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The definition of bipolar multiplicative metric space is introduced in this article, and in this space some properties are derived. Multiplicative contractions for covariant and contravariant maps are defined and fixed points are obtained. Also, some fixed point results of covariant and contravariant maps satisfying multiplicative contraction conditions are proved for bipolar multiplicative metric spaces. Moreover, Banach contraction principle and Kannan fixed point theorem are generalized.

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Common fixed point theorems for three self mappings satisfying generalized contractive conditions in cone metric spaces are derived. Also, some common fixed point results for two self mappings are deduced. Moreover, these all results generalize some important familiar results. Given example to illustrate our main result. Furthermore, an existence theorem for the common solution of the two Urysohn integral equations obtained by using our main result.

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In 1971 R. L. Carpenter proved that every derivation T on a semisimple commutative Frechet algebra A with identity is continuous. By relaxing the commutativity assumption on A and adding the surjectivity assumption on T, we derive a corresponding continuity result, for a new concept of almost derivations on Frechet algebras in this article. Also, it is further proved that every surjective almost derivation T on non commutative semisimple Frechet Q-algebras A with an additional condition on A, is continuous. Moreover, an example is provided to illustrate our main result.