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By Andreev P. D.

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Example text

0 F o is said to be s-homogeneous (a) Y if F o ( t U ) ffi t a Fo(U) holds for each (b) Let t ~ 0 F o be an a-homogeneous homogeneous F 8 tn u e X. operator. F with respect to t n ~ O, Un---A Us, (c) and all Fo if F (Un/t n) is said to be a-strongly is said to be a-quasi- ~ g ~ Y~Fo(U quasihomogeneous o) = g • with respect to Fo, if t n ~ O, Un---Au O > t~ F(un/t n) R e m a r k . The definition essentially [141). in > Fo(U O) e Y. g. was i n t r o d u c e d ~36~. 1. Let X and Y be two Banach s p a c e s , >Y, ~Y.

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. Let v i /l i=l P ~-i=l P ~-- mi(T(x)) i=l miCT(x)) liT(x) - v i II < £.

The definition essentially [141). in > Fo(U O) e Y. g. was i n t r o d u c e d ~36~. 1. Let X and Y be two Banach s p a c e s , >Y, ~Y. If F i s a - h o m o g e n e o u s and s t r o n g l y a-quasihomogeneous with respect ('b) F : X If F is a-homogeneous to and strongly closed, then F is F. 2. F o : X--~Y. Let F to Then Fo that X and be an u e X Y to F. with respect be two Banach s~aces, a-strongly to Fo, Un---~u o >Y, operator continuous. it is t a F(u/t) = Fo(U) in X in (Un) Y. ) Let with respect For - Fo i s a - h o m o g e n e o u s .

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A. D. Alexandrovs Problem for CAT(0)-Spaces by Andreev P. D.


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