Abstract: In the present paper, the singular perturbations for the higher-order scalarnonlinear boundary value problem
ε²y=f(t,ε,y,y',…,y^(n-2),εy(n-1)),t∈[0,1] H₁(y(0,ε),…,y^(n-3)(0,ε),εy^(n-2))(0,ε),εy^(n-1))(0,ε),ε)=0,H₂(y(0,ε),…,y^(n-1)(0,ε),y(1,ε),…,y^(n-1)(1,ε),ε)=0 are studied, where ε > 0 is a small parameter, n ≥ 2. Under some mild assumptions,we prove the existence and local uniqueness of the perturbed solution and give out the uniformly valid asymptotic expansions up to its nth-order derivative function by employing the Banach/Picard fixsed-point theorem. Then the existing results are extended and improved.

Key words: singular perturbation, uniformly valid asymptotic expansion, Green function, Banach/Picard fixed-point theorem