Let \(K\) be a bounded closed convex subset of a Banach space \(X\), having normal structure (i.e. for every closed convex \(H\subset K\) there is an \(x\in H\) such that \(\sup \{\| y-x\|:y\in H\}<\text{diam}\, H)\). The first theorem in the paper under review states that if \(K\) is weakly compact then under some ``nonexpansivity'' type assumption two selfmappings of \(K\) have a common fixed point. Using this theorem and the fact that every bounded closed convex subset \(K\) of a uniformly convex Banach space has normal structure the authors prove a theorem on the existence and uniqueness of a common fixed point of two selfmappings of \(K\). This result generalizes those of \textit{R. Kannan} [Fundam. Math. 70, 169--177 (1971; Zbl 0246.47065)], \textit{L. B. Ćirić} [Publ. Inst. Math., Beograd, n. Ser. 19(33), 43--50 (1975; Zbl 0325.47040)] and \textit{B. E. Rhoades} [Mat. Semin. Notes, Kobe Univ. 5, No. 1, 69--74 (1977; Zbl 0362.47025)].