Summary: In this article, we point out that some recent results proved in [

*G. V. R. Babu* and

*G. N. Alemayehu*, Appl. Math. Lett. 24, No. 6, 975–981 (2011;

Zbl 1261.54033)] are corollaries of the main result of [

*J. Ali* and

*M. Imdad*, Sarajevo J. Math. 4(17), No. 2, 269–285 (2008;

Zbl 1180.54052)]. However, combining the notions of

$R$-weakly commuting property of type

$\left(Ag\right)$ (due to

*H. K. Pathak* et al. [Bull. Korean Math. Soc. 34, No. 2, 247–257 (1997;

Zbl 0878.54032)]) together with sequential continuity of type

$\left(Ag\right)$ (to be introduced in this note) (also, alternately,

$R$-weakly commuting property of type

$\left(Af\right)$ (due to Pathak et al.) together with sequential continuity of type

$\left(Af\right)$ (to be introduced in this note)), these results are improved to yield possibly new results in metric fixed point theory.