Book contents
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- 5 The Finite Dimensional MPT
- 6 The Topological MPT
- 7 The Classical MPT
- 8 The Multidimensional MPT
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
6 - The Topological MPT
Published online by Cambridge University Press: 04 September 2009
- Frontmatter
- Contents
- Introduction
- 1 Retrospective
- I First Steps Toward the Mountains
- II Reaching the Mountain Pass Through Easy Climbs
- 5 The Finite Dimensional MPT
- 6 The Topological MPT
- 7 The Classical MPT
- 8 The Multidimensional MPT
- III A Deeper Insight in Mountains Topology
- IV The Landscape Becoming Less Smooth
- V Speculating about the Mountain Pass Geometry
- VI Technical Climbs
- A Background Material
- Bibliography
- Index
Summary
It is with the help of the methods of topology, that we shall seek answers to the fundamental question connected with the study of nonlinear equations. However, this last assertion does not imply a negative role for other methods of investigations. In fact, topological methods become powerful only by virtue of their combination with other approaches.
M.A Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, Pergamon Press, 1964.This chapter is devoted to a pure topological version of the MPT due to Katriel [516]. It constitutes a natural “upgrade” of the finite dimensional MPT presented in the former chapter to locally compact topological spaces. It will also help us to clarify more our vision of the situation and will make us see the MPT under another angle. In fact, the topological considerations constitute an inherent part of the different faces of the MPT as will be clarified later. This is the case for all variational analysis results.
The MPT we will see is topological, in the sense that no differential structure on X is needed or used. So, we do not get critical points since this notion does not (yet) make any sense in such spaces. But we get some particular points from a pure topological point of view that should be critical in the presence of a differential structure.
- Type
- Chapter
- Information
- The Mountain Pass TheoremVariants, Generalizations and Some Applications, pp. 57 - 64Publisher: Cambridge University PressPrint publication year: 2003