Gaoxun Zhang* | Honglei Zhang
OPEN ACCESS
The notion of probabilistic normed space has been redefined by C. Alsina, B. Schweizer and A. Sklar [2]. But the results about the continuous operator in this space are not many. In this paper, we study B-contractions, H-contractions and strongly ε-continuous mappings and their respective relation to the strongly continuous mappings, and give some fixed-point theorems in this space.
Probabilistic Normed (PN) Space, Fixed-point theorem, Strongly ε-continuous.
This work is supported by the Doctoral Program Research Foundation of Southwest University of Science and Technology (15zx7139).
1. A. N. Šerstnev, On the motion of a random normed space, 1963, Dokl. Akad. Nauk. SSSR., no. 149, pp. 280-283.
2. C. Alsina, B. Schweizer, A. Sklar, On the definition of a probabilistic normed space, 1993, Aequationes Math., vol. 46, pp. 91-98.
3. C. Alsina, B. Schweizer, A. Sklar, Continuity properties of probabilistic norms, 1997, J. Math. Anal. Appl., vol. 208, pp. 446-452.
4. B. Lafuerza Guillén, C. Sempi, G. Zhang, M. Zhang, Countable products of probabilistic normed spaces, 2009, Nonlinear Analysis., vol. 71, pp. 4405-4414.
5. B. Lafuerza Guillén, J.A. Rodríguez Lallena, C. Sempi, Completion of probabilistic normed spaces, 1995, Internat. J. Math. and Math. Sci., vol. 18, no.4, pp. 649-652.
6. B. Lafuerza Guillen, J.A. Rodríguez Lallena, C. Sempi, A study of boundedness in probabilistic normed spaces, 1999, J. Math. Anal. Appl., vol. 232, pp. 183-196.
7. B. Schweizer, A. Sklar, Probabilistic metric spaces, 1983, New York, Elsevier North-Holland.
8. G. Zhang, M. Zhang, On the normability of generalized Šerstnev PN spaces, 2006, J. Math. Anal. Appl., vol. 340, pp.1000-1011.
9. M. Rafi, M.S.M. Noorani, Approximate fixed-point theorem in probabilistic normed (Metric) spaces, 2006, Proceedings of the 2end IMT-GT Regional Conference on Mathematics, Statistics and Applicatons Universiti Sains, 2006, Malaysia, Penang, pp. 13-15.
10. D. Guo, Nonlinear functional analysis, 1985, Shandong Sci. and Tech. Press.