ABSTRACT
As already mentioned, the function 1(s) defined by (2.35) is upper semi-continuous; consequently, for any e > 0, the set D. of all s E [a,b] such that 1(s) > sup {7(0') - e : a ~ 0' ~ b} is non-empty and open. Choose a point s. E D. where the function (2.40) vanishes, Le. 6K(S.) = O. For any s E [a, b], we have then
Ic(s)1 +Var[G,b]9(S,.) ~ sup 1(s) ~ 1(S.) +e caS-Sb
= Ic(s.) +oK(s.)1-16K(S.)1 +Var[G.b]9(s.,.) +e, hence, by Lemma 2.1,
GS-Sb Since e > 0 was arbitrary, (2.42) is proved. • As was shown by DIALLO-ZABREJKO [1987], every integral operator or stamping operator (in Krasnosel'skij's sense) is infra-stamping. In this way, Theorem 2.4 generalizes and extends the results by DAUGAVET [1963], KRASNOSEL'SKIJ [1967], and LozANOVSKIJ [1966]. On the other hand, there exist infra-stamping operators which are neither integral operators nor stampingj just consider the operator Kz(s) = z(s/2) on the space C = C([O,l]). Theorem 2.4 is mentioned in DIALLO-ZABREJKO [1987] even for the space C = C(Q) of continuous functions on an arbitrary compact metric space Q. In contrast to Lemma 2.1, however, the continuity of the function c = c(s) is required in Theorem 2.4. As will be shown in the next chapter, this requirement is superfluous if K is considered as a bounded linear operator from C into Loo , and the function c, rather than being continuous, has merely the property that the operator (2.34) acts in the space C. This situation is typical in many applications of Lemma 2.1.
