By Ortega R., Tarallo M.
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Extra resources for Almost periodic equations and conditions of Ambrosetti-Prodi type
Let g be a non-zero bounded positive linear functional on X , with 44 g(x) :: 1 1 for x 0 E 4 0 and II :x 0 t l. Let (s i) i el be a family II ' in Y, bounded above by M. Define Si : g(x)si, with M defined similarly. If x), 0 , then (X S i )x - s i )g(x) so that RE S i . Let T be the supremum of (sdi which exists el 0, in L(X,Y), by assumption. As T S i, we have Tx. >, Six, On the other band, if t defining T' si. y:1--og(x)t, we have T' S i . Thus t T l x0 Txo. e. P. P. Let f be a non-zero, bounded positive linear functional on Y, with f(yo) • 11 Yo l = 1 far 3r 0 If g,h>,m,n with G : all these elements of 30.
Then there is a linear functional T from E into V such that: T(x)< P (x) (NxEE) Q(x)
C I so that 0 = ko EM. So in either case Yl is closed, and the result proven. 1 may be repeated. 12. 4. K(X,Y) is positvely generated wtonever X. is base-normed, if and only if each relatively compact subset of Y is order-bounded. 53 Proof. If X is base-nonmed, with base B, let T E K(X,Y). It follows that T(B) is a relatively compact subset of Y. Hence so also is co( CO1 k-1 T(B)). Let e be an upper bound for this latter set. : 0 f 113 1. 12 and define S e K(X,Y) by S : xi--+f(x)e. Clearly if b E S o Sb,Tb, O.
Almost periodic equations and conditions of Ambrosetti-Prodi type by Ortega R., Tarallo M.