The aim of this chapter is to put together most of the frequently used auxiliary notions and results which are needed for a good understanding of the whole book. Therefore, we included here basic facts about Banach spaces, the Bochner integral and usual function spaces, compactness theorems including the infinite dimensional version of Arzelà-Ascoli Theorem, C 0-semigroups, m-dissipative operators and the nonlinear evolutions governed by them, m-dissipative possibly nonlinear partial differential operators, differential and integral inequalities.

Unlike Chapter 1, which was mainly concerned with a general background, our aim here is to gather some concepts and results which, although general, are essentially focused on the specific topic of the book. After proving the Brezis-Browder Ordering Principle, we discuss a basic lemma ensuring the existence of projections. Then, we introduce and study the concept of tangent set at a point to a given set and continue with an excursion to various types of tangent cones: Bouligand, Federer, Clarke and Bony. Further on, we state and prove some fundamental results on l.s.c. and u.s.c. multi-functions and add several technical results referring to measures of noncompactness. Finally, we prove some infinite variants and consequences of Scorza Dragoni type theorems.

The aim of this chapter is to introduce the reader to the fundamentals of the viability theory for ordinary differential equations in general Banach spaces. We notice that here we confine ourselves to consider only ordinary differential equations driven by continuous right-hand sides. After explaining what viability exactly means, we prove a necessary condition for viability expressed in the terms of the Nagumo Tangency Condition. We then pass to the statement of the main sufficient (in fact necessary and sufficient) conditions for viability. Next, we prove a technical lemma ensuring the existence of a sequence of approximate solutions and continue with the complete proofs of the sufficient conditions for viability. We show how to get viability in the nonautonomous case, by using the already established theory in the autonomous one. We conclude with several results concerning noncontinuable and global solutions.

Here we introduce the reader to the basic problems on (local) invariance referring mainly to ordinary differential equations governed by continuous functions. After some simple preliminary results, we focus our attention on two sufficient conditions for local invariance expressed in terms of certain comparison inequalities coupled either with the viability of the set in question or with the Nagumo Tangency Condition. We continue with the main general sufficient condition for local invariance based on the so-called Exterior Tangency Condition. In the specific case of proximal sets, we show that viability combined with a very general comparison condition implies invariance. Next, we answer the question: “when does tangency imply exterior tangency'” and we conclude with some results on the relationship between local invariance and monotonicity.

In this chapter we extend to the case of Carathéodory solutions the viability results in Chapter 3, referring to classical solutions. We notice that in this, a fortiori nonautonomous, case, due to some reasons explained below, we will confine ourselves to consider only cylindrical sets. After showing that an a.e. Nagumo-type tangency condition is necessary for Carathéodory viability, we state and prove that the very same a.e. Nagumo-type tangency condition is also sufficient under some natural Carathéodory-type extra-conditions combined with appropriate either compactness or Lipschitz conditions on the right hand-side. Finally, we focus our attention on the existence of noncontinuable or even global Carathéodory solutions.

The aim of this chapter is to present the main results on viability and invariance in the case of differential inclusions. We first introduce the notions of exact solution and almost exact solution and we study their relationship. We next consider the autonomous case and we prove some necessary and necessary and sufficient conditions for almost exact viability expressed in terms of the set-tangency concept introduced in Chapter 2. The nonautonomous case is reduced to the autonomous one by using the usual trick of introducing a new unknown function. Some problems concerning the continuation of (almost) exact solutions and the existence of global (almost) exact solutions are also considered. Finally, we establish a sufficient condition of invariance and a necessary condition in the specific case of a finite-dimensional problem.

In order to illustrate the effectiveness of the abstract developed theory, here we gather several applications. We first show that the viability of a set with respect to a function, defined on a larger open set, implies the viability of the relative closure of that set with respect to that function. We next deal with the viability of an epigraph and we prove a necessary condition in order that a given function be a comparison function. We study the existence problem of monotone solutions for both ordinary differential equations and inclusions. Next, by using viability and invariance arguments, we prove a variant of the well-known Banach Fixed Point Theorem. Further, taking advantage of the infinite-dimensional version of the Nagumo Viability Theorem, we deduce the existence of positive solutions for a pseudo-parabolic semilinear partial differential equation. We continue with the proofs of two well-known results in the classical theory of ordinary differential equations, i.e., Hukuhara and Kneser Theorems. We conclude with an application to the characteristics method for a class of first-order partial differential equations.

