Topological degree of pseudopositive mappings (Q1078501)

Let E and F be two real separable Banach spaces, and D be a nonvoid open subset of E. Let \(f: \bar D\to F\) be proper and continuous on the closure \(\bar D\) of D and twice continuously differentiable on D. Then f is said to be pseudopositive on \(\bar D\) if \(f'(x)\) is a Fredholm operator of index 0 and the dimension of the kernel of \(f'(x)\) is even for every x in D. Using the technique of \textit{M. W. Hirsch} [Proc. Am. Math. Soc. 14, 364-365 (1963; Zbl 0113.167)], we established a definition of topological degree for the pseudopositive mappings. The values of this topological degree are positive integers and might be used to estimate the numbers of solutions of equations involved by the pseudopositive mappings.