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Singular perturbation analysis of boundary value problems for differential-difference equations. IV: A nonlinear example with layer behavior. (English) Zbl 0725.34064
[For parts I-III see SIAM J. Appl. Math. 42, 502-531 (1982; Zbl 0515.34058), ibid. 45, 687-707 (1985; Zbl 0623.34050), ibid. 45, 708-734 (1985; Zbl 0623.34051).] The authors study the model $\epsilon y''(x,\epsilon)+[y(x- 1,\epsilon)y(x,\epsilon)]'=f(x)$, $0<x<1$; $0<\epsilon \ll 1$; $y(x,\epsilon)=\phi (x)$ on -1$\le x\le 0$, and $y(1,\epsilon)=\gamma$. Integrating both sides of the equation and using singular perturbation techniques an exact solution for the constant case is constructed. Also in this case some discussion regarding the existence and uniqueness of the solution is presented. By a combination of singular perturbation analysis and numerical computations the authors give a comprehensive treatment of the full spectrum of solution behavior of the model with $\phi\ne 0$ on [-1,0].
Reviewer: H.S.Nur (Fresno)

34E15Asymptotic singular perturbations, general theory (ODE)
65L07Numerical investigation of stability of solutions of ODE
65L12Finite difference methods for ODE (numerical methods)
34D15Singular perturbations of ODE
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34K10Boundary value problems for functional-differential equations