Harris and Sibuya, Asymptotic solutions of nonlinear difference equations 133

11 . Estimates oï u{t, e)

The inequalities (9. 5) and (10. 5) imply

( H . l ) \\u{t,8)\\£K(e,)i,e^'\l'^

for I г 1 ^ £o and arbitrary values of t in (6. 2). On the other hand, since v(r, e) = 0(т*") as T -^ 0, we have

( 11 . 2 ) ii(t,s)^ 0(1 ^1^)

as t tends to zero in (6. 2). Since m is arbitrary, we have

( 11 . 3 ) u(t,e) ^0 as t tends to zero in (6. 2).

If we define u{t) by (8. 4) i. е., u(t) = u{t, 1), we get a solution of the system of integral equations (6. 6) which satisfies the following conditions :

( i ) u{t) is analytic in (6.2);

( ii ) u(t) is of exponential order as t tends to infinity in (6. 2);

( iii ) u(t) ^0 as t tends to zero in (6. 2).

12 . Proof of Theorem 1

Put

( 12 . 1 ) w{t) = e^'u{t),

where u(t) is the vector determined in Section 11.

Since u{t) is a solution of (6. 6) w{t) is a solution of (6. 4) which satisfies the following conditions :

( i ) w(t) is analytic;

( ii ) w(t) is of exponential order as t tends to infinity

( iii ) w{t) ^0 as t tends to zero in the sector

larg^l ^ ^, Q sufficiently small.

13 . Proof of Theorem 2

In order to prove Theorem 2, we cannot use the Borel-Ritt theorem. Therefore, we have to treat the system y(x + i) = f(x,y) directly.

Assume that f{x, y) is analytic for \ x\^ R and \\y \\ ^ Ôq, Put

( 13 . 1) f(x, y) = iM + A{x)y + Г' f^(x) y\

where /o(a;) and f^{x) are n-dimensional vectors with components analytic ioT \x\^ R and A (x) in an ^ by /г matrix with components analytic for \x\^ R, Assume that

( 13 . 2 ) ||(e-H„-^(oc))-4|^L

Journal fur Mathematik. Band 222. Heft 3/4 18