Abstract

The Morse Theory of critical points was extended by Palais and Smale to a certain class of functions on Hilbert manifolds. However, there are many variational problems in a nonlinear setting which for technical reasons are posed not on Hilbert but on Banach manifolds of mappings. The difficulty in extending the previous methods is in giving a proper definition of a nondegenerate critical point and in showing the existence of a vector field which is transverse to a handle near a critical point. In Hilbert manifolds Riesz representation theorem may guarantee the existence of a gradient vector field and give a suitable definition for nondegeneracy such that the gradient vector field is transverse to a handle near a critical point. But this does not work in a Banach manifold setting.

An extension has already been given by Tromba in 1977 by assuming the existence of a gradient-like vector field near critical points. But the existence of a gradient-like vector field gives as inner product structure on the model space.

In this paper we introduce a concept of a multi-valued gradient vector field for a function defined on a Banach manifold. Using this concept we generalize the Morse theory to some kind of Banach manifolds.

This paper contains three parts. The first chapter gives a definition of nondegeneracy of critical points for a real valued function defined on a reflexive Banach manifold, and then we are able to obtain a handle-body decomposition theorem and Morse inequalities for this manifold. The second chapter proves the existence of solutions for a differential inclusion for a so-called accretive multi-valued mapping on a Finsler manifold. So we can in the third chapter introduce a definition of nondegeneracy of critical points for a real valued function defined on a general Banach manifold, and furthermore, generalize the Morse handle-body decomposition theorem and the Morse inequalities to the Banach manifold.