¯ u ∈ E and µ ∈ P , and t n ≤ ¯ t < ω ¯ u n ,µ n ,λ n forsome ¯ t > and each n ≥ , then u ( t n ; ¯ u n , µ n , λ n ) → b u ( t ; ¯ u , µ ) in E as n → ∞ . Moreover max {k u ( t ; ¯ u n , µ n , λ n ) − b u ( t ; ¯ u , µ ) k | t ∈ [0 , ¯ t ] } −→ as n → + ∞ . (2)To prove it we shall need three auxillary facts. Lemma 2.3 (See [5, Proposition 3.1])
Suppose that { R n ( t, s ) } t ≥ s ≥ , n ≥ , are evolutionsystems on E with M ≥ and ω ∈ R such that k R n ( t, s ) k ≤ M e ω ( t − s ) , for any t, s ≥ with t ≥ s, and there is an evolution system { R ( t, s ) : E → E } t ≥ s ≥ such that lim n → + ∞ R n ( t, s )¯ u = R ( t, s )¯ u for all ¯ u ∈ E. Let { ¯ u n } n ≥ ⊂ E be relatively compact and { w n } n ≥ ⊂ L ([0 , l ] , E ) be uniformly integrable ( ) . Put u n : [0 , l ] → E , n ≥ , by u n ( t ) := R n ( t, u n + Z t R n ( t, s ) w n ( s ) d s, t ∈ [0 , l ] . Then the following conditions are equivalent (a) { u n ( t ) } n ≥ is relatively compact for a.e. t ∈ [0 , l ] ; (b) { u n } n ≥ is relatively compact in the space C ([0 , l ] , E ) (with the uniform convergencenorm). Lemma 2.4
Let ( H ) be satisfied and Q ⊂ E be compact. Then, for any ( T n ) in (0 , + ∞ ) with T n → + ∞ and ( µ n ) in P with µ n → µ , the convergence lim n → + ∞ T n Z T n F ( τ + h, ¯ w, µ n ) d τ = b F ( ¯ w, µ ) is uniform with respect to ¯ w ∈ Q and h > . Proof:
Put P := cl { µ n | n ≥ } . Take an arbitrary ε > . By the compactness of Q and P there is a set { ¯ w k } n ε k =1 ⊂ Q with δ , . . . , δ n ε ∈ (0 , ε ) such that Q ⊂ n ε [ k =1 B ( ¯ w k , δ n k ) and, for any τ > , µ ∈ P and ¯ w ∈ B ( ¯ w k , δ k ) , k ∈ { , . . . , n ε } , k b F ( ¯ w, µ ) − b F ( ¯ w k , µ ) k < ε/ k F ( τ, ¯ w, µ ) − F ( τ, ¯ w k , µ ) k < ε/ . Due to (1), there exists n ≥ such that, for any n ≥ n , k = 1 , . . . , n ε and h > , (cid:13)(cid:13)(cid:13)(cid:13) T n Z T n F ( τ + h, ¯ w k , µ n ) d τ − b F ( ¯ w k , µ ) (cid:13)(cid:13)(cid:13)(cid:13) < ε/ . Taking n ≥ n , ¯ w ∈ Q and h > , we get ¯ w ∈ B ( ¯ w k , δ k ) for some k = 1 , . . . , n ε and,consequently, (cid:13)(cid:13)(cid:13)(cid:13) T n Z T n F ( τ + h, ¯ w, µ n ) d τ − b F ( ¯ w, µ ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ T n Z T n k F ( τ + h, ¯ w, µ n ) − F ( τ + h, ¯ w k , µ n ) k d τ + (cid:13)(cid:13)(cid:13)(cid:13) T n Z T n F ( τ + h, ¯ w k , µ n ) d τ − b F ( ¯ w k , µ ) (cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) b F ( ¯ w k , µ ) − b F ( ¯ w, ¯ µ ) (cid:13)(cid:13)(cid:13) ≤ ε/ ε/ ε/ ε, which completes the proof. (cid:3) i.e., for any ε > , there is δ > such that, for any measurable J ⊂ [0 , l ] with the Lebesgue measure µ ( J ) ≤ δ and any n ≥ , R J k w n ( t ) k dt ≤ ε Lemma 2.5
Assume that ( H ) and ( H ) hold. Let ( T n ) be a sequence in (0 , + ∞ ) , ( k n ) asequence of positive integers, ( λ n ) a sequence in (0 , + ∞ ) and ( µ n ) in P such that T n → + ∞ , k n → ∞ , λ n → , k n λ n T n → t for some t > , µ n → µ for some µ ∈ P , as n → + ∞ , and k n λ n T n ≤ t for almost all integers n ≥ . Then, for any continuousfunction w : [0 , t ] → E , λ n T n k n − X k =0 R ( µ n ,λ n ) ( k n λ n T n , kλ n T n ) w ( kλ n T n ) → Z t b S ( µ ) ( t − s ) w ( s ) ds as n → + ∞ . Proof:
First note that, for any n ≥ , λ n T n k n − X k =0 R ( µ n ,λ n ) ( k n λ n T n , kλ n T n ) w ( kλ n T n ) = σ ,n + σ ,n + σ ,n where σ ,n := λ n T n k n − X k =0 [ R ( µ n ,λ n ) ( k n λ n T n , kλ n T n ) − b S ( µ ) ( k n λ n T n − kλ n T n )] w ( kλ n T n ) ,σ ,n := λ n T n k n − X k =0 [ b S ( µ ) ( k n λ n T n − kλ n T n ) − b S ( µ ) ( t − kλ n T n )] w ( kλ n T n ) ,σ ,n := λ n T n k n − X k =0 b S ( µ ) ( t − kλ n T n ) w ( kλ n T n ) . It follows from ( H ) and the compactness of w ([0 , t ]) that, for any ε > , there exists n ≥ such that, for all n ≥ n , ¯ w ∈ w ([0 , t ]) and s ′ , s ∈ [0 , t ] with s ′ ≥ s , k R ( µ n ,λ n ) ( s ′ , s ) ¯ w − b S ( µ ) ( s ′ − s ) ¯ w k ≤ ε. This implies σ ,n → as n → + ∞ , since k n λ n T n → t . Again, by the compactness of w ([0 , t ]) and the strong continuity of the semigroup b S ( µ ) , we gather that σ ,n → as n → + ∞ . Finally, by the continuity of [0 , t ] ∋ s b S ( µ ) ( t − s ) w ( s ) and the fact that λ n T n → as n → + ∞ , we infer that σ ,n → Z t b S ( µ ) ( t − s ) w ( s ) d s, which ends the proof. (cid:3) Proof of Theorem 2.2:
