On weak exponential expansiveness of skew-evolution semiflows in Banach spaces

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1 Yue et al. Jounal of Inequalities and Alications :165 htt:// R E S E A R C H Oen Access On weak exonential exansiveness of skew-evolution semiflows in Banach saces Tian Yue 1 Xiao-qiu Song 1* and Dong-qing Li 1 * Coesondence: sxqcumt@163.com 1 College of Sciences China Univesity of Mining and Technology Xuzhou Jiangsu 1008 China Full list of autho infomation is available at the end of the aticle Abstact The aim of this ae is to give seveal chaacteizations fo weak exonential exansiveness oeties of skew-evolution semiflows in Banach saces. Vaiants fo weak exonential exansiveness of some well-known esults in unifom exonential stability theoy (Datko (1973)) and exonential instability theoy (Lua (010) Megan et al. (008)) ae obtained. MSC: Pimay 93D0; seconday 34D0 Keywo: skew-evolution semiflow; weak exonential exansiveness; Baeia-Valls weak exonential exansiveness; Lyaunov function 1 Intoduction It is well known that in ecent yeas the exonential stability theoy of one aamete semigous of oeatos and evolution oeatos has witnessed significant develoment. A numbe of long-standing oen oblems have been solved and the theoy seems to have obtained a cetain degee of matuity. One of the most imotant esults of the stability theoy is due to Datko who oved in 1970 in [1] thatastonglycontinuoussemigou of oeatos {T(t)} t 0 is unifomly exonentially stable if and only if fo each vecto x fom the Banach sace X the function t T(t)x lies in L (R + ). Late Pazy genealizes theesultin[] fol (R + ) 1. In 1973 Datko [3] genealized the esults above and oved that an evolutionay ocess U = {U(t s)} t s 0 with unifom exonential gowth is unifomly exonentially stable if and only if thee exists an exonent 1suchthat su s 0 s U(t s)x dt < foeachx X. This esult was imoved by Rolewicz in 1986 (see [4]). In [5] and[6]theauthosgenealizedtheesultsaboveinthecaseofc 0 - semigous and evolutionay ocess esectively and esented a unified teatment in tems of Banach function saces. In [7] the oetyof nonunifom exonential stability has been studied by L. Baeia and C. Valls. In addition the weak exonential stability of evolution oeatos in Banach saces has been investigated and seveal imotant esults have been obtained by Lua Megan and Poa in [8]. Since the existence oblem of exonential exansiveness of evolution equations is distinct comaed to the studies devoted to stability and to dichotomy esectively exonential exansiveness is a oweful tool when eole analyze the asymtotic behavio of dynamical systems. In the last few yeas new concets of exonential exansiveness and in aticula of exonential instability have been intoduced and chaacteized (see [9 0]). Fo instance in [16] Megan and his atnes obtained some necessay and sufficient conditions fo unifom exonential instability of linea skew-oduct semiflows in 014 Yue et al.; licensee Singe. This is an Oen Access aticle distibuted unde the tems of the Ceative Commons Attibution License (htt://ceativecommons.og/licenses/by/.0) which emits unesticted use distibution and eoduction in any medium ovided the oiginal wok is oely cited.

