Topological Nonlinear Analysis: Degree, Singularity, and Variations - Softcover

Inhaltsangabe

Variational Methods and Nonlinear Problems: Classical Results and Recent Advances.- • Introduction.- • Lusternik-Schnirelman Theory.- • Applications to Nonlinear Eigenvalues.- • Unbounded Functionals.- • Elliptic Dirichlet Problems.- • Singular Potentials.- • References.- to Morse Theory: A New Approach.- • Introduction.- • Contents.- • The Abstract Theory.- • The Morse Index.- • The Poincaré Polynomial.- • The Conley Blocks.- • The Morse Relations.- • Morse Theory for Degenerate Critical Points.- • Some Existence Theorems.- • An Application to Riemannian Geometry.- • Riemannian Manifolds.- • Geodesies.- • The Morse Theory for Geodesics.- • The Index Theorem.- • An Application to Space-Time Geometry.- • Introduction.- • Some Examples of Lorentzian Manifolds.- • Morse Theory for Lorentzian Manifolds.- • Preliminary Lemmas.- • Proof of The Morse Relations For Static Space-Time.- • Some Application to a Semilinear Elliptic Equation.- • Introduction.- • The Sublinear Case.- • The Superlinear Case Morse Relations for Positive Solutions.- • The Functional Setting.- • Some Hard Analysis.- • The Photography Method.- • The Topology of The Strip.- • References.- Applications of Singularity Theory to the Solutions of Nonlinear Equations.- • The Full Lyapunov-Schmidt Reduction.- • Mather's Theory of C?-Stability of Mappings - Global Theory.- • Mather's Local Theory as Paradigm.- • Singularity Theory with Special Conditions.- • The Structure of Nonlinear Fredholm Operators.- • Multiplicities of Solutions to Nonlinear Equations.- • The Theory for Topological Equivalence.- • Bibliography.- Fixed Point Index Calculations and Applications.- • The Fixed Point Index.- • Some Remarks on Convex Sets.- • A Basic Index Calculation.- • Index Calculations in Product Cones.- • Applications of Index Formulae - I.- • Applications of Index Formulae - II.- • Some Global Branches.- • Monotone Dynamical Systems.- • Preliminaries.- • Connecting Orbits and Related Results.- • Generic Convergence.- • References.- Topological Bifurcation.- • Abstract.- • Introduction.- • Preliminaries.- • One Parameter Bifurcation.- • Local Bifurcation.- • Global Bifurcation.- • Special Nonlinearities.- • Multiparameter Bifurcation.- • Sufficient Conditions for Local Bifurcation.- • Necessary Conditions for Linearized Local Bifurcation.- • Multiparameter Global Bifurcation.- • A Summation Formula and A Generalized Degree.- • Structure and Dimension of Global Branches.- • O-EPI Maps.- • Dimension.- • Application to Bifurcation Problems.- • Equivariant Bifurcation.- • Preliminaries.- • Consequences of the Symmetry.- • ?-EPI Maps.- • ?-Degree.- • The Equivariant J-Homomorphism and Sufficient Conditions.- • Necessary and Sufficient Conditions for Equivariant Bifurcation.- • Bibliography.- Critical Point Theory.- • Introduction.- • The Mountain Pass Theorem.- • The Saddle Point Theorem.- • Linking and A General Critical Point Theorem.- • Periodic Solutions of Hamiltonian Systems.- • Introduction.- • The Technical Framework.- • Periodic Solutions of Prescribed Energy.- • Periodic Solutions of Prescribed Period.- • Connecting Orbits.- • Introduction.- • Homoclinic Solutions.- • Heteroclinic Solutions.- • References.- Symplectic Topology: An Introduction.- • The Classical Uncertainty Principle, Symplectic Rigidity.- • Construction of Symplectic Invariants.- • Generating Functions.- • Historical Remarks.- • Appendix: Rigidity for Finite Dimensional Lie Groups.

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