WORTH property, García-Falset coefficient and Opial property of infinite sums

• Autorzy: Hardtke J.-D.

Identyfikatory

DOI

10.14708/cm.v55i1.718

• Języki publikacji: EN

Abstrakty

EN

We prove some results concerning the WORTH property and the García-Falset coefficient of absolute sums of infinitely many Banach spaces. Also, the Opial property/uniform Opial property of infinite ℓ^p-sums is studied, and some properties analogous to the Opial property/uniform Opial property are discussed for Lebesgue–Bochner spaces L^pμ; X.

Słowa kluczowe

EN

Opial property; uniform Opial property; WORTH property; García-Falset coefficient; absolute sums; fixed-point property; Lebesgue–Bochner spaces

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2015
• Tom: Vol 55, No. 1
• Strony: 23--44
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Hardtke J.-D.

• Department of Mathema􀦞cs, Freie Universität Berlin, Germany

Bibliografia

• [1] D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423–424, DOI 10.2307/2043954.
• [2] L. P. Belluce and W. A. Kirk, Nonexapnsive mappings and fixed-points in Banach spaces, Illinois J. Math. 11 (1967), no. 3, 474–479.
• [3] M. Besbes, Points fixes et théorèmes ergodiques dans les espaces de Banach, Doctoral Thesis, Université de Paris 6, Paris, 1991 (french).
• [4] J. M. Borwein and B. A. Sims, Nonexpansive mappings on Banach lattices and related topics, Houston J. Math 10 (1984), no. 3, 339–356.
• [5] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490, DOI 10.1016/0022-247X(83)90147-6.
• [6] S. Dhompongsa, A. Kaewcharoen, and A. Kaewkhao, Fixed point property of direct sums, Nonlinear Anal. 63 (2005), no. 5–7, 2177–2188, DOI 10.1016/j.na.2005.02.020.
• [7] T. Domínguez Benavides, Weak uniform normal structure in direct sum spaces, Studia Math. 103(1992), no. 3, 283–289.
• [8] J. Gao and K. S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math. 99 (1991), no. 1, 41–56.
• [9] J. Gao, Normal structure and modulus of u-convexity in Banach spaces, Func􀦞on Spaces, Differential Operators and Nonlinear Analysis (Paseky nad Jizerou, 1995), Prometheus, Prague, 1996, 195–199.
• [10] J. García-Falset, Stability and fixed points for nonexpansive mappings, Houston J. Math. 20 (1994), no. 3, 495–506.
• [11] J. García-Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (1997), 532–542, DOI 10.1006/jmaa.1997.5657.
• [12] K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge 1990, DOI 10.1017/CBO9780511526152.
• [13] K. Goebel and W. A. Kirk, Classical theory of nonexpansive mappings, Handbook of Metric Fixed Point Theory (W. A. Kirk and B. Sims, eds.), Kluwer Academic Publishers, Dordrecht–Boston––London, 2001, 49–91.
• [14] J.D. Hardtke, Absolute sums of Banach spaces and some geometric properties related to rotundity and smoothness, Banach J. Math. Anal. 8 (2014), no. 1, 295–334.
• [15] M. Kato and T. Tamura, Weak nearly uniform smoothness and worth property of ψ-direct sums of Banach spaces, Comment. Math. 46 (2006), no. 1, 113–129.
• [16] M. Kato and T. Tamura, Weak nearly uniform smoothness of the ψ-direct sums (X1 … XN)ψ, Comment. Math. 52 (2012), no. 2, 171–198.
• [17] M. A. Khamsi, On uniform Opial condition and uniform Kadec-Klee property in Banach and metric spaces, J. Nonlinear Anal: Theory, Methods Appl. 26 (1996), no. 10, 1733–1748, DOI 10.1016/0362-546X(94)00216-5.
• [18] D. Kutzarova, S. Prus, and B. Sims, Remarks on orthogonal convexity of Banach spaces, Houston J. Math. 19 (1993), no. 4, 603–614.
• [19] K.S. Lau, Best approximation by closed sets in Banach spaces, J. Approx. Theory 23 (1978), 29–36.
• [20] H.J. Lee, M. Mar􀦠n, and J. Merí, Polynomial numerical indices of Banach spaces with absolute norms, Linear Algebra Appl. 435 (2011), 400–408, DOI 10.1016/j.laa.2011.01.037.
• [21] P. K. Lin, K. K. Tan, and H. K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, J. Nonlinear Anal: Theory, Methods Appl. 24 (1995), no. 6, 929–946, DOI 10.1016/0362-546X(94)00128-5.
• [22] A. Maji and P. D. Srivastava, On some geometric properties of generalized Musielak-Orlicz sequence spaces and corresponding operator ideals, Banach J. Math. Anal. 9 (2015), no. 4, 14–33, DOI 10.15352/bjma/09-4-2.
• [23] E.M. Mazcuñán-Navarro, On the modulus of u-convexity of Ji Gao, Abstr. Appl. Anal. 2003 (2003), no. 1, 49-54, DOI 10.1155/S1085337503204127.
• [24] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), no. 4, 591–597.
• [25] S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. Theory, Methods Appl. 18 (1992), no. 8, 697–704, DOI 10.1016/0362-546X(92)90165-B.
• [26] S. Prus, Geometrical background of metric fixed point theory, Handbook of Metric Fixed Point Theory (W. A. Kirk and B. Sims, eds.), Kluwer Academic Publishers, Dordrecht–Boston–London, 2001, 93–132.
• [27] S. Saejung, On the modulus of U-convexity, Abstr. Appl. Anal. 2005 (2005), no. 1, 59–66, DOI 10.1155/AAA.2005.59.
• [28] S. Saejung, Another look at Cesàro sequence spaces, J. Math. Anal. Appl. 366 (2010), 530–537, DOI 10.1016/j.jmaa.2010.01.029.
• [29] B. A. Sims, Orthogonality and fixed points of nonexpansive mappings, Proc. Centre Math. Anal. Austral. Nat. Univ. 20 (1988), 178–186.
• [30] B. A. Sims, A class of spaces with weak normal structure, Bull. Austral. Math. Soc. 50 (1994), 523–528, DOI 10.1017/S0004972700016634.
• [31] B. Tanbay, Direct sums and the Schur property, Turkish J. Math. 22 (1998), 349–354.