Existence of solutions and asymptotic analysis of a class of singularly perturbed ODEs with boundary conditions

Abstract

We study the existence of solutions of a class of singularly perturbed BVPs $\varepsilon y'' +2y' +f(y) =0, ~y(0)=0, ~y(A)=0$ for some $A>0$ and $f>0$. Given an $f$ satisfying certain conditions, we will show that for each $\varepsilon>0$, there exists $A(\varepsilon)$ such that, if $0 < A < A(\varepsilon)$, then the problem has exactly two solutions. As an example we consider $f(y)=e^y$, where we show that $A(\varepsilon)>1$ for $\varepsilon>0$ sufficiently small. Both solutions have the same outer solution on $(0,1]$, but have different boundary behavior near $x=0$. One of them is bounded, the other unbounded as $\varepsilon \to 0$. A uniform expansion for the smaller solution has been obtained from matched asymptotic expansions. We prove rigorously that these asymptotics are correct up to $O(\varepsilon).$ We also prove that for a class of functions, $\underset{\varepsilon \to 0}{\lim} A(\varepsilon) = 2\int_0^{\infty} dy/f(y)$.