Impulsive differential inclusions. A fixed point approach (Q2841474)

The monograph is devoted to impulsive differential equations and inclusions. Its content is mainly based on results obtained by the authors during the last 10 years. The selection of the topics reflects the particular interests of the authors. The proofs mostly use some single-valued or multi-valued version of a fixed point theorem or a suitable version of a nonlinear alternative. The presented theoretical results are illustrated by examples.NEWLINENEWLINEChapter 1 describes some common models in which impulsive constructions arise. The authors show the Kruger-Thiemer model dealing with adjusting the distribution of medicines absorbed orally in the gastro-intestinal system of the human body, the Lotka-Volterra model for population growth, the pulse vaccination model and they list references for other motivation models in Economics and Biomathematics.NEWLINENEWLINEChapter 2 presents notations, definitions, lemmas and theorems needed in the next chapters. These include fixed point theorems, properties of set-valued mappings, measure of noncompactness results, semigroups and extrapolation spaces.NEWLINENEWLINEChapter 3 deals with functional differential equations with infinite delay. In the first section of this chapter the authors establish sufficient conditions for the existence and uniqueness of solutions to an initial value problem for first order infinite delay differential equation with impulses at fixed moments \(0<t_1<\dots<t_m<b\). The problem has the form NEWLINE\[NEWLINE y'(t)=f(t,y_t), \quad a.e.\;t\in [0,b], \eqno(1) NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t_k+)-y(t_k)=I_k(y(t_k)),\;k=1,\dots, m, \eqno (2) NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t)=\phi(t),\quad t\in (-\infty,0], \eqno(3) NEWLINE\]NEWLINE where \(f: [0,b]\times B\to\mathbb R^n\) is a Carathéodory function, the impulse functions \(I_k:\mathbb R^n\to\mathbb R^n\) are continuous, \(\phi\in B\) and \(B\) is a suitable phase space. In the second section of Chapter 3 the authors study problems of the type (1)--(3), but now the differential equation has multiple delays, that is NEWLINE\[NEWLINE y'(t)=f(t,y_t)+\sum_{i=1}^{n_*} y(t-T_i), \eqno (4) NEWLINE\]NEWLINE where \(n_*\) is a positive integer. Problem (4), (2), (3) is first investigated on the compact \([0,b]\) and then local existence and uniqueness results are extended to the interval \([0,\infty)\) with infinitely many impulse points. In addition, stability results are proved. The last section of Chapter 3 is concerned with the following second order infinite delay impulsive initial value problem NEWLINE\[NEWLINE y''(t)=f(t,y_t), \quad a.e.\;t\in [0,b], \eqno(5) NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t_k+)-y(t_k-)=I_k(y(t_k-)),\;y'(t_k+)-y'(t_k-)=\bar I_k(y(t_k-)),\;k=1,\dots, m, \eqno (6) NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t)=\phi(t),\;y'(0)=\eta \in \mathbb R^n, \quad t\in (-\infty,0]. \eqno(7) NEWLINE\]NEWLINE Here, local existence and uniqueness results as well as global existence and uniqueness and stability results are presented.NEWLINENEWLINEChapter 4 gives theorems about existence and uniqueness of solutions of the boundary value problem NEWLINE\[NEWLINE y'(t)=f(t,y_t), \quad a.e.\;t\in [0,\infty), \eqno (8) NEWLINE\]NEWLINE NEWLINE\[NEWLINE y(t_k+)-y(t_k-)=I_k(y_k(t_k-)),\;k=1,\dots, \eqno (9) NEWLINE\]NEWLINE NEWLINE\[NEWLINEAy(t)-\lim_{t\to \infty}y(t)=\phi(t),\quad t\in (-\infty,0],\;A>1. \eqno(10) NEWLINE\]NEWLINE The main object of Chapter 5 is to prove a Filippov type theorem and a Filippov-Wazewski type theorem for impulsive differential inclusions \(y'(t)\in F(t,y(t))\) and impulsive functional differential inclusions NEWLINE\[NEWLINE y'(t)\in F(t,y_t) \eqno (11) NEWLINE\]NEWLINE on a bounded interval. Here, \(F\) is a multivalued mapping. The authors study existence of solutions, compactness of solution sets, relaxation problems or upper semicontinuity without convexity. In addition, impulsive differential inclusions are also considered on noncompact intervals.NEWLINENEWLINEChapter 6 provides existence results for problems (11), (2), (3) and (11), (9), (10). The convex and nonconvex cases are discussed.NEWLINENEWLINEChapter 7 generalizes the theory of some of the previous chapters to initial value problems of functional differential equations and functional differential inclusions under impulses for which the impulse effects vary with time. Such impulses, called state-dependent impulses, can be written as NEWLINE\[NEWLINE y(t+)=I_k(y(t)),\quad t=\tau_k(y(t)),\;k=1,\dots, m, NEWLINE\]NEWLINE where \(\tau_k:\mathbb R^n\to\mathbb R\). Chapter 7 is completed by results for impulsive neutral functional differential equations with infinite delay.NEWLINENEWLINEChapter 8 extends the results of Chapter 5 which are devoted to a Filippov type theorem and to a relaxed problem in the context of neutral differential inclusions.NEWLINENEWLINEChapter 9 deals with topological and geometric properties of solution sets for impulsive differential equations and inclusions for periodic and terminal problems. The authors provide Aronszajn type results and consider contractible sets, \(\mathbb R_\delta\)-sets, proximate retracts and viable solutions.NEWLINENEWLINE Chapter 10 is concerned with first and second order impulsive semilinear functional differential inclusions in Banach spaces. Questions addressed include exact controllability and controllability in extrapolation spaces.NEWLINENEWLINEChapter 11 considers impulsive stochastic differential equations, impulsive sweeping processes and integral inclusions of Volterra type in Banach spaces.NEWLINENEWLINEThe text is finished by an appendix and an extensive bibliography.