MIHAIL MEGAN and LARISA BIRIŞ
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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIV, 2008, f.2 POINTWISE EXPONENTIAL TRICHOTOMY OF LINEAR SKEW-PRODUCT SEMIFLOWS BY MIHAIL MEGAN and LARISA BIRIŞ Abstract. The aim of this paper is to give necessary and sufficient conditions for pointwise exponential trichotomy of linear skew-product semiflows. The results obtained here are generalizations of the theorems proved for the case of pointwise exponential dichotomy. Mathematics Subject Classification 2000: 34D05, 34D20. Key words: Linear skew-product semiflow, exponential trichotomy. 1. Preliminaries. Let V be a Banach space and let (X, d) be a metric space. We denote by B(V ) the Banach algebra of all bounded linear operators from V into itself. The norm on V and on B(V ) will be denoted by. Definition 1.1. A continuous mapping ϕ : R + X X is said to be a semiflow on X, if it has the following properties: (i) ϕ(0, x) = x, for all x X; (ii) ϕ(t + s, x) = ϕ(t, ϕ(s, x), for all (t, s, x) R 2 + X. Definition 1.2. A pair S = (Φ, ϕ) is called a linear skew-product semiflow on E = X V if ϕ is a semiflow on X and Φ : R + X B(V ) satisfies the following conditions: Communicated at the Conference on Mathematical Analysis and Applications, CAMA 07-Iaşi, October, 2007
2 254 MIHAIL MEGAN and LARISA BIRIŞ 2 (c 1 ) Φ(0, x) = I (the identity operator on V ), for all x X; (c 2 ) Φ(t + s, x) = Φ(t, ϕ(s, x))φ(s, x), for all (t, s, x) R 2 + X (the cocycle identity); (c 3 ) there are M, ω > 0 such that Φ(t, x) Me ωt, for all (t, x) R + X. If, in addition, the function Φ(, x)v is continuous, for every (x, v) E, then S is called a strongly continuous linear skew-product semiflow. If S = (Φ, ϕ) is a linear skew-product semiflow, then the mapping Φ is called the cocycle associated with the linear skew-product semiflow S. Definition 1.3. An application P : X B(V ) is said to be a projection family on V if P 2 (x) = P (x), for all x X. Definition 1.4. A projection family P : X B(V ) is said to be compatible with the linear skew-product semiflow S if Φ(t, x)p (x)=p (ϕ(t, x))φ(t, x), for all (t, x) R + X. Definition 1.5. Let P 1, P 2, P 3 be three projection families compatible with S. We say that these determine a decomposition of the identity if P 1 (x) + P 2 (x) + P 3 (x) = I, P i (x)p j (x) = 0, for all i j and x X. Definition 1.6. A linear skew-product semiflow S = (Φ, ϕ) is said to be uniformly exponentially trichotomic if there exist three projection families P 1, P 2, P 3 compatible with S which determine a decomposition of the identity and the constants N 1, N 2, N 3, N 4 1, ν 1, ν 2, ν 3, ν 4 > 0 such that: (i) N 1 e tν 1 Φ(t, x)p 1 (x)v P 1 (x)v, for all (t, x, v) R + E; (ii) N 2 e tν 2 P 2 (x)v Φ(t, x)p 2 (x)v, for all (t, x, v) R + E; (iii) P 3 (x)v N 3 e tν 3 Φ(t, x)p 3 (x)v, for all (t, x, v) R + E; (iv) Φ(t, x)p 3 (x)v N 4 e tν 4 P 3 (x)v, for all (t, x, v) R + E. Definition 1.7. A projection family P : R + B(V ) is said to be compatible with the linear skew-product semiflow S at the point x X, if Φ(t, ϕ(t 0, x))p k (t 0 ) = P k (t + t 0 )Φ(t, ϕ(t 0, x)), for all (t, t 0 ) R 2 +.
