Davis: Present Status of Integral Equations 15 
kernel was assumed to be a bilinear form in an infinite number of 
variables. _ 
It may be shown that the genre of the Fredholm determinant 
D(x) is at most one for the case of continuous kernels . 162 It is 
important to notice that, for finite bilinear kernels, the genre is 
zero since D(x) reduces to a polynomial. 
6. Generalizations. Generalizations of many kinds followed. 
E. Schmidt , 358 T. Lalesco , 287 A. Vergerio , 359 and G. Bratu 349 studied 
inclusive types of non-linear equations and stated existence theorems 
with more or less generality. 
In his work on the non-linear problem, Schmidt studied equations 
of the form 
u(x) + p K (x,t) u(t) dt + s f X . J n ~ ‘K(x,t„ u, . . t n ) u(t,) ai 
h(t t ) Pl . u(t n ^fh(t n ) gn dt 1 . .dt n = 0, 
where the sum is extended over a set of integral values «i, pi, 
one at least of which is assumed to be different from zero. 
Lalesco’s equation was similar in character, involving powers of 
the unknown function under the integral sign. Thus he considered 
existence theorems for the equation, 
u(x) = f(x) + f [K 1 (x,t)u(t)+K 2 (x,t)u 2 (t) + +K n (x,t) u n (t)] dt. 
J o 
Vergerio introduced the notion of linear operators into his 
equation which he wrote in the form, 
u(x) = f(x) + X f K(x,t) L(u) dt. 
*■ a 
where L( ) designates a general operator satisfying the linearity 
conditions : 
L(u + v) =• L(u) + L(v), 
c L(u) = L(cu).* 
Bratu, while considering very general types of non-linear equa- 
tions, also made fundamental contributions to the study of the 
important special cases 
u(x) = f(x) + X f K(x,t) u 2 (t) dt, 
J o 
and 
u(x) = f(x) + X f K(x,t) e u(t) dt. 
J o 
*The degree of generality of this equation can be readily appreciated from the work of S. Pincherle 
and C. Bo nr let who have shown that L(u) is equivalent to a differential operator of infinite order. 
Pincherle: Memoire sur le calcul fonctionnel distributif. Math. Annalen, vol. 49 (1897), pp. 325-381; 
Bourlet: Sur les operations en general et les equations differentielles lineaires d’ordre infini. Annales 
de l’ecole normale superieure, 3d ser., vol. 14 (1897), pp. 133-190. 
