International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 2, Pages 245-254
Nonsmooth analysis approach to Isaac's equation
Institute of Mathematics, Agricultural & Pedagocical University, Siedlce 08-110, Poland
Received 14 November 1991; Revised 17 October 1992
Copyright © 1993 Leszek S. Zaremba. 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.
We study Isaacs' equation ( is a highly nonlinear
function) whose natural solution is a value of a suitable differential game. It has been felt
that even though may be a discontinuous function or it may not exist everywhere,
is a solution of in some generalized sense. Several attempts have been made to overcome this
difficulty, including viscosity solution approaches, where the continuity of a prospective solution or
even slightly less than that is required rather than the existence of the gradient . Using
ideas from a very recent paper of Subbotin, we offer here an approach which, requiring literally no
regularity assumptions from prospective solutions of , provides existence results. To prove the
uniqueness of solutions to , we make some lower- and upper-semicontinuity assumptions on a
terminal set . We conclude with providing a close relationship of the results presented on Isaacs'
equation with a differential games theory.