Approximation of abstract quasilinear evolution equations in the sense of Hadamard (Q946841)

The equations are of the form \[ u'(t) = A(u(t))u(t), \quad 0 \leq t \leq T , \quad u(0) = u_0 \] where \(\{A(w); w \in D\}\) is a family of closed linear operators in a Banach space \(X\) satisfying conditions involving the domains \(D(A(w))\) and strong continuity of \(A(\cdot)\) in a subspace \(Y \subset X.\) The initial value problem is \textit{well posed in the sense of Hadamard} if solutions exist for initial values in a subspace \(D_0 \subset X_0 \subset X\) and satisfy \[ \| u(t) - v(t)\| _X \leq M\| u(0) - v(0)\| _{X_0}. \] The results are on approximation of solutions of the equation by Euler's method, \[ {u_n(t + h_n) - u_n(t) \over h_n} = A_n(u_n(t))u_n(t) \] where \(A_n(w)\) is a suitable approximation to \(A(w).\)