Solvability and properties of solutions of nonlinear by Skripnik

By Skripnik

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7. ,~ i A" (0) u. ]Lw, ]P-2. 9). 8. F o r N-> N3 the field r on SN(0, r) is l i n e a r l y homotopic to the field N = f A" (o) i=1 To compute the r o t a t i o n of the field X~ ) (u) we introduce the a u x i l i a r y o p e r a t o r A : W~,~(a) - - W ~ , ~ ( a ) defined by =IA'(O)u. Lvdx. 4 is applicable to the o p e r a t o r A if r 0 is defined by < r0u, v > = ! [ L - - A ' (0)1 u. Lvdx. 4 thus implies the following result. 7. Let conditions 1) and 2) be s a t i s f i e d , and suppose that z e r o is a nondegenerate c r i t i c a l point of the o p e r a t o r A defined by Eq.

We shall denote the rotation of Au on S by "y(Au, S). 9. In [99] it was shown that the concept of rotation introduced above can be extended to b r o a d e r c l a s s e s of o p e r a t o r s , and s o m e a s s u m p t i o n s on the space X can also be dropped. In p a r t i c u l a r , it is possible to drop the a s s u m p t i o n s r e g a r d i n g the s e p a r a b i l i t y and reflexivity of the s p a c e X. It is possible to define a s i n g l e valued r o t a t i o n of the v e c t o r field Au on S for a bounded, demicontinuous, pseudomonotone o p e r a t o r A if 0 ~ AS.

T h e n t h e o p e r a t o r A 1 :-D ~ X * , A l u = 5(u)Au + A ' u s a t i s f i e s c o n d i t i o n (S)+. H e r e 5(u) is d e f i n e d by Eq. 1). Proof. <0. W e m a y a s s u m e t h a t 5(Un) ~ 50, and w e c o n s i d e r n~cc two c a s e s : a) 50 = 0; b) 50 > 0. I n the f i r s t c a s e w e i m m e d i a t e l y o b t a i n A ' u 0 = 0, s o t h a t f r o m t h e h y p o t h e s e s of t h e t h e o r e m i t f o l l o w s t h a t u 0 = 0. F r o m t h i s and 5(u n) - - 0 w e find I-~-( Attn, a~ } r , w h i c h e n s u r e s by c o n d i t i o n n~co (S)+ t h e s t r o n g c o n v e r g e n c e of Un to z e r o .

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