Resolvent family for the evolution process with memory

Abstract

AbstractIn this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, , that is, called the resolvent family for the linear evolution process with memory, the is called the memory effect family. In this paper, we prove that the families and are exponentially bounded, and the family associate with an operator pair that is called generator of the resolvent family. Using , we derive associated differential equation with memory and representation of via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on . The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations.