Higher Order Abstract Cauchy Problems: Their Existence and Uniqueness Families

Let X,Y be Banach spaces. Of concern are the higher order abstract Cauchy problem (ACP n ) in X and its inhomogeneous version (IACP n ). A new operator family of bounded linear operators from Y to X is introduced, called an existence family for (ACP n ), so that the existence and continuous dependence on initial data of the solutions of (ACP n ) and (IACP n ) can be studied, and some basic results in a quite general setting can be obtained. A sufficient and necessary condition ensuring that (ACP n ) possesses an exponentially bounded existence family, in terms of Laplace transforms, is presented. As a partner of the existence family, for (ACP n ), a uniqueness family of bounded linear operators on X is defined to guarantee the uniqueness of solutions. These two operator families for (ACP n ) are generalizations of the classical strongly continuous semigroups and sine operator functions, the C ‐regularized semigroups and sine operator functions, the existence and uniqueness families for (ACP 1 ), and the C ‐propagation families for (ACP n ). They have a special function in treating those ill‐posed (ACP n ) and (IACP n ) whose coefficient operators lack commutativity.