[For part I see ibid. 42, 502-531 (1982; Zbl 0515.34058).]
This paper is the second in a series of three papers on this subject. The first dealt with problems involving boundary and interior layer phenomena.
In this paper the authors concentrate on problems with solutions exhibiting rapid oscillations. In particular they study two types of resonance phenomena, ”global” and ”local” resonance. The authors point out the novel features that are present in D.D.E. which are not present in the problems without the difference terms. A modified WKB method together with matched asymptotic expansion is used to study the solutions of the system \[ \epsilon y''(x,\epsilon)+\phi (x)y(x,\epsilon)+\alpha (x)y'(x-1,\epsilon)+\beta (\alpha)y(x-1,\epsilon)=\psi (x), \] 0\(<x<\ell\), \(0<\epsilon \ll 1\), \(y(x,\epsilon)=\phi (x)\) for -1\(\leq x\leq 0\), \(y(\ell,\epsilon)=\gamma\). The authors split the interval (0,\(\ell)\) into two segments (0,1) and (1,\(\ell)\). Different methods are used to obtain the solutions on each of the two segments.