 
Summary: JOURNAL OF INTEGRAL
EQUATIONS AND APPLICATIONS
Volume 1, Number 1, Winter 1988
THE DISCRETE GALERKIN METHOD FOR
NONLINEAR INTEGRAL EQUATIONS (*)
KENDALL ATKINSON AND FLORIAN POTRĄ
ABSTRACT. Let K be a completely continuous nonlinear
integral operator, and consider solving x = K(x) by Galerkin's
method. This can be written as xn = PnK(xn),Pn an or
thogonal projection; the iterated Galerkin solution is defined
by xn = K(xn). We give a general framework and error anal
ysis for the numerical method that results from replacing all
integrals in Galerkin's method with numerical integrals. A
special high order formula is given for integral equations aris
ing from solving nonlinear twopoint boundary value prob
lems.
1. Introduction. Consider the problem of solving the nonlinear
Urysohn integral equation
(1.1) x(i)= I K{t,s,x(s))ds, ted.
Jn
