Resolvent family for the evolution process with memory

In 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, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ 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 (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations.