Journal of Applied Mathematics and Stochastic Analysis
Volume 12 (1999), Issue 2, Pages 169-178

A note on multivalued differential equations on proximate retracts

Donal O'Regan

National University of Ireland, Department of Mathematics, Galway, Ireland

Received 1 October 1997; Revised 1 July 1998

Copyright © 1999 Donal O'Regan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper discusses viable solutions for differential inclusions in Banach spaces. Existence will be established in two steps. In step 1, a nonlinear alternative of Leray-Schauder type [8] for maps with closed graphs will be used to establish a variety of existence principles for the Cauchy differential inclusion. Step 2 involves using the results in step 1 together with some tricks involving the Bouligand cone (and sometimes the Urysohn function) so that new existence criteria can be established for multivalued differential equations on proximate retracts.