The author considers the following problem (*) \(\varepsilon y''+b(x,y)y'- c(x,y)=0\) for \(x\in[-1,1]\), \(y(-1)=A\), \(y(1)=B\) where \(0<\varepsilon\ll 1\) is the perturbation parameter, \(b\) and \(c\in C^ 2([- 1,1]\times{\mathbb{R}})\), \(b(0,y)=0\), \(b_ x(0,y)<0\) for \(| y|\leq r+1\), \(b(x,y)\neq 0\) for \(| y|\leq r+1\) and \(x\neq 0\), \(c_ y(x,y)\geq c_ 0>0\) on \([-1,1]\times{\mathbb{R}}\) where \(r\) is a finite number and \(c_ 0\) a constant. Using the maximum principle it is shown that the solution of (*) is bounded.
Then the problem is reduced to a nonlinear initial value problem for two differential equations of the first order. This initial value problem is numerically solved by a special difference scheme and it is shown that the approximate solution converges to a solution of (*) in the maximum norm for arbitrary \(\varepsilon>0\). A numerical example illustrates the theoretical results.