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In the present work we solve the problem of finding the fuzzy distance between two subsets of a fuzzy metric space for which we use a non-self fuzzy contraction mapping from one set to the other. It is a fuzzy extension of the proximity point problem which is by its nature a problem of global optimization. The contraction is defined here by two control functions. We define a geometric property of the fuzzy metric space. The main result is illustrated with an example. Our result extends a fuzzy version of the Banach contraction mapping principle.

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In this paper we investigate Hyers-Ulam-Rassias stability of certain nonlinear functional equations. Considerations of such stabilities in different branches of mathematics have been very extensive. Again the fuzzy concepts along with their several extensions have appeared in almost all branches of mathematics. Here we work on intuitionistic fuzzy real Banach spaces, which is obtained by combining together the concepts of fuzzy Banach spaces with intuitionistic fuzzy sets. We establish that pexiderized quadratic functional equations defined on such spaces are stable in the sense of Hyers-Ulam-Rassias stability. We adopt a fixed point approach to the problem. Precisely, we use a generxalized contraction mapping principle. The result is illustrated with an example.

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