Degeneracy Results for Fully Nonlinear Integral Operators

It is shown that integral operators of the fully nonlinear type $K(x)(t)={\int }_{\mathrm{\Omega }}k(t,s,x(t),x(s))ds$ exhibit similar degeneracy phenomena in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In particular, $K$ is Fr'echet differentiable in $L_p$ 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 $L_p$ 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 Fr'echet differentiability or locally compact.