Existence and Comparison Results for Differential Equations of Sobolev Type
Vatsala, A. S.
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Recently a new class of differential equations, called differential equations of Sobolev type was studied in  in which an existence theorem of Picards type was investigated as well as a variation of constants formula. In , 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  for Sobolev-differential equations and show that solutions can be extended to the entire square under consideration. This result extends results found in  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 . This in turn proves the existence of extremal solutions. For special cases of the above results see [2,3,7]