3 3 POINTWISE EXPONENTIAL TRICHOTOMY 255 Definition 1.8. A linear skew-product semiflow S = (Φ, ϕ) is said to be uniformly exponentially trichotomic at the point x X if there exist three projection families P 1, P 2, P 3 compatible with S which determine a decomposition of the identity and the constants N 1, N 2, N 3, N 4 1, ν 1, ν 2, ν 3, ν 4 > 0 such that: (t x 1 ) N 1e tν 1 Φ(t, ϕ(t 0, x))p 1 (t 0 )v P 1 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 2 ) N 2e tν 2 P 2 (t 0 )v Φ(t, ϕ(t 0, x))p 2 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 3 ) P 3(t 0 )v N 3 e tν 3 Φ(t, ϕ(t 0, x))p 3 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 4 ) Φ(t, ϕ(t 0, x))p 3 (t 0 )v N 4 e tν 4 P 3 (t 0 )v, for all (t, t 0, v) R 2 + V. Remark 1.1. If S is uniformly exponentially trichotomic, then S is uniformly exponentially trichotomic at every point x X. Indeed, this fact follows if we consider P k (t) = P k (ϕ(t, x)). Remark 1.2. Generally, if the linear skew-product semiflow S is uniformly exponentially trichotomic at every point x X it does not result that S is uniformly exponentially trichotomic, as it is shown in the following Example 1.1. Let C = C(R +, R + ) be the space of all continuous functions x : R + R +, which is metrizable with respect to the metric d(x, y) = n=1 1 d n (x, y) 2 n 1 + d n (x, y), where d n (x, y) = sup t [0,n] x(t) y(t). For every n N let f n C be a decreasing function such that there exists lim t f n (t) = 1 2n+1. For every n N let X n = {fn s : s R + }, where fn(t) s := f n (t + s), for all s 0. We denote by X = n=1 X n. Then the mapping ϕ : R + X X, ϕ(t, x)(s) := x(t + s), is a semiflow on X. Let V = R 3 with the norm (v 1, v 2, v 3 ) = v 1 + v 2 + v 3. We define Φ : R + X B(V ), Φ(t, x)v = (e t 0 x(τ)dτ v 1, e t 0 x(τ)dτ v 2, e t 0 x(τ)dτ v 3 ) and we have that S = (Φ, ϕ) is a linear skew-product semiflow on E.
4 256 MIHAIL MEGAN and LARISA BIRIŞ 4 Let P i : R 3 R 3, i = 1, 3, P 1 (v 1, v 2, v 3 ) = (v 1, 0, 0), P 2 (v 1, v 2, v 3 ) = (0, v 2, 0), P 3 (v 1, v 2, v 3 ) = (0, 0, v 3 ). For every x X, S is uniformly exponentially trichotomic at the point x X relative to the projection family (P (x)) x X, where P (x) = P, but S is not uniformly exponentially trichotomic. 2. Main results Proposition 2.1. The linear skew-product semiflow S = (Φ, ϕ) is uniformly exponentially trichotomic at the point x X if and only if there exist three projection families P 1, P 2, P 3 compatible with S which determine a decomposition of the identity and the constants N 1, N 2, N 3, N 4 1, ν 1, ν 2, ν 3, ν 4 > 0 such that: (t x 1 ) N 1e tν 1 Φ(t + t 1, ϕ(t 0, x))p 1 (t 0 )v Φ(t 1, ϕ(t 0, x))p 1 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 2 ) N 2e tν 2 Φ(t 1, ϕ(t 0, x))p 2 (t 0 )v Φ(t + t 1, ϕ(t 0, x))p 2 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 3 ) Φ(t 1, ϕ(t 0, x))p 3 (t 0 )v N 3 e tν 3 Φ(t + t 1, ϕ(t 0, x))p 3 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 4 ) Φ(t + t 1, ϕ(t 0, x))p 3 (t 0 )v N 4 e tν 4 Φ(t 1, ϕ(t 0, x))p 3 (t 0 )v, for all (t, t 0, v) R 2 + V. Proof. Necessity. We shall prove the conditions (t x 1 ). Similarly we can prove the other statements. We observe that Φ(t + t 1, ϕ(t 0, x))p 1 (t 0 )v = Φ(t, ϕ(t 1, ϕ(t 0, x)))φ(t 1, ϕ(t 0, x))p 1 (t 0 )v = Φ(t, ϕ(t 1 + t 0, x))φ(t 1, ϕ(t 0, x))p 1 (t 0 )v = Φ(t, ϕ(t 1 + t 0, x))p 1 (t 0 + t 1 )Φ(t 1, ϕ(t 0, x))v 1 N 1 e tν 1 P 1 (t 0 + t 1 )Φ(t 1, ϕ(t 0, x))v = 1 N 1 e tν 1 Φ(t 1, ϕ(t 0, x))p 1 (t 0 )v, for all (t, t 0, v) R 2 + V. Sufficiency. It results immediately for t 1 = 0.
