In this paper we give necessary and sufficient conditions for uniform exponential stability of linear skew-product semiflows in Banach spaces. We give theorems of characterization for uniform exponential stability of linear skew-product semiflows in terms of boundedness of some operators acting

Andrea A. Minda

Mihaela A. Tomescu

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 241    Andrea A. Minda, Mihaela Tomescu Perron Conditions for Stability of Linear Skew-Product Semiflows in Banach Spaces  In this paper we give necessary and sufficient conditions for uniform exponential stability of linear skew-product semiflows in Banach spaces. We give theorems of characterization for uniform exponential stability of linear skew-product semiflows in terms of boundedness of some operators acting on ( )00 R ,C X+ , ( )R ,bC X+ , ( )R ,+ ΧcC and ( )R ,pL X+ , respectively. 1. Linear skew-product semiflow  Let X be a Banach space, let ( ),dΘ  be a metric space and let E X= × Θ . We shall denote by ( )B X  the Banach algebra of all bounded linear operators from X  into itself. Throughout the paper, de norm on X  and on ( )B X  will be denoted by ⋅ .  Definition 1.1  A continuous mapping : Rσ +Θ × → Θ  is said  to be a semiflow on Θ , if it has the following properties:  (i)  ( ,0)σ θ θ= , θ∀ ∈ Θ   (ii) ( )( , ) ( , ),s t s tσ θ σ σ θ+ =  for 2( , , ) Rθ +∀ ∈ Θ×s t   Definition 1.2  A pair  ( , )π σ= Φ  is called linear skew-product semiflow on E X= × Θ  if σ  is a semiflow on Θ  and : R ( )B X+Φ Θ × →  satisfies the follow-ing conditions: (i)  ( ,0)θΦ = I , the identity operator on X ,  θ∀ ∈Θ  (ii) ( )( ) ( )( , ) , , ,t s s t sθ σ θ θΦ + = Φ Φ  for all 2( , , ) Rθ +∈ Θ×t s  (the cocycle identity)  (iii) 1∃ ≥M  and 0ω >  such that ( , ) tt Me ωθΦ ≤  for ( , ) Rθ +∀ ∈ Θ×t  ANALELE UNIVERSIT ĂłII   “EFTIMIE MURGU” RE ŞIłA  ANUL XIII, NR. 1, 2006, ISSN 1453 - 7397   242 If, in addition, (iv) for every ( , )θ ∈x E  the mapping ( , )t t xθ→ Φ  is continuous then π  is called a strongly continuous linear skew-product semiflow.  Remark 1.1  Statement (iii) is equivalent with the following  (iii)’ there exist a nondecreasing function *: R R+ +→f  such that ( ) ( ),t f tθΦ ≤  for all ( ), Rθ θ +∈ ×t .  Example 1.1 Let X  be a Banach space. We consider +(R ,R)C  the space of all continuous function with the topology of uniform convergence on compact subsets on R+ . This space is metrizable with the metric  1( , )1( , )1 ( , )2nnn nd x yd x yd x y∞== ⋅+∑  where [ ]0,( , ) sup ( ) ( )nt nd x y x t y t∈= − . On the Banach space X, we consider the nonautonomous differential equation  ( ) ( ) ( ) , 0x t a t x t t= ≥&  where +: R Ra +→  be a uniformly continuous function such that there exists : lim ( )ta tα→∞= < ∞ . If we denote by ( ) ( )sa t a t s= +  and by { }: Rsa s +Θ = ∈  then  ( , )( ) : ( )t s t sσ θ θ= +  is a semiflow on Θ , for : R ( )B X+Φ Θ × → , ( ) ( )00,tdt x e xθ τ τθ ∫Φ =  we have that ( ),π σ= Φ  is a linear skew-product semiflow on E X= × Θ .  