Maximal and Minimal Solutions and Comparison Results for Differential Equations in Abstract Cones
Mitchell, Roger W.
Mitchell, A. Richard
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As is well known, an important technique in the theory of differential equations is concerned with estimating a function satisfying a differential inequality by means of the extremal solutions of the corresponding differential equation. This comparison principle has been widely employed in studying the qualitative theory of differential equations (see ). If we desire to develop a similar comparison result in abstract spaces we must consider cones. These results could be of great value in applications to the theory of differential equations in abstract spaces. First, we must consider existence results for maximal and minimal solutions in cones which can then be utilized to prove comparison results. This approach would unify various comparison theorems for scalar, finite, and infinite systems of differential equations. Naturally the notion of quasimonotone functions must be introduced for abstract spaces. In this paper, employing the properties of abstract cones and the Kuratowski measure of non-compactness of a set, we prove existence of extremal solutions, discuss comparison theorems and, as an application of the comparison technique, consider a general uniqueness theorem.