T-normed vector spaces and fixed point theorems via a binary relation

In this paper, we introduce the notion of [Formula: see text]-normed vector spaces by extending normed vector spaces. This concept can be considered the first generalization of normed vector spaces satisfying the [Formula: see text]-separation axiom. Using this axiom, some fixed point theorems are proved via a binary relation in the setting of [Formula: see text]-normed vector spaces without using neither the compactness nor the uniform convexity. Furthermore, some examples are given to show the superiority of the proven results.