In this paper, we investigate a common fixed point problem for two pairs $\{f,S\}$ and $\{g,T\}$ of weakly compatible selfmaps of a complete b-metric $(X,d;s)$, satisfying a contractive condition of Ćirić type. This contraction and some of its variants were used in the paper [29] published in 2016 by V. Ozturk and S. Radenović, requiring the (E.A.)-property for the pairs $\{f,S\}$ and $\{g,T\}$. The aim of this paper is to provide some improvements to the main result of [29]. Our main theorem will improve certain results published in 2015, by V. Ozturk and D. Turkoglu (see [30] and [31]). We improve also results from other related papers (see the references herin). Indeed, we remove the (E.A.)-property and weaken certain assumptions imposed in these papers. So, our work aims to extend and unify, in one go, several common fixed point results known in a recent literature. We furnish two illustrative examples and we prove that the fixed point problem, considered here, for the pairs $\{f,S\}$ and $\{g,T\}$ is well-posed. We compare our main result with a recent result obtained in 2018 by N. Hussain, Z. D. Mitrović and S. Radenović in [19].