一类奇摄动泛函微分方程边值问题

A Kind of Singularly Perturbed Boundary Value Problemsfor Functional Differential Equations

该文研究一类泛函微分方程边值问题εx″( t) =f ( t,x( t) ,x( t-τ( t) ) ,x′( t) ,ε) ,t∈ ( 0 ,1 ) ,x( t) =φ( t,ε) ,t∈ [-τ,0 ],x( 1 ) =A(ε) ,其中ε>0为小参数 ,τ( t)≥τ0 >0 ,τ=maxt∈ [0 ,1] τ( t) <1 .利用微分不等式理论证明了边值问题解的存在性 ,并给出了解的一致有效渐近展开式 .

In this paper ,we study a kind of boundary value problems for functional differential equations εx″(t)=f(t,x(t),x(t-τ(t)),x′(t),ε),t∈(0,1),x(t)=φ(t,ε),t∈\,x(1)=A(ε), where ε>0 is asmall parameter,τ(t)≥τ\-0>0,τ= maxt∈\ τ(t)<1.Using the theory of differential inequality,we p rove the existence of the solution and uniformly valid asymptotic expansions of the solution is given as well.