Impulsive Differential Inclusions
Description
Differential equations with impulses arise as models of many evolving processes that are subject to abrupt changes, such as shocks, harvesting, and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume these perturbations act instantaneously or in the form of impulses. As a consequence, impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. There are also many different studies in biology and medicine for which impulsive differential equations provide good models.
During the last 10 years, the authors have been responsible for extensive contributions to the literature on impulsive differential inclusions via fixed point methods. This book is motivated by that research as the authors endeavor to bring under one cover much of those results along with results by other researchers either affecting or affected by the authors' work. The questions of existence and stability of solutions for different classes of initial value problems for impulsive differential equations and inclusions with fixed and variable moments are considered in detail. Attention is also given to boundary value problems. In addition, since differential equations can be viewed as special cases of differential inclusions, significant attention is also given to relative questions concerning differential equations. This monograph addresses a variety of side issues that arise from its simpler beginnings as well.
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Content
- Intro
- Notations
- 1 Introduction and Motivations
- 1.1 Introduction
- 1.2 Motivational Models
- 1.2.1 Kruger-Thiemer Model
- 1.2.2 Lotka-Volterra Model
- 1.2.3 Pulse Vaccination Model
- 1.2.4 Management Model
- 1.2.5 Some Examples in Economics and Biomathematics
- 2 Preliminaries
- 2.1 Some Definitions
- 2.2 Some Properties in Fréchet Spaces
- 2.3 Some Properties of Set-valued Maps
- 2.3.1 Hausdorff Metric Topology
- 2.3.2 Vietoris Topology
- 2.3.3 Continuity Concepts and Their Relations
- 2.3.4 Selection Functions and Selection Theorems
- 2.3.5 Hausdorff Continuity
- 2.3.6 Measurable Multifunctions
- 2.3.7 Decomposable Selection
- 2.4 Fixed Point Theorems
- 2.5 Measures of Noncompactness: MNC
- 2.6 Semigroups
- 2.6.1 C0-semigroups
- 2.6.2 Integrated Semigroups
- 2.6.3 Examples
- 2.7 Extrapolation Spaces
- 3 FDEs with Infinite Delay
- 3.1 First Order FDEs
- 3.1.1 Examples of Phase Spaces
- 3.1.2 Existence and Uniqueness on Compact Intervals
- 3.1.3 An Example
- 3.2 FDEs with Multiple Delays
- 3.2.1 Existence and Uniqueness Result on a Compact Interval
- 3.2.2 Global Existence and Uniqueness Result
- 3.3 Stability
- 3.3.1 Stability Result
- 3.4 Second Order Impulsive FDEs
- 3.4.1 Existence and Uniqueness Results
- 3.5 Global Existence and Uniqueness Result
- 3.5.1 Uniqueness Result
- 3.5.2 Example
- 3.5.3 Stability
- 4 Boundary Value Problems on Infinite Intervals
- 4.1 Introduction
- 4.1.1 Existence Result
- 4.1.2 Uniqueness Result
- 4.1.3 Example
- 5 Differential Inclusions
- 5.1 Introduction
- 5.1.1 Filippov's Theorem
- 5.1.2 Relaxation Theorem
- 5.2 Functional Differential Inclusions
- 5.2.1 Filippov's Theorem for FDIs
- 5.2.2 Some Properties of Solution Sets
- 5.3 Upper Semicontinuity without Convexity
- 5.3.1 Nonconvex Theorem and Upper Semicontinuity
- 5.3.2 An Application
- 5.4 Inclusions with Dissipative Right Hand Side
- 5.4.1 Existence and Uniqueness Result
- 5.5 Directionally Continuous Selection and IDIs
- 5.5.1 Directional Continuity
- 6 Differential Inclusions with Infinite Delay
- 6.1 Existence Results
- 6.2 Boundary Differential Inclusions
- 7 Impulsive FDEs with Variable Times
- 7.1 Introduction
- 7.1.1 Existence Results
- 7.1.2 Neutral Functional Differential Equations
- 7.2 Impulsive Hyperbolic Differential Inclusions with Infinite Delay
- 7.3 Existence Results
- 7.3.1 Phase Spaces
- 7.3.2 The Nonconvex Case
- 8 Neutral Differential Inclusions
- 8.1 Filippov's Theorem
- 8.2 The Relaxed Problem
- 8.2.1 Existence and Compactness Result: an MNC Approach
- 9 Topology and Geometry of Solution Sets
- 9.1 Background in Geometric Topology
- 9.2 Aronszajn Type Results
- 9.2.1 Solution Sets for Impulsive Differential Equations
- 9.3 Solution Sets of Differential Inclusions
- 9.4 s-selectionable Multivalued Maps
- 9.4.1 Contractible and Rd -contractible
- 9.4.2 Rd-sets
- 9.5 Impulsive DIs on Proximate Retracts
- 9.5.1 Viable Solution
- 9.6 Periodic Problems
- 9.6.1 Poincaré Translation Operator
- 9.6.2 Existence Result
- 9.7 Solution Set for Nonconvex Case
- 9.7.1 Continuous Selection and AR of Solution Sets
- 9.8 The Terminal Problem
- 9.8.1 Existence and Solution Set
- 10 Impulsive Semilinear Differential Inclusions
- 10.1 Nondensely Defined Operators
- 10.2 Integral Solutions
- 10.3 Exact Controllability
- 10.3.1 Controllability of Impulsive FDIs
- 10.3.2 Controllability of Impulsive Neutral FDIs
- 10.4 Controllability in Extrapolation Spaces
- 10.5 Second Order Impulsive Semilinear FDIs
- 10.5.1 Mild Solutions
- 10.5.2 Filippov's Theorem
- 10.5.3 Filippov-Wazewski's Theorem
- 11 Selected Topics
- 11.1 Stochastic Differential Equations
- 11.1.1 Itô Integral
- 11.1.2 Definition of a Mild Solution
- 11.1.3 Existence and Uniqueness
- 11.1.4 Global Existence and Uniqueness
- 11.2 Impulsive Sweeping Processes
- 11.2.1 Preliminaries in Nonsmooth Analysis
- 11.2.2 Uniqueness Result
- 11.3 Integral Inclusions of Volterra Type in Banach Spaces
- 11.3.1 Resolvent Family
- 11.3.2 Existence results
- 11.3.3 The Convex Case: an MNC Approach
- 11.3.4 The Nonconvex Case
- 11.4 Filippov's Theorem
- 11.4.1 Filippov's Theorem on a Bounded Interval
- 11.5 The Relaxed Problem
- Appendix
- A.1 CM ech Homology Functor with Compact Carriers
- A.2 The Bochner Integral
- A.3 Absolutely Continuous Functions
- A.4 Compactness Criteria in C([a,b]), Cb([0, 8), E), and PC([a,b],E)
- A.5 Weak-compactness in L1
- A.6 Proper Maps and Vector Fields
- A.7 Fundamental Theorems in Functional Analysis
- Bibliography
- Index
