Topics on Stability and Periodicity in Abstract Differential Equations

The following sections are included:

• Preface

• Contents

The following sections are included:

• Banach Spaces and Linear Operators
  • Banach Spaces
  • Linear Operators
  • Spectral Theory of Linear (Closed) Operators

• Strongly Continuous Semigroups of Operators
  • Definition and Basic Properties
  • Compact Semigroups and Analytic Strongly Continuous Semigroups
  • Spectral Mapping Theorems
  • Commuting Operators

• Spectral Theory and Almost Periodicity of Functions
  • Introduction
  • Spectrum of a Bounded Function
  • Uniform Spectrum of a Bounded Function
  • Almost Periodic Functions
  • Sprectrum of an Almost Periodic Function
  • A Spectral Criterion for Almost Periodicity of a Function
  • Almost Automorphic Functions

The following sections are included:

• Perron Theorem

• Evolution Semigroups and Perron Theorem

• Stability Theory
  • Exponential Stability
  • Strong Stability

• Comments and Further Reading Guide
  • Further Reading Guide
  • Comments

The following sections are included:

• Evolution Semigroups and Almost Periodic Solutions of Periodic Equations
  • An Example
  • Evolution Semigroups
  • The Finite Dimensional Case
  • The Infinite Demensional Case
  • Almost Periodic Solutions and Applications
    • Invariant functions spaces of evolution semigroups
    • Monodromy operators
    • Unique solvability of the inhomogeneous equations in
    • Unique solvability in AP(𝕏) and exponential dichotomy
    • Unique solvability of the inhomogeneous equations in
    • Unique solvability of nonlinearly perturbed equations
    • Example 1
    • Example 2

• Evolution Semigroups and Sums of Commuting Operators
  • Invariant Function Spaces
  • Differential Operator and Notions of Admissibility
  • Admissibility for Abstract Ordinary Differential Equations
  • Higher Order Differential Equations
  • Abstract Functional Differential Equations
  • Examples and Applications

• Decomposition Theorem and Periodic, Almost Periodic Solutions
  • Spectral Decomposition
  • Spectral Criteria For Almost Periodic Solutions

• Comments and Further Reading Guide
  • Further Reading Guide
  • Comments

The following sections are included:

• The Inhomogeneous Linear Equation

• Method of Invariant Subspaces and Almost Automorphic Solutions of Second-Order Differential Equations

• Existence of Almost Automorphic Solutions to Semilinear Differential Equations

• Method of Sums of Commuting Operators and Almost Automorphic Functions

• Almost Automorphic Solutions of Second Order Evolution Equations
  • Mild Solutions of Inhomogeneous Second Order Equations
  • Operators
  • Nonlinear Equations

• The Equations x'=f(t,x)

• Comments and Further Reading Guide

The following sections are included:

• Periodic Solutions of Nonlinear equations
  • Nonlinear Equations Without Delay
  • Nonlinear Equations With Finite Delay
  • Nonlinear Equations With Infinite Delay
  • Non-Densely Defined Equations

• Evolution Semigroups and Almost Periodic Solutions
  • Evolution Semigroups
  • Almost periodic solutions

• Comments and Further Reading Guide
  • Further Reading Guide
  • Comments

The following sections are included:

• Appendices
  • Lipschitz Operators
  • Fixed Point Theorems
  • Invariant Subspaces
  • Semilinear Evolution Equations

• Bibliography

• Index