Let R n := R ( µ n ,λ n ) and u n : [0 , ¯ t ] → E , n ≥ , be given by u n ( t ) := u ( t ; ¯ u n , µ n , λ n ) for t ∈ [0 , ¯ t ] , n ≥ . (3)Obviously u n ( t ) = R n ( t, u n + Z t R n ( t, s ) w n ( s ) d s for t ∈ [0 , ¯ t ] (4)with w n : [0 , ¯ t ] → E defined as w n ( s ) := F ( µ n ,λ n ) ( s, u ( s ; ¯ u n , µ n , λ n )) for s ∈ [0 , ¯ t ] . By ( H ) ,there are C , C > such that k u n ( t ) k ≤ C and k w n ( t ) k ≤ C for all t ∈ [0 , ¯ t ] and n ≥ . (5)In the rest of the proof we shall argue as follows: we take any subsequence of ( u n ) , denote itagain by ( u n ) and show that it contains a subsequence converging to b u ( · ; ¯ u , µ ) | [0 , ¯ t ] in thespace C ([0 , ¯ t ] , E ) ; having this we will conclude that the original ( u n ) converges uniformlyto b u ( · ; ¯ u , µ ) on [0 , ¯ t ] and the assertion will follow.Start with an observation that, in view of ( H ) , for each t ∈ [0 , ¯ t ] , the set { u n ( t ) } n ≥ isrelatively compact. And due to ( H ) , ( H ) and Lemma 2.3, ( u n ) contains a subsequenceconverging uniformly on [0 , ¯ t ] to some b v . Denote that subsequence again by ( u n ) . It followsdirectly from ( H ) that R n ( t, u n = R ( µ n ,λ n ) ( t, u n → b S ( µ ) ( t )¯ u as n → + ∞ (6)uniformly with respect to t ∈ [0 , ¯ t ] .Now take an arbitrary t ∈ (0 , ¯ t ] and any sequences ( T n ) in (0 , + ∞ ) and ( k n ) of positiveintegers such that T n → + ∞ , k n → + ∞ , k n λ n T n → t , as n → + ∞ , and k n λ n T n ≤ t forany n ≥ (e.g. T n := λ − / n , k n := [ t/λ n T n ] ). First observe that Z t R n ( t, s ) w n ( s ) d s = I ,n + I ,n + I ,n where I ,n := Z k n λ n T n R n ( k n λ n T n , s ) w n ( s ) d s,I ,n := Z k n λ n T n ( R n ( t, s ) − R n ( k n λ n T n , s )) w n ( s ) d s,I ,n := Z tk n λ n T n R n ( t, s ) w n ( s ) d s. It is immediate to see that, by ( H ) and (5), I ,n → as n → + ∞ .To deal with ( I ,n ) , we claim thatthe set e Q := { w n ( s ) | s ∈ [0 , t ] , n ≥ } is relatively compact . (7)Indeed, to see this take any sequence ( ¯ w k ) in e Q . Then, for each k ≥ , there are an integer n k ≥ and s k ∈ [0 , t ] such that ¯ w k = w n k ( s k ) = F ( s k /λ n k , u n k ( s k ) , µ n k ) . We may assume that s k → s as k → + ∞ and, by the uniform convergence of ( u n ) , that u n k ( s k ) → e u for some e u ∈ E . In view of ( H ) , for any ε > one finds n ≥ such that,for all n ≥ n , k ¯ w k − F ( s k /λ n k , e u, µ ) k = k F ( s k /λ n k , u n k ( s k ) , µ n k ) − F ( s k /λ n k , e u, µ ) k < ε. Now, since, by ( H ) , the set { F ( s, e u, µ ) | t ≥ } is relatively compact, it follows that ( ¯ w k ) contains a convergent subsequence, which proves (7).It can be easily seen that ( H ) along with (7) implies that, for any s ′ , s ∈ [0 , t ] with s ′ ≥ s and ¯ w ∈ e Q , R n ( s ′ , s ) ¯ w → b S ( µ ) ( s ′ − s ) ¯ w as n → + ∞ uniformly with respet to s, s ′ and ¯ w. (8)In consequence, for any ε > there exists n ≥ such that, for all n ≥ n , k R n ( t, s ) w n ( s ) − b S ( µ ) ( t − s ) w n ( s ) k ≤ ε/ t, for s ∈ [0 , t ] , (9) k b S ( µ ) ( k n λ n T n − s ) w n ( s ) − R n ( k n λ n T n , s ) w n ( s ) k ≤ ε/ t, for s ∈ [0 , k n λ n T n ] . (10)Moreover, by the strong conitnuity of the semigroup b S ( µ ) and the relative compactnes of e Q , there is n ≥ n such that, for all n ≥ n and s ∈ [0 , t ] , k b S ( µ ) ( t − s ) w n ( s ) − b S ( µ ) ( k n λ n T n − s ) w n ( s ) k < ε/ t. (11)Since it follows from (9), (10) and (11) that, for all n ≥ n and s ∈ [0 , t ] , k ( R n ( t, s ) − R n ( k n λ n T n , s )) w n ( s ) k ≤ ε/ t + ε/ t + ε/ t = ε/t we infer, for n ≥ n , k I ,n k ≤ k n λ n T n ( ε/t ) ≤ ε , i.e., I ,n → as n → + ∞ .Now our aim is to show that I ,n → Z t b S ( µ ) ( t − s ) b F ( b v ( s ) , µ ) ds as n → + ∞ . To this end observe that, for sufficiently large n ≥ , I ,n = Z k n λ n T n R n ( k n λ n T n , s ) w n ( s ) d s = k n − X k =0 Z ( k +1) λ n T n kλ n T n R n ( k n λ n T n , s ) w n ( s ) d s = k n − X k =0 Z T n R n ( k n λ n T n , kλ n T n + λ n τ ) w n ( kλ n T n + λ n τ ) λ n d τ = J ,n + J ,n + J ,n + J ,n with J ,n := k n − X k =0 Z T n [ R n ( k n λ n T n ,kλ n T n + λ n τ ) − R n ( k n λ n T n ,kλ n T n )] w n ( kλ n T n + λ n τ ) λ n d τJ ,n := k n − X k =0 R n ( k n λ n T n , kλ n T n ) Z T n ( w n ( kλ n T n + λ n τ ) − F ( kT n + τ, b v ( kλ n T n ) , µ n )) λ n d τJ ,n := k n − X k =0 R n ( k n λ n T n , kλ n T n ) 1 T n Z T n ( F ( kT n + τ, b v ( kλ n T n ) , µ n ) − b F ( b v ( kλ n T n ) , µ )) λ n T n d τJ ,n := k n − X k =0 R n ( k n λ n T n , kλ n T n ) b F ( b v ( kλ n T n ) , µ )) λ n T n . First we note that, for all n ≥ , τ ∈ [0 , T n ] and k = 0 , , . . . , k n − , k [ R n ( k n λ n T n ,kλ n T n + λ n τ ) − R n ( k n λ n T n ,kλ n T n )] w n ( kλ n T n + λ n τ ) k≤ k [ R n ( k n λ n T n ,kλ n T n + λ n τ ) − b S ( µ ) ( k n λ n T n − kλ n T n − λ n τ )] w n ( kλ n T n + λ n τ ) k + k [ b S ( µ ) ( k n λ n T n − kλ n T n − λ n τ ) − b S ( µ ) ( k n λ n T n − kλ n T n )] w n ( kλ n T n + λ n τ ) k + k [ b S ( µ ) ( k n λ n T n − kλ n T n ) − R n ( k n λ n T n ,kλ n T n )] w n ( kλ n T n + λ n τ ) k . Hence, in view of the uniform convergence in (8) and the uniform equicontinuity of functions [0 , t ] ∋ s b S ( µ ) ( s ) ¯ w , ¯ w ∈ e Q , we deduce that, for any ε > , there is n ≥ , such thatfor all n ≥ n , τ ∈ [0 , T n ] and k = 0 , , . . . , k n − , k [ R n ( k n λ n T n ,kλ n T n + λ n τ ) − R n ( k n λ n T n ,kλ n T n )] w n ( kλ n T n + λ n τ ) k < ε/t k J ,n k < k n λ n T n ( ε/t ) ≤ ε. This means that J ,n → as n → + ∞ .As for J ,n , take any ε > and note that, by the uniform convergence of ( u n ) , the set e Q ′ := cl [ n ≥ u n ([0 , t ]) is compact and b v ([0 , t ]) ⊂ e Q ′ . Therefore, by ( H ) , we find η > such that, for any τ ≥ , µ ∈ cl { µ n | n ≥ } , ¯ v , ¯ v ∈ e Q ′ with k ¯ v − ¯ v k ≤ η , k F ( τ, ¯ v , µ ) − F ( τ, ¯ v , µ ) k ≤ ε/tM e | ω | t . Furthermore, again by the uniform convergence of ( u n ) , there is δ > and n ≥ suchthat, for any n ≥ n , k u n ( τ ) − b v ( τ ) k < η for any τ , τ ∈ [0 , t ] with | τ − τ | < δ. Let n ≥ n be such that λ n T n < δ for each n ≥ n . Then, for any n ≥ n , τ ∈ [0 , T n ] and k ∈ { , . . . , k n − } , k u n ( kλ n T n + λ n τ ) − b v ( kλ n T n ) k ≤ η, and, consequently, for any n ≥ n , τ ∈ [0 , T n ] and k = 0 , , . . . , k n − , k w n ( kλ n T n + λ n τ ) − F ( kT n + τ, b v ( kλ n T n ) , µ n ) k = k F ( kT n + τ, u n ( kλ n T n + λ n τ ) , µ n ) − F ( kT n + τ, b v ( kλ n T n ) , µ n ) k ≤ ε/tM e | ω | t . Therefore, for any n ≥ n , one has k J ,n k ≤ M e | ω | t λ n k n − X k =0 Z T n k w n ( kλ n T n + λ n τ ) − F ( kT n + τ, b v ( kλ n T n ) , µ ) k d τ< M e | ω | t λ n k n T n ( ε/tM e | ω | t ) = k n λ n T n ε/t ≤ ε, which yields J ,n → as n → + ∞ .Next observe that, in view of Lemma 2.4, lim n → + ∞ T n Z T n F ( h + τ, ¯ w, µ n ) d τ = b F ( ¯ w, µ ) uniformly with respect to ¯ w ∈ e Q ′ and h > . Hence k J ,n k ≤ M e | ω | t λ n k n − X k =0 (cid:13)(cid:13)(cid:13)(cid:13) T n Z T n F ( kT n + τ, b v ( kλ n T n ) , µ n ) d τ − b F ( b v ( kλ n T n ) , µ ) (cid:13)(cid:13)(cid:13)(cid:13) and we find that J ,n → as n → + ∞ .Finally, due to Lemma 2.5, J ,n → Z t b S ( µ ) ( t − s ) b F ( b v ( s ) , µ ) d s. Summing up, we have already showed that Z t R n ( t, s ) w n ( s ) d s → Z t b S ( µ ) ( t − s ) b F ( b v ( s ) , µ ) d s n → + ∞ in (4), we arrive at b v ( t ) = b S ( µ ) ( t ) b v (0) + Z t b S ( µ ) ( t − s ) b F ( b v ( s ) , µ ) d s, for any t ∈ [0 , ¯ t ] . In particular, b v is a mild solution of ˙ u = b A ( µ ) u + b F ( u, µ ) and, in view of ( H ) , b v = b u ( · ; ¯ u , µ ) | [0 , ¯ t ] . Hence the original sequence ( u n ) converges uniformly to b u ( · ; ¯ u , µ ) on [0 , ¯ t ] , which together with (3), implies u n ( t n ; ¯ u n , µ n , λ n ) → b u ( t ; ¯ u , µ ) and as well as (2). (cid:3) Remark 2.6 (a) For single-valued F , Theorem 2.2 is an extension of [11, Theorem 5.4.1] ina few aspects: firstly, the periodicity assumption of F is relaxed; secondly, the case where A depends on time is included; thirdly, the sublinear growth condition is not requiredand, finally, no explicit compactness assumptions on F are imposed (which may be ofimportance, see e.g. [3] where one could not apply [11, Theorem 5.4.1]).(b) Note also that Theorem 2.2 covers [3, Theorem 2.4] as well.We end this section with an example showing the availability of assumptions ( H ) and ( H ) and derive Theorem 1.1. Proposition 2.7
Let E be a separable Banach space and { A ( µ ) ( t ) } t ≥ , µ ∈ P , be such thatthe family { R ( µ ) } µ ∈ P of the corresponding evolution systems is continuous and satisfies ( H ) and ( H ) . Assume that a continuous F : [0 , + ∞ ) × E × P → E satisfies the followingconditionsfor any ¯ v ∈ E , there exist L > and δ > such that k F ( t, ¯ v , µ ) − F ( t, ¯ v , µ ) k ≤ L k ¯ v − ¯ v k for any t ≥ , ¯ v , ¯ v ∈ B (¯ v, δ ) , µ ∈ P ; (12) there is c > such that k F ( t, ¯ v, µ ) k ≤ c (1 + k ¯ v k ) for any ( t, ¯ v, µ ) ∈ [0 , + ∞ ) × E × P ; (13) and for any C > there is k ≥ such that β ( F ([0 , + ∞ ) × Q × P )) ≤ k · β ( Q ) for any Q ⊂ B (0 , C ) where β stands for the Hausdorff or Kuratowski measure of noncompactness ( see e.g. [7] or [11]).(14) Then ( H ) − ( H ) are satisfied. We shall use the following general properties involving the measures of noncompactness.
Lemma 2.8 (see [7, Prop. 9.3] or [11])
Let E be a separable Banach space, W ⊂ L ([0 , l ] , E ) , l > , be countable and integrably bounded ( i.e. there exists c ∈ L ([0 , l ]) such that k w ( t ) k ≤ c ( t ) for all w ∈ W and a.e. t ∈ [0 , l ] ) and φ : [0 , l ] → R be given by φ ( t ) := β ( { u ( t ) | u ∈ W } ) . Then φ ∈ L ([0 , l ]) and β (cid:18)(cid:26)Z l u ( τ ) d τ | u ∈ W (cid:27)(cid:19) ≤ Z l φ ( τ ) d τ. Lemma 2.9 (see [4, Lemma 5.4])
Let T n : E → E , n ≥ , be bounded linear operators ona Banach space E such that, for any ¯ u ∈ E , ( T n ¯ u ) is a Cauchy sequence. Then, for anybounded set { ¯ u n } n ≥ ⊂ E , β ( { T n ¯ u n } n ≥ ) ≤ (cid:18) lim sup n → + ∞ k T n k (cid:19) β ( { ¯ u n } n ≥ ) . Proof of Proposition 2.7:
It is standard to see that the local Lipschitzianity of F impliesthe local existence, i.e., ( H ) holds. It can be also easily deduced that the sublinear growthand the Gronwall inequality yield ( H ) .In order to verify ( H ) , take any { λ n } n ≥ ⊂ (0 , + ∞ ) and relatively compact sets { ¯ u n } n ≥ ⊂ E , { µ n } n ≥ ⊂ P and suppose that < t < ω ¯ u n ,µ n ,λ n for any n ≥ . Define φ : [0 , t ] → R by φ ( τ ) := β ( { u ( τ ; ¯ u n , µ n , λ n ) } n ≥ ) and w n : [0 , t ] → E by w n ( τ ) := F ( τ /λ n , u ( τ ; ¯ u n , µ n , λ n ) , µ n ) , τ ∈ [0 , t ] . By ( H ) , there is C > such that sup n ≥ sup τ ∈ [0 ,t ] k w n ( τ ) k < C , and, in view of Lemma 2.8, φ is integrable.Taking into account ( H ) and ( H ) and using Lemmas 2.9 and 2.8, for any r ∈ [0 , t ] , onegets φ ( r ) ≤ β ( { R ( µ n ,λ n ) ( r, u n } n ≥ ) + β (cid:26)Z r R ( µ n ,λ n ) ( r, τ ) w n ( τ ) d τ (cid:27) n ≥ ! ≤ M e | ω | t β ( { ¯ u n } n ≥ ) + Z r β ( { R ( µ n ,λ n ) ( r, τ ) w n ( τ ) } n ≥ ) d τ ≤ M e | ω | t Z r β ( { w n ( τ ) } n ≥ ) d τ. Hence, by use of (14), there is k > such that φ ( r ) ≤ M e | ω | t k Z r φ ( τ ) d τ, which implies φ ( r ) = 0 and completes the proof of ( H ) . (cid:3) Remark 2.10
