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# GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES

Abstract

Given two (real) normed (linear) spaces $X$ and $Y$ , let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$ , where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$ . It is known that $X\otimes _{1}Y$ is $2$ -UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$ -dimensional and $Y$ is $k$ -UR, then $X\otimes _{1}Y$ is $(m+k)$ -UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$ -UR, then both $X$ and $Y$ are $(k-1)$ -UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$ , we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$ . It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$ -uniform rotundity relative to every $k$ -dimensional subspace we show that this result holds with a weaker condition on $X$ .

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Some of the results in this article are part of the author’s PhD thesis written at the Indian Institute of Technology Madras, Chennai, India with the financial support of the Council of Scientific and Industrial Research, New Delhi, India.

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[1] Alspach, D. E., ‘A fixed point free nonexpansive map’, Proc. Amer. Math. Soc. 82(3) (1981), 423424.
[2] Amir, D. and Ziegler, Z., ‘Relative Chebyshev centers in normed linear spaces. II’, J. Approx. Theory 38(4) (1983), 293311.
[3] Casini, E., ‘Degree of convexity and product spaces’, Comment. Math. Univ. Carolin. 31(4) (1990), 637641.
[4] Clarkson, J. A., ‘Uniformly convex spaces’, Trans. Amer. Math. Soc. 40(3) (1936), 396414.
[5] Day, M. M., James, R. C. and Swaminathan, S., ‘Normed linear spaces that are uniformly convex in every direction’, Canad. J. Math. 23 (1971), 10511059.
[6] Garkavi, A. L., ‘On the optimal net and best cross-section of a set in a normed space’, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 87106.
[7] Geremia, R. and Sullivan, F., ‘Multidimensional volumes and moduli of convexity in Banach spaces’, Ann. Mat. Pura Appl. (4) 127 (1981), 231251.
[8] Goebel, K., ‘On the structure of minimal invariant sets for nonexpansive mappings’, Ann. Univ. Mariae Curie-Skłodowska Sect. A 29 (1975), 7377.
[9] Goebel, K. and Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, 28 (Cambridge University Press, Cambridge, 1990).
[10] Karlovitz, L. A., ‘Existence of fixed points of nonexpansive mappings in a space without normal structure’, Pacific J. Math. 66(1) (1976), 153159.
[11] Kirk, W. A., ‘A fixed point theorem for mappings which do not increase distances’, Amer. Math. Monthly 72 (1965), 10041006.
[12] Kuczumow, T., Reich, S. and Schmidt, M., ‘A fixed point property of 1 -product spaces’, Proc. Amer. Math. Soc. 119(2) (1993), 457463.
[13] Lin, P. K., ‘ k-uniform rotundity is equivalent to k-uniform convexity’, J. Math. Anal. Appl. 132(2) (1988), 349355.
[14] Llorens-Fuster, E., ‘The fixed point property for renormings of 2 ’, Arab. J. Math. 1(4) (2012), 511528.
[15] Milman, V. D., ‘Geometric theory of Banach spaces. II. Geometry of the unit ball’, Uspekhi Mat. Nauk 26(6(162)) (1971), 73149.
[16] Sullivan, F., ‘A generalization of uniformly rotund Banach spaces’, Canad. J. Math. 31(3) (1979), 628636.
[17] Tan, K. K. and Xu, H. K., ‘On fixed point theorems of nonexpansive mappings in product spaces’, Proc. Amer. Math. Soc. 113(4) (1991), 983989.
[18] Veena Sangeetha, M., ‘On relative -uniform rotundity, normal structure and fixed point property for nonexpansive maps’, Preprint.
[19] Veena Sangeetha, M. and Veeramani, P., ‘Uniform rotundity with respect to finite-dimensional subspaces’, J. Convex Anal. 25(4) (2018), 12231252.
[20] Wiśnicki, A., ‘On the fixed points of nonexpansive mappings in direct sums of Banach spaces’, Studia Math. 207(1) (2011), 7584.
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Bulletin of the Australian Mathematical Society
• ISSN: 0004-9727
• EISSN: 1755-1633
• URL: /core/journals/bulletin-of-the-australian-mathematical-society
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