Strongly nonlinear integral equations of hammerstein type.
This paper studies the solution of the nonlinear Hammerstein equation u(x) + k(x,y)f[y,u(y)]mu(dy) = h(x) in the singular case, i.e., where the linear operator K with kernel k(x,y) is not defined for all the range of the nonlinear mapping F given by Fu(y) = f[y,u(y)] over the whole class X of functions u which are potential solutions of the equation. An existence theorem is derived under relatively minimal assumptions upon k and f, namely that (Ku,u) >/= 0, that K maps L(1) into L(1) (loc) and is compact from L(1) [unk] L(infinity) into L(1) (loc), that f(y,s) has the same sign as s for s >/= R, and that for each constant r > 0, f(y,s) </= g(r)(y) for s </= r where g is bounded and summable. The proof is obtained by combining a priori bounds, a truncation procedure, and a convergence argument using the Dunford-Pettis theorem.
Browder
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Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637.