Note that Theorem 1.1 is a consequence of Theorem 2.2. Indeed, theLipshitzianity on bounded sets implies that (14) holds and, if { A ( t ) } t ≥ satisfy ( A and ( A , then, in view of Proposition 2.7, assumptions ( H ) − ( H ) are fulfilled. Let V be a Banach space densely and continuously embedded into a Banach space E .If a linear operator A : D ( A ) → E generates a C semigroup { S A ( t ) } t ≥ of bounded linearoperators on E , then V is said to be A -admissible provided V is an invariant subspace foreach S A ( t ) , t ≥ , and the family of restrictions { S A ( t ) V : V → V } t ≥ ( S A ( t ) V ¯ v := S A ( t )¯ v , ¯ v ∈ V ) is a C semigroup on V . Define the part of A in the space V as a linear opera-tor A V : D ( A V ) → V given by D ( A V ) := { ¯ v ∈ D ( A ) ∩ V | A ¯ v ∈ V } , A V ¯ v := A ¯ v for ¯ v ∈ D ( A V ) . In view of [14, Ch. 4, Theorem 5.5], if V is A -admissible then A V is thegenerator of the C semigroup { S A ( t ) V } t ≥ .3Now let { A ( t ) } t ≥ be a family of linear operators in E satisfying the following conditions ( Hyp ) { A ( t ) } t ≥ is a stable family of infinitesimal generators of C semigroups on E , i.e.,there are M ≥ and ω ∈ R such that k S A ( t n ) ( s n ) . . . S A ( t ) ( s ) k L ( E,E ) ≤ M e ω ( s + ... + s n ) , whenever ≤ t ≤ . . . ≤ t n and s , . . . , s n ≥ , where { S A ( t ) ( s ) } s ≥ is the C semigroup generated by A ( t ) ; ( Hyp ) V is A ( t ) -admissible for each t ≥ and the family { A V ( t ) } t ≥ is a stable family ofgenerators of C semigroups on V with constants M V ≥ and ω V ∈ R ; ( Hyp ) V ⊂ D ( A ( t )) and A ( t ) ∈ L ( V, E ) , for any t ≥ , and the mapping [0 , + ∞ ) ∋ t A ( t ) ∈ L ( V, E ) is continuous.These are so called hyperbolic conditions and they determine a unique evolution system. Proposition 3.1 (see [14, Ch. 5, Theorem 3.1])
Let { A ( t ) } t ≥ be a family of linear opera-tors in a Banach space E satisfying ( Hyp ) − ( Hyp ) . Then there exists a unique evolutionsystem { R ( t, s ) } t ≥ s ≥ on E with the following properties (i) k R ( t, s ) k ≤ M e ω ( t − s ) for s ≥ ; (ii) ∂ + ∂t R ( t, s ) v (cid:12)(cid:12)(cid:12) t = s = A ( s ) v for v ∈ V , s ≥ ; (iii) ∂∂s R ( t, s ) v = − R ( t, s ) A ( s ) v for v ∈ V , ≤ s ≤ t . We shall consider parameterized evolution systems.
Proposition 3.2
Let P be a metric space of parameters. Suppose that families { A ( µ ) ( t ) } t ≥ , µ ∈ P , satisfy conditions ( Hyp ) − ( Hyp ) with constants M, M V , ω, ω V independent of µ .Let R ( µ ) = { R ( µ ) ( t, s ) } t ≥ s ≥ be the corresponding evolution systems on E determined byProposition . (i) For any µ, ν ∈ P , ¯ v ∈ V and t, s ≥ with t ≥ s , k R ( ν ) ( t, s )¯ v − R ( µ ) ( t, s )¯ v k ≤ MM V e ( | ω | + | ω V | ) t k ¯ v k V Z ts k A ( ν ) ( r ) − A ( µ ) ( r ) k L ( V,E ) d r. (ii) If lim ν → µ Z T k A ( ν ) ( r ) − A ( µ ) ( r ) k L ( V,E ) d r = 0 , for any µ ∈ P and T > , (15) then { R ( µ ) } µ ∈ P is a continuous family of evolution systems on E . Proof: (i) We use the construction from [14, Ch. 5, Theorem 3.1]. Fix
T > . Recallthat, for any µ ∈ P and ¯ u ∈ E , R ( µ ) ( t, s )¯ u = lim n → + ∞ R ( µ ) n ( t, s )¯ u for ≤ s ≤ t ≤ T, n ≥ , the operator R ( µ ) n ( t, s ) : E → E is given by ( ) R ( µ ) n ( t, s ) := S ( µ ) j ( t − s ) if s, t ∈ [ t nj , t nj +1 ] , s ≤ t,S ( µ ) k ( t − t nk ) k − Q j = l +1 S ( µ ) j ( T /n ) ! S ( µ ) l ( t nl +1 − s ) if s ∈ [ t nl , t nl +1 ] , t ∈ [ t nk , t nk +1 ] and k > l ≥ , with t nj := ( j/n ) T , S ( µ ) j := S A ( µ ) ( t nj ) , for j = 0 , , . . . , n . Moreover { R ( µ ) n ( t, s ) } ≤ s ≤ t ≤ T , µ ∈ P , are evolution systems such that k R ( µ ) n ( t, s ) k L ( E,E ) ≤ M e ω ( t − s ) , R ( µ ) n ( t, s ) V ⊂ V, k R ( µ ) n ( t, s ) k L ( V,V ) ≤ M V e ω V ( t − s ) , (16)whenever ≤ s ≤ t ≤ T and, for any ¯ v ∈ V , ∂∂t R ( µ ) n ( t, s )¯ v = A ( µ ) n ( t ) R ( µ ) n ( t, s )¯ v for t
6∈ { t n , t n , . . . , t nn } , s ≤ t, (17) ∂∂s R ( µ ) n ( t, s )¯ v = − R ( µ ) n ( t, s ) A ( µ ) n ( s )¯ v for s
