Abstract
This paper concerns quasi-linear implicit differential equations of form 0=A/sub 1/(x)x/spl dot/-g/sub 1/(x), 0=g/sub 2/(x), where A/sub 1/: U/spl rarr/L(R/sup n/,R/sup n-m/)/spl isin/C/sup 1/, g/sub l/: U/spl rarr/R/sup n-m//spl isin/C/sup 1/, g/sub 2/: U/spl rarr/R/sup m//spl isin/C/sup 2/, U/spl sube/R/sup n/ is open, n, m/spl isin/N, and m<n. In particular, (1) is considered about impasse points x/sub 0//spl isin/U, i.e., points x/sub 0/ beyond which solutions are not continuable. Under appropriate assumptions, it is shown that there is a diffeomorphism that transforms solutions of the implicit differential equation (1) near such points into solutions of the normal form x/sub 1//sup r/x/spl dot//sub 1/=/spl sigma/, x/spl dot//sub 2/=0,...,x/spl dot//sub n-m/=0, x/sub n-m+1/=0,...,x/sub n/=0, near 0, and vice versa, where /spl sigma/=/spl plusmn/1=const. In particular, standard impasse points in the sense of RABIER and RHEINBOLDT lead to (2) with r=1. A practical example for r=2 is also given.
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