5 5 POINTWISE EXPONENTIAL TRICHOTOMY 257 Proposition 2.2. The linear skew-product semiflow S = (Φ, ϕ) is uniformly exponentially trichotomic at the point x X if and only if there exist three projection families P 1, P 2, P 3 compatible with S which determine a decomposition of the identity and two nondecreasing functions f, g : R + R + with lim t f(t) = lim t g(t) = such that: (t x 1 ) f(t) Φ(t, ϕ(t 0, x))p 1 (t 0 )v P 1 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 2 ) f(t) P 2(t 0 )v Φ(t, ϕ(t 0, x))p 2 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 3 ) P 3(t 0 )v g(t) Φ(t, ϕ(t 0, x)p 3 (t 0 )x, for all (t, t 0, v) R 2 + V ; (t x 4 ) Φ(t, ϕ(t 0, x))p 3 (t 0 )v g(t) P 3 (t 0 )v, for all (t, t 0, v) R 2 + V. Proof. Necessity. It is obvious from Definition 1.8. Sufficiency. To prove (t x 1 ) (tx 1 ), we denote n = [t], t 0. Then there exists s N and s [0, r) such that t = ns + r. We obtain that Φ(t, ϕ(t 0, x))p 1 (t 0 )v = Φ(ns + r, ϕ(t 0, x))p 1 (t 0 )v = Φ(r, ϕ(ns, ϕ(t 0, x)))φ(ns, ϕ(t 0, x))p 1 (t 0 )v = Φ(r, ϕ(t 0 + ns, x))φ(ns, ϕ(t 0, x))p 1 (t 0 )v) 1 f(r) Φ(ns, ϕ(t 0, x))p 1 (t 0 )v) = 1 f(r) Φ(s, ϕ((n 1)s, ϕ(t 0, x)))φ((n 1)s, ϕ(t 0, x))p 1 (t 0 )v) = 1 f(r) Φ(s, ϕ(t 0 + (n 1)s, ϕ(t 0, x)))φ((n 1)s, ϕ(t 0, x))p 1 (t 0 )v) 1 1 f(r) f(s) Φ((n 1)s, ϕ(t 0, x))p 1 (t 0 )v) 1 1 f(r) f(s) n P 1(t 0 )v 1 N 1 e tν 1 P 1 (t 0 )v, where N 1 = f(s) f(r) > 1 and ν 1 = ln f(s) > 0. Similarly we can prove the other implications. Definition 2.1. Two projection families Q 1, Q 2 : R + B(V ) are said to be compatible with the linear skew-product semiflow S at the point x X if (i) Q 1 (t)q 2 (t) = Q 2 (t)q 1 (t) = 0, for all t R + ;
6 258 MIHAIL MEGAN and LARISA BIRIŞ 6 (ii) Φ(t, ϕ(t 0, x))q k (t 0 ) = Q k (t + t 0 )Φ(t, ϕ(t 0, x)), for all (t 0, t) R 2 +. Proposition 2.3. The linear skew-product semiflow S = (Φ, ϕ) is uniformly exponentially trichotomic at the point x X if and only if there exist two projection families Q 1, Q 2 compatible with S and the constants N 1, N 2, N 3, N 4 1, ν 1, ν 2, ν 3, ν 4 > 0 such that: (t x 1 ) N 1e tν 1 Φ(t, ϕ(t 0, x))q 1 (t 0 )v Q 1 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 2 ) N 2e tν 2 Q 2 (t 0 )v Φ(t, ϕ(t 0, x))q 2 (t 0 )v, for all (t, t 0, v) R 2 + V ; (t x 3 ) [I Q 1(t 0 )]v N 3 e tν 3 Φ(t, ϕ(t 0, x))[i