6∈ { t n , t n , . . . , t nn } , s ≤ t, (18)with A ( µ ) n ( t ) := A ( µ ) ( t nk ) if t nk ≤ t < t nk +1 for k = 0 , . . . , n − and A ( µ ) n ( T ) := A ( µ ) ( T ) . ( Hyp ) implies that, for any µ ∈ P , one has k A ( µ ) n ( t ) − A ( µ ) ( t ) k L ( V,E ) → as n → + ∞ uniformly with respect to t ∈ [0 , T ] . (19)Fix any ¯ v ∈ V , µ, ν ∈ P , n ≥ and s, t ∈ [0 , T ] with s < t and define φ : [ s, t ] → E by φ ( r ) := R ( µ ) n ( t, r ) R ( ν ) n ( r, s )¯ v . In view of (16), (17) and (18), the map φ is differentiable on [ s, t ] except the finite number of points and R ( ν ) n ( t, s )¯ v − R ( µ ) n ( t, s )¯ v = φ ( t ) − φ ( s ) = Z ts φ ′ ( r ) d r = Z ts R ( µ ) n ( t, r )( A ( ν ) n ( r )) − A ( µ ) n ( r )) R ( ν ) n ( r, s )¯ v d r. This together with (16) yields k R ( ν ) n ( t, s )¯ v − R ( µ ) n ( t, s )¯ v k ≤ M M V e ( | ω | + | ω V | ) t k ¯ v k V Z ts k A ( ν ) n ( r ) − A ( µ ) n ( r ) k L ( V,E ) d r. Passing to the limit with n → + ∞ , we get k R ( ν ) ( t, s )¯ v − R ( µ ) ( t, s )¯ v k ≤ M M V e ( | ω | + | ω V | ) t k ¯ v k V Z ts k A ( ν ) ( r ) − A ( µ ) ( r ) k L ( V,E ) d r. (ii) It follows immediately from (i) that R ( ν ) ( t, s )¯ v → R ( µ ) ( t, s )¯ v for any ¯ v ∈ V , as ν → µ and the convergence is uniform with respect to s, t from [0 , T ] . To see it for anarbitrary ¯ u ∈ E , note that, for any ¯ v ∈ V , k R ( ν ) ( t, s )¯ u − R ( µ ) ( t, s )¯ u k ≤ k R ( ν ) ( t, s )¯ v − R ( µ ) ( t, s )¯ v k + 2 M e | ω | T k ¯ u − ¯ v k , which, in view of the density of V in E , implies (ii). (cid:3) Here we use the convention that Q nj = m T j := T n ◦ T n − ◦ . . . ◦ T m , for integers m, n with m < n and asequence T m , T m +1 , . . . , T n of bounded operators on E . { A ( µ ) ( t ) } t ≥ , µ ∈ P , are as in Proposition 3.2. Define A ( µ,λ ) ( t ) : D ( A ( µ,λ ) ( t )) → E , for λ > and t ≥ , by A ( µ,λ ) ( t ) := A ( µ ) ( t/λ ) . Note that, for each µ ∈ P and λ > , the family { A ( µ,λ ) ( t ) } t ≥ also satisfies ( Hyp ) − ( Hyp ) with the same constants (independent of µ , λ ) as for { A ( µ ) ( t ) } t ≥ , µ ∈ P . Thefollowing linear averaging principle holds. Theorem 3.3
Let { A ( µ ) ( t ) } t ≥ , µ ∈ P , be as in Proposition . If, additionally, for each µ ∈ P , there is a generator b A ( µ ) of a C semigroup { b S ( µ ) ( t ) : E → E } t ≥ such that V is b A ( µ ) -admissible, V ⊂ D ( b A ( µ ) ) and lim T → + ∞ , ν → µ T Z T k A ( ν ) ( t + h ) − b A ( µ ) k L ( V,E ) d t = 0 uniformly with respect to h ≥ , (20) then, for any µ ∈ P and λ > , k R ( µ,λ ) ( t, s ) k ≤ M e ω ( t − s ) whenever ≤ s ≤ t (21) and, for any t ≥ , s ∈ [0 , t ] , µ ∈ P and ¯ u ∈ E , lim λ → + , ¯ v → ¯ u, ν → µ R ( ν,λ ) ( t, s )¯ v = b S ( µ ) ( t − s )¯ u where the convergence is uniform for t and s from bounded intervals. Lemma 3.4
Under the assumption (20) , for any µ ∈ P and t, s ≥ with t ≥ s , lim λ → + , ν → µ Z ts k A ( ν,λ ) ( r ) − b A ( µ ) k L ( V,E ) d r = 0 . Proof:
Take any µ ∈ P and let ( λ n ) in (0 , + ∞ ) and ( µ n ) in P be arbitrary sequences suchthat λ n → and µ n → µ in P . Let a sequence ( k n ) of positive integers and ( T n ) in (0 , + ∞ ) be such that k n → + ∞ , T n → + ∞ , k n λ n T n → t − s as n → + ∞ and k n λ n T n ≤ t − s foreach n ≥ . Observe that Z ts k A ( µ n ,λ n ) ( r ) − b A ( µ ) k d r = Z t − s k A ( µ n ,λ n ) ( s + ρ ) − b A ( µ ) k d ρ = Z k n λ n T n k A ( µ n ,λ n ) ( s + ρ ) − b A ( µ ) k d ρ + Z t − sk n λ n T n k A ( µ n ,λ n ) ( s + ρ ) − b A ( µ ) k d ρ. Clearly the second term tends to as n → + ∞ . Furthermore Z k n λ n T n k A ( µ n ,λ n ) ( s + ρ ) − b A ( µ ) k d ρ = k n − X k =0 Z ( k +1) λ n T n kλ n T n k A ( µ n ,λ n ) ( s + ρ ) − b A ( µ ) k d ρ = λ n T n k n − X k =0 T n Z T n k A ( µ n ) ( s/λ n + kT n + τ ) − b A ( µ ) k d τ and, in view of (20), it also converges to as n → + ∞ . (cid:3) Proof of Theorem 3.3:
