# Article

 Title: The fixed point property in Musielak-Orlicz sequence spaces (English) Author: Thompson, H. Bevan Author: Cui, Yunan Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 42 Issue: 2 Year: 2001 Pages: 299-309 . Category: math . Summary: In this paper, we give necessary and sufficient conditions for a point in a Musielak-Orlicz sequence space equipped with the Orlicz norm to be an {\bf H}-point. We give necessary and sufficient conditions for a Musielak-Orlicz sequence space equipped with the Orlicz norm to have the {\it Kadec-Klee} property, the uniform {\it Kadec-Klee} property and to be nearly uniformly convex. We show that a Musielak-Orlicz sequence space equipped with the Orlicz norm has the fixed point property if and only if it is reflexive. (English) Keyword: nearly uniformly convex Keyword: uniform Kadec-Klee property Keyword: Kadec-Klee property Keyword: Musielak-Orlicz sequence space Keyword: fixed point property MSC: 46B20 MSC: 46B45 MSC: 46E30 MSC: 47H10 idZBL: Zbl 1056.46021 idMR: MR1832148 . Date available: 2009-01-08T19:10:00Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119244 . Reference: [1] Chen S.: Geometry of Orlicz spaces.Dissertation Math., Warsaw, 1996. Zbl 1089.46500, MR 1410390 Reference: [2] Cui Y.A., Hudzik H.: Maluta coefficient and Opial property in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm.Nonlinear Anal. Theory Methods & Appl., to appear. MR 1656529 Reference: [3] Cui Y.A., Hudzik H., Nowak M., Pluciennik R.: Some geometric properties in Orlicz sequence spaces equipped with the Orlicz norm.J. Convex Anal. 6 (1999), 91-113. MR 1713953 Reference: [4] Denker M., Hudzik H.: Uniformly non-$l_n^{(1)}$ Musielak-Orlicz sequence spaces.Proc. Indian. Acad. Sci. 101.2 (1991), 71-86. MR 1125480 Reference: [5] Diestel J.: Sequence and Series in Banach Spaces.Graduate Texts in Math. 92, Springer-Verlag, 1984. MR 0737004 Reference: [6] Dowling P.R., Lennard C.J., Turett B.: Reflexivity and the fixed-point property for nonexpansive maps.J. Math. Anal. Appl. 200 (1996), 653-662. Zbl 0863.47038, MR 1393106 Reference: [7] Dulst D., Sims B.: Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK).Banach Space Theory and its Applications (Bucharest, 1981), pp.35-43; Lecture Notes in Math. 991, Springer, Berlin-New York, 1983. Zbl 0512.46015, MR 0714171 Reference: [8] Goebel K., Sekowski T.: The modulus of non-compact convexity.Ann. Univ. Maria Curie-Sklodowska, Sect. A 38 (1984), 41-48. MR 0856623 Reference: [9] Goebel R., Kirk W.A.: Topics in Metric Fixed Point Theory.Cambridge University Press, 1990. MR 1074005 Reference: [10] Hudzik H., Kaminska A.: Some remarks on convergence in Orlicz spaces.Comment. Math. 21 (1979), 81-88. MR 0577673 Reference: [11] Hudzik H., Ye Y.: Support functionals and smoothness in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm.Comment. Math. Univ. Carolinae 31.4 (1990), 661-684. MR 1091364 Reference: [12] Huff R.: Banach spaces which are nearly uniformly convex.Rocky Mountain J. Math. 10 (1980), 473-749. Zbl 0505.46011, MR 0595102 Reference: [13] Kadec M.I.: Relations between some properties of convexity of the ball of a Banach spaces.Functional Anal. Appl. 16 (1982), 93-100. Reference: [14] Kaminska A.: Uniform rotundity of Musielak-Orlicz sequence spaces.J. Approx. Theory 47.4 (1986), 302-322. Zbl 0606.46003, MR 0862227 Reference: [15] Kaminska A.: Flat Orlicz-Musielak sequence spaces.Bull. Acad. Polon. Sci. Math. 30 (1982), 347-352. Zbl 0513.46008, MR 0707748 Reference: [16] Kantorovic L.V., Akilov G.P.: Functional Analysis (in Russian).2nd edition, Moscow, 1978. MR 0511615 Reference: [17] Musielak J.: Orlicz spaces and modular spaces.Lecture Notes in Math. 1034, Springer Verlag, Berlin, 1983. Zbl 0557.46020, MR 0724434 Reference: [18] Pluciennik R., Wang T., Zhang Y.: H-point and denting points in Orlicz spaces.Comment. Math. 33 (1993), 135-151. MR 1269408 Reference: [19] Rao M.M., Ren Z.D.: Theory of Orlicz spaces.Marcel Dekker Inc., New York, Basel, HongKong, 1991. Zbl 0724.46032, MR 1113700 Reference: [20] Wu Congxin, Sun Huiying: Norm calculations and complex rotundity of Musielak-Orlicz sequence spaces.Chinese Math. Ann. 12A (Special Issue) 98-102. .

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