In this chapter we reconsider some problems already studied in the case of ordinary differential equations, within the more general frame of semilinear evolution equations governed by single-valued continuous perturbations of infinitesimal generators of C 0-semigroups. As in the previous case we just mentioned, we start with the autonomous case. So, after introducing the concept of mild viability and that one of A-tangent vector to a set at a given point, we prove a necessary condition for mild viability expressed in terms of a tangency condition which, whenever A ≡ 0, reduces to the Nagumo Tangency Condition. We next state and subsequently prove several necessary and sufficient conditions for mild viability. Further, we show how the quasi-autonomous case reduces to the autonomous one and we rephrase all the results already obtained in the autonomous case within this more general frame. We prove some necessary and sufficient conditions for mild viability in the specific case of a class of semilinear reaction-diffusion systems. We conclude the chapter with some results concerning the existence of noncontinuable, as well as of global mild solutions.

In this chapter we focus our attention on the case of semilinear evolution equations governed by multi-valued perturbations of infinitesimal generators of C 0-semigroups. We first consider the autonomous case and start with the definition of the A-quasi-tangent set at a point to a given set. Using this new concept, we next prove the main necessary condition for mild viability. Then, we state and prove several necessary and sufficient conditions for mild viability. We extend the previous results to the quasi autonomous case and we conclude with some facts concerning the existence of noncontinuable or even global mild solutions.

In this chapter we reconsider some problems, already touched upon in Chapter 8 in the semilinear case, within the (partly) more general frame of fully nonlinear evolution equations governed by continuous perturbations of infinitesimal generators of nonlinear semigroups of contractions. We begin with the definition of the C 0-viability and with the one of A-tangent vector at a point to a given set, in the case of an m-dissipative, possibly nonlinear operator A. We prove a necessary condition for C 0-viability expressed in terms of this tangency concept and we continue with the statements and proofs of several necessary and sufficient conditions for C 0-viability. We extend the results to the quasi-autonomous case and next, we focus our attention on the problem of the existence of C 0-noncontinuable or even global solutions. We then consider a class of fully nonlinear reaction-diffusion systems and we prove several necessary and sufficient conditions for C 0-viability. We conclude with some sufficient conditions for C 0-local invariance.

Here we extend our study in Chapter 10 to the more general case of nonlinear evolutions equations governed by multi-valued perturbations of m-dissipative operators. More precisely, we start with the autonomous case by defining the concept of A-quasi-tangent set at a point to a given set, in the case in which A is m-dissipative and possible nonlinear. Then, we prove the main necessary condition for C 0-viability expressed in terms of this new tangency concept. We next show that, under various natural extra-assumptions, the already established necessary condition is also sufficient. We extend the results to the quasi-autonomous case and we conclude with some results on the existence of noncontinuable or even global C 0-solutions.

In this chapter we reconsider some problems, already touched upon in Chapter 5 and Chapter 10. More precisely, here, we deal with the (partly) more general frame of fully nonlinear evolutions equations governed by single-valued Carathéodory perturbations of m-dissipative operators in separable Banach spaces. We begin by proving a necessary condition for C 0-viability and continue with the statements and proofs of several necessary and sufficient conditions for C 0-viability. Finally, we focus our attention on the problem of the existence of C 0-noncontinuable or even global solutions.

Here we collect several applications illustrating the effectiveness of the abstract developed theory. We begin with a sufficient condition for a set K, which is invariant with respect to the infinitesimal generator, A, of a C 0-semigroup, to be viable with respect to A + f with f : K → X continuous. From this, we deduce the existence of orthogonal solutions of a first-order system of partial differential equations of hyperbolic type. Further, we deduce a necessary and sufficient condition in order that a first-order partial differential equation of hyperbolic type have a unique solution taking values in a certain closed subset in ℝ. Next, using viability techniques, we show how to get necessary and sufficient conditions for a pair of functions to be a Lyapunov pair for a semilinear evolution equation. We notice that the existence of such a pair implies the asymptotic stability of the null solution of the semilinear evolution equation in question. We continue with several comparison results: for a semilinear diffusion equation, for a semilinear pray-predator system, for a nonlinear diffusion inclusion and for a fully nonlinear reaction-diffusion system. We next prove a null controllability result for a class of semilinear evolution equations, and we conclude with an existence result for periodic solutions to a fully nonlinear evolution equation.

In the present chapter, perturbation results for various linear operators in a Hilbert space are presented. These results are used in the next chapters to derive bounds for the spectral radiuses and stability conditions.