Singular Perturbation of Initial Value Problem for a Nonlinear Second Order Systems

In this paper,the singular perturbation of initial value problem for nonlinear second order vector differential equations ε'x"=f(i,x,x',ε) x(0,ε)=α,x'(0,ε)=β is discussed,where r>0 is an arbitrary constant,ε>0 is a small parameter,x,f,a and Under suitable assumptions,by using the method of many-parameter expansion and the technique of diagonalization,the existence oj the solution of perturbation problem is proved and its uniformly valid asymptotic expansion of higher order is derived.