Differential equations with operator coefficients

This book is the first systematic and self-contained presentation of a theory of arbitrary order ordinary differential equations with unbounded operator coefficients in a Hilbert or Banach space, developed over the last ten years by the authors. It deals with conditions of solvability, classes of uniqueness, estimates for solutions and asymptotic representations of solutions at infinity. The authors show how the classical asymptotic theory of ordinary differential equations with scalar coefficients can be extended to very general equations with unbounded operator coefficients. By contrast with previous work the authors' approach enables them to obtain asymptotic formulae for solutions under weak conditions on the coefficients of equations. Exposition of abstract results is accompanied by many new applications to the theory of partial differential equations. In Appendix a systematic treatment of the theory of holomorphic operator functions is given.