Abstract
In this paper, we define almost convex space. Let \(T:X\rightarrow Y\) be a linear bounded operator. This paper shows that: (1) If X is almost convex and 2-strictly convex, Y is a Banach space, D(T) is closed, N(T) is an approximatively compact Chebyshev subspace of D(T) and R(T) is a 2-Chebyshev hyperplane of Y, then there exists a homogeneous selection \({T^\sigma }\) of \({T^\partial }\) such that continuous points of \(T^\sigma \) is dense on Y. (2) If X is locally uniformly convex, Y is reflexive, D(T) is closed, N(T) is a proximinal subspace of D(T) and R(T) is a closed hyperplane of Y, then \(T^{\partial }\) is single-valued, homogeneous and continuous on Y. The results are a perfect answer to the open problem posed by Nashed and Votruba (Bull Am Math Soc 80:831–835, 1974).
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This research is supported by “China Natural Science Fund under Grant 11871181” and “China Natural Science Fund under Grant 11561053”.
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Communicated by Mikhail Ostrovskii.
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Shang, S., Cui, Y. Almost convexity and continuous selections of the set-valued metric generalized inverse in Banach spaces. Banach J. Math. Anal. 15, 11 (2021). https://doi.org/10.1007/s43037-020-00098-3
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DOI: https://doi.org/10.1007/s43037-020-00098-3