It is shown that integral operators of the fully nonlinear type $K(x)(t)=\int_\Omega k(t,s,x(t),x(s))\,ds$ exhibit similar degeneracy phenomena in a large class of spaces as superposition operators $F(x)(t)=f(t,x(t))$. In particular, $K$ is Frechet 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 Frechet differentiability or locally compact.

中文翻译：

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完全非线性积分算子的简并结果

结果表明，完全非线性类型的积分算子 $K(x)(t)=\int_\Omega k(t,s,x(t),x(s))\,ds$ 在大类空间作为叠加运算符 $F(x)(t)=f(t,x(t))$。特别地，仅当 $K$ 与 "$x(t)$" 参数是仿射的时，它才在 $L_p$ 中是 Frechet 可微的。如果 $K$ 满足局部 Lipschitz 或紧致性条件，则类似的简并结果成立。还考虑了向量函数、无限测度空间和比仅 $L_p$ 更丰富的函数空间类。作为附带结果，在这个更一般的设置中获得了叠加算子的简并断言，补充了标量函数的已知结果。作为一个特定的例子，