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46) also has the solution with a discontinuous derivative (Fig. 2). k=0,+1,+2, ... The derivative of this solution is bounded: IA(t)l~1/2; the value of the derivative k ( 0 ) is not defined, and lim x(t) = 0. 41 there is a series of interesting examples of this type. 5. Still more basic difficulties connected with the determination of the solution to equations of advanced type. 41 DIFFERENTIAL EQUATIONS WITH DE V IATIN G ARGUMENTS C o n s i d e r t h e e q u a t i o n of t h e f i r s t o r d e r x ( t ) = f ( t , x ( t - - r ) , 2 ( t - - r ) ), t’tO, where t h e f u n c t i o n f ( t , x , y ) i s i n f i n i t e l y d i f f e r e n t i a b l e with r e s p e c t t o a l l i t s arguments, w i t h a s i n g l e c o n s t a n t d e v i a t i o n T > O , and on t h e i n i t i a l t h e r e i s given an i n f i n i t e l y d i f f e r e n t i a b l e set Et 0 i n i t i a l f u n c t i o n + ( t ) .

1 6 ) f o r e a c h s t e p d i f f e r e n t i a l e q u a t i o n s w i t h o u t d e v i a t i n g a r g u m e n t s o f t h e form k ( t ) = f(t,x(t),ok(t-r(t,x(t)))). 3. I n t e g r a b l e Types o f E q u a t i o n s w i t h a Deviatincr Aruument The method o f s t e p s i s one o f t h e b a s i c methods of i n t e g r a t i n g d i f f e r e n t i a l e q u a t i o n s w i t h a deviat i n g argument. A p p l i c a t i o n o f t h i s method t o a n e q u a t i o n o f t h e form k ( t ) = f ( t , x ( t ) , x ( t - .

I n t h e mentioned t h e o r e m , t h e most r e s t r i c t i v e c o n d i t i o n i s ( 5 2 ) . However, t h i s c o n d i t i o n i s neede d i n t h e s e n s e t h a t i n t h e case 8 = 1 , t h e r e e x i s t examples c o n t r a d i c t i n g t h e c o n c l u s i o n o f t h e theorem. I n f a c t , t h e i n i t i a l v a l u e problem c o n s i d e r e d i n Example 1 s a t i s f i e s a l l t h e c o n d i t i o n s o f t h e theorem e x c e p t ( 5 2 ) . H e r e L =L =L =1 a n d , conseY Z Q I n t h i s case, i f EfO-the solution quently, a = l .

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