Approximation and convergence theorems for nonlinear semigroups associated with semilinear evolution equations (Q2747831)

The article is devoted to an approximation theory for nonlinear semigroups in a Banach space \(X\) associated with semilinear problems NEWLINE\[NEWLINE{d\over dt} u(t)= (A+B) u(t),\quad t> 0,\quad u(0)= x\in D\subset X,NEWLINE\]NEWLINE where \(A\) is a generator of a \((c_0)\)-semigroup \(T= \{T(t): t\geq 0\}\) and \(B\) a nonlinear operator from \(D\) into \(X\). Under definite stability and consistency conditions semilinear versions of the Lax equivalence theorem and Neven-Trotter-Kato theorems are given. Moreover, for a locally Lipschitzian semigroup an approximation-solvability theorem is proven.