Well-posedness of differential-operator problems. II: The Cauchy problem for complete second-order equations in Banach spaces
[For Part I, see the review Zbl 0922.34055 above. ] Consider the Cauchy problem for the complete second-order equation \[ u'' = B u' + Au,\quad t \geq 0,\qquad u(0)=u_0,\quad u'(0) = u_1, \] where \(A\) and \(B\) are closed linear operators on a Banach space \(X\). The authors introduce the notion of the \(\omega\)-well-posedness of the above Cauchy problem and construct a theory of \(M,N\)-functions for commuting operators \(A,B\). Necessary and sufficient conditions for the \(\omega\)-well-posedness of the above problem are given in terms of conditions on the resolvent of the operator pair \(A,B\), which is \[ R(\lambda^2) = (\lambda^2-\lambda B - A)^{-1}. \]
scientific article; zbMATH DE number 1292311
scientific article