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Given a probability space (Ω, A, P ), a separable metric space X, and a random-valued vector function f : X × Ω → X, we obtain some theorems on the existence and on the uniqueness of continuous solutions ϕ : X → R of the equation ϕ(x) = Ω ϕ(f (x, ω))P (dω). 2000 Mathematics Subject Classification. 39B12, 39B52, 60B12.doi:10.1553/sunda2004ssii113 fatcat:zp4yzmez4jc3lku3fj2nynjdva