Solvability of nonlinear operator equations
W. J. Cramer, W. Harmon Ray · Pacific Journal of Mathematics · 1981
Let X and Y be Banach spaces, P: X-» Y a Gateaux differentiable operator having closed graph.Suppose (i) for each R>0 there is a δ>0 such that dPΛB(O;l))^B(O;δ) whenever (ii) P'^K) is bounded whenever el(K)QY is compact; then P is an open mapping of X onto Y. Similar results are obtained for compact Gateaux differentiable operators using a local version of (i); the same local version gives a domain invariance theorem for Gateaux differentiable operators having closed graph.Related results deal with M. Altman's theory of contractor directions and theory of normal solvability as developed by F. E. Browder and others.