On Coddington and Levinson's results for a nonlinear boundary value problem involving a small parameter

Chang, K. W.

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 54 (1973) no. 4, pp. 536-543

Résumé

Supposto $f,g \in C^{1}$, $f \ge \mu > 0$, $f(t,y)y' + g(t,y) = 0$ abbia una soluzione $y = \bar{y}(t)$ tale che $\bar{y}(1) = \beta$, Coddington e Levinson hanno dimostrato che per $\epsilon$ sufficientemente piccolo, il sistema non lineare $\epsilon y'' + f(t,y)y' + g(t,y) = 0$, $y(0) = \alpha$, $y(1) = \beta$ ha una soluzione $y = y(t,\epsilon)$ in $[0,1]$ e inoltre $y(t,\epsilon) \to \bar{y}(t)$, $y'(t,\epsilon) \to \bar{y}'(t)$ per $0 \delta \le t \le 1$. Con un nuovo metodo si dimostra che se $f,g \in C^{2}$ allora $y(t,\epsilon) = \bar{y}(t) + 0(\epsilon) + 0(e^{-\mu t/\epsilon})$, $y'(t,\epsilon) = \bar{y}'(t) + 0(\epsilon) + 0(\epsilon^{-1}e^{-\mu t/\epsilon})$ per $0 \le t \le 1$

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