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Abstract
The purpose of our article is to extend the classical notion of Fréchet differentiability to multifunctions. To this end we define the notion of affinity for multifunctions and study the basic properties of affine multifunctions. Then using affine multifunctions as local approximations and the Hausdorff distance for defining an approximation mode, we introduce the notion of Fréchet differentiability for multifunctions mapping points of a finite-dimensional normed space to compact convex subsets of another finite-dimensional normed space. We characterize Fréchet differentiability of multifunctions through the differentiable properties of their support functions and discuss the relationship of our notion of differentiability with other ones which were studied by Blagodatskikh (Blagodatskikh, V.I., 1984, Maximum principle for differential inclusions. Trudy Matematicheskogo Instituta AN SSSR, 166, 23–43 (in Russian)), Rubinov (Rubinov, A.M., 1985, The conjugate derivative of a multivalued mapping and differentiability of the maximum function under connected constraints. Sibirskii Matematicheskii Zhurnal, 26(3), 147–155 (in Russian)), Tyurin (Tyurin, Yu. N., 1965, A mathematical formulation of a simplified model of industrial planning. Ekonomika i Matematicheskie Metody, 1(3), 391–409 (in Russian)), Banks and Jacobs (Banks, H.T. and Jacobs, M.Q., 1970, On differential calculus of multifunctions. Journal of Mathematical Analysis and Applications, 29(3), 246–272).
Notes
The article is dedicated to Professor V.F. Demyanov on his 65th birthday.