Fuzzy principles and characterization of trustworthiness
Recently, Q. J. Zhu showed that some different basic principles of subdifferential calculus for the \(\beta\)-subdifferentials of lower semicontinuous functions on Banach spaces are equivalent. In the present paper, the author points out that these equivalences can be extended to more general subdifferentials but also to associated \(\varepsilon\)-versions of these notions. Moreover, it is shown that for special subdifferentials the list of mentioned principles can be complemented by further conditions. In detail, the author introduces a more general notion of subdifferential which includes different kinds of limiting subdifferentials and also all modifications of approximate subdifferentials. In the first main theorem it is shown that the following principles are equivalent: 5 versions of the Local Basic Fuzzy Principles of Ioffe (1983, 1984, 1990), Fabian (1989), Borwein/Ioffe (1996) and Borwein/Zhu (1996), the Nonlocal Basic Fuzzy Principle of Borwein/Treiman/Zhu (1996), the Multi-Directional Mean-Value Inequality of Clarke/Ledyaev (1994) and the Extremal Principle of Kruger/Mordukhovich (1984). In the second and third theorem it is pointed out that these equivalences remain true also for the associated \(\varepsilon\)-subdifferentials and -- using the \(\beta\)-subdifferential with an arbitrary bornology \(\beta\) -- that all the conditions are equivalent to the dense subdifferentiability of l.s.c. functions and to the trustworthiness of the space in the sense of Ioffe (1983, 1984).
scientific article; zbMATH DE number 1268116