By W W. Rouse 1850-1925 Ball

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Diff. Urav. Otkl. Argumentom, Vols. 1-8 (1962-1973) and the survey articles of Zverkin, Kamenskii, Norkin, and El'sgol'tz [1], Kamenskii, Norkin, and El'sogl'tz [1], and Mishkis and El'sgol'tz [1]. A very significant contribution to this question was made by Driver [2], who gave a formulation that has also been generalized by Melvin [1, 2]. 4). Suppose 1> is a given absolutely continuous function on [a-r, a]. 4) almost everywhere on [a, a+ A). Of course, in order for this initial-value problem to make sense, the function f must satisfy the following property: If x is any given absolutely continuous function on [a - r, a + A) and if F(t) = f(t,x(t),x(t- r),x(t- r)), a:::; t

The theory of this chapter will be generalized even more in subsequent pages with the emphasis being on qualitative and geometric properties of the solutions. 1). 1). The theory of this equation will not be developed in this book, but there is an excellent presentation in Chapters 12 and 13 of Bellman and Cooke [1]. 1) to have negative real parts. 8 Supplementary remarks 35 LPJ(t)e>-jt j where the Aj are the roots of the characteristic function and the PJ are polynomials. Some results on this question are contained in Chapter 7.

1) y(t) = Dq,f(t,xt(a,,f))yt. )). Proof. 3 that the solution x x(a, ¢,f) of the RFDE(f) through (a,¢) is unique. Let the maximal interval of existence of x be [a- r, a+ w) and fix b < w. Our first objective is to show that x(a, ¢, f)(t) is continuously differentiable with respect to¢ on [a- r, a+ b]. There is an open neighborhood U of¢ such that x(a, 'lj;, f)(t), 'lj; E U, is defined for t E [a- r, a+ b]. If W = {(t,xt) : t E [a, a+ b]}, then W is compact. 2. Choose a so that M a ~ j3 and ka < 1, where k is a bound of the derivative of f with respect to¢ on f.?.

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