221

+ J е-^+' I fit, У(Ь), y'(t)] -fit, уо(<), Voit)] I (К < I +

- - - - 00

00

+ / e-l^-'l . I Л«, 2/(0, y'm -f[t, Voit), уМ I d< < I + I fi = е.

- - - - 00

It follows from the relations (11) and (12)

Pn [ Ty { x ) — Туо{х)] < e, if ргпЫх) — уо(х)] < д and thus continuity of the operator (9) is proved.

The compactness of the operator (9) will be proved by the application of the AscoH-Arzela theorem. It is therefore sufficient to show that Ty{x) are equi-bounded and equi-continuous on each of the intervals

<—w , n).

Let ^j(x) 6 if. Then

Ty { x )

\ J ^-'^-'' -fit, y(t), y'(t)] dt S~ f e-l^-^l dt=:K

—00 _oo

X

'\й\ f e-^+« .f[t, y(t), y'(t}] à.t\ + Ku2K.

I •/

iTy ( x ) ]

Hence TM с M and at the same time also the set TM is equibounded. The equi-continuity of the functions from TM on the interval (—n, n} will be proved in the following way: Let e > 0 be an arbitrary number and let

Then

\xi — X2\<-^, a?i < a;2, Xi, XzE (—n,n}, Ty(x,) — Ty(x2) \=~\ (e-^^ — e-^^) j é .f[t, y(t), y\t)] at +

- - - - 00

x^

00

+ (e^' — e=^^) j e-'. f[t, y(t), y'{t)] àt