Singularly perturbed nonlinear boundary value problems on infinite interval (Q1262442)

The author considers the B.V.P. \(\{ \epsilon y'' = f(x,y,y',\epsilon), y(-\infty,\epsilon) = \infty, y(\infty,\epsilon) = \beta \}\) with the assumption that the degenerate problem \(\{0 = f(x,y,y',0)\}\) has weakly discontinuous solution u, and \(u(-\infty)=\infty\), \(u(\infty)=\beta\). With the use of the usual Nagumo condition, and other inequalities related to differential equations, the author proves the existence and uniqueness of the solution to the problem and obtain the following asymptotic estimate \(| y(x,\epsilon)-u(x)| \leq c,\sqrt{\epsilon}e^{-\mu_ 0| x-x_ 0|}\) \(-\infty <x<\infty\).