Title: On the curl of singular completely continous vector fields in Banach spaces

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Henryka Czarnecka
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(1973). On the curl of singular completely continous vector fields in Banach spaces.

1 Title: On the curl of singular completely continous vector fields in Banach spaces Author: Adam Bielecki, Tadeusz Dłotko Citation style: Bielecki Adam, Dłotko Tadeusz. (1973). On the curl of singular completely continous vector fields in Banach spaces. "Prace Naukowe Uniwersytetu Śląskiego w Katowicach. Prace Matematyczne" (Nr 3 (1973), s ).

2 P R A C E N A U K O W E U N IW E R S Y T E T U Ś L Ą S K IE G O W K A T O W IC A C H N R 30 P R A C E M A T E M A T Y C Z N E III, 1973 A D A M B IE L E C K I, T A D E U S Z D Ł O T K O O N THE CURL O t S IN G U L A R C O M PLE TELY CONTINO U S VECTOR FIELD S IN B A N A C H SPACES Let us consider the Banach space X with the sphere S { x : x e X, x = R ). By h (x) we denote the application of this sphere into X and assume that h is completely continous. The application 0 (x) = x h (x) is called a completely continous vector field on x = R. The condition 0 (x) j= 0 is essential in the definition of the curl v(0,r) of the vector field (see [1] p. 112). A vector field 0 (x ) on x = R is called non singular on S if 0 (x)#0 for x = R. But in special cases it is admissible to introduce y (0, R) for singular completely continous vector fields. Therefore let us change the definition of a curl y ( 0, R ) in a suitable manner. It is easy to prove that if 0 (x) = x h (x), ip (x) = x H (x) are two completely continous vector fields on the sphere x = R, 0 (x) is non singular on S and (1) h (x ) H (x) H< x h (x) for x = R, then op(x) is also non singular on S and y (ip, R) = y (0, R) (see [1] p. 170). In fact, if then H. (x ) = A H (x) + (l A) h (x), 0 < A < 1, Il x = R, x - H x (x)! = x A H ( x ) - ( l - A ) h (x) > > x h (x) - j X 11 H ( x ) - h (x) > so > x h (x) (1 A) > 0, x H x (x) = 0 for x = R, Ae[0,l]. In particular for A = 1 and x = R; x H {x)^= 0. W e proved that for 7 P race m atem atyczne I I I 97

> (1 - X) x - h (x) > 0. From this considerations w'e have a natural generalisation y(h,r) = y (H 1, R) = y (h, R ) D E FIN IT IO N.

3 l e [0,1] the integer number function T (A) = y (H y (x), R) is defined and continous. Just for that reason T (I) const. W e have y (h (x), R ) = = Y (Ha ( x ), R) = y (H (x), R), I e [0,1] I f w e replace (1) by a w eakly condition (2) l H (x ) h (x ) < x h(x), then we have x H (x) > 0 for 2 e [0,l), because x Hx (x) > (1 - X) x - h (x) > 0. From this considerations w'e have a natural generalisation y(h,r) = y (H 1, R) = y (h, R ) D E FIN IT IO N. I f xjj (x) = x ~ H (x) is a com pletely continous vector field on x = R and if there exist a completely continous vector field $ (x) = x h (x) non singular and satisfied (2), then y (H, R ) «y (h, R). It is interesting, that by such generalization of thé curl y (H, R) the following theorem is true. THEOREM. I f the com pletely continous vector field ip (x) = x H (x) defined for x ^ R has the generalized curl y (H, R) and y (H, R) T- 0 for x = R, then the equation x = H (x) has at the least one solution x D such, that x D R. Demonstration. In accord with our assumptions it exist a absolutely continous function h (x) such, that H (x ) h (x)!! x h (x) and x ~ h (x) > 0 for x = R. Let us define two prolongations follows h* and H* of h and H on x [ = R as

4 The functions h* and H* are completely continuous. Let x = R + e, e > 0, then II H*(ar) h * (l) < II R R) I! = i f f»x_ TS1 *(ifir)1< 11 x ~ h ' (x" If en = 1 In, then from considerations mentioned above it exist a sequence x, x n R + l/ n and x A = H *(xn). Let us consider the sets {xn} and (H *(x n) } for n = 1,2,... The last set is compact, so we can take a subsejuence H *(xn) which tend to x 0 as k mx>. So we have and for n >-oo H *(xnj ) H *(x0) = x a. But x 0 II < R, and H *(x0) = H (x a) = x 0 which finished the demonstration REM ARK. L e t us assumé that 0 (x ) = x h (x ) and ^ (x) = x H (x) are two completely continous vector fields on x = R, 0 (x) is non singular and y (h, R ) = 0. I f exist such an element x, x = R that x-n/ç H (x) = 2 h (x ) 5, then from condition (1) does not result that y (H, R) exist and Y (H, R ) = y (h, R). Instead if w e assume the generalized definition od y (h, R) and condition (2) is true, then from the theorem above y(h,r) = y(h, R). REFERENCES. [1J. M. A. K r a snosielskij: Topological methods in the theory o f nonlinear integral equations. M oskwa 1956 (russ.). [2], M. A. K r a s n o s i e 1 s k i j, P.P. Zabreiko, J. B. Rutickij, W. J Otjecenko: A p roxim a te solution o f operator equations. M oskwa 1969 (russ.).

SUR L A R O T A T IO N DU C H A M P V E C T O R IE L S IN G U L IE R C O M PLE TE M E N T C O N T IN U D A N S L E SPACE DE B A N A C H Résumé Dans cette note on considéré un champ vectoriel q> (x) =

5 SUR L A R O T A T IO N DU C H A M P V E C T O R IE L S IN G U L IE R C O M PLE TE M E N T C O N T IN U D A N S L E SPACE DE B A N A C H Résumé Dans cette note on considéré un champ vectoriel q> (x) = x F (x) pour x = R > 0 de l espace de Banach X, et l opération F : X -* X est complètement continue. I l peut arriver, qu il existe xçx, x = R et cp(*) = 0. On généralise la définition de la rotation du champ vectoriel complètement continu et non singulier en un champ singulier, en conservant le théorèm e fondam ental: Si la rotation généralisée est différente de zéro sur la sphère j x = R, alors l équation x = H (x ) possède au moins une solution, telle que x ^ R. Cela consiste un supplément à la théorie des champs vectoriels complètement continus. A D A M B IE L E C K I I TA D E U S Z D Ł O T K O O O BRO CIE O S O B LIW E G O P E Ł N O C IĄ G Ł E G O P O L A W E K TO R O W E G O W P R Z E S T R Z E N I B A N A C H A Streszczenie W pracy rozpatruje się pole w ektorow e <p(x) = x F (x ) na sferze x = R > 0 przestrzenni Bancha X, przy czym operacja F : X _ > X jest pełnociągła. Dopuszcza się istnienie miejsc zerowych pola tp na sferze II x II = R. Uogólnia się definicję obrotu pełnociągłego nieosobliwego pola w ektorowego na pole osobliwe, przy tym zachowuje się podstawowe twierdzenie: Jeżeli uogólniony obrót jest różny od zera na sferze x = R, to równanie x = H (x ) ma przynajm niej jedno rozwiązanie zawarte w kuli domkniętej x < R. Stanowi to uzupełnienie teorii pełnociągłych pól w ektorowych.

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