First, observe that the families of operators { A ( µ,λ ) ( t ) } t ≥ , µ ∈ P , λ > , also satisfy ( Hyp ) − ( Hyp ) and, by the very construction, the correspondingevolution system { R ( µ,λ ) ( t ) } t ≥ admits the same growth condition as { R ( µ ) ( t, s ) } t ≥ s ≥ , i.e.,(21) holds. Next we intend to prove that, for any t, s ≥ with t ≥ s , µ ∈ P and ¯ u ∈ E ,the limit lim λ → + ,ν → µ, ¯ v → ¯ u R ( ν,λ ) ( t, s )¯ v exists. To this end fix any T > and note thatit follows from Proposition 3.2 (i) that, for C := M M V e ( | ω | + | ω V | ) T , any µ , µ , µ ∈ P , λ , λ > , ¯ v ∈ V and t, s ∈ [0 , T ] with t ≥ s , one has k R ( µ ,λ ) ( t, s )¯ v − R ( µ ,λ ) ( t, s )¯ v k ≤ C k ¯ v k V Z T k A ( µ ,λ ) ( r ) − A ( µ ,λ ) ( r ) k L ( V,E ) d r ≤ C k ¯ v k V (cid:18)Z T k A ( µ ,λ ) ( r ) − b A ( µ ) k L ( V,E ) d r ++ Z T k b A ( µ ) − A ( µ ,λ ) ( r ) k L ( V,E ) d r (cid:19) . Now it is immediate, in view of Lemma 3.4, that the limit lim λ → + , ν → µ R ( µ,λ ) ( t, s )¯ v existsand is uniform with respect to t, s from bounded intervals. If one takes any ¯ u ∈ E , thenthe existence of the limit lim λ → + , ν → µ R ( ν,λ ) ( t, s )¯ u and its uniformness with respect to t, s ∈ [0 , T ] come from the inequality k R ( µ ,λ ) ( t, s )¯ u − R ( µ ,λ ) ( t, s )¯ u k ≤ k R ( µ ,λ ) ( t, s ) ¯ w − R ( µ ,λ ) ( t, s ) ¯ w k + 2 M e | ω | T k ¯ u − ¯ w k holding for any µ , µ ∈ P , λ , λ > and arbitrary ¯ w ∈ V . Finally, by the growthcondition (21), for µ , µ ∈ P , λ , λ > and ¯ v , ¯ v ∈ E , k R ( µ ,λ ) ( t, s )¯ v − R ( µ ,λ ) ( t, s )¯ v k ≤ k R ( µ ,λ ) ( t, s )¯ v − R ( µ ,λ ) ( t, s )¯ u k + k R ( µ ,λ ) ( t, s )¯ u − R ( µ ,λ ) ¯ u k + k R ( µ ,λ ) ¯ u − R ( µ ,λ ) ( t, s )¯ v k≤ M e | ω | T k ¯ v − ¯ u k + k R ( µ ,λ ) ( t, s )¯ u − R ( µ ,λ ) ¯ u k + M e | ω | T k ¯ v − ¯ u k , which implies that lim λ → + , ν → µ, ¯ v → ¯ u R ( ν,λ ) ( t, s )¯ v exists and is uniform with respect to t, s from bounded intervals.Let operators b R ( µ ) ( t, s ) : E → E , t, s ≥ with t ≥ s , µ ∈ P , be defined by b R ( µ ) ( t, s )¯ u := lim λ → + R ( µ,λ ) ( t, s )¯ u, for any ¯ u ∈ E. To complete the proof we need to show that b R ( µ ) ( t, s ) = b S ( µ ) ( t − s ) for any t, s ≥ with t ≥ s and µ ∈ P . For fixed µ ∈ P , λ > , n ≥ , ¯ v ∈ V , T > and t, s ∈ [0 , T ] with t ≥ s ,define ϕ λ : [ s, t ] → E by ϕ λ ( r ) := b S ( µ ) ( t − r ) R ( µ,λ ) n ( r, s )¯ v where R ( µ,λ ) n are approximating evolution systems from the construction of R ( µ,λ ) (see theproof of Proposition 3.2). Since R ( µ,λ ) n ( r, s )¯ v ∈ V ⊂ D ( b A ( µ ) ) , one has, for a.e. r ∈ [ s, t ] , ϕ ′ λ ( r ) = ∂∂r (cid:16) b S ( µ ) ( t − r ) R ( µ,λ ) n ( r, s )¯ v (cid:17) = [ − b A ( µ ) b S ( µ ) ( t − r )] R ( µ,λ ) n ( r, s )¯ v + b S ( µ ) ( t − r )[ A ( µ,λ ) n ( r ) R ( µ,λ ) n ( t, r )¯ v ]= b S ( µ ) ( t − r )( A ( µ,λ ) n ( r ) − b A ( µ ) ) R ( µ,λ ) n ( r, s )¯ v (cf. (17)). Hence, using the estimates for R ( µ,λ ) n as those in (16), one gets k R ( µ,λ ) n ( t, s )¯ v − b S ( µ ) ( t − s )¯ v k = k ϕ λ ( t ) − ϕ λ ( s ) k ≤ Z ts k ϕ ′ λ ( r ) k d r ≤ Z ts k b S ( µ ) ( t − r ) k L ( E,E ) k A ( µ,λ ) n ( r ) − b A ( µ ) k L ( V,E ) k R ( µ,λ ) n ( r, s ) k L ( V,V ) k ¯ v k V d r ≤ c M e | b ω | t M V e | ω V | t k v k V Z ts k A ( µ,λ ) n ( r ) − b A ( µ ) k L ( V,E ) d r where c M ≥ and b ω ∈ R are such that k b S ( µ ) ( τ ) k L ( E,E ) ≤ c M e b ωτ for any τ ≥ . Letting n → + ∞ , one obtains k b S ( µ ) ( t − s )¯ v − R ( µ,λ ) ( t, s )¯ v k ≤ c M M V e ( | b ω | + | ω V | ) t k v k V Z ts k A ( µ,λ ) ( r ) − b A ( µ ) k L ( V,E ) d r (cf. (19)). In view of Lemma 3.4, if we pass to the limit with λ → + then b S ( µ ) ( t − s )¯ v = b R ( µ ) ( t, s )¯ v . Since ¯ v ∈ V was arbitrary and V is dense in E , we find that b S ( µ ) ( t − s ) = b R ( µ ) ( t, s ) . (cid:3) Consider the following system of equations ∂u∂t ( x, t ) = N X j =1 a j ( x, t ) ∂u∂x j ( x, t ) + b ( x, t ) u ( x, t ) + f ( x, t ) , x ∈ R N , t > with continuously differentiable a j : R