ON THE ORDER OF EXPONENTIAL GROWTH OF THE SOLUTION OF THE LINEAR DIFFERENCE EQUATION WITH PERIODIC COEFFICIENT IN BANACH SPACE

An equation of the form y − A ( t ) y = f ( t ) is considered, where Δ y = y ( t + δ ) − y ( t ) δ , and the necessary and sufficient criteria for the exponential growth of the solution of this equation is obtained.

Let E be a complex Banach space.Denote by A(t) t >-o} a family of linear bounded operators from E into itself.We assume that A(t) is periodic and strongly continuous in t [o, ).
Let II II be the norm in E. Denote by E the set of all elements f(t) E such that sup II f(t)ll exp (-t) < (R).

RESULTS.
Let Ay '(t+6)- where c) is the zero of E.
Let us assume that f E The solution of equation ( 2.1) can be written in the form t-6 t-6 y(t) 6 S A(i) y(i) + 6 s f(i) (2.3) =o =o where t [n6], [a] denotes the greatest positive integer a and 6 is a positive integer.
Without loss of generality we suppose that 6 I.
Putting t I, 2 n in (2.3), one obtains n-1 j y(t) z where is the unit operator.Let w be the period of A(t).
r=l j=l The last equation can be written in the fore t " t-rw w-1 y z w (B-I) -I .fj((r-1)w+j-l) + f((r-l) w + w-l)} (2.6) where Y is a contour which circumscribes all the specter of the operator B, [1].
It can be seen that if f E then (B-I) -1 f E for every y.
From equation (2.6) we obtain a necessary and sufficient criterion for the exponen- tial growth of the solution with an index B. Let o B denote the specter of the I operator B. Assume that o OB Set o = In IoI.
The following theorem holds- THEOREM.If f Era, then the solution y of equation ( 2.1) belongs to E B such that B > n, when o B o' when n < no.
PROOF.To prove the sufficiency, we consider the following three, cases: 1 (1) If n > lnlI then y(t) defined by (2.6) belongs to E.
This means that y E B where B > n.