DUNKEL. LINEAR DIFFERENTIAL EQUATIONS. 355 



Lemma IV. Ifb is a continuous function ofx in the interval < x ^ c, 

 and if: 



X XX 



g t (x) = I -dx . . . / -dx I x~ r b dx 



c c c 



t integrations 



where r is real and greater than zero, then : 



X 



(38) \x^g t {x)\<~j\b\\dx\ (0<x^c). 



c 



When t = 1 we have : 



X X 



I 3 fg 1 (x) | ^ x r Cx- r \b\ \dx\<> (\b\\dx\. 



c c 



Assume that IV. is true for t = t x . Then : 



X XX 



I * r ff h+1 (*) I ^ *f- 1 ^ (*) I i cfe | < ^if*- 1 -" \f\ b\\dx\j l M 



c C C 



^^[/^l^l][/l s ll & l] = ^[ 1 -fi) r ]/ lsll,fe l 



a; 

 c 



Therefore IV. is true for t = t x + 1, and the lemma is proved. 



Lemma V. Ifb, r, and g t are defined as in IV. and it is further as- 

 sumed that the absolute value of b is integrable up to x — then : 



(39) limit x r g t (x) = 0. 



x=0 



To prove this, let us choose a constant s such that < s < r. Then : 



XXX 



\x"g,(x)\=x*\ x*-* \ l -dx . . . C^-dx f x~<~> x^bdx I 



c c c x 



S(^rip/r'l»ll*|. 



c 



as we see from IV. by replacing rbyr — s and £ by x~ s b. 



