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Weak nearly uniform smoothness of the ψ-direct sums (X1 O…O XN)ψ

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We shall characterize the weak nearly uniform smoothness of the ψ-direct sum (X1 O…O XN)ψof N Banach spaces X1,..., XN, where ψ is a convex function satisfying certain conditions on the convex set [formula]. To do this, a class of convex functions which yield l1-like norms will be introduced. We shall apply our result to the fixed point property for nonexpansive mappings (FPP). In particular, an example which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square will be presented.
Rocznik
Strony
171--198
Opis fizyczny
Bibliogr. 28 poz.
Twórcy
autor
  • Department of Mathematics, Faculty of Engineering, Shinshu University, Nagano 380-8553, Japan
autor
  • Graduate School of Humanities and Social Sciences, Chiba University Chiba 263-8522, Japan
Bibliografia
  • 1] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Note Ser. 10 (1973).
  • [2] S. Chen, Y. Cui, H. Hudzik and B. Sims, Geometric properties related to fixed point theory in some Banach function lattices, In: Handbook of Metric Fixed Point Theory, eds. W. A. Kirk and B. Sims, Kluwer, Dordrecht, 2001, pp. 339-389.
  • [3] S. Dhompongsa, A. Kaewcharoen and A. Kacwkhao, Fixed point property of direct sums, Nonlinear Anal. 63(2005), e2177-e2188.
  • [4] S. Dhompongsa, A. Kaewkhao and S. Saejung, Uniform smoothness and U-convexity of ψ-direct sums, J. Nonlinear Convex Anal. 6 (2005), 327-338.
  • [5] P. N. Dowling, On convexity properties of ψ-direct sums of Banach spaces, J. Math. Anal. Appl. 288 (2003), 540-543.
  • [6] P. N. Dowling and S. Saejung , Non-squareness and uniform non-squareness of Z-direct sums, 3. Math. Anal. Appl. 369 (2010), 53-59.
  • [7] P. N. Dowling and B. Tvirett, Complex strict convexity of absolute norms on Cn and direct sums of Banach spaces, J. Math. Anal. Appl. 323 (2006), 930-937.
  • [8] J. Garcia-Falset, Stability and fixed points for nonexpansive mappings, Houston J. Math. 20 (1994), 495 GOG.
  • [9] J. Garcia-Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (1997), 532-542.
  • [10] J. Garcia-Falset, E. Llorens-Fuster and E. M. Mazcufian-Navarro, Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings, J. Funct. Anal. 233 (2006), 494-514.
  • [11] M. Kato, K.-S. Saito and T. Tamura, On the ψ-direct sums of Banach spaces and convexity, J. Aust. Math. Soc. 75 (2003), 413-422.
  • [12] M. Kato, K.-S. Saito and T. Tamura, Uniform non-squareness of ψ-direct sums of Banach spaces X Oψ Y, Math. Inequal. Appl., 7(2004), 429-437.
  • [13] M. Kato. K.-S. SaltO illld T. Trillium, Uniform non-t^-nesa of t/j-direct sums of Banach spaces, J. Nonlinear Convex Anal. 11 (2010), 113-133.
  • [14] M. Kato and T. Tamura, Weak nearly uniform smoothness and WORTH property of ψ-direct sums of Banach spaces, Comment. Math. Prace Mat. 46 (2006), 113-129.
  • [15] M. Kato and T. Tamura, Uniform non-l-ness of l1-sums of Banach spaces, Comment. Math. Prace Mat. 47 (2007), 161-169.
  • [16] M. Kato and T. Tamura, Uniform non-[formula] -sums of Banach spaces, Comment. Math. Prace Mat. 49 (2009), 179-187.
  • [17] M. Kato and T. Tamura, On a class of convex functions which yield l1like norms, in preparation.
  • [18] D. Kutzarova, S. Prus and B. Sims, Remarks on orthogonal convexity of Banach spaces, Houston J. Math. 19 (1993), 603-614.
  • [19] E. Llorens-Fuster, Some moduli and constants related to metric fixed point theory, In: Handbook of Metric Fixed Point Theory, eds. W. A. Kirk and B. Sims, pp. 133-175, Kluwer, Dordrecht, 2001.
  • [20] K-I. Mitani, S. Oshiro and K.-S. Saito, Smoothness ofip-direct sums of Banach spaces, Math. Inequal. Appl. 8 (2005), 147-157.
  • [21] L. Nikolova, L.E. Persson and S. Varosanec, The Beckenbach-Dresher inequality in the indirect sums of spaces and related results, J. Inequal. Appl. 2012:7, 14 pp.
  • [22] L. Nikolova and T. Zachariades, On ψ-interpolation spaces, Math. Inequal. Appl. 12 (2009), 827-838.
  • [23] S. Prus, Nearly uniformly smooth Banach spaces, Boll. U. M. I., (7)3-B (1989), 507-521.
  • [24] K.-S. Saito and M. Kato, Uniform convexity of ψ-direct sums of Banach spaces, J. Math. Anal. Appl. 277 (2003), 1-11.
  • [25] K.-S. Saito, M. Kato and Y. Takahashi, Von Neumann-Jordan constant of absolute normalized norms on C2, J. Math. Anal. Appl. 244 (2000), 515-532.
  • [26] K.-S. Saito, M. Kato and Y. Takahashi, On absolute norms on C", J. Math. Anal. Appl. 252 (2000), 879-905.
  • [27] Y. Takahashi, M. Kato and K.-S. Saito, Strict convexity of absolute norms on C2 and direct sums of Banach spaces, J. Inequal. Appl. 7 (2002), 179-186.
  • [28] T. Zachariades, On t^ spaces and infinite ip-direct sums of Banach spaces, Rocky Mount. J. Math. 41 (2011), 971-997.
Typ dokumentu
Bibliografia
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