In this paper we present the star formation properties of $\rm{I_{subaru}\leq}$ 23 X-ray detected AGN and non-AGN galaxies in the green valley with far-IR (FIR) emission, using data from the COSMOS field. We measured star formation rates (SFR) using FIR Herschel/PACS data and we observed the location of AGN and non-AGN galaxies on the main-sequence of star formation. We went a step further in analysing the importance of AGN in quenching star formation in the green valley, the region proposed to be the transitional phase in galaxy evolution where galaxies are moving from later- to earlier-types. We found that most of our green valley X-ray detected AGN with far-IR emission have SFRs higher than the ones of inactive galaxies at fixed stellar mass ranges, the result that is different when considering optical data. These FIR AGN have still very active star formation, being located either on or above the main-sequence of star formation (in total 82\% of our sample). Therefore, they do not show signs of star formation quenching, but rather its enhancement. Our results may suggest that for X-ray detected AGN with FIR emission if there is an influence of AGN feedback on the star formation in the green valley the scenario of AGN positive feedback seem to take place, rather than the negative one.

Accepted for publication in MNRAS