March  2020, 28(1): 327-346. doi: 10.3934/era.2020019

Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces

1. 

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

2. 

Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China

* Corresponding author: Shaoqiang Shang

Received  December 2019 Revised  February 2020 Published  March 2020

In this paper, some criteria for weakly approximative compactness and approximative compactness of weak$ ^{*} $ hyperplane for Musielak-Orlicz-Bochner function spaces are given. Moreover, we also prove that, in Musielak-Orlicz-Bochner function spaces generated by strongly smooth Banach space, $ L_{M}^{0}(X) $ (resp $ L_{M}(X) $) is an Asplund space if and only if $ M $ and $ N $ satisfy condition $ \Delta $. As a corollary, we obtain that $ L_{M}^{0}(R) $ (resp $ L_{M}(R) $) is an Asplund space if and only if $ M $ and $ N $ satisfy condition $ \Delta $.

Citation: Shaoqiang Shang, Yunan Cui. Weak approximative compactness of hyperplane and Asplund property in Musielak-Orlicz-Bochner function spaces. Electronic Research Archive, 2020, 28 (1) : 327-346. doi: 10.3934/era.2020019
References:
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A. Canino, B. Sciunzi and A. Trombetta, On the moving plane method for boundary blow-up solutions to semilinear elliptic equations, Adv. Nonlinear Anal., 9 (2020), no. 1, 1–6. doi: 10.1515/anona-2017-0221.  Google Scholar

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S. Chen, Geometry of Orlicz Spaces, Dissertationes Math., (Rozprawy Mat.) Vol. 356, Warszawa, 1996,204 pp.  Google Scholar

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M. Denker and H. Hudzik, Uniformly non-$l_{n}^{(1)}$ Musielak-Orlicz sequence spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), no. 2, 71–86. doi: 10.1007/BF02868018.  Google Scholar

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L. L. Fang, W. T. Fu and H. Hudzik, Uniform Gateaux differentiablity and weak uniform rotundity in Musielak-Orlicz function spaces, Nonlinear Anal., 56 (2004), no. 8, 1133–1149. doi: 10.1016/j.na.2003.11.007.  Google Scholar

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F. A. Hoeg and P. Lindqvist, Regularity of solutions of the parabolic normalized p-Laplace equation, Adv. Nonlinear Anal., 9 (2020), no. 1, 7–15. doi: 10.1515/anona-2018-0091.  Google Scholar

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H. Hudzik and W. Kurc, Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices, J. Approx. Theory, 95 (1998), no. 3,353–368. doi: 10.1006/jath.1997.3226.  Google Scholar

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H. Hudzik, W. Kurc and M. Wisła, Strongly extreme points in Orlicz function spaces, J. Math. Anal. Appl., 189 (1995), no. 3,651–670. doi: 10.1006/jmaa.1995.1043.  Google Scholar

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H. Hudzik, W. Kowalewski and G. Lewicki, Approximate compactness and full rotundity in Musielak-Orlicz space and Lorentz-Orlicz space, Z. Anal. Anwend., 25 (2006), no. 2,163–192. doi: 10.4171/ZAA/1283.  Google Scholar

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H. Hudzik and W. Kowalewski, On some local geometric properties in Musielak-Orlicz function spaces, Z. Anal. Anwend., 23 (2004), no. 4,683–712. doi: 10.4171/ZAA/1216.  Google Scholar

[11]

C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), no. 1,845–867. doi: 10.1515/anona-2016-0197.  Google Scholar

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P. Kolwicz and R. Pluciennik, P-convexity of Orlicz-Bochner function spaces, Proc. Am. Math. Soc., 126 (1998), no. 8, 2315–2322. doi: 10.1090/S0002-9939-98-04290-7.  Google Scholar

[13]

W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J. Approx. Theory 69 (1992), no. 2,173–187. doi: 10.1016/0021-9045(92)90141-A.  Google Scholar

[14]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), no. 1,613–632. doi: 10.1515/anona-2020-0016.  Google Scholar

[15]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), no. 4,737–764. doi: 10.1515/ans-2016-0023.  Google Scholar

[16]

D. Preiss, R. Phelps and I. Namioka., Smooth Banach spaces, weak asplund spaces and monotone operators or usco mappings, Israel J. Math., 72 (1990), no. 3,257–279. doi: 10.1007/BF02773783.  Google Scholar

[17]

S. Shang and Y. Cui, Uniform rotundity and k-uniform rotundity in Musielak-Orlicz-Bochner function spaces and applications, J. Convex Anal., 22 (2015), no. 3,747–768.  Google Scholar

[18]

S. Shang, Y. Cui and Y. Fu, Smoothness and approximative compactness in Orlicz function spaces, Banach J. Math. Anal., 8 (2014), no. 1, 26–38. doi: 10.15352/bjma/1381782084.  Google Scholar

[19]

S. Shang, Y. Cui and Y. Fu, Extreme points and rotundity in Musielak-Orlicz-Bochner function spaces endowed with Orlicz norm, Abstr. Appl. Anal., (2010), Art. ID 914183, 13 pp. doi: 10.1155/2010/914183.  Google Scholar

[20]

S. Shang, Y. Cui and H. Hudzik, Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm, Nonlinear Anal., 75 (2012), no. 6, 3009–3020. doi: 10.1016/j.na.2011.11.012.  Google Scholar

