# Weak exponential dichotomy for skew evolution semiflows in Banach spaces.

1. INTRODUCTION

The study of asymptotic properties, such as exponential dichotomy, considered basic concept that appear in the theory of dynamical systems, plays an important role in the study of stable, instable and central manifolds. Some of the original results concerning stability and instability were published in (Megan et al., 2007), for a particular case of skew-evolution semiflows .

Concerning previous results, (Latcu & Megan 1991) presents the weak exponential dichotomy for evolutionary processes. (Stoica & Megan, 2008) gave the definition for the following notions: evolution semiflow, evolution cocycle over the semiflow and skew-evolution semiflow. Caracterizations for the weak exponential stability, instability for skew-evolution semiflows on Banach spaces were obtained by (Minda et al., 2009) and (Tomescu et al., 2009).

In this paper we extend the asymptotic properties of weak exponential dichotomy for the skew-evolution semiflows defined on Banach spaces. The results concerning the nonuniform exponential dichotomy are generalizations of some theorems proved for evolution operators.

2. DEFINITIONS AND PRELIMINARY RESULTS

Let (X,d) be a metric space, U a Banach space, B(U) the space of all bounded linear operations from U into itself and let us denote Y = X x U

Definition 1 A mapping [phi] : [R.sup.2.sub.+] x X [right arrow] X is called evolution semiflow on X if it satisfies the following properties:

(i) [phi](t,t,x) = x, [for all] (t,x) [member of] [R.sub.+] x X

(ii) [phi](t, s, [phi], [t.sub.0], x)) = [phi], (t, [t.sub.0] x),

[for all]t [greater than or equal to] s [greater than or equal to] [t.sub.0] [greater than or equal to] 0, [for all]x [member of] X .

Definition 2 A mapping E : [R.sup.2.sub.+] x X [right arrow] B(U) is called evolution cocycle over the semiflow [phi]: [R.sup.2.sub.+] x X [right arrow] X if : (ce1) E(t,t,x) = I, [for all](t,x) [member of] [R.sub.+] x X

(ce2) E(t, s,[phi](s,[t.sub.0], x)) E(s,[t.sub.0], x) = E(t, [t.sub.0], x)

[for all]t [greater than or equal to] s [greater than or equal to] [t.sub.0] [greater than or equal to] 0, [for all]x [member of] X .

Definition 3 The mapping C : [R.sup.2.sub.+] x Y [right arrow] Y, C(t, s, x,u) = ([phi](t, s, x)), E(t, s, x)u) where E is an evolution cocycle over an evolution semiflow [phi], is called skew evolution semiflow on Y.

Definition 4 A skew-evolution semiflow C = ([phi],E) is said to have exponential growth if there exists M [greater than or equal to] 1, [omega] > 0 such that, for every (x, u) [member of] Y there exists [t.sub.0] [greater than or equal to] 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Definition 5 The skew-evolution semiflow C = ([phi], E) is said to be weak exponentially stable is there exists N [greater than or equal to] 1, v > 0 such that, for every (x, u) [member of] Y there is [t.sub.0] [greater than or equal to] 0 such that

[parallel]E(t,[t.sub.0],x)u[parallel] [less than or equal to] [Ne.sup.-v(t-s)] [parallel]E(s, [t.sub.0],x)u[parallel] for all t [greater than or equal to] s [greater than or equal to] [t.sub.0] + 1.

Definition 6 A skew-evolution semiflow C = ([phi],E) is said to have exponential decay if there exists M [greater than or equal to] 1, [omega] > 0 such that, for every (x, u) [member of] Y there exists [t.sub.0] [greater than or equal to] 0 such that

[parallel]E(s,[t.sub.0],x)u[parallel] [less than or equal to] [Me.sup.[omega](t-s)][parallel] E(t,[t.sub.0],x)u [parallel](3)

Definition 7 The skew-evolution semiflow C = ([phi], E) is said to be weak exponentially instable if there exists N [greater than or equal to] 1, v > 0 such that, for every (x, u) [member of] Y there is [t.sub.0] [greater than or equal to] 0 such that

[parallel]E(s,[t.sub.0], x)u[parallel] [less than or equal to] [Ne.sup.-v(t-s)][parallel]E(t,[t.sub.0], x)u[parallel] (4)

for all t [greater than or equal to] s [greater than or equal to] [t.sub.0] +1

3. WEAK EXPONENTIAL DICHOTOMY

Definition 8 Two projector families [{[P.sub.k]}.sub.ke{12}] are said to be compatibile with a skew-evolution semiflow C = ([phi],E) if

(1) [P.sub.1] (x) + [P.sub.2](x) = I, [P.sub.1] (x) [P.sub.2] (x) = [P.sub.2] (x) [P.sub.1] (x) = 0

(2) [P.sub.k]([phi](t,s,x))E(t,s,x)u = E(t,s,x) [P.sub.k] (x)u, k [member of] {1,2} for all t [greater than or equal to] s [greater than or equal to] [t.sub.0] > 0 and all (x,u) [member of] Y.

