Existence and Regularity for Boundary Cauchy Problems with Infinite Delay

The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers. 1. Introduction Consider the following problem: where represents the density of the population of age at time , is the death rate, and is the number of newborns at time . Such models were introduced by Lotka in 1925 and have been studied by many authors. For a detailed discussion, we refer the reader to [1, 2]. The problem (1) can be transformed into the following abstract boundary Cauchy problem: where is an unbounded operator on a Banach space of functions on with domain , for each , is the operator defined by for , and is a “boundary space.” For each , is a bounded linear operator from to？？ . Equation (2) can be further transformed into a Cauchy problem. To do this, suppose that the domain of and are Banach spaces such that is dense and continuously embedded in . and . We make the following hypothesis.(S1) generates a -semigroup on where denotes the kernel of .(S2) is a surjection from to？？ . has a continuous inverse for any (the resolvent set of ).If assumptions (S1) and (S2) hold, then the operator is continuous from to？？ , and for all the operator satisfies At least formally, we can rewrite (2) as It is easy to see that (4) is a form of the following abstract Cauchy problem: where is the infinitesimal generator of a -semigroup on a general Banach space , is a bounded linear operator satisfying certain conditions, and . In this way, the problem of solving (1) or (2) is transformed to that of solving (5). Equations of the form like (5) were considered in [3–5]. An important tool used is the multiplicative perturbation which was first studied by Desch and Schappacher [3] in 1989 for -semigroup. In recent years, this type of perturbations has been further developed and applied by many authors (cf., e.g., Engel and Nagel [6], Piskar？v and Shaw [7]). In this paper, our proof will also be based on an application of the multiplicative perturbation theorem. Equation (2) has been considered in [8, 9] for the cases and , respectively. Suppose that is a linear space of functions from to？？ . Then these two cases can be viewed as a function from to？？ . That says that depends on the “history” of . Thus, for such functions , (2) becomes a retarded Cauchy problem. The following abstract retarded Cauchy problem has been considered