Definition 1.3  A linear skew-product semiflow ( ),π σ= Φ  on E X= × Θ  is said to be   stable  if there exists  0N >  such that ( ) ( ), , , Rt N tθ θ +Φ ≤ ∀ ∈ Θ ×   Definition 1.4  A linear skew-product semiflow ( ),π σ= Φ  on E X= × Θ  is uniformly  exponentially  stable  if there are  , 0N ν >  such that ( ) ( ), , , Rtt Ne tνθ θ− +Φ ≤ ∀ ∈ Θ ×   Example 1.2  Let Rλ +∈ . Consider the linear skew product semiflow ( ),λ λπ σ= Φ  where ( ) ( ), ,tt e tλλ θ θ−Φ = Φ  and ( )σπ ,Φ=  is the linear skew product semiflow given in example 1.1 . For λ α> , λπ is uniformly exponentially stable and for  0,λ α∈     and ( )0θ τ α= , for all 0τ ≥  we have :  243  ( )0,,x ift xe x ifλ α λλ αθλ α− =Φ = < , so  λπ  is not  uniformly  exponentially stable.   Let  ( )R ,+ ΧbC  be the linear space of all bounded continous functions : R+ → Χu  and  ( ) ( ) ( ){ }0 R , R , : lim 0btC X u C X u t+ + →∞= ∈ =  ( ) ( ) ( ) ( ){ }00 R , R , : 0 lim 0btC X u C X u u t+ + →∞= ∈ = =   Endowed with sup - norm ( )tuut 0sup≥=  , ( )0 R ,C X+ and ( )R ,bC X+ are Banach spaces.   Let ( )R ,+ ΧcC  be the space of all  continous functions : R+ → Χu  with  compact  support  contained in  ( )∞,0 .              Denote by F the linear space of all Bochner measurable functions : R+ → Χu  identifying the functions which are equal almost everywhere. For every [ )∞∈ ,1p  the linear space    ( ) ( )0R , :ppL u F u t dt∞+  Χ = ∈ < ∞   ∫   is a  Banach space with respect to the norm:  ( )10:pppu u t dt∞  =  ∫    2. Perron conditions for exponential stability Definition 2.1  We say that a strongly continous linear skew-product semiflow ( )σπ ,Φ=  is  ( ) ( )( )( ), , , ,c bC R X B C R X+ +Θ - stable if:  (i)  the linear operator ( ) ( )( ): , , ,c bP C R X B C R X+ +→ Θ      ( ) ( ) ( ) ( )( ) ( )0, ,tP u t t u dθ τ θ τ τ τ τ= Φ −∫          is well defined .  (ii)  there exist  K>0 such that :                         ( )( ) ( ), , , ,b cB C R XPu K u u C R X+ +Θ ≤ ∀ ∈   244 Definition  2.2   Let ( ) ( ) ( ){ }00, , , ,bE R X C R X C R X+ + +∈  , we say that a strongly continous linear skew-product semiflow  ( ),π σ= Φ  is  ( ) ( )( )( ), , , ,bE R X B C R X+ +Θ -stable if ( ) ( )( ): , , ,bS E R X B C R X+ +→ Θ  with:             ( ) ( ) ( ) ( )( ) ( )0, ,tS u t t u dθ σ θ τ τ τ τ= Φ −∫  is well defined .  Proposition 2.1 If π  is ( ) ( )( )( ), , , ,bE R X B C R X+ +Θ -stable , then  S is a bounded operator.  Proof Let ( ) ( ), ,nu E R X+⊂ ( ),u E R X+∈  and ( )( ), ,bf B C R X+∈ Θ  such that:   uunn ∞→→   in  ( ),E R X+   and  ( )nnS u f→∞→ in ( )( ), ,bB C R X+Θ  We have that:                                       ( ) ( ) ( ) ( ) ( ) ( ) ( ), ,nnS u t f t t Rθ θ θ +→∞→ ∀ ∈ Θ ×  Using uunn ∞→→  uniformly on R+  we obtain ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )0 0, , , ,t tn nnS u t t u d t u dθ σ θ τ τ τ τ σ θ τ τ τ τ→∞= Φ − → Φ − =∫ ∫  ( ) ( ) ( ) ( ) ( ), ,S u t t Rθ θ += ∀ ∈ Θ ×   We have  ( )f S u=  , so  S is a closed linear operator  and S  is bounded.   Proposition 2.2  Let ( ),π σ= Φ be a linear skew-product semiflow on  E X= × Θ . If there are 0 0t >  and ( )0,1c ∈  such that  ( )0,t cθΦ ≤  for θ∀ ∈ Θ , than  π  is uniformly exponentially unstable.  