We denote for any k [member of] {1,2}: [E.sub.k](t,[t.sub.0], x) = E(t, [t.sub.0], x)[P.sub.k](x) and [C.sub.k] (t, s, x, u) = ([phi](t, s, x)), [E.sub.k] (t, s, x)u) [for all]t [greater than or equal to] s [greater than or equal to] [t.sub.0] [greater than or equal to] 0, [for all]x [member of] X

Definition 9 A skew-evolution semiflow C = ([phi],E) is said to be weak exponentially dichotomic if there are [N.sub.1], [N.sub.2],v > 0 such that:

(ed1) for every u [member of] U there is [t.sub.0] > 0 such that

[parallel][E.sub.1](t,[t.sub.0],x)u[parallel] [less than or equal to] [N.sub.1][e.sup.-v(t-s)] [parallel][E.sub.1](s,[t.sub.0],x)u[parallel]

for all t [greater than or equal to] s [greater than or equal to] [t.sub.0] +1

(ed2) for every u [member of]U there is [t.sub.0] > 0 such that

[parallel][E.sub.2](t,[t.sub.0],x)u[parallel] [greater than or equal to] [N.sub.2][e.sup.v(t- s)][parallel][E.sub.2](s,[t.sub.0],x)u[parallel]

for all t [greater than or equal to] s [greater than or equal to] [t.sub.0] +1

Theorem 1 The skew-evolution semiflow C = ([phi],E) is weak exponentially dichotomic if and only if there are two nondecreasing continous functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Necessity is obvious.

Sufficiency is an immediate consequence of Lemma 1.1 from (Minda et al. 2009) and Lemma 1.1 from (Tomescu et al.2009)

Theorem 2 The skew-evolution semiflow C = ([phi],E) is weak exponentially dichotomic if and only if there exist two projectors [P.sub.1] and [P.sub.2] compatibile with C such that [C.sub.1] has exponential growth and [C.sub.2] has exponential decay and there are [M.sub.1], [M.sub.2] > 0 such that :

(1) for every (x, u) [member of] 7 there is [t.sub.1] [greater than or equal to] 0 such that :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(2) for every (x,u) [member of] Y there is [t.sub.2] > 0 such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof Necessity is immediate by direct verification. Sufficiency: [C.sub.1] has exponential growth so, similary as in Theorem 4.1 from (Stoica & Megan 2008), there exists a nondecreasing function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which shows that the condition (1) from Theorem 1 is satisfied. [C.sub.2] has exponential decay so, similary as in Theorem 4.1 from (Stoica & Megan 2008), there exists a nondecreasing function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similary, from the condition (2) from Theorem 2 it follows that for all t [greater than or equal to] s [greater than or equal to] [t.sub.2] + 1 we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which shows that the condition (2) from Theorem 1 is satisfied. By Theorem 1 we have that C is weak exponentially dichotomic.

4. CONCLUSION

In order to compete the study of this weak asymptotic properties of skew-evolution semiflows, after weak exponential stability, instability and dichotomy, in the future we want to study the new concept of weak exponential trichotomy.

5. REFERENCES

Latcu, D. & Megan, M. (1991) Exponential dichotomy of evolution operators in Banach spaces, University of Timisoara, Preprint Series in Mathematics, Report S.T.S 57, Oktober 1991

Megan, M.; Stoica,C. & Buliga, L.(2007) On asymptotic behaviors for linear skew-evolution semiflows in Banach spaces, Carpathian Journal of Mathematics, Volume 23, No.1-2, pp 117-125, ISSN 1584-2851

Minda,A.; Tomescu, M.; Anghel, C.; & Stoica, D. (2009) Exponential stability of skew-evolution semiflows in Banach spaces, ICFCC 2009 Proceedings, pp 256-258, ISBN 978-1-4244-3754-2, Kuala Lumpur, april, 2009

Stoica, C. & Megan,M. (2008) Nonuniform behaviors for skew- evolution semiflows in Banach spaces, Preprint Univ. Bordeaux, arXiv: 0806.1409, pp 1-16

Tomescu, M.; Minda,A.; Anghel, C.; & Popovici, P. (2009) Exponential instability of skew-evolution semiflows in

Banach spaces, ICSPS 2009 Proceedings,pp 976-979,ISBN 978-0-7695-3654-5, Singapore, may 2009
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Author: Printer friendly Cite/link Email Feedback Minda, Andrea; Tomescu, Mihaela; Ramneantu, Luminita Annals of DAAAM & Proceedings Report 4EUAU Jan 1, 2009 1561 Calculus of regenerative losses coefficient in Stirling engines. Virtual laboratory communication. Banach spaces Opposites Polarity