A system of abstract measure delay differential equations

Measure Delay Differential Equations 3 3 Statement of the problem Let μ be a σ-finite real measure on X. Given a p ∈ ca(X,M) with p << μ, consider the abstract measure delay differential equation (in short AMDDE), involving the delay w, dp dμ = f(x, p(Sx), p(Sxw)), a.e. [μ] on x0z, p(E) = q(E), E ∈M0, (4) where q is a given known vector measure, dp/dμ is a Radon-Nikodym derivative of p with respect to μ and f : Sz × IR n × IR → IR is such that f(x, p(Sx), p(Sxw)) is μ-integrable for each p ∈ ca(Sz,Mz). Definition 3.1 Given an initial real measure q on M0, a vector p ∈ ca(Sz,Mz) (z > x) is said to be a solution of AMDDE (4) if (i) p(E) = q(E), E ∈M0, (ii) p << μ on x0z, (iii) p satisfies (4) a.e. [μ] on x0z. Remark 3.1 The AMDDE (4) is equivalent to the abstract measure integral equation