The aim of this paper is to consider a new approach for obtaining common fixed point theorems in metric spaces by subjecting the triangle inequality to a Lipschitz type condition. For values of the Lipschitz constant k < 1/3 the condition reduces to a Banach type contractive condition and we get the results known so far. However, values of k ≥ 1/3 yield new result. It may be observed that in the setting of metric spaces k ≥ 1/3 generally does not ensure the existence of fixed points and there is no known method for dealing these cases. In Theorem 1 and Theorem 2 we provide results under a new condition. In the last section of this paper (Theorem 3 and Theorem 4) by using the (E.A) property introduced by Aamri and Moutawakil [2] we extend the results obtained in Theorem 1 and Theorem 2.
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In the present paper we obtain a common fixed point theorem under a new contractive condition which is independent of the known contractive definitions. In the second fixed point theorem we study the dynamics of a class of functions induced by real numbers and then apply the result to obtain general tests for divisibility of numbers.
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