特邀嘉宾---Prof. Dr. Peter Kloeden


School of Mathematics & Statistics, Huazhong University of Science & Technology, Wuhan, China


Biography: Professor Peter Kloeden is Professor of Mathematics at the Huazhong University of Science & Technology under the \1000" Expert Program. Previously he was Professor of Mathematics at the Goethe University in Frankfurt am Main and held various positions in universities in Australia. Proessor Kloeden is a Fellow of SIAM and a Fellow of the Australian Mathematcial Society. He has broad interests in applied analysis, dynamical systems, fuzzy sets, setvalued analysis and stochastic analysis and is the author of over 300 research papers and 10 books including P. E. Kloeden and E. Platen, The Numerical Solution of Stochastic Di erential Equations, Springer (1992)
P. Diamond und P. E. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scienti c (1994).
His interests in fuzzy sets focuses mainly on fuzzy di erential equations and fuzzy dynamical systems.

Speech Title: A Peano theorem for fuzzy di erential equations with evolving membership grade
Abstract: Classical fuzzy di erential equations de ned in terms of the Hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets. Here Hullermeier's suggestion of de ning fuzzy di erential equations at each level set via di erential inclusions is combined with ideas of Aubin on morphological equations, which allow nonlocal set evolution, to remove the assumption of fuzzy convexity and thus to allow fuzzy di erential equations to be de ned for non-convex level sets.
This approach uses reachable sets as a more general form of set integration and, in contrast to the Aumann set integral, does not necessarily give rise to convex sets. The results presented are even more general since they concern fuzzy sets that need be only closed without additional assumptions of convexity, compactness or even normality. In particular, an existence and uniqueness theorem is established under the assumption that the right-hand sides satisfy a one-sided Lipschitz condition rather than a much stronger Lipschitz condition.
The Peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy di erential equations. The diculty appears to be due to the standard use of the supremum metric dist1 de ned by the supremum over the Hausdor metric between the level sets of the fuzzy sets. Another may have been the classical formulation of fuzzy di erential equations in terms of the Hukuhara derivative of the level sets.
A Peano theorem is established here for fuzzy di erential equations formulated by combining Hullermeier's suggestion of de ning fuzzy di erential equations at each level set via di erential inclusions with Aubin's morphological equations, which allow non-local set evolution. A major di erence from previous publications is the use of the endograph metric distendo, essentially the Hausdor metric between the endographs in Rn[0; 1] of fuzzy sets, instead of the supremum metric dist1 Another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy di erential equations. The result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity, compact supports or even normality.