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Singular perturbation analysis of boundary-value problems for differential-difference equations. V: Small shifts with layer behavior. (English) Zbl 0796.34049

The authors study the two systems (1) $\epsilon {y}^{\text{'}\text{'}}\left(x;\epsilon \right)+a\left(x\right){y}^{\text{'}}\left(x-\delta \left(\epsilon \right);\epsilon \right)+b\left(x\right)y\left(x;\epsilon \right)=f\left(x\right)$, on $0, $0<\epsilon \ll 1$, and $0\le \delta \left(\epsilon \right)\ll 1$, subject to the interval and boundary conditions $y\left(x;\epsilon \right)=\varphi \left(x\right)$ on $-\delta \left(\epsilon \right)\le x\le 0$, $y\left(1;\epsilon \right)=\gamma$, respectively, where $a\left(x\right)$, $b\left(x\right)$, $f\left(x\right)$, $\delta \left(\epsilon \right)$, and $\varphi \left(x\right)$ are smooth functions and $\gamma$ is a constant, and (2) ${\epsilon }^{2}{y}^{\text{'}\text{'}}\left(x;\epsilon \right)+\alpha \left(x\right)y\left(x-\delta \left(\epsilon \right);\epsilon \right)+\omega \left(x\right)y\left(x;\epsilon \right)+\beta \left(x\right)y\left(x+\eta \left(\epsilon \right);\epsilon \right)=f\left(x\right)$, (note the coefficient of ${y}^{\text{'}\text{'}}$ is ${\epsilon }^{2}$ and not $\epsilon$ as in (1)) on $0, $0<\epsilon \ll 1$, $0\le \delta \left(\epsilon \right)\ll 1$, and $0\le \eta \left(\epsilon \right)\ll 1$, subject to the interval conditions $y\left(x;\epsilon \right)=\varphi \left(x\right)$ on $-\delta \left(\epsilon \right)\le x\le 0$, $y\left(x;\epsilon \right)=\psi \left(x\right)$ on $1\le x\le 1+\eta \left(\epsilon \right)$, where $\alpha \left(x\right)$, $\omega \left(x\right)$, $\beta \left(x\right)$, $f\left(x\right)$, $\delta \left(\epsilon \right)$, $\eta \left(\epsilon \right)$, $\varphi \left(x\right)$, $\psi \left(x\right)$ are smooth functions. They examine the solutions when the shifts are not zero and determine when the shifts can be ignored to leading order and what their sizes are when they begin to influence the qualitative features of the solutions. Also they study layer behavior using Laplace transform which in turn requires values of the roots of several exponential polynomials. Some details of that is given in the paper.

[For part IV, see Stud. Appl. Math. 84, No. 3, 231-273 (1991; Zbl 0725.34064), for part VI, see the review below (Zbl 0796.34050)].

Reviewer: H.S.Nur (Fresno)

##### MSC:
 34K10 Boundary value problems for functional-differential equations 34K25 Asymptotic theory of functional-differential equations 44A10 Laplace transform 30C15 Zeros of polynomials, etc. (one complex variable) 92C20 Neural biology