ABSTRACT
A general theory of equations with a causal operator has been developed by Corduneanu [2002]. A short description of functional equation with a causal operator is as follows.
Let E = E([0, T ),Rn) be a functional space with a norm. Let the operator V : E → E. The V is said to be a causal operator if the following conditions are satisfied: for each pair of elements from the space E such that for x(s) = y(s) when 0 ≤ s ≤ t the correlation (V x)(s) = (V y)(s) holds true when 0 ≤ s ≤ t for arbitrary t < T . An example causal operator is
(V x)(t) =
K(t, s, x(s)) ds, t ∈ [0, T ),
whereK(t, s, x) is the function with values in Rn determined for 0 ≤ s ≤ t < T and x ∈ Rn or
(V x)(t) = f(t) +
K(t, s, x(s)) ds,
where f(t) is a continuous function on [0, T ). In this chapter we use the comparison principle for the set of differential
equations with a robust causal operator to prove several stability theorems.
