具非线性边界条件的泛函微分方程边值问题奇摄动(英文)

Singularly Perturbed Boundary Value Problems for Functional Differential Equations with Nonlinear Boundary Conditions

研究一类具非线性边界条件的泛函微分方程边值问题εx″( t) =f ( t,x( t) ,x( t-τ) ,x′( t) ,ε) , t∈ ( 0 ,1 ) ,x( t) =φ( t,ε) , t∈ [-τ,0 ], h( x( 1 ) ,x′( 1 ) ,ε) =A(ε) .我们利用微分不等式理论证明了边值问题解的存在性 。

A kind of boundary value problems for functional differential equations with nonlinear boundary condition as follows εx″(t)=f(t,x(t),x(t-τ),x′(t),ε), t∈(0,1), x(t)=φ(t,ε), t∈[-τ,0], h(x(1),x′(1),ε)=A(ε) is studied. Using the theory of differential inequality, we prove the existence of the solution, and an uniformly valid asymptotic expansions of the solution is given as well.