This paper aims at demonstrating, for the first time, very probable universal peculiarities of the evolution of stars in the lower red giant branch (RGB) of Galactic globular clusters (GCs), reflected in two corresponding dips in the luminosity functions (LFs). By relying on the database of Hubble Space Telescope photometry of GCs, we analyze the lower RGB LFs of a sample of 18 GCs in a wide metallicity range, Delta[Fe/H] approximate to 1.9 dex. We first show that in the F555W-(F439W-F555W) color-magnitude diagrams (CMDs), the lower RGB of GCs, except for the most metal-poor of them, frequently shows an apparent "knee." It reveals itself as a fairly abrupt change of the RGB slope. At the same luminosity level, the RGB LFs show a feature in the form of a more or less pronounced dip. We find that the magnitude difference between the RGB base and the given feature is, on average, around Delta F555W(base)(dip) approximate to 1.4 mag. It shows a marginal variation with metallicity, if any, comparable to the error. At the same time, the magnitude difference between the dip and the RGB bump, Delta F555W(dip)(bump), decreases with increasing metallicity and falls within the range 0.8 less than or similar to Delta F555W(dip)(bump) less than or similar to 1.7 mag. Generalized LFs (GLFs) have been obtained for three subsamples of GCs within limited metallicity ranges and with different horizontal branch (HB) morphology. They reproduce the "knee-related" dip that is statistically significant in two of the GLFs. This feature turns out to be more pronounced in the GLFs of GCs with either the blue or red HB morphology than with the intermediate one. The same GLFs also reveal an additional probable universal dip. It shows up below the RGB bump at. F555W slightly increasing from similar to 0.3 to similar to 0.5 mag with increasing metallicity. Also, the statistical significance of this "prebump" dip increases, on average, toward higher metallicity. Except for the well known RGB bump, no other universal features corresponding to those found here were so far empirically revealed or theoretically predicted in the lower RGB of GCs.