By Jozef Banas
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Extra resources for Measures of noncompactness in Banach spaces.
Sample text
The s e t 0 0 • can _ u s e the procedure closely patterned on that from Denote w ( x , � , C. ) the Kolmogcirov modulus of continui with respect to p , where fl i s de fined as in chapter 7 , Ch apter 7. e. ) if j) = 0. Obvious ly 51 l im e- o w � e J ( x , fl , e. ) I o. Next , for X e 'Y't(,Lp , put w (x, p , e } sup [ w (x , � . l ( X , e } ( the last equality holds for any Then we have E l im w (x, p , e J e. - o �) . 1 . 2 . The fun ction s . ) are measure s of noncompac tness �i th maximum prope rty .
2 . l im sup h - 0 The fun ation { sup ( IS x-x l : x E h . xJ} a regu l a r measul'e of non aompaatne ss· and r o o f: P roving . that � 0 is a me asure i s standard. First inequal i ty comes from. the inequal i ties 55 shown above . The last inequality comes from an e as i ly proved fact tha t in part 1 0 . 1 . the . S tek l ov modulus " for x with respect to J' as We can also define some measures s imi l arly as Le t us de fine " ( x , J'> , E. :»( x , fi , f> = e, ) : X E: x] , O . 3. The function '1 ( X, j3 , l) is a mea s ure of n oncompactness wi th maximum prope rty .
N L: i•O Denote x (i) = [ x 1x< l l 1 (i ): x e: we have the following T H E 0 R E M in C = C�a, b> then i s a measure i n P r o o f: � n C L max [ • 1-0 x ] , i=0 , 1 , 2 , 8. 1. 1 . t ) l : t e: { a � b) ] . , n , (x ( O ) = x) . If µ i s a measure of non compactness �i th A s imple consequence of the definition . For example the functions w ( x C n) ) 2 'Xi c (X ( n ) ) and (n) w (X ; � , &) are such measures . same is true for the spaces � ( ( a ,b) , Rn) of vector-valued differentiable functions Theor� · S .
