Generation of nonlinear semigroups by a partial differential equation (Q912375)

In the literature on nonlinear semigroups of operators, the emphasis has been mainly on sufficient conditions for generation as necessary conditions are in general hard to find. The author's contribution is in this area. Let C be a closed subset of a Banach space X and let \(T=\{T(t):\geq 0\}\) be a strongly continuous semigroup of continuous functions from C to C. Let Q be the Banach space of bounded continuous functions from C to Y, another Banach space. Then there exists a linear operator \(A: Dom(A)\subset Q\to Q\) such that Dom(A) is pointwise dense in Q (i.e. if \(f\in Q\) there is an \(f_ n\in Dom(A)\), \(n=1,2,..\). such that \(f_ n(x)\to f(x)\) for all \(x\in C\). Moreover, for all \(\lambda >0\), \(\| (I-\lambda A)^{-1}\| \leq 1;\) and if \(f\in Dom(A)\) and \(u(t,x)=f(T(t)x),\) then \[ \partial u(t,x)/\partial t=(A(u(t,.)))(x),\quad u(t,x)=\lim_{n\to \infty}((I-(t/n)A)^{- n}f)(x)\quad for\quad t\geq 0,\quad x\in C. \]