×

2-strict convexity and continuity of set-valued metric generalized inverse in Banach spaces. (English) Zbl 1472.47002

Summary: Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if \(X\) is approximately compact and \(X\) is 2-strictly convex, then metric generalized inverses of bounded linear operators in \(X\) are upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, a sufficient condition for set-valued mapping \(T^\partial\) to be continuous mapping is given.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A06 Linear relations (multivalued linear operators)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Megginson, R. E., An Introduction to Banach space theory (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0910.46008
[2] Jefimow, N. W.; Stechkin, S. B., Approximative compactness and Cheby-shev sets, Soviet Mathematics, 2, 1226-1228 (1961)
[3] Aubin, J.-P.; Frankowska, H., Set-Valued Analysis, 2 (1990), Boston, Mass, USA: Birkhäuser Boston, Boston, Mass, USA
[4] Chen, S.; Hudzik, H.; Kowalewski, W.; Wang, Y.; Wisła, M., Approximative compactness and continuity of metric projector in Banach spaces and applications, Science in China A, 51, 2, 293-303 (2008) · Zbl 1153.46008
[5] Oshman, E. V., Characterization of subspaces with continuous metric projection into a normed linear space, Soviet Mathematics, 13, 1521-1524 (1972) · Zbl 0268.46017
[6] Shang, S. Q.; Cui, Y. A.; Fu, Y. Q., Nearly dentability and approximative compactness and continuity of metric projector in Banach spaces, Scientia Sinica Mathematica, 41, 9, 815-825 (2011) · Zbl 1488.46035
[7] Sullivan, F., A generalization of uniformly rotund Banach spaces, Canadian Journal of Mathematics, 31, 3, 628-636 (1979) · Zbl 0422.46011
[8] Skowski, T.; Stachura, A., Noncompact smoothness and noncompact convexity, Atti del Seminario Matematico e Fisico dell’Università di Modena, 36, 2, 329-338 (1988) · Zbl 0681.46023
[9] Nashed, M. Z.; Votruba, G. F., A unified approach to generalized inverses of linear operators. II. Extremal and proximal properties, Bulletin of the American Mathematical Society, 80, 831-835 (1974) · Zbl 0289.47011
[10] Wang, Y. W., Generalized Inverse of Operator in Banach Spaces and Applications (2005), Beijing, China: Science Press, Beijing, China
[11] Wang, Y.; Liu, J., Metric generalized inverse for linear manifolds and extremal solutions of linear inclusion in Banach spaces, Journal of Mathematical Analysis and Applications, 302, 2, 360-371 (2005) · Zbl 1061.47003
[12] Huang, Q.; Ma, J., Perturbation analysis of generalized inverses of linear operators in Banach spaces, Linear Algebra and Its Applications, 389, 355-364 (2004) · Zbl 1072.47012
[13] Chen, G.; Xue, Y., Perturbation analysis for the operator equation \(T x = b\) in Banach spaces, Journal of Mathematical Analysis and Applications, 212, 1, 107-125 (1997) · Zbl 0903.47004
[14] Wang, Y.; Zhang, H., Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces, Linear Algebra and its Applications, 426, 1, 1-11 (2007) · Zbl 1133.47011
[15] Hudzik, H.; Wang, Y.; Zheng, W., Criteria for the metric generalized inverse and its selections in Banach spaces, Set-Valued Analysis, 16, 1, 51-65 (2008) · Zbl 1148.47003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.