The following fixed point theorem has been proved in [\textit{J. Achari} and \textit{B. K. Lahiri}, Riv. Mat. Univ. Parma, IV. Ser. 6, 161-165 (1980; Zbl 0463.47038)]. Theorem: Let \(X\) be a reflexive Banach space and \(K\) be a non-empty closed convex bounded subset of \(X\). Suppose that \(T\) is a mapping of \(K\) into itself such that (A) \(\| Tx-Ty\| \leq \max \{\| x-y\|;\| x-Tx\|;\| y- Ty\| \}\) for every \(x,y\in K,\) (B) \(K\) has normal structure, (C) \(\sup_{y\in F}\| y-Ty\| \leq \delta(F)\) for every non-empty closed convex subset \(F\) of \(K\), containing more than one element and mapped into itself by \(T\). Then \(T\) has a fixed point in \(K\). In this note we prove the above theorem by relaxing the condition (C) by \((C^ 1)\sup_{y\in F}\| y-Ty\| <\delta (F)\) for every non-empty closed convex subset \(F\) of \(K\), containing more than one element and mapped into itself by \(T\).