2 Yue et al. Jounal of Inequalities and Alications :165 Page of 11 htt:// tems of Banach sequence saces and Banach function saces. In [1] and[15] the cases of unifom exonential instability has been consideed fo evolution families and linea skew-oduct flows esectively. Additionally in [19] Lua consideed a weake notion of instability fo evolution oeatos thus some necessay and sufficient chaacteizations fo weak exonential instability of evolution oeatos wee obtained. The concet of skew-evolution semiflows intoduced and chaacteized by Stoica and Meganin[17] by means of evolution semiflows and cocycles seems to be moe aoiate fo the study of asymtotic behavios of evolution equations. They deend on thee vaiables contay to a skew-oduct semiflow o an evolution oeato fo which they ae genealizations and which deend only on two. The exonential instability and unifom exonential stability fo skew-evolution semiflows ae studied by Stoica and Megan in [17] and[1] esectively. In the esent ae we intoduce the concet of weak exonential exansiveness fo skew-evolution semiflows which is an extension of classical concet of exonential exansiveness. Ou main objective is to give some chaacteizations fo weak exonential exansiveness oeties of skew-evolution semiflows in Banach saces and vaiants fo weak exonential exansiveness of some well-known esults in unifom exonential stability theoy (Datko [3]) and exonential instability theoy (Lua [19] Megan and Stoica [17]) ae obtained. As alications we obtain chaacteizations of the concets in tems of Lyanov functions. We note that in ou oof we don t need to assume the stong continuity of skew-evolution semiflows. Peliminaies Let (X d)beameticsacev a Banach saces. The nom on V and on the sace B(V )of all bounded oeatos on V will be denoted by.wedenotet = {(t ) R + : t 0} and Y = X V. I is the identity oeato. Definition.1 (see [1]) A maing ϕ : T X X is called evolution semiflow on X if following oeties ae satisfied: (es1) ϕ(t t x)=x (t x) R + X; (es) ϕ(t ϕ( x)) = ϕ(t x) (t ) ( ) T x X. Definition. (see [1]) A maing : T X B(V ) is called evolution cocycle ove an evolution semiflow ϕ if it satisfies following oeties: (ec1) (t t x)=i t 0 x X; (ec) (t ϕ( x)) ( x)= (t x) (t ) ( ) T x X. Definition.3 (see [1]) A maing C : T Y Y defined by C(t x v)= ( ϕ(t x) (t x)v ) (1) whee is an evolution cocycle ove an evolution semiflow ϕ is called a skew-evolution semiflow on Y. Examle.4 Let us denote C = C(R + R) the set of all continuous functions x : R + R endowed with the toology of unifom convegence on bounded sets. The set C is metiz-

3 Yue et al. Jounal of Inequalities and Alications :165 Page 3 of 11 htt:// able with esect to the metic d(x y)= n=1 1 d n (x y) n 1+d n (x y) wheed n(x y)= su x(t) y(t). t [ nn] Afunctionf : R + [1 ) definedbyf (t) =e t +1t 0. We denote f t (τ) =f (t + τ) t τ R +.LetX be the closue of the set {f t t R + } in C.Then(X d) is a metic sace and the maing ϕ : T X X ϕ(t x)=x t is an evolution semiflow on X.LetV = R. We conside : T X B(V )givenby (t x)v = e x(τ ) dτ v which is an evolution cocycle and C =(ϕ )isaskew-evolutionsemiflowony. Remak.5 The skew-evolution semiflows ae genealizations of the evolution oeatos and of the skew-oduct semiflows (cf. [1] Examle and 3). Definition.6 A skew-evolution semiflow C =(ϕ ) is called with exonential decay if thee ae M ω >0suchthat (t t0 x 0 )v 0 Me ω(t ) ( t0 x 0 )v 0 () fo all (t ) ( ) T and all (x 0 v 0 ) Y. Definition.7 A skew-evolution semiflow C =(ϕ ) is said to be unifomly exansive if thee exists a constant N >0 such that (t t0 x 0 )v 0 N ( t0 x 0 )v 0 (3) fo all (t ) ( ) T and all (x 0 v 0 ) Y. Definition.8 A skew-evolution semiflow C =(ϕ ) is said to be unifomly exonentially exansive if thee ae N α >0 such that (t x 0 )v 0 Ne α(t ) ( x 0 )v 0 fo all (t ) ( ) T and all (x 0 v 0 ) Y. Remak.9 It is obvious that a skew-evolution semiflow C =(ϕ ) isunifomlyexonentially exansive if and only if thee ae N α >0suchthat (t t0 x 0 )v 0 Ne α(t ) v 0 (4) fo all (t x 0 v 0 ) T Y.

