Existence and Comparison Results for Differential Equations of Sobolev Type
Abstract
**Please note that the full text is embargoed** ABSTRACT: Recently a new class of differential equations, called
differential equations of Sobolev type was studied in [4]
in which an existence theorem of Picards type was investigated
as well as a variation of constants formula. In [5], existence
and comparison results for a class of Volterra integral equation
of Sobolev-type were discussed. In this paper we recall the Peano
type existence result from [5] for Sobolev-differential equations
and show that solutions can be extended to the entire square
under consideration. This result extends results found in [1] for
nonlinear Volterra integral equations. Our results include a
comparison result in addition to the usual type of differential
inequalities, and a differential inequality theorem such as
Müller's [6]. This in turn proves the existence of extremal
solutions. For special cases of the above results see [2,3,7]