DUNKEL. — LINEAR DIFFERENTIAL EQUATIONS. 355 



Lemma IV. If bis a continuous function ofx in the interval <x^c, 

 and if: 



XXX 



g^{x) = I -dx . . . I -dx I x-^'bdx 



c c c 



t integrations 



where r is real mid greater than zero, then : 



X 



(38) \^9.{^)\'S^J\^\<i^\ {0<x^c). 



c 

 When t = 1 we have : 



X X 



\!rrg^{x)\^x'-rx~'-\h\\dx\S 1 1 ^ | I ^-^ I • 

 c c 



Assume that IV. is true for i = ti. Then : 



X 3; X 



c c c 



^c c c 



X 



<\f\b\\dx\. 



r \j 

 c 



Therefore IV. is true for t = ti -\- 1, and the lemma is proved. 



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

 sumed that the absolute value of b is integrable up to x = 0, then : 



(39) limit x'-^r, (.x) = 0. 



a;=0 



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



lar"^, (a:)l = x'W-' I -dx . . . j - dx j x'^'-'^ x'^ b dx\ 



X 



(.J^J-'l'li''-l. 



c c X 



as we see from IV. by replacing r by r — s and i by a; ' b. 



