F(r\CTIO\s IN THK TIIKORY OF INTEGRAL EQUATIONS. 



141 



11. Let Xi (t) be a function of t defined in an interval (0, TJ) (77 > 0) which 

 possesses a limited derivative of the second order ; further, let the function lie such 

 that 



We propose to prove that (a s 

 p 



(22) 



In the first place, we observe that, if n is any positive numher less than unity 

 and p > 1, 



Since the right-hand member converges to zero as \ tends to oo, it follows that 



..... (23) 



Again, by a known theorem of the differential calculus, 



C.) (o ==<==>?), 



where t^ is a point of the interval (0, t). Denoting by c the upper limit of | x\ (0 1 

 in (0, t) we see that 



. 



-Xi ()] * S cp e-" |/[*- Xl (I)] | < dfc 



Hence we have 



-"'/[s-X. (01 X. (0 *-f 

 Since 



for all values of p and t which are not negative, the right-hand member is not greater 

 than 



and this, by a known property of Lebesgue integrals, converges ]to zero as p tends 

 to oo. Referring to (23) we thus see that 



P f -"/[>-x ()] * SEEE 



Jo 

 For values of p which are sufficiently great, 



(24) 



