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Siimiiirry, \Ye cliarwterize the preconipact sets i n spaces of vector valued continuous fiinrtioiis iriicl iise tltc restilting criteria t o investigate asymptotic belmviour of stirh ftinctions defined on a Iialfline. This problem arose in the conte.xt of I) qiiuiitative stidy of soltitions to the abstract ('AUCHU problem. We give partic:itlar coiwirleratioii to the relationship !Jetween vector valued iwymptotically almost periodic functions on a wbinterwl [u, -) of the red line irnd precompactiiess of the set of its tmnslatcx. Our compactnes.. criteria arc also applied to a qwstion concerning the approximstion proi~~"uy for rpuiw of vertor valued continuoiw fiinrt ions with topologies inrfitced hy weighted matogties of the riipremiirn norm. ati well as to ohtnin nonlinear variants on factorization of rompact opraton through reftesive B.~-AcH spaces.Ilitrodnction. Evalution equatiom can be used to model a wide range of time dependent physical processes, and it is frequently the cam, at lesst in practice, th%t such a process can be represented by a (semi-) dynamicrrl system n: R + x ;3 Math. Sachr. 185 (1988) D 43 Essen 1 BRD Lkpartrnmi of Batheniatics Cniverereity o/ Arkansas Fayeitetdle, drknnsns 727701 c.x.4.

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