Integrodifferential systems with infinitely many delays


In this paper we consider the initial-value problems: (P1)X(t)=(AX)(t) for t>0, X(0+)=I, X(t)=0 for t<0 and (P2) Y(t)=(QY)(t) for t>0, Y(0+)=I, Y(t)=0 for t<0, where A and Q are linear specified operators, I and0 — the identity and null matrices of order n, and X(t), Y(t) are unknown functions whose values are square matrices of order n. Sufficient conditions are established under which the problems (P1) and (P2) have the same unique solution, locally summable on the half-axis t ⩾0. Using this fact and some properties of the Laplace transform we find a new proof for the variation of constants formula given in[1, 2]. On the basis of this formula we derive certain results concerning a class of integrodifferential systems with infinite delay.


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Entrata in Redazione il 2 marzo 1977.

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Luca, N. Integrodifferential systems with infinitely many delays. Annali di Matematica 116, 177–187 (1978).

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  • Infinite Delay
  • Constant Formula
  • Integrodifferential System