Does equip ability trigger summoning sickness if the equipped creature was already on the battlefield?

Summoning sickness (the restriction on attacking and using tap-symbol abilities) is determined solely by whether a creature has been under your continuous control since the beginning of your most recent turn. This is governed by CR 302.6, which states a creature cannot attack or use tap-symbol activated abilities unless it has been under its controller's control continuously since the start of their most recent turn — or has haste.

Equipping a creature (CR 702.6) simply attaches an Equipment permanent to a creature. The act of equipping does not cause the creature to 'enter the battlefield' again, does not change who controls it, and does not reset any continuous-control clock. Therefore, using an equip ability on a creature that is already on the battlefield has absolutely no effect on that creature's summoning sickness status.

Conversely, if a creature is already summoning-sick (it entered the battlefield this turn), equipping it with a sword or shield does not cure that sickness unless the Equipment somehow grants haste. The creature still cannot attack or use tap-symbol abilities until your next turn (unless it has haste from another source).

Concrete example: You cast a Grizzly Bears on your turn. It enters with summoning sickness. You pay the equip cost and attach a Longsword to it. Grizzly Bears still cannot attack this turn, because it entered the battlefield this turn regardless of the Equipment attached to it.

Summary: equipping never interacts with summoning sickness rules (CR 302.6, CR 702.6b). The only ways to bypass summoning sickness are having the haste ability (CR 702.10) or having been on the battlefield under your control since the start of your turn.