CONTRACTION MAPPING AND ITS APPLICATION IN MENGER SPACE

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.