A Peano theorem for fuzzy differential equations with evolving membership grade

Abstract

The Peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations. The difficulty appears to be due to the standard use of the supremum metric ▪∞ defined by the supremum over the Hausdorff metric between the level sets of the fuzzy sets. Another may have been the classical formulation of fuzzy differential equations in terms of the Hukuhara derivative of the level sets. Here a Peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining Hüllermeier's suggestion of defining fuzzy differential equations at each level set via differential inclusions with Aubin's morphological equations, which allow non-local set evolution. A major difference from previous publications is the use of the endograph metric ▪end, essentially the Hausdorff metric between the endographs in Rn×[0,1] of fuzzy sets, instead of the supremum metric ▪∞. Another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations. The result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity, compact supports or even normality.