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Fixed point iterations of some contraction mappings with variations in cone metric space (CMS) domains have been discussed in this paper. More specifically, an increasing sequence of subsets of an order complete CMSs has been considered such that one member of the sequence is mapped into the next member of the sequence by a map for which a contraction condition is satisfied. The usual iteration method has been considered to establish generalized fixed point results in CMSs. Furthermore, some examples are provided to illustrate our main results.

In this article, some fixed point theorems of contraction mappings on cone metric spaces are obtained. Moreover, the concept of weak convergence in metric spaces is extended to cone metric spaces, and some results on weak order convergence of fixed point iterations of various types of contraction mappings on cone metric spaces are obtained.

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In 1971 R. L. Carpenter proved that every derivation on a semisimple commutative Frechet algebra with identity is continuous. The concept of almost derivations on Frechet algebras is introduced in this article. Also, R. L. Carpenter result motivates us to ask an open question: Is every almost derivation on semisimple commutative Frechet algebras continuous?. Moreover, a partial answer to this open question is derived in the sense that every almost derivation T on semisimple commutative Frechet Q-algebras A, with an additional condition on A, is continuous. Furthermore, an example is provided to illustrate our main result.