On a nonlinear integral equation without compactness
F. Isaia
Abstract.
The purpose of this paper is to obtain an existence result for the integral
equation
u(t) = j(t, u(t))
+
b
ò
a
y(t, s, u(s))ds,
t Î[a, b].
where j : [a, b] ´ R ® R
and y : [a, b] ´ [a, b] ´ R
® R are continuous functions which satisfy some special
growth conditions. The main idea is to transform the integral equation into
a fixed point problem for a condensing map T : C[a, b] ® C[a, b].
The "a priori estimate method" (which is a
consequence of the invariance under homotopy of the degree defined for
a-condensing perturbations of the identity) is used in order to prove
the existence of fixed points for T. Note that the assumptions on
functions j and y do not generally assure the compactness of
operator T, therefore the Leray-Schauder degree cannot be used (see K.
Deimling).
Keywords.
Nonlinear integral equation, condensing map,
topological degree, a priori estimate method.
Compressed Postscript 
