Abstract
We give relationships between some Banach-space geometric properties that guarantee the weak fixed point property. The results extend some known results of Dalby and Xu.
Open Access
A note on properties that imply the fixed point property
Received07 Jan 2005
Accepted04 Mar 2005
Published19 Feb 2006
References
A. G. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Universitext, Springer, New York, 1990.
View at: Zentralblatt MATH | MathSciNetJ. A. Clarkson, “The von Neumann-Jordan constant for the Lebesgue spaces,” Annals of Mathematics. Second Series, vol. 38, no. 1, pp. 114–115, 1937.
View at: Google Scholar | Zentralblatt MATH | MathSciNetT. Dalby, “Relationships between properties that imply the weak fixed point property,” Journal of Mathematical Analysis and Applications, vol. 253, no. 2, pp. 578–589, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. M. Day, R. C. James, and S. Swaminathan, “Normed linear spaces that are uniformly convex in every direction,” Canadian Journal of Mathematics, vol. 23, pp. 1051–1059, 1971.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. Dhompongsa, P. Piraisangjun, and S. Saejung, “Generalised Jordan-von Neumann constants and uniform normal structure,” Bulletin of the Australian Mathematical Society, vol. 67, no. 2, pp. 225–240, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetT. Dominguez Benavides, “A geometrical coefficient implying the weak fixed point property and stability results,” Houston Journal of Mathematics, vol. 22, no. 4, pp. 835–849, 1996.
View at: Google ScholarJ. Gao and K.-S. Lau, “On the geometry of spheres in normed linear spaces,” Australian Mathematical Society. Journal. Series A, vol. 48, no. 1, pp. 101–112, 1990.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. Gao and K.-S. Lau, “On two classes of Banach spaces with uniform normal structure,” Studia Mathematica, vol. 99, no. 1, pp. 41–56, 1991.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. García-Falset, “Stability and fixed points for nonexpansive mappings,” Houston Journal of Mathematics, vol. 20, no. 3, pp. 495–506, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. García-Falset, “The fixed point property in Banach spaces with the NUS-property,” Journal of Mathematical Analysis and Applications, vol. 215, no. 2, pp. 532–542, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. García-Falset, E. Llorens-Fuster, and E. M. Mazcuñán-Navarro, “Banach spaces which are -uniformly noncreasy,” Nonlinear Analysis, vol. 53, no. 7-8, pp. 957–975, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. García-Falset and B. Sims, “Property and the weak fixed point property,” Proceedings of the American Mathematical Society, vol. 125, no. 10, pp. 2891–2896, 1997.
View at: Google Scholar | Zentralblatt MATH | MathSciNetA. L. Garkavi, “On the optimal net and best cross-section of a set in a normed space,” Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, vol. 26, pp. 87–106, 1962 (Russian), translated in American Mathematical Society Transactions Series \textbf {39} (1964), 111–131.
View at: Google Scholar | Zentralblatt MATH | MathSciNetK. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.
View at: Zentralblatt MATH | MathSciNetN. J. Kalton, “-ideals of compact operators,” Illinois Journal of Mathematics, vol. 37, no. 1, pp. 147–169, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNetM. Kato, L. Maligranda, and Y. Takahashi, “On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces,” Studia Mathematica, vol. 144, no. 3, pp. 275–295, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetW. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965.
View at: Google Scholar | Zentralblatt MATH | MathSciNetW. A. Kirk, “Nonexpansive mappings in product spaces, set-valued mappings and -uniform rotundity,” in Nonlinear Functional Analysis and Its Applications (Berkeley, Calif, 1983), F. E. Browder, Ed., vol. 45, Part 2 of Proceedings of Symposia in Pure Mathematics, pp. 51–64, American Mathematical Society, Rhode Island, 1986.
View at: Google Scholar | Zentralblatt MATH | MathSciNetP.-K. Lin, K.-K. Tan, and H. K. Xu, “Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings,” Nonlinear Analysis, vol. 24, no. 6, pp. 929–946, 1995.
View at: Google Scholar | Zentralblatt MATH | MathSciNetE. M. Mazcuñán-Navarro, Geometry of Banach spaces in metric fixed point theory, M.S. thesis, Department of Mathematical Analysis, Universitat de Valencia, Valencia, 1990.
Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 73, pp. 591–597, 1967.
View at: Google Scholar | Zentralblatt MATH | MathSciNetB. Sims, “Ultra”-Techniques in Banach Space Theory, vol. 60 of Queen's Papers in Pure and Applied Mathematics, Queen's University, Ontario, 1982.
View at: Zentralblatt MATH | MathSciNetB. Sims, “Orthogonality and fixed points of nonexpansive maps,” in Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), vol. 20 of Proceedings of the Centre for Mathematics and its Applications, Australian National University, pp. 178–186, Australian National University, Canberra, 1988.
View at: Google Scholar | Zentralblatt MATH | MathSciNetB. Sims, “A class of spaces with weak normal structure,” Bulletin of the Australian Mathematical Society, vol. 49, no. 3, pp. 523–528, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNetB. Sims, “Banach space geometry and the fixed point property,” in Recent Advances on Metric Fixed Point Theory (Seville, 1995), vol. 48 of Ciencias, pp. 137–160, Universidad de Sevilla, Seville, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNetF. Sullivan, “A generalization of uniformly rotund Banach spaces,” Canadian Journal of Mathematics, vol. 31, no. 3, pp. 628–636, 1979.
View at: Google Scholar | Zentralblatt MATH | MathSciNetH. K. Xu, G. Marino, and P. Pietramala, “On property and its generalizations,” Journal of Mathematical Analysis and Applications, vol. 261, no. 1, pp. 271–281, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2006 S. Dhompongsa and A. Kaewkhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.