  The perturbation correction of general relativity has obtained the general
relationship between entropy and extremality. The relation was found to exactly
hold for the charged AdS black hole, and extended to the rotating and massive
gravity black holes. In this paper, We confirm the universal relation for the
four dimensional RN-AdS black hole surrounded by the quintessence, and find a
new extremality relation between black hole mass and the normalization factor
$a$ related to the density of quintessence. Besides, we also find that when the
black hole entropy increases, the correction constant is negative, the black
hole mass and the mass-charge ratio decreases, hence this black hole displays
WGC-like behavior. Moreover, we derive the general relationship on nonlinear
electrodynamic source couples to AdS spacetime with quintessence field. We find
another new extremality relation between black hole mass and the magnetic
monopole charge $\beta$.
