Positive Solutions of Advanced Differential Systems

We study asymptotic behavior of solutions of general advanced differential systems [Formula: see text] , where F : Ω → ℝ(n) is a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument and Ω is a subset in ℝ × C (r) (n), C (r) (n) : = C([0, r], ℝ(n)), y (t)∈C (r) (n), and y (t)(θ) = y(t + θ), θ ∈ [0, r]. A monotone iterative method is proposed to prove the existence of a solution defined for t → ∞ with the graph coordinates lying between graph coordinates of two (lower and upper) auxiliary vector functions. This result is applied to scalar advanced linear differential equations. Criteria of existence of positive solutions are given and their asymptotic behavior is discussed.