Proof Let 1M ≥  and 0ω >  be given by definition  1.2. and ν  be a pozitive number such that   0tc e ν−= .  Let θ ∈Θ  be  fixed. For 0t ≥  there are Nn ∈  and [ ]00,r t∈  such that 0t nt r= + . Than we obtain :  ( ) ( )( ) ( ) 0 00 0, , , , t n t tt nt r nt Me e Neω ν νθ σ θ θ − −Φ ≤ Φ Φ ≤ ≤    Where ( ) 0tN Me ω ν+= . So, π  is uniformly exponentially stable.  245    Theorem 2.1   Let ( )σπ ,Φ=  be a  strongly continous linear skew-product semiflow  on  Θ×Χ=E . Then the following assertions are equivalent:  (i)  π  is  uniformly exponentially stable; (ii) π  is ( ) ( )( )( ), , , ,b bC R X B C R X+ +Θ - stable; (iii) π  is  ( ) ( )( )( )00 , , , ,bC R X B C R X+ +Θ - stable; (iv) π  is  ( ) ( )( )( )00 00, , , ,C R X B C R X+ +Θ - stable;  Proof (i) ( )ii⇒  Let  N,v > 0 with:       ( ) ( ), , Rvtt Ne tθ θ− +Φ ≤ ∀ ∈Θ × . Let ( )R ,+∈ Χbu C  . For all ( ), Rtθ +∈Θ ×  we have  ( ) ( ) ( ) ( )( )0 0t tt s NS u t Ne u d N u e ds uν τ νθ τ τν− − −≤ ≤ ≤∫ ∫  So  S(u) ∈  B(Θ , Cb(R+, X)).  It follows that π  is ( ) ( )( )( ), , , ,b bC R X B C R X+ +Θ -stable. The implication (ii) ( )iii⇒  is obvious, ( ) ( )i iv⇒ is a simple exercise, ( ) ( )iv iii⇒  is obvious . Suppose that  (iii) holds and there  is  0Κ >  such that    uuP Κ≤θ          (2.1)  for all   ( ) ( ), R ,ou Cθ +∈ Χ × Θ . Consider  1≥Μ  and 0ω >  given by definition 1.2. Let  Θ∈θ  and Χ∈x . If :R 0,2α + →     is a continous functions  with  the  support  contained in ( )1,0  and with the property :  ( )101s dsα =∫  Then we consider the function                      ( ) ( ) ( ):R , ,u X u t t t xα θ+ → = Φ  So ( )0 ,u C R X+∈ and ( )0,1sup 2tu u t Me xω∈  = ≤   246 For 1≥t  we have that:           ( ) ( ) ( ) ( ) ( )( ) ( ) ( )0, , , ,tP u t s s t s s xds t xθ α σ θ θ θ= Φ − Φ = Φ∫  Then using,  (2.1)we have:           ( ) ( ), 2t x P u KMe xωθΦ ≤ ≤                                 (2.2)                 For [ ]1,0∈t  we have :                             ( ),t MeωθΦ ≤                                     (2.3) So, denoting by ( )2 1L K Meω= +  and using (2.2) and (2.3) we obtain:                                                    ( ),t LθΦ ≤  for all ( ) +×Θ∈ Rt,θ  Consider 4eLKν = and ( ),0:tsR R t se dsνϕ ϕ −+ +→ = ∫  The function ϕ  is strictly increasing on  +R  with ( ) 21limttϕν→∞=  So, we can choose 0>δ such that ( )212ϕ δν> . Let Θ∈θ  and Xx ∈ . Define the function                ( ) ( ): , ,tv R X v t te t xν θ−+ → = Φ  Then  ( )00 ,v C R X+∈  and                                    0sup ttLv L x te xeνν−≥≤ =   We observe that ( ) ( ) ( ) ( ) ( ),P v xθ δ ϕ δ θ δ= Φ   and it follows that :             ( ) ( ) ( ) ( )2 2 1, 2 , 2 22LKx x P v x xeθ δ ν ϕ δ θ δ ν νΦ ≤ Φ ≤ ≤ =  It results that:      ( ) 1,2θ δΦ ≤  for all Θ∈θ .  From proposition 2.2 we have that π  is uniformly exponentially stable.  Definition 2.3  Let ( )1,p ∈ ∞ . We say that a strongly continous linear skew-product semiflow ( )σπ ,Φ=  is  ( ) ( )( )( ), , , ,c bC R X B C R X+ +Θ - p-stable if:  (i)  the linear operator  ( ) ( )( ): , , ,c bP C R X B C R X+ +→ Θ    ( ) ( ) ( ) ( )( ) ( )0, ,tP u t t u dθ τ θ τ τ τ τ= Φ −∫   247         is well defined .  (ii)  there exist  K>0 such that : ( )( ) ( ), , , ,b cB C R X pPu K u u C R X+ +Θ ≤ ∀ ∈  Definition  2.4   Let ( )1,p ∈ ∞   , we say that a strongly continous linear skew-product semiflow  ( ),π σ= Φ  is ( ) ( )( )( ), , , ,p bL R X B C R X+ +Θ -stable if ( ) ( )( ): , , ,p bQ L R X B C R X+ +→ Θ  with:              ( ) ( ) ( ) ( )( ) ( )0, ,tQ u t t u dθ σ θ τ τ τ τ= Φ −∫  is well defined . Proposition 2.3  If π  is ( ) ( )( )( ), , , ,p bL R X B C R X+ +Θ -stable , then  Q is a bounded operator.  Theorem 2.2   Let ( )1,p ∈ ∞ and ( )σπ ,Φ=  be a  strongly continous linear skew-product semiflow  on  Θ×Χ=E . Then the following assertions are equivalent:  (i)    π  is  uniformly exponentially stable; (ii) π  is ( ) ( )( )( ), , , ,p bL R X B C R X+ +Θ - stable; (iii) π  is  ( ) ( )( )( ), , , ,c bC R X B C R X+ +Θ -p- stable; Proof (i) ⇒  (ii)  Let  N,v > 0  such that:     ( ) ( ), , Rvtt Ne tθ θ− +Φ ≤ ∀ ∈Θ × . Let  ( )R ,pu L X+∈  and ,1ppp=−.  For all  ( ), Rtθ +∈Θ× we have: ( ) ( ) ( ) ( )( )0tv tQ u t N e u dτθ τ τ− −≤ ≤∫  ( )( ),,,1/ 1 /( )1/,0 0p pt tpvp tppNN e d u d uvpτ τ τ τ− −   ≤ ≤         ∫ ∫  So  ( ) ( )( ), ,bQ u B C R X+∈ Θ  (ii) ⇒  (iii) follows from proposition 2.3. (iii) ⇒  (i) see [7], theorem 2.4.4.    248 References  [1] Chicone, C., Latushkin, Y., Evolution semigroups in Dynamical Sys-tems and Differential Equations, Mathematical Surveys and Monographs 70 American Mathematical Society, 1999 [2] Chow, S. N., Leiva, H. Existence and roughness of exponential di-chotomy for linear skew-product semiflow in Banach spaces, J. Differen-tial Equations 120 (1995), 429-477 [3] Chow, S. N., Leiva, H.,Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proceeding of the American Mathematical Society, volume 124, number 4, 1996, 1071-1081 [4] Megan, M., Sasu, A.L., On uniform exponential stability of linear skew-product semiflows in Banach spaces, Bulletin Belgian Mathematical Society Simon Stevin 9 (2002) 143-154 [5] Megan, M., Sasu, A.L., Sasu, B., Perron conditions and uniform ex-ponential sability of linear skew-product semiflows on locally compact spaces, Acta Math. Univ. Comenianae, Vol LXX, 2(2001), 229-240 [6] Megan, M., Sasu, A.L.,. Sasu, B, On uniform exponential unstability of linear skew-product semiflows, Seminar on Mathematical Analysis and Applications in Control Theory, University of the West, Timişoara, 2002 [7] Sasu, A. L, Admisibilitate şi proprietăŃi asimptotice ale cociclilor, Editura Politehnica Timişoara, 2005  Addresses: • Assoc. Prof. Drd. Andrea A. Minda,  “Eftimie Murgu” University of Re-şiŃa,Romania, PiaŃa “Traian Vuia”, nr. 1-4, ReşiŃa, andreaminda@yahoo.com • Asist.univ. drd. Mihaela Tomescu, University of  Petrosani , mihaela_tomescu2000@yahoo.com;   

Perron Conditions for Stability of Linear Skew- Product Semiflows in Banach Spaces

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Perron Conditions for Stability of Linear Skew- Product Semiflows in Banach Spaces

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