Let $F$ and $G$ be continuous, commuting mappings of a complete metric space $(X,d)$ into $B(X)$ satisfying the inequality \begin{align*} ẹlta(F^px,G^py)eq\max\{cẹlta&(F^rx,G^sy),frac12ẹlta(F^rx,F^{r'}x),frac12ẹlta(G^sy,G^{s'}y): &0eq r, seq p; 0eq r', s'<p\} \end{align*} for all $x,y$ in $X$, where $0\leq c<1$ and $p$ is a fixed positive integer. It is proved that if $F$ and $G$ also map $B(X)$ into itself, then $F$ and $G$ have a unique common fixed point $z$.