奇摄动三阶微分方程的边值问题

The Boundary Value Problem of Singularly Perturbed Third Order Differential Equation

研究含小参数ε>0的三阶微分方程边值问题:在f(t,x,y,ε),A(ε),B(ε),C(ε)适当光滑,f_x(t,x,y,ε)≤0,f_y(t,x,y,ε)≥m>0以及退化问题0=f(t,x,x′,0),x(0)=A(0)于0≤t≤1上有解的条件下,证明了解的存在性,并且给出了解的一致有效估计。

In this paper we study the boundary value problem of third order differential equation with a small parameter ε〉0: x(0) =A(ε), x'(0)=B(ε), x'(1) =C(ε). Under the assumptions that f(t,x,y,ε),A(ε),B(ε),C(ε) are all properly smooth, f_x(t,x,y,ε)≤0, f_y(t,x,y,ε)≥m〉0 and the reduced problem 0=f(t,x,x',0),x(0)=A(0) has a solution on [0,1], we prove the existence of the solution and give a uniformly valid estimate of the solution.