Q 1 (t 0 )]v, for all (t, t 0, v) R 2 + V ; (t x 4 ) Φ(t, ϕ(t 0, x))[i Q 2 (t 0 )]v N 4 e tν 4 [I Q 2 (t 0 )]v, for all (t, t 0, v) R 2 + V. Proof. Necessity. We consider Q 1 (t) = P 1 (t) and Q 2 (t) = P 2 (t), for all t R +. Then the conditions (t1 x ), (tx 2 ) are obvious. For (tx 3 ), we observe that [I Q 1 (t 0 )v] 2 = (P 2 (t 0 ) + P 3 (t 0 ))v 2 = P 2 (t 0 )v 2 + P 3 (t 0 )v 2 1 N2 2 e 2tν 2 Φ(t, ϕ(t 0, x))p 2 (t 0 )v 2 + N3 2 e 2tν 3 Φ(t, ϕ(t 0, x))p 3 (t 0 )v 2 N 2 e 2tν 3 Φ(t, ϕ(t 0, x))[i Q 1 (t 0 )]v 2, for all (t, t 0, v) R 2 + V, where N = max{ 1 N 2, N 3 }. So (t x 3 ) follows. The proof of (t x 4 ) is similar. Sufficiency. We consider P 1 (t 0 ) = Q 1 (t 0 ), P 2 (t 0 ) = Q 2 (t 0 ) and P 3 (t 0 ) = I P 1 (t 0 ) P 2 (t 0 ). We shall prove statements (t x 3 ) and (tx 4 ) from Definition 1.8. The other statements are obvious. It follows that P 3 (t 0 ) = (I Q 1 (t 0 ))(I Q 2 (t 0 )) and P 3 (t 0 )v = (I Q 1 (t 0 ))(I Q 2 (t 0 ))v N 3 e tν 3 Φ(t, ϕ(t 0, x))(i Q 1 (t 0 ))(I Q 2 (t 0 ))v = N 3 e tν 3 Φ(t, ϕ(t 0, x))p 3 (t 0 )v, for all (t, t 0, v) R 2 + V. So statement (t x 3 ) is proved. Similarly, for (tx 4 ), we obtain Φ(t, ϕ(t 0, x))p 3 (t 0 )v = Φ(t, ϕ(t 0, x))(i Q 1 (t 0 ))(I Q 2 (t 0 ))v = Φ(t, ϕ(t 0, x))(i Q 2 (t 0 ))(I Q 1 (t 0 ))v N 4 e tν 4 P 3 (t 0 )v,
7 7 POINTWISE EXPONENTIAL TRICHOTOMY 259 for all (t, t 0, v) R 2 + V. Definition 2.2. Four projection families R 1, R 2, R 3, R 4 : R + B(V ) are said to be compatible with the linear skew-product semiflow S at the point x X if (i) R 1 (t) + R 3 (t) = R 2 (t) + R 4 (t) = I, for all t R + ; (ii) R 1 (t)r 2 (t) = R 2 (t)r 1 (t) = 0 and R 3 (t)r 4 (t) = R 4 (t)r 3 (t), for all t R +. Proposition 2.4. The linear skew-product semiflow S = (Φ, ϕ) is uniformly exponentially trichotomic at the point x X if and only if there exist four projection families R 1, R 2, R 3, R 4 compatible with S and the constants N 1, N 2, N 3, N 4 1, ν 1, ν 2, ν 3, ν 4 > 0 such that: (i) N 1 e tν 1 Φ(t, ϕ(t 0, x))r 1 (t 0 )v R 1 (t 0 )v, for all (t, t 0, v) R 2 + V ; (ii) N 2 e tν 2 R 2 (t 0 )v Φ(t, ϕ(t 0, x))r 2 (t 0 )v, for all (t, t 0, v) R 2 + V ; (iii) R 3 (t 0 )v N 3 e tν 3 Φ(t, ϕ(t 0, x))r 3 (t 0 )v, for all (t, t 0, v) R 2 + V ; (iv) Φ(t, ϕ(t 0, x))r 4 (t 0 )v N 4 e tν 4 R 4 (t 0 )v, for all (t, t 