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The book is an almost self-contained presentation of the most
important concepts and results in viability and invariance. The
viability of a set K with respect to a given function (or
multi-function) F, defined on it, describes the property that, for
each initial data in K, the differential equation (or inclusion)
driven by that function or multi-function) to have at least one
solution. The invariance of a set K with respect to a function (or
multi-function) F, defined on a larger set D, is that property
which says that each solution of the differential equation (or
inclusion) driven by F and issuing in K remains in K, at least for
a short time.
The book includes the most important necessary and sufficient
conditions for viability starting with Nagumo's Viability Theorem
for ordinary differential equations with continuous right-hand
sides and continuing with the corresponding extensions either to
differential inclusions or to semilinear or even fully nonlinear
evolution equations, systems and inclusions. In the latter (i.e.
multi-valued) cases, the results (based on two completely new
tangency concepts), all due to the authors, are original and extend
significantly, in several directions, their well-known classical
counterparts.
- New concepts for multi-functions as the classical tangent vectors
for functions
- Provides the very general and necessary conditions for viability
in the case of differential inclusions, semilinear and fully
nonlinear evolution inclusions
- Clarifying examples, illustrations and numerous problems,
completely and carefully solved
- Illustrates the applications from theory into practice
- Very clear and elegant style
Filling a gap in the literature, Delay Differential Evolutions
Subjected to Nonlocal Initial Conditions reveals important results
on ordinary differential equations (ODEs) and partial differential
equations (PDEs). It presents very recent results relating to the
existence, boundedness, regularity, and asymptotic behavior of
global solutions for differential equations and inclusions, with or
without delay, subjected to nonlocal implicit initial conditions.
After preliminaries on nonlinear evolution equations governed by
dissipative operators, the book gives a thorough study of the
existence, uniqueness, and asymptotic behavior of global bounded
solutions for differential equations with delay and local initial
conditions. It then focuses on two important nonlocal cases:
autonomous and quasi-autonomous. The authors next discuss
sufficient conditions for the existence of almost periodic
solutions, describe evolution systems with delay and nonlocal
initial conditions, examine delay evolution inclusions, and extend
some results to the multivalued case of reaction-diffusion systems.
The book concludes with results on viability for nonlocal evolution
inclusions.
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