Some new results about Hammerstein equations

Abstract

A detailed discussion and a complete bibliography about equation (1) can be found in [3]. The new feature about the results we present here is the fact that we do not assume any coercivity for F. When Fis monotone and K maps L(Q) into Z,°°(£i), there is no growth restriction on F either (cf. Theorem 1). The monotonicity of F can be weakened when Kis compact (cf. Theorem 4). Also some of these results are valid for systems in the case where F is the gradient of a convex function (cf. Theorem 5). Assume (2) Kis a monotone hemicontinuous mapping from //(fi) into L°°(Q) which maps bounded sets into bounded sets, (3) f(x, r):CixR-^R is continuous and nondecreasing in r for a.e. x eQ, and is integrable in x for all r e R.