By Richard. Abbatt

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**Extra resources for A treatise on the calculus of variations**

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Then 2"P ::; {3(Up) ::; 2q. L) is not purely atomic, we have that THEOREM 32 METRIC FIXED POINT THEORY Proof. Suppose that Xl, X2, ••• ,Xn are n vectors r-separated in Up. 8, so that T = 1 - 1/n and the inequality becomes n 1 ~ -lIx-J ~ n2 -xkll q j,k=l The conditions on Ilxnll and ( 1 ~ 2 1- -n IIxn - xmll )2- ~ -llx-llq· 1 q n ~n j=l combine with this to give (1)2- 1 q ~ 2 nn(n -1)-r n2 n or r<2ii1 - The inequality (3(Up) ~ J q (n-l)Y -n 1 2ii results from letting n ---+ +00. 1 In order to obtain the equality, we must find a 2ii-separated sequence in Up.

Y. Hence, any a-minimal subset B of B' verifies the required inequality. 9. 8(A). LEMMA Proof. 8(A) = a(A). 8(A) = a(A) which contradicts the a-minimality of A. 8. 10. Let (X, d) be a complete metric space. 8 is minimalizable but, in general, not strictly minimalizable. THEOREM Proof. Let A be a bounded subset of X. 8(A) and for every infinite subset B of A we know that B is a-minimal and so a(B) = a(A). 8(A). 8(A)-c. 8(B) by the a-minimality of B. 8 is minimalizable. 8 is not strictly minimalizable in general and conclude the proof, we need only consider the following example: Example 7: Let X = £00(£2) C £2 x £2 X ...

The set C of all condensing mappings in ~l(C) is residual in 42 METRIC FIXED POINT THEORY ° Proof. Without loss of generality we can assume that E C. Let N be the set of all k-set-contractive mappings (k E [0,1)) in ~l(C), that is, N = Uk