A New Method Based on the RKHSM for Solving Systems of Nonlinear IDDEs with Proportional Delays

and Applied Analysis 3 For any u(x) ∈ W 2 [0, +∞), let ⟨u(x), ψ i (x)⟩ = 0, i = 1, 2, . . ., which means that ⟨u (x) , (L ∗ φ i (x)⟩ = ⟨Lu (⋅) , φ i (⋅)⟩ = (Lu) (x i ) = 0. (12) Note that {x i } ∞ i=1 is dense in [0, +∞); therefore,Lu(x) = 0. It follows that u(x) ≡ 0 from the existence ofL. 3.1. Construction the of Iterative Sequence ?̃? n (x) and Ṽ n (x). Next we construct the iterative sequence ?̃? n (x) and Ṽ n (x), putting LM n (x) = F (x, ?̃? n−1 (x) , Ṽ n−1 (px)) , ?̃? n (x) = P n M n (x) , LN n (x) = G (x, Ṽ n−1 (x) , ?̃? n−1 (qx)) , Ṽ n (x) = P n N n (x) , (13) where M n (x),N n (x) ∈ W 2 [0, +∞) and P n : W 2 → Span{ψ 1 , ψ 2 , . . . , ψ n } is a orthogonal projection operator. Then by (13) it followed that: