Contraction maps on probabilistic metric spaces. (English) Zbl 0773.54033
The well-known Banach contraction principle states that every contraction mapping on a complete metric space has a unique fixed point. However, an exact analogue of this result is not true in general in probabilistic metric space (PM-space). In this paper, by imposing a growth condition on the distance distribution function, it is proved that for a large class of triangle functions, contraction mappings on complete PM-spaces have a unique fixed point.
Reviewer: S.N.Mishra (Umtata / Transkai)
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E70 | Probabilistic metric spaces |
| 47H10 | Fixed-point theorems |
Keywords:
Banach contraction principle; unique fixed point; distance distribution function; triangle function; contraction mappings; complete PM-spacesReferences:
| [1] | Apostol, T. M., Mathematical Analysis (1957), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0126.28202 |
| [2] | Feller, W., (Introduction to Probability Theory and Its Applications, Vol. 2 (1971), Wiley: Wiley New York) · Zbl 0219.60003 |
| [3] | Hadzic, O., A generalization of the contraction principle in PM spaces. Review of research, Zb. Rad. (Kragujevac), 10, 13-21 (1980) · Zbl 0499.47038 |
| [4] | Hicks, T. L., Fixed point theory in probabilistic metric spaces, (Review of Research Faculty of Science, Vol. 13 (1983), University of Novi Sad: University of Novi Sad Novi Sad), 63-72 · Zbl 0574.54044 |
| [5] | Knopp, K., Theory and Application of Infinite Series (1944), Black: Black London · JFM 54.0222.09 |
| [6] | Moynihan, R., Infinite \(τ_T\) products of probability distribution functions, J. Austral. Math. Soc. Ser. A, 26, 227-240 (1978) · Zbl 0385.60036 |
| [7] | Radu, V., Some fixed point theorems in probabilistic metric spaces, (Lecture Notes in Math., Vol. 1233 (1987), Springer-Verlag: Springer-Verlag New York), 125-133 · Zbl 0622.60008 |
| [8] | Schweizer, B.; Sherwood, H.; Tardiff, R., Comparing two contraction notions on probabilistic metric spaces, Stochastica, 12 (1988) · Zbl 0689.60019 |
| [9] | Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), North-Holland: North-Holland New York · Zbl 0546.60010 |
| [10] | Sehgal, V. M.; Bharucha-Reid, A. T., Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory, 6, 97-102 (1972) · Zbl 0244.60004 |
| [11] | Sherwood, H., Complete probabilistic metric spaces, Z. Wahrsch. Verw. Geb., 20, 117-128 (1971) · Zbl 0212.19304 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
