Topological degree and A-proper operators (Q1089606)

Generalized degree theory for the class of A-proper mappings is described and used to state some results on bifurcation theory. One result shows that an idea of \textit{J. F. Toland} [Quart. J. Math. Oxford, II. Ser. 24, 241-250 (1973; Zbl 0256.47049)] extends to the A-proper setting and allows one to obtain asymptotic bifurcation results from results on bifurcation from zero. Some applications of the bifurcation results to non-semilinear problems are indicated. Particular attention is paid to the existence of periodic solutions of second-order ordinary differential equations of the form \(x''=g(x,x',x'')\), which cannot be ''solved'' for the highest order derivative. Several sets of possible hypotheses used to prove the A- proper property are discussed. Some of these are shown to allow the problem to be reduced to one of the form \(x''=f(x,x')\) involving compact maps. More general hypotheses are given which do not apparantly allow such a reduction but fit the A-proper theory.