## Document Type

Report

## Source Publication Title

Technical Report 88

## Abstract

In our recent paper [3] we have studied the existence of maximal and minimal solutions to the IVP in a Banach space [see pdf for notation]. (1) [see pdf for notation] where [see pdf for notation] maps [see pdf for notation] into [see pdf for notation], with [see pdf for notation] and [see pdf for notation] a cone. The essential hypotheses have been that [see pdf for notation]f is quasimonotone with respect to [see pdf for notation] and that [see pdf for notation] and [see pdf for notation] have some natural properties. If such extremal solutions exist then it is trivial to prove the usual comparison theorems known from the finite-dimensional case. For example, if [see pdf for notation] is the minimal solution of (1) on some interval [see pdf for notation] and if [see pdf for notation] satisfies [see pdf for notation] and [see pdf for notation] then [see pdf for notation]. In the present paper we shall establish existence and comparison theorems for (1) without the hypothesis that [see pdf for notation] be quasimonotone, but under conditions which have been considered in case [see pdf for notation] in the classical paper of M. Muller [10] in 1926. This is not the first attempt to extend Muller's results to infinite dimensions, since recently P. Volkmann [12] tried to do this. We shall improve the existing results considerably.

## Disciplines

Mathematics | Physical Sciences and Mathematics

## Publication Date

7-1-1978

## Language

English

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Lakshmikantham, V. and Deimling, K., "Existence and Comparison Theorems for Differential Equations in Banach Spaces" (1978). *Mathematics Technical Papers*. 101.

https://mavmatrix.uta.edu/math_technicalpapers/101