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Ergodicity and its applications in regularity and solutions of pseudo-almost periodic equations. (English) Zbl 1004.34033

A relation between ergodicity and regularity in pseudo-almost periodic equations is established. For this purpose the following operator is considered

L:C ' () n C() n ,yLyy ' +A(t)y·

It is shown that the following three statements are equivalent:

(1) The operator L is regular.

(2) An solution to the homogeneous equation Ly=0 exhibits an exponential dichotomy.

(3) For every fPAP 0 () n , the inhomogeneous equation Ly=f has a unique solution in C() n .

Here, C() n and C ' () n are the nth powers of C(), respectively, where C() is the space of the bounded continuous functions on the real line supplied with the sup norm and C ' () is the space of differentiable functions ϕ with ϕ ' C() (with the norm ϕ C ' () =ϕ C() +ϕ ' C() ). PAP() is the space of pseudo-almost periodic functions and PAP 0 ()PAP() consists of those ϕPAP() for which M(ϕ)lim T 1 2T -T T |ϕ|dt=0.

The main result states that, if the matrix A(t) is such that a ij =0 for all i>j and a ii , i=1,n, are ergodic, then the operator L is regular if and only if M(Rea ii )0, i=1,,n. Furthermore, if A(t) and f are in PAP() n then the unique solution y to Ly=f is again in PAP() n .

34C27Almost and pseudo-almost periodic solutions of ODE
34D09Dichotomy, trichotomy