N × [0 , + ∞ ) → M , j = 1 , . . . , N ( N ≥ ), andcontinuous b : R N × [0 , + ∞ ) → M with M being the space of all square matrices of order M ≥ with the usual maximum norm denoted by | · | and u : R N × [0 , + ∞ ) → R M .We assume that a j ( x, t ) is symmetric for any j = 1 , . . . , N and ( x, t ) ∈ R N × [0 , + ∞ ) .By B ( R N , M ) denote the space of all bounded continuous ϕ : R N → M with the norm k ϕ k B ( R N , M ) := sup x ∈ R N | ϕ ( x ) | . And let B ( R N , M ) denote the space of all continuouslydifferentiable functions ϕ : R N → M such that both ϕ and its derivative are bounded andequip it with the norm k ϕ k B ( R N , M ) := sup x ∈ R N (cid:16) | ϕ ( x ) | + k ϕ ′ ( x ) k L ( R N , M ) (cid:17) .We shall also assume that the maps t a j ( · , t ) ∈ B ( R N , M ) , j = 1 , . . . , N , and t b ( · , t ) ∈ B ( R N , M ) are well defined, continuous and bounded. By use of [16, Section4.6] we conclude that the family of operators { A ( t ) } t ≥ in E := L ( R N , R M ) given by D ( A ( t )) := u ∈ E | ∂u∂x j exists for each j = 1 , . . . , N, N X j =1 a j ( · , t ) ∂u∂x j ∈ E , where weak derivatives are considered, and [ A ( t ) u ]( x ) := N X j =1 a j ( x, t ) ∂u∂x j ( x ) + b ( x, t ) u ( x ) , for each u ∈ D ( A ( t )) , a.e. x ∈ R N , satisfies ( Hyp ) − ( Hyp ) with V := H ( R N , R M ) , M = M V = 1 and some constants ω, ω V .Consequently, due to Proposition 3.1, the family { A ( t ) } t ≥ determines a unique evolutionsystem { R ( t, s ) } t ≥ s ≥ on E . Therefore, if we assume that f : R N × [0 , + ∞ ) → R M is suchthat the map [0 , + ∞ ) ∋ t f ( · , t ) ∈ L ( R N , R M ) is well defined and continuous, then,for λ > , we may consider ˙ u ( t ) = A ( t/λ ) u ( t ) + F ( t/λ ) (22)8with F : [0 , + ∞ ) → E given by F ( t ) := f ( · , t ) for each t ≥ .Now suppose that there are b a j ∈ B ( R N , M ) , j := 1 , . . . , N , with symmetric valuesand b b ∈ B ( R N , M ) such that lim T → + ∞ T Z T k a j ( · , τ + h ) − b a j k B ( R N , M ) d τ = 0 (23)and lim T → + ∞ T Z T k b ( · , τ + h ) − b b k B ( R N , M ) d τ = 0 (24)uniformly with respect h > . We claim that lim T → + ∞ T Z T k A ( τ + h ) − b A k L ( V,E ) d τ = 0 (25)uniformly with respect to h > , where b A : D ( b A ) → E is defined by D ( b A ) := u ∈ E | ∂u∂x j exists for each j = 1 , . . . , N, N X j =1 b a j ∂u∂x j ∈ E , b Au := N X j =1 b a j ∂u∂x j + b bu. Indeed, for any u ∈ V and t ≥ , k A ( t ) u − b Au k E ≤ N X j =1 Z R N | a j ( x, t ) ∂u∂x j ( x ) − b a j ( x ) ∂u∂x j ( x ) | dx / + (cid:18)Z R N | b ( x, t ) u ( x ) − b b ( x ) u ( x ) | dx (cid:19) / ≤ N X j =1 sup x ∈ R N | a j ( x, t ) − b a j ( x ) | + sup x ∈ R N | b ( x, t ) − b b ( x ) | k u k V and this yields, for any h > , T Z T k A ( τ + h ) − b A k L ( V,E ) d τ ≤ N X j =1 T Z T k a j ( · , τ + h ) − b a j k B ( R N , M ) d τ + 1 T Z T k b ( · , τ + h ) − b b k B ( R N , M ) d τ, which, by use of (23) and (24), gives (25) uniformly with respect to h > . Consequently,if { R ( λ ) ( t, s ) } t ≥ s ≥ , λ > , denote the evolution system generated by { A ( t/λ ) } t ≥ and { b S ( t ) } t ≥ the semigroup generated by b A , then, in view of Theorem 3.3, we get that, forany ¯ u ∈ E , µ ∈ P and t, s ≥ with t ≥ s , lim λ → + , ¯ v → ¯ u R ( λ ) ( t, s )¯ v = b S ( t − s )¯ u uniformly with respect to t, s from bounded intervals.At last suppose that Z R N | f ( x + y, t ) − f ( x, t ) | d x → as y → , uniformly in t ≥ , Z R N \ B (0 ,R ) | f ( x, t ) | d x → as R → + , uniformly in t ≥ , and that there is b f ∈ L ( R N , R M ) such that lim T → + ∞ Z R N (cid:12)(cid:12)(cid:12)(cid:12) T Z T f ( x, t + h ) d t − b f ( x ) (cid:12)(cid:12)(cid:12)(cid:12) d x = 0 , uniformly with respect to h > . Then the set F ([0 , + ∞ )) is relatively compact in L ( R N , R M ) and lim T → + ∞ T Z T F ( t + h ) d t = b f in L ( R N , R M ) uniformly with respect to h > . 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