# Article

Full entry | PDF   (0.2 MB)
Keywords:
fixed point; modular spaces; $\rho$-nonexpansive mapping; $\rho$-normal structure; $\rho$-uniform normal structure; $\rho _r$-uniformly convex
Summary:
In this paper, we extend several concepts from geometry of Banach spaces to modular spaces. With a careful generalization, we can cover all corresponding results in the former setting. Main result we prove says that if $\rho$ is a convex, $\rho$-complete modular space satisfying the Fatou property and $\rho _r$-uniformly convex for all $r>0$, C a convex, $\rho$-closed, $\rho$-bounded subset of $X_\rho$, $T:C\rightarrow C$ a $\rho$-nonexpansive mapping, then $T$ has a fixed point.
References:
[1] Aksoy A. G., Khamsi M. A.: Nonstandard methods in fixed point theory. Spinger-Verlag, Heidelberg, New York 1990. MR 1066202 | Zbl 0713.47050
[2] Ayerbe Toledano J. M., Dominguez Benavides T., and López Acedo G.: Measures of noncompactness in metric fixed point theory: Advances and Applications Topics in metric fixed point theory. Birkhäuser-Verlag, Basel, 99 (1997). MR 1483889
[3] Chen S., Khamsi M. A., Kozlowski W. M.: Some geometrical properties and fixed point theorems in Orlicz modular spaces. J. Math. Anal. Appl. 155 No. 2 (1991), 393–412. MR 1097290
[4] Dominguez Benavides T., Khamsi M. A., Samadi S.: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Analysis 40 No. 2 (2001), 267–278. MR 1849794
[5] Goebel K., Kirk W. A.: Topic in metric fixed point theorem. Cambridge University Press, Cambridge 1990. MR 1074005
[6] Goebel K., Reich S.: Uniform convexity, Hyperbolic geometry, and nonexpansive mappings. Monographs textbooks in pure and applied mathematics, New York and Basel, 83 1984. MR 0744194 | Zbl 0537.46001
[7] Khamsi M. A.: Fixed point theory in modular function spacesm. Recent Advances on Metric Fixed Point Theorem, Universidad de Sivilla, Sivilla No. 8 (1996), 31–58. MR 1440218
[8] Khamsi M. A.: Uniform noncompact convexity, fixed point property in modular spaces. Math. Japonica 41 (1) (1994), 1–6. MR 1305537 | Zbl 0820.47063
[9] Khamsi M. A.: A convexity property in modular function spaces. Math. Japonica 44, No. 2 (1990). MR 1416264
[10] Khamsi M. A., Kozlowski W. M., Reich S.: Fixed point property in modular function spaces. Nonlinear Analysis, 14, No. 11 (1990), 935–953. MR 1058415
[11] Kumam P.: Fixed Point Property in Modular Spaces. Master Thesis, Chiang Mai University (2002), Thailand.
[12] Megginson R. E.: An introduction to Banach space theory. Graduate Text in Math. Springer-Verlag, New York 183 (1998). MR 1650235 | Zbl 0910.46008
[13] Musielak J.: Orlicz spaces and Modular spaces. Lecture Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York 1034 (1983). MR 0724434 | Zbl 0557.46020
[14] Musielak J., Orlicz W.: On Modular spaces. Studia Math. 18 (1959), 591–597. MR 0101487 | Zbl 0099.09202
[15] Nakano H.: Modular semi-ordered spaces. Tokyo, (1950).

Partner of