By Jozef Banas

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**Example text**

3). We have I i m sup (I R X U n n-oo - a )� n The family � satysfies all the axioms 1° - 5° of the definition of kernel. Conditions 1° 4° are easily verified. The condition 5° is a consequence of the continuity of the function � with respect to the Hausdorff metric. This measure is even lipschit ziar.. Indeed, if Xe K(Y,a) and YC K(X,a)" then RnXc �y + aJ\iK(9,l) and vice versa. Hence - and finally 25 which next illlplies 1-> {X) where B = sup[IR l n - {Y) I� B D ( X , Y ) . ,1,2, • • • ]. J implies the maximum property.

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 . 1 . 1 is a special case of a more general result . Let E be an arbitrary Banach space and let A be a l inear closed operator with the domain EA dense in E and with values covering � • . 38 = another Banach space plete on E A F.

L xc K(X0, r+e) • There exists a fini• There exists an index n such to any xe X coresponds x0e: X0 such that Hence (6. 5 ) which means that. 3. E com·�" 2 (TX} � k 'XJ E e. "'ll i ly (with 'l'(,1 oo the s is ter. Obv i ously this space has no basis. cs. ples of such measures: µ1 n-oo measures of noncompactness norm). so:rie non lim supllRnTXO. tand ard in sup a delicate mat we ca n define The fo l l ow ing formulas give some exam 1 im sup [sup lxn-al) , n. -oo I im [ sup (sup[l•xn-xml: n,m} p]J] , p -<><> XE X lim sup diam Xn n-= where x n [xn: xe x ] The kernels of P1 th ese - the family ces If a 1'2 - the of = 0 then sets all bo un ded con verg ing to consisting a with the same nrate".

### Measures of noncompactness in Banach spaces. by Jozef Banas

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