We prove an infinite-dimensional version of Sard's theorem for Fréchet manifolds. Let M (respectively, N ) be a bounded Fréchet manifold with compatible metric d M (respectively, d N ) modeled on Fréchet spaces E (respectively, F ) with standard metrics. Let f W M ! N be an M C k -Lipschitz-Fredholm map with k > maxfInd f; 0g: Then the set of regular values of f is residual in N:
In this paper, we develop the geometry of bounded Fréchet manifolds. We prove that a bounded Fréchet tangent bundle admits a vector bundle structure. But, the second order tangent bundle T 2 M of a bounded Fréchet manifold M becomes a vector bundle over M if and only if M is endowed with a linear connection. As an application, we prove the existence and uniqueness of an integral curve of a vector field on M . 2010 AMS Mathematics subject classification. Primary 58A05, 58B25, Secondary 37C10.Keywords and phrases. Bounded Fréchet manifold, second order tangent bundle, connection, vector field.
We provide sufficient conditions for existence of a global diffeomorphism between tame Fréchet spaces. We prove a version of the mountain pass theorem which plays a key ingredient in the proof of the main theorem.
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 .
We prove a Lusternik-Schnirelmann type theorem for a <em>C</em><sup>1</sup>- function φ : M → R, where M is a connected infinite dimensional Frechet manifold of class <em>C</em><sup>1</sup>. To this end, in this context we prove the so-called Deformation Lemma and by using it we derive the result generalizing the Minimax Principle.
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