关键词: nonlinear boundary value problem, singular perturbation, uniformly efficient asymptotic expansion, higher orderdifferential inequalities, boundary layer correction

Abstract: In this paper, it has been studied that the singular perturbations for the higherorder nonlinear boundary value problem of the formε²y(n)=f(t, ε, y.…, y^(n-2))pj(ε)y⁽¹⁾(0, ε)-qj(ε)y^(j+1)(0.ε)=Aj(ε) (0≤j≤n-3)a₁(ε)y^(n-2)(0.ε)-a₂(ε)y^(n-1)(0, ε)=B(ε)b₁(ε)y^(n-2)(1, ε)+b₂(ε)y^(n-1),(1. ε)=C(ε)by the method of higher order differential inequalities and boundary layer corrections.Under some mild conditions, the existence of the perturbed solution is proved and itsuniformly efficient asymptotic expansions up to its n-th order derivative function aregiven out. Hence, the existing results are extended and improved.

Key words: nonlinear boundary value problem, singular perturbation, uniformly efficient asymptotic expansion, higher orderdifferential inequalities, boundary layer correction