Some fixed point therems with R transitive binary relation on Branciari metric spaces.

What is it about?

In this paper, by using certain auxiliary functions, we prove some fixed point theorems in the context of Branciari metric spaces (X,d) where X is equipped with a transitive binary relation R.

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Why is it important?

At the first glance, Branciari metric space seems a generalization of standard metric, but it is not. Indeed, the topology of Branciari metric space is not compatible with the topology of standard metric. This is the most interesting part of this space but not the least. For example, due to weakness of the topology of Branciari metric space, uniqueness of the limit is not necessarily true; or continuity of the distance function is not true in general. In such a weak abstract space, we also consider a transitive binary relation R on the space. The standard example of the transitive binary relation R is partial order. Thus, our framework becomes more interesting. In this frame, we prove some fixed point theorems via auxiliary functions.

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