4 Yue et al. Jounal of Inequalities and Alications :165 Page 4 of 11 htt:// Remak.10 If a skew-evolution semiflow is unifomly exonentially exansive then it is unifomly exansive. The convese is not necessaily valid. To show this we conside the following examle. Examle.11 We conside X = R + V = R and a non-deceasing and bounded function f : R + [1 ). It is obvious that the maing ϕ : T R + R + defined by ϕ(t x 0 )=t + x 0 (t x 0 ) T R + is an evolution semiflow on R + and the maing : T R + B(R)givenby (t x 0 )= f (t + x 0) (t x 0 ) T R + f (x 0 ) is an evolution cocycle on R. Then the skew-evolution semiflow C =(ϕ ) isunifomly exansive with N =1. On the othe hand if we assume that C =(ϕ ) is unifomly exonentially exansive then thee ae constants N α >0 such that f (t + x 0 ) f (x 0 ) Ne α(t ) (t x 0 ) T R +. Fom this fo =0weobtainf (t + x 0 ) Ne αt f (x 0 ) which fo t gives a contadiction and hence C is not unifomly exonentially exansive. Definition.1 A skew-evolution semiflow C =(ϕ ) is called weakly exonentially exansive if thee ae N α > 0 such that fo all (x 0 v 0 ) Y thee exists 0with (t t0 x 0 )v 0 Ne α(t ) ( t0 x 0 )v 0 (5) fo all t. Remak.13 If a skew-evolution semiflow is unifomly exonentially exansive then it is weakly exonentially exansive. The following examle shows that the convese is not valid. Examle.14 We conside the metic sace X and an evolution semiflow ϕ on X defined as in Examle.11.LetV = R with the Euclidean nom and the evolution oeato U(t )=P(t )Q( ) (also see [19] Examle 11) whee ( ) e t sin t e (t t0) cos t P(t )= e t cos t e (t t0) sin t and Q( )= ( ) cos sin. sin cos Then the maing U : T X B(R )givenby U (t x 0 )=U(t )isanevolution cocycle on R ove the evolution semiflow ϕ. Fo evey v 0 R thee exist ρ 0and [0 π) suchthatv 0 =(ρ cos ρ sin ) T.It is easy to see that U (t x 0 )v 0 = P(t )Q( )v 0 = P(t )(ρ0) T = ( ρe t sin t ρe t cos t ) T

5 Yue et al. Jounal of Inequalities and Alications :165 Page 5 of 11 htt:// and hence U (t x 0 )v 0 = ρe t = e t U ( x 0 )v 0 fo all t which oves that the skew-evolution semiflow C =(ϕ ) isweaklyexonentially exansive. On the othe hand we obseve that fo y 0 =(sin cos ) T U (t x 0 )y 0 = P(t )(0 1) T = ( e (t ) cos t e (t ) sin t ) T and hence U (t x 0 )y 0 = e (t ) = e (t ) U ( x 0 )y 0 which shows that C is not unifomly exonentially exansive. Definition.15 A skew-evolution semiflow C =(ϕ ) is called weakly exonentially exansive in the Baeia-Valls sense if thee ae N α >0andβ 0 such that fo all (x 0 v 0 ) Y thee exists 0with e β (t t0 x 0 )v 0 Ne α(t ) ( t0 x 0 )v 0 (6) fo all t. Definition.16 (see [17]) A skew-evolution semiflow C =(ϕ ) is called stongly measuable if the maing t (t x 0 )v 0 is measuable on [ ) fo all ( x 0 v 0 ) R + Y. Definition.17 A maing L : T Y R is said to be a Lyaunov function fo the skewevolution semiflow C =(ϕ ) if thee is a constant a 0 such that fo all (x 0 v 0 ) Y thee exists 0with L(t x 0 v 0 )+ fo all t. e a(t s) (s t0 x 0 )v 0 L( t0 x 0 v 0 ) (7) 3 Themainesults Poosition 3.1 A skew-evolution semiflow C =(ϕ ) is weakly exonentially exansive if and only if thee exists a deceasing function f :[0 ) (0 ) with lim t f (t) =0 such that fo evey (x 0 v 0 ) Ytheeis 0 with the oety ( x 0 )v 0 f (t ) (t x 0 )v 0 (8) fo all t.

