16 
Indiana University Studies 
One of the most general functional equations involving integrals 
that has appeared in the literature is due to H. Galajikian, 353 who has 
made some progress in the existence theorem for the equation 
u(x) - g(x, 9 J, <p 2 ), 
where <pi = J* h t x A u(t)J dt. 
However, in a problem of this generality, the structure of the 
theorems has become so tenuous, that they are of little value in 
applied mathematics. Much important work, however, remains 
to be done in the case of special equations and the nature of these 
problems might profitably be indicated by the functional equations 
that arise in application. 
A. Kneser, 342 W. A. Hurwitz, 341 and G. Andreoli 338 are responsible 
for the development of the theory of the mixed integral equations 
(called by Kneser “belastete integralgleichungen”) and the latter has 
shown how it can be included under the theory of the general 
Fredholm equation if certain singularities are introduced into the 
kernel. In an important memoir which appeared in 1914 and in 
subsequent shorter papers, Andreoli has developed many of the 
properties of the general equation associated with his name. 336 
Several special kinds of equations which involve the unknown 
function under signs of integration have been studied in addition to 
the standard types already mentioned. A few of these and the 
names of those chiefly responsible for their development are tabulated 
below.* 
Equations 
u(x) = f(x) + S h i (x)u(i)(x)+ 2 f b Ki (x,t)u«)(t)dt 
; = 1 i = n J n 
Authors 
E. Bounitzky, U. Crudeli 
G. Fubini, J. Horn, G. 
Lauricella, S. Pincherle, 
C. Platrier. 
P. J. Browne. 
E. Cesaro. 
h(x)u(ax) = f(x)-fx n g(x)+X f K(x,t)u(xt)dt 
^ a 
g(0) =*= 0, h(0) = c. 
u(x) = f(x) + - 
j/ x' 
r: 
u(x 
1 
■-) 
t 2 
: dt 
|/x 
P. Nalli, C. Popovici. u(x) = A[g(x) u(ax) + f N(x,t) u(t) dt -j- 
~ o 
/»ax 
I P(x,t) u(t) dt] + f(x) 
J o 
r ° 0 
f(x) = I u(t) u(x -t) dt. 
J _00 
G. Polya, C. Runge. 
""For references see Bibliography, X. 
