R is of bounded variation where ttlimo fm (r) = f(r), : r E [a, b] I. The generalized Perron integral 44 then the sequence Um : [a, b] x [a, b] -+ R given by U. g(t), T, t E [a, b] f o r m = 1, 2, ... (g(Q) - g(a)) for J = [a, Q] and for a given e > 0 the corresponding superadditive interval function -t can be given in the form -t(J) = e varfl g.
47) a Proof. 34 for the case when V(r,t) = f(T)g(t). 37 Remark. 34 can be also used to deduce the following known result. If f : [a, b] -4R, If(s) I _< c for s E [a, b] where c is a constant, g : [a, b] -+ JR is of bounded variation on [a, b] and the integral fa f (s) dg(s) exists then b f(s)dg(s) c varQ g. Indeed, it is easy to see that It -,r 1. (vara - vary) for every t, r E [a, b] and the statement follows immediately from the known fact that the integral fa c d(vara) = c. 34. 38 Lemma. Let h : [a, b] - JR be a nonnegative nondecreasing function which is continuous from the left on (a, b].
Generalized Ordinary Differential Equations (Series in Real Analysis) by Stefan Schwabik
Categories: Differential Equations