Degeneracy results for fully nonlinear integral operators

Abstract

It is shown that integral operators of the fully nonlinear type K(x)(t)=∫Ωk(t,s,x(t),x(s))ds exhibit similar degeneracy phenomena in a large class of spaces as superposition operators F(x)(t)=f(t,x(t)). In particular, K is Fréchet differentiable in Lp only if it is affine with respect to the “x(t)” argument. Similar degeneracy results hold if K satisfies a local Lipschitz or compactness condition. Also vector functions, infinite measure spaces, and a much richer class of function spaces than only Lp are considered. As a side result, degeneracy assertions for superposition operators are obtained in this more general setting, complementing the known results for scalar functions. As a particular example, it is shown that the operators arising in continuous limits of coupled Kuramoto oscillators fail everywhere to be Fréchet differentiable or locally compact.