Extremality results proved in this Monograph for an abstract
operator equation provide the theoretical framework for developing
new methods that allow the treatment of a variety of discontinuous
initial and boundary value problems for both ordinary and partial
differential equations, in explicit and implicit forms. By means of
these extremality results, the authors prove the existence of
extremal solutions between appropriate upper and lower solutions of
first and second order discontinuous implicit and explicit ordinary
and functional differential equations. They then study the
dependence of these extremal solutions on the data. The authors
begin by developing an existence theory for an abstract operator
equation in ordered spaces and offer new tools for dealing with
different kinds of discontinuous implicit and explicit differential
equation problems. They present a unified approach to the existence
of extremal solutions of quasilinear elliptic and parabolic
problems and extend the upper and lower solution method to elliptic
and parabolic inclusion of hemivariation type using variational and
nonvariational methods. Nonlinear Differential Equations in Ordered
Spaces includes research that appears for the first time in book
form and is designed as a source book for pure and applied
mathematicians. Its self-contained presentation along with numerous
worked examples and complete, detailed proofs also make it
accessible to researchers in engineering as well as advanced
students in these fields.
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