Abstract:
Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton–Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.
\Bibitem{Sub91}
\by A.~I.~Subbotin
\paper On a~property of the subdifferential
\jour Math. USSR-Sb.
\yr 1993
\vol 74
\issue 1
\pages 63--78
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\crossref{https://doi.org/10.1070/SM1993v074n01ABEH003335}
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This publication is cited in the following 16 articles:
M. I. Gomoyunov, N. Yu. Lukoyanov, “Minimax solutions of Hamilton–Jacobi equations in dynamic optimization problems for hereditary systems”, Russian Math. Surveys, 79:2 (2024), 229–324
Dmitry V. Khlopin, “On two-sided unidirectional mean value inequality in a Fréchet smooth space”, Ural Math. J., 9:2 (2023), 132–140
Anton Plaksin, “Viscosity Solutions of Hamilton–Jacobi Equations for Neutral-Type Systems”, Appl Math Optim, 88:1 (2023)
M.I. Gomoyunov, A.R. Plaksin, “Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations”, Journal of Functional Analysis, 285:11 (2023), 110155
A. G. Chentsov, “Differential Approach–Evasion Game: Alternative Solvability and the Construction of Relaxations”, Diff Equat, 57:8 (2021), 1088
A. G. Chentsov, “Relaksatsii igrovoi zadachi sblizheniya, svyazannye s alternativoi v differentsialnoi igre sblizheniya-ukloneniya”, Vestnik rossiiskikh universitetov. Matematika, 25:130 (2020), 196–244
N. Yu. Lukoyanov, A. R. Plaksin, “Inequalities for subgradients of a value functional in differential games for time-delay systems”, Dokl. Math., 101:1 (2020), 76–79
Anton Plaksin, “Minimax and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Time-Delay Systems”, J Optim Theory Appl, 187:1 (2020), 22
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