0, v) R 2 + V. Proof. Necessity. It is immediate for R 1 = Q 1, R 2 = Q 2, R 3 = I Q 1, R 4 = I Q 2. Sufficiency. If we denote P 1 = R 1, P 2 = R 2 and P 3 = R 3 R 4 then P 1 + P 2 + P 3 = R 1 + R 2 + (I R 1 )(I R 2 ) = I. Also P 1 P 2 = R 1 R 2 = 0 P 1 P 3 = R 1 (I R 1 )(I R 2 ) = 0 P 2 P 3 = R 2 (I R 1 )(I R 2 ) = 0 It is sufficient to prove statement (t x 3 ) and (tx 4 ) from Definition 1.8. For (tx 3 ) it follows that P 3 (t 0 )v = R 3 (t 0 )R 4 (t 0 )v N 3 e tν 3 Φ(t, ϕ(t 0, x))r 3 (t 0 )R 4 (t 0 )v = N 3 e tν 3t Φ(t, ϕ(t 0, x))p 3 (t 0 )v, for all (t, t 0, v) R 2 + V. Similarly for (t x 4 ).
8 260 MIHAIL MEGAN and LARISA BIRIŞ 8 REFERENCES 1. Chow, S.N.; Leiva, H. Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations 120 (1994), Elaydi, S.; Hajek, O. Exponential trichotomy of differential systems, J. Math. Anal. Appl. 129 (1988), Megan, M.; Buliga, L. On uniform exponential trichotomy of linear skew-product semiflows in Banach spaces, Proceedings of the 9th National Conference of the Romanian Mathematical Society Lugoj, Editura Universitatea de Vest din Timişoara (2005), Megan, M.; Sasu, A.L.; Sasu, B. Perron conditions for pointwise and global exponential dichotomy of linear skew-product flows, Integr. equ. oper. theory, 50 (2004), Megan, M.; Stoica, C. Equivalent definitions for uniform exponential trichotomy of evolution operators in Banach spaces, Integr. equ. oper. theory, 60(2008), Megan, M.; Stoica, C.; Buliga, L. Trichotomy for linear skew-product semiflows, Applied Analysis and Differential Equations, World Scientific, (2007), Megan, M.; Stoica, C.; Buliga, L. On asymptotic behaviours of evolution operators in Banach spaces, Seminar of Mathematical Analysis and Applications in Control Theory, West University of Timişoara, (2006), Papaschinopoulos, G. On exponential trichotomy of linear difference equations, Applicable Analysis 40 (1991), Sacker, R.S.; Sell, G.R. Dichotomies for linear evolutionary equations in Banach spaces, J. Diff. Equations, 12 (1994), Received: 10.XII.2007 Faculty of Mathematics and Computer Science, West University of Timişoara, ROMANIA megan@rectorat.uvt.ro Faculty of Mathematics and Computer Science, West University of Timişoara, ROMANIA larisa.biris@math.uvt.ro
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