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Contraction Mapping And Its Application In Menger Space

Authors: Piyush Kumar Tripathi*, Suyash Narayan Mishra;

Contraction Mapping And Its Application In Menger Space

Abstract

The probabilistic analogue of the Banach contraction principle as given by Sehgal and Bharucha Reid states that a contraction mapping on a complete Menger Space (X, F, min) has a unique fixed point ((X, , min) = (X, F, t) where t(a,b)= min{a,b}). Later it was realized that t-norm ‘min’ could be replaced by weaker t-norms. Sherwood showed that the above result is an exception rather than a rule: specifically for any Archimedean t-norm, there exists a complete Menger space and a contraction by Sehgal on (X, F) which has no fixed point. In this paper some fixed point theorem established in Menger space by using new concept of dual contraction.

Keywords

Contraction Mapping, Application In Menger Space.

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