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# On Fixed Point Theorem of Self Map in Fuzzy Metric Spaces

 Prapoorna Manthena , M. Rangamma Department of Mathematics, University College of Science, Osmania University, Hyderabad, India Related article at Pubmed, Scholar Google

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## Abstract

In this paper, we prove a fixed point theorem for a self map in fuzzy metric space where the map satisfies a different condition.

### Keywords

t – norm, fuzzy metric space, cauchy sequence, complete fuzzy metric space.

### INTRODUCTION

The fundamental work for the fuzzy theory was first given by Zadeh [9] in 1965, who introduced the concept of fuzzy set. Kramosil and Michalek [4] developed the fuzzy metric space and later George and Veeramani [1] modified the notion of fuzzy metric spaces by introducing the concept of continuous t – norm. Many researchers have enormously developed the theory by defining different concepts and blending many properties. Fuzzy set theory has its significance in various fields such as communication, gaming, signal processing, modelling theory, image processing, etc. The purpose of this work is to prove the existence of a fixed point of a self map in fuzzy metric space (in the context of George and Veeramani [1]) by using certain condition on the mapping.

### CONCLUSION

In this article, we have proved the existence of a fixed point in a fuzzy metric space and showed that the fixed point obtained will be unique if it further satisfies an additional condition.

### References

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4. Kramosil, I and Michalek, J, “Fuzzymetric and Statisticalmetricspaces”, Kybernetica , Vol. 11, pp. 326 – 334, 1975.

5. MaggieAphane, “Onsomeresults of analysis in metricspaces and fuzzymetricspaces”, dissertation, University of South Africa, 2009.

6. Mihet, D, “A Banachcontractiontheorem in fuzzymetricspaces”, Fuzzy Sets and Systems, Vol. 144, pp. 431 – 439, 2004.

7. Schweizer, B and Sklar, A, “Statisticalmetricspaces”, Pacific Journal of Mathematics, Vol. 10, pp. 313 – 334, 1960.

8. YonghongShen, Dong Qiu and WeiChen, “Fixedpointtheorems in fuzzymetricspaces”, AppliedMathematicsLetters, Vol. 25,pp. 138 – 141, 2012.

9. Zadeh, L. A, “Fuzzy sets” , Inform. Control, Vol. 8, pp. 338 – 353, 1965.