By Svatopluk Fučík, Jindřich Nečas, Jiří Souček, Vladimír Souček (auth.)

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**Sample text**

Vp~. 1. Let E ~ - n e t of K. be a subset of is said to be completely X. A continuous if : transfor- - (i) T is continuous, (ii) if M 36 - is bounded subset of Now let K {Vl, ... , Vp~ E, then be a compact set and be an £ - n e t of K K. For T(M) is compact. its closure. 2. Let T [Ix - vill ~ ~- , if tlx - v itl > £ • be a completely continuous mapping with M, a bounded subset of defined on K X, and let as described above. If liT(x) ~roof. if l~T(x) - F~ T ( x ) - F[ T ( x ) ~ I < ~ F£ be x e M, then • I% = mi(T(x)) iffil T(M) = K.

For an arbitrary oo x ~ H, where x :L aiY i , define oo oO u(x) = ~- aiU~Y i) ~ }-- i=-~ so t h a t the domain of phism of H a:i+l i=-~o U is onto itself. H. I t follows On t h e o t h e r that hand, U U is is a homeomor- a homeomorphism of the set S=ix~H, onto itself as well. Now d e f i n e f : xl It >1/2 c a n be shown t h a t Now we show t h a t xo e B such that f f is llx~l --I~ the mapping f (i - LLxl[ ) Yo + U(x) on H . a h o m e o m o r p h i s m a n d maps has no f i x e d xo = f(Xo).

I E x a m p 1 e if Xn---~ x o Xn, x o e if in M. Xn---~ x o in • Xn, x o e M. on M if xn ~ xo in X F ( x o) = y . continuous on subset M M if F D c M, is conti- F(D) is com- if for each bounded i s a bounded s u b s e t o f (6) (3) in the other. then some relations X subset Y. I. (Example of a completely which is not strongly Z2 X I. M M in Y. (Class However, Let for for imply Xn----~ x O Xn, X o e on and for each bounded neither one is contained Theorems Y closed The two classes of mappings F M if for T is said to be bounded on D ~ M, F(D) below.