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 $0 \le t \le 1$

Bibliographie

[1] Coddington E. A. and Levinson N., A boundary value problem for a nonlinear differential equation with a small parameter, «Proc. Amer. Math. Soc.», 3, 73-81 (1952). | DOI | MR | Zbl

[2] Erdélyi A., On a nonlinear boundary value problem involving a small parameter, «J. Austral. Math. Soc.», 2, 425-439 (1962).

[3] Erdélyi A., Approximate solutions of a nonlinear boundary value problem, «Arch. Rational Mech. Anal.», 29, 1-17 (1968). | DOI | MR | Zbl

[4] O'Malley R. E., On a boundary value problem for a nonlinear differential equation with a small parameter, «SIAM J. Appl. Math.», 17, 569-581 (1969). | DOI | MR

[5] Wasow W., Singular perturbations of boundary value problems for nonlinear differential equations of the second order, «Comm. Pure Appl. Math.», 9, 95-113 (1956). | DOI | MR | Zbl

[6] Willet D., On a nonlinear boundary value problem with a small parameter multiplying the highest derivative, «Arch. Rational Mech. Anal.», 23, 276-287 (1966). | DOI | MR