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 (rm ACP_n) in X and its inhomogeneous version (rm IACP_n) . A new operator family of bounded linear operators from Y to X is introduced, called an existence family for (rm ACP}_n) , so that the existence and continuous dependence on initial data of the solutions of (rm ACP_n) and (rm IACP_n) can be studied, and some basic results in a quite general setting can be obtained. A sufficient and necessary condition ensuring that (rm ACP_n) possesses an exponentially bounded existence family, in terms of Laplace transforms, is presented. As a partner of the existence family, for (rm ACP_n) , a uniqueness family of bounded linear operators on X is defined to guarantee the uniqueness of solutions. These two operator families for (rm 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 (rm ACP _n) . They have a special function in treating those ill-posed (rm ACP_n) and (rm IACP_n) whose coefficient operators lack commutativity.