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By Robert E. 1941- Gaines

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Liege 38 (1969) 308-398) where it is proved using Cesari's method discussed in Chapter IT. More direct proofs are given by Strygin (Math. Notes Acad. Sci. USSR 8 (1970) 600-602) and by Mawhin (Bull. Ac. R. Bel6ique~ CI. Sci. (5) 55(1969) 934-947 ; E _ ~ Diff 70, Marseille, 1970; J. Differential E~uations 10 (1971) 240-261) 35 for periodic solutions of ordinary and functional differential equations, Mawhin's proof being in the spirit of the one given here. The case of operator equations in Banach spaces is given in Mawhin, Rapp.

Let G = {(t,x,y): Suppose Ix I < R, IY] < h(x)}. (to, xo, Yo) E ~G and (to, xo) @ 3GI. Then [xol < R and IYoI = h(to). Define V(t,x,y) = y2 _ (h(x))2 . I are clearly satisfied. grad V • li Yo = f(t0 ,x0 ,Y0 = -2h(xo)h'(xo)yo 2h(x0)h'(x0 L2y0 We have ILi I " Y0 = f(t0 ,x0 ,Y0 ) + 2Ay0f(to,x0,y0) = 2y0[ly01 ~ + Xf(t0,xo,Yo)]. lyot By hypothesis is a Nagumoset (b) the latter quantity is nonzero for ~ E (0,I). ~) for ~ E (0,1). Define E {x : x E CI[0,T], (t,x(t), x'(t)) E O ~ g r t E [0,T]}. 10). Suppose x(t) e BO.

We use this approach to unify recent results of several authors. Specific bibliographical in- formation concerning these results will appear at the end of Part V. In Part VI, we use projection methods to obtain a theory of approximation for the solutions whose existence is established in Part V. The mate- rial presented here partially meets objective c) and is related to the book of Krasnosel'skii, Vainikko, Zabreiko, Rutitskii, Operator, Equations, (La Recherche de (Approximate Solution of 1972, Noordhoff) and the recent theses of Strasberg Solutions P~riodique d'Equations Diff~rentielles Non Lin@aires, Univ.