MIHAIL MEGAN and LARISA BIRIŞ

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, 26-27 October, 2007

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 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.

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 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 + ;

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 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 ).

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), 429-477. 2. Elaydi, S.; Hajek, O. Exponential trichotomy of differential systems, J. Math. Anal. Appl. 129 (1988), 362-374. 3. 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), 220-229. 4. 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), 489-504. 5. Megan, M.; Stoica, C. Equivalent definitions for uniform exponential trichotomy of evolution operators in Banach spaces, Integr. equ. oper. theory, 60(2008), 499-506. 6. Megan, M.; Stoica, C.; Buliga, L. Trichotomy for linear skew-product semiflows, Applied Analysis and Differential Equations, World Scientific, (2007), 227 236. 7. 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), 1-21. 8. Papaschinopoulos, G. On exponential trichotomy of linear difference equations, Applicable Analysis 40 (1991), 89-109. 9. Sacker, R.S.; Sell, G.R. Dichotomies for linear evolutionary equations in Banach spaces, J. Diff. Equations, 12 (1994), 721-735. 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