Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDEs

1. Fixed Point Index Theory. Multiple Fixed Point Theorems
1.1 Measures of Noncompactness
1.2 The Brower Fixed Point Theorem. The Schauder Fixed Point
Theorem
1.3 Fixed Points of Strict Set Contractions
1.4 The Kronecker Index
1.5 The Brower Degree
1.5.1 Smooth Mappings
1.5.2 Homotopy Invariance
1.5.3 Continuous Mappings. Basic Properties
1.5.4 The Case f : D _ Rn !Sn1
1.6 The Leray-Schauder Degree
1.7 The Fixed Point Index for Completely Continuous Mappings
1.8 The Fixed Point Index for Strict Set Contractions
1.9 Multiple Fixed Point Theorems
2. Applications to ODEs
2.1 Periodic Solutions for First Order ODEs
2.2 BVPs for First Order ODEs
2.3 BVPs for Second Order ODEs
2.4 BVPs with Impulses
3. Applications to FDEs
3.1 Global existence for a class fractional-differential equations
3.2 Multiple Solutions for a BVP of Nonlinear Riemann-Liouville
Fractional Differential Equations .
3.3 Multiple Solutions for a BVP of Nonlinear Caputo Fractional
Differential Equations
4. Applications to Parabolic Equations
4.1 Differentiability of the Classical Solutions with Respect to the
Initial Conditions of an IVP
4.2 Local Existence of Classical Solutions for an IBVP
4.3 Periodic Solutions
4.4 Multiple Solutions for an IBVP with Robin Boundary Conditions
5. Applications to Hyperbolic Equations
5.1 Differentiability of the Classical Solutions with Respect to the
Initial Conditions for an IVP for a Class Hyperbolic Equations
5.2 Multiple Solutions for an IBVP with Robin Boundary Conditions
5.3 Periodic Solutions
6. Applications to Elliptic Equations
6.1 Multiple Solutions for an BVP with Robin Boundary Conditions
6.2 Existence and Smoothness of Navier-Stokes Equations
References