Range of Gateaux differentiable operators and local expansions

Abstract

for y e Y, a considerable number of which involve local or infinitesimal assumptions on the operator P, by showing that P is surjective. However, in many cases, in general P need not be surjective, although for some y G y, the equation Px = y is solvable. For example, let P be a Gateaux differentiable operator having closed graph such that for each x G X,

Homeomorphisms and monotone vector fields

“…This result (and different variations of it) is widely used to prove existence and uniqueness theorems for operator equations, partial differential equations and variational inequalities (see [19]). Surprisingly, in the finite dimensional case this result boils down just to the continuity and expansivity of A + I, beeing a particular case (it is not trivial to show) of a classical homeomorphism theorem of Browder [4, Theorem 4.10] (connected to this subject see also [1]- [3], [6], [13]- [15].) We shall generalize this result for a complete connected Riemannian manifold M .…”