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The main purpose of this book is to present the basic theory and some recent de­ velopments concerning the Cauchy problem for higher order abstract differential equations u(n)(t) + ~ AiU(i)(t) = 0, t ~ 0, { U(k)(O) = Uk, 0 ~ k ~ n-l. where AQ, Ab . . . , A - are linear operators in a topological vector space E. n 1 Many problems in nature can be modeled as (ACP ). For example, many n initial value or initial-boundary value problems for partial differential equations, stemmed from mechanics, physics, engineering, control theory, etc. , can be trans­ lated into this form by regarding the partial differential operators in the space variables as operators Ai (0 ~ i ~ n - 1) in some function space E and letting the boundary conditions (if any) be absorbed into the definition of the space E or of the domain of Ai (this idea of treating initial value or initial-boundary value problems was discovered independently by E. Hille and K. Yosida in the forties). The theory of (ACP ) is closely connected with many other branches of n mathematics. Therefore, the study of (ACPn) is important for both theoretical investigations and practical applications. Over the past half a century, (ACP ) has been studied extensively.

Banach [Espaces de ] --- Banach [Ruimten van ] --- Banach spaces --- Cauchy [Probleem van ] --- Cauchy [Problème de ] --- Cauchy problem --- Differentiaalvergelijkingen --- Differential equations --- Equations différentielles --- Hilbert [Espace d' ] --- Hilbert [Ruimte van ] --- Hilbert space --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Calculus --- Differential equations. --- Ordinary Differential Equations. --- 517.91 Differential equations