... (Pε) exists and is unique and satisfies (2.1)uniformly for 0≤ k ≤ N,wherey(0)k and y( j)kare the solutions of (2.2 )and( 2.4), respectively.More precisely, for all n≥ 0 and all 0 ≤ k ... <ε0,thesolution(yk(ε)) of (3.1) exists and is unique, and satisfiesyk(ε) =∞j=0εjy( j)k,0≤ k ≤ N, (3.5)uniformly for 0≤ k ≤ N,wherey(0)k and y( j)kare the solutions of (2.2 )and( 2.4), respectively.More ... direct consequence of (3.2)andTheorem 2.1.12 Perturbation method for difference equations[6] M. S. Krishnarayalu, Singular perturbation methods for one-point two-point and multi-pointboundary...