6 Yue et al. Jounal of Inequalities and Alications :165 Page 6 of 11 htt:// Poof Necessity. It is a simle veification fo f (t) = 1 N e αt wheen and α ae given by Definition.1. Sufficiency. Accoding to the oety of function f thee exists a constant δ >0such that f (δ) < 1. Fom the hyothesis we find that fo evey (x 0 v 0 ) Y thee is 0satisfying elation (8). Fo evey t thee ae n N and l [0 δ) suchthatt = nδ + l. Then the following inequalities: ( x 0 )v 0 f (l) (t nδ x 0 )v 0 f (l)f (δ) ( t (n 1)δ t0 x 0 ) v0 f (l) [ f (δ) ] n (t t0 x 0 )v 0 f (0) [ ] n+1 f (δ) (t t0 x 0 )v 0 f (δ) 1 N e α(t ) (t x 0 )v 0 hold fo all t whee we have denoted N = f (δ) f (δ) and α = ln. f (0) δ Finally it follows that C =(ϕ ) is weakly exonentially exansive. Coollay 3. A skew-evolution semiflow C =(ϕ ) is weakly exonentially exansive if and only if thee exists a non-deceasing function g :[0 ) (0 ) with lim t g(t)= + such that fo evey (x 0 v 0 ) Ytheeis 0 with the oety g(t ) ( x 0 )v 0 (t x 0 )v 0 (9) fo all t. Theoem 3.3 Let C =(ϕ ) be a stongly measuable skew-evolution semiflow with exonential decay. Then C is weakly exonentially exansive if and only if thee ae >0and L >0such that fo evey (x 0 v 0 ) Ytheeis 0 with (s x 0 )v 0 L (t x 0 )v 0 (10) fo all t. Poof Necessity. If C is weakly exonentially exansive then fom Definition.1 it follows that thee ae N α > 0 with the oety that fo all (x 0 v 0 ) Y thee exists 0such that (s t0 x 0 )v 0 N e α(t s) (t x 0 )v 0 L (t t0 x 0 )v 0 fo all t whee >0isfixedandL = N α.

7 Yue et al. Jounal of Inequalities and Alications :165 Page 7 of 11 htt:// Sufficiency. We assume that thee ae >0andL > 0 such that fo evey (x 0 v 0 ) Y thee is 0 satisfying inequality (10). Let t.ift +1wehave L (t t0 x 0 )v 0 and fo t [ +1)wehave (s t0 x 0 )v 0 M e ω(s ) ( x 0 )v 0 = M e ωτ dτ ( t0 x 0 )v M e ωτ dτ ( t0 x 0 )v 0 0 = 1 e ω ω M ( x 0 )v 0 (t x 0 )v 0 e ω M ( x 0 )v 0 (s t0 x 0 )v 0 whee M ω > 0 ae given by Definition.6. Hence (t x 0 )v 0 K ( x 0 )v 0 (11) fo all t wheek = M [e ω +(1 e ω )/ωl]. On the othe hand L (t t0 x 0 )v 0 (s t0 x 0 )v 0 (s t0 x 0 )v 0 K(t ) ( t0 x 0 )v 0 (1) fo all t. Adding u (11)and(1)we obtain ( 1+L 1/ ) (t t0 x 0 )v 0 K 1/ [ 1+(t ) 1/] ( t0 x 0 )v 0 fo all t. Accoding to Coollay 3. C is weakly exonentially exansive which en the oof. Theoem 3.4 Let C =(ϕ ) be a stongly measuable skew-evolution semiflow with exonential decay. Then C is weakly exonentially exansive if and only if thee ae L : T Y R a Lyaunov function fo C and a constant b >0such that L(t x 0 v 0 ) b (t t0 x 0 )v 0 (13) fo all t.

