The authors study the two systems (1) , on , , and , subject to the interval and boundary conditions on , , respectively, where , , , , and are smooth functions and is a constant, and (2) , (note the coefficient of is and not as in (1)) on , , , and , subject to the interval conditions on , on , where , , , , , , , 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)].