Summary: In this paper, the concept of cone \(s\)-generalized \(b\)-metric space over Banach algebra and generalized \(T\)-contractions are introduced as a generalization of cone \(b\)-metric space over Banach algebra and \(T\)-contractions respectively. Also a similar generalization for \(b\)-metric space is obtained.
We give an affirmative answer to open problem posed by Z. Kadelburg et al. [Gulf J. Math. 3, No. 3, 57–67 (2015; Zbl 1389.54093)], by proving fixed-point theorems for generalized \(T\)-contractive mappings in such space. In addition some examples are given to illustrate that our generalization is a real generalization.