Abstract
In Chapter 30 we considered first-order evolution equations of the form
, with the operators A(t): V → V* and b(t) ∈ V* for all t ∈ ]0,T[. In this connection, “V ⊆ H ⊆ V*” is an evolution triple.
The analytical theory of semigroups is a recent addition to the ever-growing list of mathematical disciplines .... I hail a semigroup when I see one and I seem to see them everywhere! Friends have observed, however, that there are mathematical objects which are not semigroups.
Einar Hille (1948)
The importance of the class of nonexpansive mappings lies neither in its trivial generalization of a Lipschitz condition, nor in a comparable durability or fruitfulness, but in two key observations: first, nonexpansive mappings are intimately tied to the monotonicity methods developed since the early 1960’s, and constitute one of the first classes of mappings for which fixed-point results were obtained by using the fine geometric structure of the underlying Banach space instead of compactness properties. Second, they appear in applications as the shift operator for initial value problems of differential inclusions of the form where the operators A(t) are in general multivalued, and in some sense positive (accretive) and only minimally continuous.
Ronald Bruck (1983)
The condition of maximal accretivity is enjoyed by many important operators in applications.
Michael Crandall (1986)
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Zeidler, E. (1990). Maximal Accretive Operators, Nonlinear Nonexpansive Semigroups, and First-Order Evolution Equations. In: Nonlinear Functional Analysis and its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0981-2_7
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DOI: https://doi.org/10.1007/978-1-4612-0981-2_7
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