CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES

doi:10.1007/s10473-022-0324-4

Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (3): 1225-1237.doi: 10.1007/s10473-022-0324-4

• Articles • Previous Articles Next Articles

CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES

Shaoqiang SHANG

College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, China

Received:2021-01-13 Revised:2021-06-28 Published:2022-06-24
Contact: Shaoqiang SHANG,E-mail:sqshang@163.com E-mail:sqshang@163.com
Supported by:
This research was supported by the "China Natural Science Fund under grant 11871181" and the "China Natural Science Fund under grant 11561053".

Abstract

Abstract: In this paper, we prove that if $X$ is an almost convex and 2-strictly convex space, linear operator $T: X \to Y$ is bounded, $N(T)$ is an approximative compact Chebyshev subspace of $X$ and $R(T)$ is a 3-Chebyshev hyperplane, then there exists a homogeneous selection ${T^\sigma }$ of ${T^\partial }$ such that continuous points of ${T^\sigma }$ and ${T^\partial }$ are dense on $Y$.

Key words: Continuous selection, 3-Chebyshev hyperplane, set-valued metric generalized inverses, 2-strictly convex space

CLC Number:

46B20

Shaoqiang SHANG. CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES[J].Acta mathematica scientia,Series B, 2022, 42(3): 1225-1237.

Trendmd

References

[1] Shang S, Cui Y. Almost convexity and continuous selections of the set-valued metric generalized inverse in Banach spaces. Banach J Math Anal, 2021, 15(1):1-22
[2] Shang S, Cui Y. Approximative compactness and continuity of the set-valued metric generalized inverse in Banach spaces. J Math Anal Appl, 2015, 422(2):1361-1375
[3] Hudzik H, Wang Y W, Zheng W. Criteria for the Metric Generalized Inverse and its Selections in Banach Spaces. Set-Valued Anal, 2008, 16(1):51-65
[4] Shang S. Nearly dentability and continuity of the set-valued metric generalized inverse in Banach spaces. Ann Funct Anal, 2016, 426(3):508-520
[5] Nashed M Z, Votruba G F. A unified approach to generalized inverses of linnear operator:II Extremal and proximinal properties. Bull Amer Math Soc, 1974, 80(1):831-835
[6] Ma H F, Henryk Hudzik, Wang Y W. Continuous Homogeneous Selections of Set-Valued Metric Generalized Inverses of Linear Operators in Banach Spaces. Acta Mathematica Sinica, English Series 2012, 28(1):45-56
[7] Shang S, Zhang J. Metric projection operator and continuity of the set-valued metric generalized inverse in Banach spaces. Journal of Function Space, 2017, 2017(1):1-11
[8] Shang S, Cui Y. Continuity points and continuous selections of the set-valued metric generalized inverse in Banach spaces. Israel Journal of Mathematics, 2019, 234:209-228
[9] Aubin J, Frankowska H. Set-Valued Analysis. Boston:Birkhauser, 1990
[10] Cheng L, Ruan Y, Teng Y. Approximation of convex functions on the dual of Banach spaces. J Approx Theory, 2002, 116:126-140
[11] Chen S T. Geometry of Orlicz spaces. Warszawa:Dissertations Math, 1996
[12] Singer I. On the set of best approximation of an element in a normed linear space. Rev Roumaine Math Pures Appl, 1960, 5(1):383-402
[13] Wang Y W. Generalized Inverse of Operator in Banach Spaces and Applications. Beijing:Science Press, 2005

[1]	Yunan CUI, Paweƚ FORALEWSKI, Joanna KOŃCZAK. ORLICZ-LORENTZ SEQUENCE SPACES EQUIPPED WITH THE ORLICZ NORM [J]. Acta mathematica scientia,Series B, 2022, 42(2): 623-652.
[2]	Yuqi SUN, Wen ZHANG. COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES [J]. Acta mathematica scientia,Series B, 2021, 41(5): 1493-1502.
[3]	Shaoqiang SHANG. THE BALL-COVERING PROPERTY ON DUAL SPACES AND BANACH SEQUENCE SPACES [J]. Acta mathematica scientia,Series B, 2021, 41(2): 461-474.
[4]	Zihou ZHANG, Vicente MONTESINOS, Chunyan LIU. SOME METRIC AND TOPOLOGICAL PROPERTIES OF NEARLY STRONGLY AND NEARLY VERY CONVEX SPACES [J]. Acta mathematica scientia,Series B, 2020, 40(2): 369-378.
[5]	Duanxu DAI. STABILITY OF ε-ISOMETRIES ON L_∞-SPACES [J]. Acta mathematica scientia,Series B, 2019, 39(6): 1733-1742.
[6]	Duanxu DAI, Bentuo ZHENG. STABILITY OF A PAIR OF BANACH SPACES FOR ε-ISOMETRIES [J]. Acta mathematica scientia,Series B, 2019, 39(4): 1163-1172.
[7]	Wanzhong GONG, Daoxiang ZHANG. MONOTONICITY IN ORLICZ-LORENTZ SEQUENCE SPACES EQUIPPED WITH THE ORLICZ NORM [J]. Acta mathematica scientia,Series B, 2016, 36(6): 1577-1589.
[8]	YI Ji-Jin, WANG Rui-Dong, WANG Xiao-Xiao. EXTENSION OF ISOMETRIES BETWEEN THE UNIT SPHERES OF COMPLEX l ^p(Γ)(p >\|1) SPACES [J]. Acta mathematica scientia,Series B, 2014, 34(5): 1540-1550.
[9]	Murat KARAKAS, Mikail ET, Vatan KARAKAYA. SOME GEOMETRIC PROPERTIES OF A NEW DIFFERENCE SEQUENCE SPACE INVOLVING LACUNARY SEQUENCES [J]. Acta mathematica scientia,Series B, 2013, 33(6): 1711-1720.
[10]	Ji Gao, Satit Saejung. NORMAL STRUCTURE AND SOME GEOMETRIC PARAMETERS RELATED TO THE MODULUS OFU-CONVEXITY IN BANACH SPACES [J]. Acta mathematica scientia,Series B, 2011, 31(3): 1035-1040.
[11]	B.Zlatanov . ON A CLASS OF KÖTHE SEQUENCE SPACES WITH NORMAL STRUCTURE [J]. Acta mathematica scientia,Series B, 2011, 31(2): 576-590.
[12]	WANG Rui-Dong. ON LINEAR EXTENSION OF 1-LIPSCHITZ MAPPING FROM HILBERT SPACE INTO A NORMED SPACE [J]. Acta mathematica scientia,Series B, 2010, 30(1): 161-165.
[13]	Ding Guanggui. THE ISOMETRIC EXTENSION OF AN INTO MAPPING FROM THE UNIT SPHERE TO THE UNIT SPHERE S(E) [J]. Acta mathematica scientia,Series B, 2009, 29(3): 469-479.
[14]	Antonio Aizpuru; Francisco J Garc. A SHORT NOTE ABOUT EXPOSED POINTS IN REAL BANACH SPACES [J]. Acta mathematica scientia,Series B, 2008, 28(4): 797-800.
[15]	Yang Changsen; Wang Fenghui. ON A GENERALIZED MODULUS OF CONVEXITY AND UNIFORM NORMAL STRUCTURE [J]. Acta mathematica scientia,Series B, 2007, 27(4): 838-844.

CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0