## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 319394, 6 pages.

### Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces

#### Abstract

Kirk and Xu studied the existence of fixed points of asymptotic pointwise nonexpansive mappings in the Banach space. In this paper, we investigate these kinds of mappings in modular spaces. Moreover, we prove the existence of fixed points of asymptotic pointwise nonexpansive mappings in modular spaces. The results improve and extend the corresponding results of Kirk and Xu (2008) to modular spaces.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 319394, 6 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357180288

Digital Object Identifier
doi:10.1155/2012/319394

Mathematical Reviews number (MathSciNet)
MR2915721

Zentralblatt MATH identifier
1318.47072

#### Citation

Wang, Xue; Chen, Ying; Chen, Rudong. Fixed Points of Asymptotic Pointwise Nonexpansive Mappings in Modular Spaces. J. Appl. Math. 2012, Special Issue (2012), Article ID 319394, 6 pages. doi:10.1155/2012/319394. https://projecteuclid.org/euclid.jam/1357180288

#### References

• H. Nakano, Modulared Semi-Ordered Spaces, Tokyo, Japan, 1950.
• J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, pp. 49–65, 1959.
• A. Razani, E. Nabizadeh, M. Beyg Mohamadi, and S. Homaei Pour, “Fixed points of nonlinear and asymptotic contraction in the modular spaces,” Abstract and Applied Analysis, vol. 2007, Article ID 40575, 10 pages, 2007.
• M. A. Khamsi, “Quasi-contraction mappings in modular spaces without ${\Delta }_{2}$-condition,” Fixed Point Theory and Applications, vol. 2008, Article ID 916187, 6 pages, 2008.
• K. Kuaket and P. Kumam, “Fixed points of asymptotic pointwise contractions in modular spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1795–1798, 2011.
• W. A. Kirk and H.-K. Xu, “Asymptotic pointwise contractions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 12, pp. 4706–4712, 2008.
• K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.
• F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.
• N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4423–4429, 2009.
• J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983.
• M. A. Khamsi and W. M. Kozlowski, “On asymptotic pointwise nonexpansive mappings in modular function spaces,” Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 697–708, 2011.
• W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type,” Israel Journal of Mathematics, vol. 17, pp. 339–346, 1974.