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Yannelis-Prabhakar Continuous Selection Theorem[1]
Consider a correspondence (set-valued function) defined on a paracompact space, taking values on a Hausdorff linear topological vector space. If such a correspondence has open lower sections (lower inverse is open) and it is convex and non-empty valued, then it admits a continuous selection.
The assumption of open lower sections implies that the correspondence is lower semicontinuous[1] and consequently it does not imply the Michael selection theorem. The usefulness of the Yannelis-Prabhakar theorem lies on the fact that the conditions imposed by Michael (are not typically satisfied in economic models), i.e., the correspondence takes closed values on a separable Banach space, are not needed in the Yannelis-Prabhakar theorem. By relaxing separability and closed value, the Yannelis-Prabhakar theorem has found useful applications in economics. The referenced articles discuss further generalizations[2], extensions[3][4], and applications[4] of the Yannelis-Prabhakar continuous selection theorem.
- ^ a b Yannelis, Nicholas C.; Prabhakar, N. D. (1983-12-01). "Existence of maximal elements and equilibria in linear topological spaces". Journal of Mathematical Economics. 12 (3): 233–245. doi:10.1016/0304-4068(83)90041-1. ISSN 0304-4068.
- ^ Wu, Xian; Shen, Shikai (1996-01-01). "A Further Generalization of Yannelis–Prabhakar's Continuous Selection Theorem and Its Applications". Journal of Mathematical Analysis and Applications. 197 (1): 61–74. doi:10.1006/jmaa.1996.0007. ISSN 0022-247X.
- ^ Hou, Ji-Cheng (2008-05-15). "A new generalization of the Yannelis–Prabhakar equilibrium existence theorem for abstract economies". Nonlinear Analysis: Theory, Methods & Applications. 68 (10): 3159–3165. doi:10.1016/j.na.2007.03.010. ISSN 0362-546X.
- ^ a b Khan, M. Ali; Uyanik, Metin (2021-04-01). "The Yannelis–Prabhakar theorem on upper semi-continuous selections in paracompact spaces: extensions and applications". Economic Theory. 71 (3): 799–840. doi:10.1007/s00199-021-01359-4. ISSN 1432-0479.