A Global Diffeomorphism Theorem for Fréchet Spaces

Abstract

We give sufficient conditions for a C 1 c -local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fréchet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.2010 Mathematics Subject Classification. 57R50;46A04;46A50 .

Some critical point results for Fréchet manifolds

“…We shall apply this theorem to generalize to Fréchet manifolds global diffeomorphism theorems for Fréchet spaces (see [1,Theorem 3.1] and [2,Theorem 4.1]). The proof is almost identical to the case of Fréchet spaces.…”

A multiplicity theorem for Fréchet spaces

“…As mentioned in the introduction, we will require the non-smooth analysis of locally Lipschitz mappings. In [2], the critical points theory for these mappings, generalizing the Clarke subdifferential, has been developed. Now, we will revisit what will be needed later on.…”

“…We denote by Lip loc p , Rq the set of locally Lipschitz functionals on . We will refer to the following definitions, which can be found in [2]. For ϕ P Lip loc p , Rq, the generalized directional derivative ϕ ˝px, yq at each x P in the direction y P is defined by ϕ ˝px, yq :" lim sup ) .…”

“…Motivated by these results and recent developments in critical point theory in Fréchet spaces ( [2,3,4]), this paper aims to derive global implicit function theorems, which involve no loss of derivative, applicable to arbitrary Fréchet space for mappings which are only continuously differentiable by employing methods of critical point theory.…”

“…[2], Theorem 3.2, Mountain Pass Theorem). Consider ϕ P Lip loc p , Rq and an open neighborhood U of x P .…”