[21]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), no. 12, 2732–2763. doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[22]

Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), no. 3, 1351–1358. doi: 10.3934/cpaa.2019065.  Google Scholar

show all references

References:
[1]

A. Canino, B. Sciunzi and A. Trombetta, On the moving plane method for boundary blow-up solutions to semilinear elliptic equations, Adv. Nonlinear Anal., 9 (2020), no. 1, 1–6. doi: 10.1515/anona-2017-0221.  Google Scholar

[2]

S. Chen, Geometry of Orlicz Spaces, Dissertationes Math., (Rozprawy Mat.) Vol. 356, Warszawa, 1996,204 pp.  Google Scholar

[3]

M. Denker and H. Hudzik, Uniformly non-$l_{n}^{(1)}$ Musielak-Orlicz sequence spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), no. 2, 71–86. doi: 10.1007/BF02868018.  Google Scholar

[4]

L. L. Fang, W. T. Fu and H. Hudzik, Uniform Gateaux differentiablity and weak uniform rotundity in Musielak-Orlicz function spaces, Nonlinear Anal., 56 (2004), no. 8, 1133–1149. doi: 10.1016/j.na.2003.11.007.  Google Scholar

[5]

P. Foralewski, H. Hudzik and P. Kolwicz, Non-squareness properties of Orlicz-Lorentz sequence spaces, J. Funct. Anal., 264 (2013), no. 2,605–629. doi: 10.1016/j.jfa.2012.10.014.  Google Scholar

[6]

F. A. Hoeg and P. Lindqvist, Regularity of solutions of the parabolic normalized p-Laplace equation, Adv. Nonlinear Anal., 9 (2020), no. 1, 7–15. doi: 10.1515/anona-2018-0091.  Google Scholar

[7]

H. Hudzik and W. Kurc, Monotonicity properties of Musielak-Orlicz spaces and dominated best approximation in Banach lattices, J. Approx. Theory, 95 (1998), no. 3,353–368. doi: 10.1006/jath.1997.3226.  Google Scholar

[8]

H. Hudzik, W. Kurc and M. Wisła, Strongly extreme points in Orlicz function spaces, J. Math. Anal. Appl., 189 (1995), no. 3,651–670. doi: 10.1006/jmaa.1995.1043.  Google Scholar

[9]

H. Hudzik, W. Kowalewski and G. Lewicki, Approximate compactness and full rotundity in Musielak-Orlicz space and Lorentz-Orlicz space, Z. Anal. Anwend., 25 (2006), no. 2,163–192. doi: 10.4171/ZAA/1283.  Google Scholar

[10]

H. Hudzik and W. Kowalewski, On some local geometric properties in Musielak-Orlicz function spaces, Z. Anal. Anwend., 23 (2004), no. 4,683–712. doi: 10.4171/ZAA/1216.  Google Scholar

[11]

C. Imbert, T. Jin and L. Silvestre, Hölder gradient estimates for a class of singular or degenerate parabolic equations, Adv. Nonlinear Anal., 8 (2019), no. 1,845–867. doi: 10.1515/anona-2016-0197.  Google Scholar

[12]

P. Kolwicz and R. Pluciennik, P-convexity of Orlicz-Bochner function spaces, Proc. Am. Math. Soc., 126 (1998), no. 8, 2315–2322. doi: 10.1090/S0002-9939-98-04290-7.  Google Scholar

[13]

W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J. Approx. Theory 69 (1992), no. 2,173–187. doi: 10.1016/0021-9045(92)90141-A.  Google Scholar

[14]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), no. 1,613–632. doi: 10.1515/anona-2020-0016.  Google Scholar

[15]

N. S. Papageorgiou and V. D. Rǎdulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud., 16 (2016), no. 4,737–764. doi: 10.1515/ans-2016-0023.  Google Scholar

[16]

D. Preiss, R. Phelps and I. Namioka., Smooth Banach spaces, weak asplund spaces and monotone operators or usco mappings, Israel J. Math., 72 (1990), no. 3,257–279. doi: 10.1007/BF02773783.  Google Scholar

[17]

S. Shang and Y. Cui, Uniform rotundity and k-uniform rotundity in Musielak-Orlicz-Bochner function spaces and applications, J. Convex Anal., 22 (2015), no. 3,747–768.  Google Scholar

[18]

S. Shang, Y. Cui and Y. Fu, Smoothness and approximative compactness in Orlicz function spaces, Banach J. Math. Anal., 8 (2014), no. 1, 26–38. doi: 10.15352/bjma/1381782084.  Google Scholar

[19]

S. Shang, Y. Cui and Y. Fu, Extreme points and rotundity in Musielak-Orlicz-Bochner function spaces endowed with Orlicz norm, Abstr. Appl. Anal., (2010), Art. ID 914183, 13 pp. doi: 10.1155/2010/914183.  Google Scholar

[20]

S. Shang, Y. Cui and H. Hudzik, Uniform Gateaux differentiability and weak uniform rotundity in Musielak-Orlicz function spaces of Bochner type equipped with the Luxemburg norm, Nonlinear Anal., 75 (2012), no. 6, 3009–3020. doi: 10.1016/j.na.2011.11.012.  Google Scholar

[21]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), no. 12, 2732–2763. doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[22]

Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), no. 3, 1351–1358. doi: 10.3934/cpaa.2019065.  Google Scholar

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