8 Yue et al. Jounal of Inequalities and Alications :165 Page 8 of 11 htt:// Poof Necessity. Let a = 0. We conside the alication L : T Y R L(t x v)= e a(t s) (s x)v 0. Then fom Definition.1 we find that thee ae N α >0andfoevey(x 0 v 0 ) Y thee is 0with L(t x 0 v 0 ) = (s t0 x 0 )v 0 b (t t0 x 0 )v 0 fo all t wheeb = 1 αn.itiseasytoseethat L(t x 0 v 0 )+ e a(t s) (s x 0 )v 0 L( t0 x 0 v 0 ) 0 fo all t.hencel is a Lyaunov function fo C such that the elation (13)istue. Sufficiency. We assume that thee ae L : T Y R alyaunovfunctionfoc and a constant b > 0 such that the elation (13)hold. Then (s x 0 )v 0 e a(t s) (s x 0 )v 0 L( x 0 v 0 ) L(t x 0 v 0 ) L(t x 0 v 0 )= L(t x 0 v 0 ) b (t t0 x 0 )v 0 fo all t wheea 0 is given by Definition.17.ByTheoem3.3 we conclude that C is weakly exonentially exansive. Poosition 3.5 A skew-evolution semiflow C =(ϕ ) is weakly exonentially exansive in the Baeia-Valls sense if and only if thee ae N >0λ > μ 0 such that fo all (x 0 v 0 ) Ytheeexists 0 with e μt (t t0 x 0 )v 0 Ne λ(t ) ( t0 x 0 )v 0 (14) fo all t. Poof Necessity. It follows by a simle veification fo μ = β and λ = α +βwheeconstants α >0andβ 0 ae given by Definition.15. Sufficiency. Fom the hyothesis thee ae N >0λ > μ 0 such that fo all (x 0 v 0 ) Y thee exists 0satisfying N ( x 0 )v 0 e μt e λ(t ) (t x 0 )v 0 = e μ e (λ μ)(t ) (t t0 x 0 )v 0 fo all t which imlies that C is weakly exonentially exansive in the Baeia- Valls sense with α = λ μ and β = μ.

9 Yue et al. Jounal of Inequalities and Alications :165 Page 9 of 11 htt:// Theoem 3.6 Let C =(ϕ ) be a stongly measuable skew-evolution semiflow with exonential decay. Then C is weakly exonentially exansive in the Baeia-Valls sense if and only if thee ae L α >0 >0and β 0 such that fo evey (x 0 v 0 ) Ytheeis 0 with e (α+β)(t s) (s t0 x 0 )v 0 Le βt (t t0 x 0 )v 0 (15) fo all t. Poof Necessity. If C isweaklyexonentiallyexansiveinthebaeia-vallssense thenby Poosition 3.5 thee ae N >0λ > μ 0 such that fo all (x 0 v 0 ) Y thee exists 0 with e (α+β)(t s) (s x 0 )v 0 N e μt e (λ α β)(t s) (t t0 x 0 )v 0 Le βt (t x 0 )v 0 fo all t whee >0isfixedβ = μ α (0 λ μ)andl = N (λ α β). Sufficiency. We assume that thee ae L α >0 >0andβ 0suchthatfoevey (x 0 v 0 ) Y thee is 0 satisfying inequality (15). Let t.ift +1then M e (α+β+ω) e (α+β)(t ) ( t0 x 0 )v 0 = = M e (α+β+ω) e (α+β)(t ) ( t0 x 0 )v 0 e (α+β+ω) e ω(s ) e (α+β)(t s) e (α+β)(s ) (s x 0 )v 0 e (α+β+ω)[(s ) 1] e (α+β)(t s) (s x 0 )v 0 e (α+β)(t s) (s t0 x 0 )v 0 Le βt (t t0 x 0 )v 0 and theefoe e (α+β) (t t0 x 0 )v 0 ML 1/ e (α+β+ω) e αt ( t0 x 0 )v 0 whee M ω > 0 ae given by Definition.6. We conside t [ +1).Then ( x 0 )v 0 M 1 e ω(t ) (t x 0 )v 0 = M 1 e (α+β+ω)(t ) e (α+β)(t ) (t t0 x 0 )v 0 M 1 e (α+β+ω) e (α+β)(t ) (t t0 x 0 )v 0.

10 Yue et al. Jounal of Inequalities and Alications :165 Page 10 of 11 htt:// Futhe we obtain e (α+β) (t x 0 )v 0 Me (α+β+ω) e (α+β)t ( x 0 )v 0 Me (α+β+ω) e αt ( x 0 )v 0. Hence C is weakly exonentially exansive in the Baeia-Valls sense which en the oof. Remak 3.7 Theoems 3.3 and 3.6 ae the vesions of the classical stability theoems and exonential instability theoems due to Datko [3] Lua [19] Megan and Stoica [17] fo weak exonential exansiveness of skew-evolution semiflows. Theoem 3.8 Let C =(ϕ ) be a stongly measuable skew-evolution semiflow with exonential decay. Then C is weakly exonentially exansive in the Baeia-Valls sense if and only if thee ae L : T Y R a Lyaunov function fo C and constants c >0d 0 such that L(t x 0 v 0 ) ce dt (t t0 x 0 )v 0 (16) fo all t. Poof Accoding to the conclusion of Theoem 3.6 the agumentation can be obtained as well as that of Theoem 3.4. Cometing inteests The authos declae that they have no cometing inteests. Authos contibutions The authos comleted the ae togethe. They also ead and aoved the final manuscit. Autho details 1 College of Sciences China Univesity of Mining and Technology Xuzhou Jiangsu 1008 China. School of Science Hubei Univesity of Automotive Technology Shiyan Hubei 4400 China. Acknowledgements The authos would like to thank the efeee fo helful suggestions and comments. This wok was suoted by the Fundamental Reseach Fun fo the Cental Univesities (No.013XK03). Received: 5 June 013 Acceted: 1 Ail 014 Published: 06 May 014 Refeences 1. Datko R: Extending a theoem of Liaunov to Hilbet saces. J. Math. Anal. Al (1970). Pazy A: Semigous of Linea Oeatos and Alications to Patial Diffeential Equations. Singe New Yok (1983) 3. Datko R: Unifom asymtotic stability of evolutionay ocesses in a Banach sace. SIAM J. Math. Anal (1973) 4. Rolewicz S:On unifom N-equistability. J. Math. Anal. Al (1986) 5. Neeven JMAM: Exonential stability of oeatos and semigous. J. Funct. Anal (1995) 6. Peda P Pogan A Peda C: Functionals on function and sequence saces connected with the exonential stability of evolutionay ocesses. Czechoslov. Math. J (006) 7. Baeia L Valls C: Stability of Nonautonomous Diffeential Equations. Singe Belin (008) 8. Lua N Megan M Poa I-L: On weak exonential stability of evolution oeatos in Banach saces. Nonlinea Anal (010) 9. Minh NV Räbige F Schnaubelt R: Exonential stability exonential exansiveness and exonential dichotomy of evolution equations on the half-line. Integal Equ. Oe. Theoy (1998) 10. Megan M Sasu AL Sasu B: Nonunifom exonential unstability of evolution oeatos in Banach saces. Glas. Mat. Se. III (001)

11 Yue et al. Jounal of Inequalities and Alications :165 Page 11 of 11 htt:// 11. Megan M Pogan A: On exonential h-exansiveness of semigous of oeatos in Banach saces. Nonlinea Anal (003) 1. Megan M Sasu AL Sasu B: Banach function saces and exonential instability of evolution families. Ach. Math (003) 13. Megan M Sasu AL Sasu B: Peon conditions fo unifom exonential exansiveness of linea skew-oduct flows. Monatshefte Math (003) 14. Megan M Sasu B Sasu AL: Exonential exansiveness and comlete admissibility fo evolution families. Czechoslov. Math. J (004) 15. Megan M Sasu AL Sasu B: Exonential stability and exonential instability fo linea skew-oduct flows. Math. Bohem (004) 16. Megan M Sasu AL Sasu B: Exonential instability of linea skew-oduct semiflows in tems of Banach function saces. Results Math (004) 17. Megan M Stoica C: Exonential instability of skew-evolution semiflows in Banach saces. Stud. Univ. Babeş-Bolyai Math. LIII 17-4 (008) 18. Sasu B: New citeia fo exonential exansiveness of vaiational diffeence equations. J. Math. Anal. Al (007) 19. Lua N: Necessay and sufficient conditions fo weak exonential instability of evolution oeatos. Int. J. Pue Al. Math (010) 0. Lua N Megan M: Exonential dichotomies of evolution oeatos in Banach saces. Monatshefte Math. (013). doi: /s y 1. Stoica C Megan M: On unifom exonential stability fo skew-evolution semiflows on Banach saces. Nonlinea Anal (010) /109-4X Cite this aticle as: Yue et al.: On weak exonential exansiveness of skew-evolution semiflows in Banach saces. Jounal of Inequalities and Alications :165

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal