Generally, Vaishnavas follow the Pancharatra Agamas, Shaivites follow the Shaiva Agamas, and Shaktas follow the Shaiva Agamas. But long ago, there was a group of people in Kashmir who followed the Shakta Agamas but adopted Shiva as their Ishta Devata. That group evolved into what we now call Kashmir Shaivism. Because it has its roots in Shakta Agamas, it subscribes to a monistic philosophy, as opposed to philosophic Shaivite sects like Shaiva Siddhanta which are more dualistic in their philosophy.
Now the most famous Kashmiri Shaivite philosopher was Abhinavagupta, who lived in the 10th century. Abhinavagupta's guru was Lakshmanagupta, whose guru was Utpaladeva, whose guru was Somananda. Now Somananda was the shishya of Vasugupta, the author of the Shiva Sutras, the defining text of Kashmiri Shaivism. But that's not the only lineage Somananda is associated with. Somananda is also claimed to be a biological descendant of Durvasa's son Tryamabaka, who is also revered as a Kashmiri Shaivite Acharya. Here is what Abhinavagupta says about Tryambaka in this excerpt from his Tantraloka:
As, thus, with the disappearance of the Shāstras the world became engrossed in spiritual darkness, Shiva-as the Deity is called-took pity on men and, appearing on the Kailāsa mountain in the form of Shrikantha, commanded the Sage Durvasas to spread in the world the knowledge of these Shāstras again. Durvasas, thus commanded, created, by the power of his mind, three sons - Tryambaka, Amardaka and Shrinātha by names - whom he charged with the mission of establishing spiritual order and of teaching men again the ancient and eternal Shaiva faith and doctrine in their three aspects of Abheda, Bheda and Bhedabheda-of Unity, Diversity and Diversity-in-unity. Tryambaka was to teach the first, Amardaka the second, while Shrinatha was to have the charge of the last, It is this Abheda or Advaya Shaiva teaching, thus retaught to the world by Tryambaka, which is spoken of as the Trika.
Trika is basically what we now call the Kashmiri Shaivism. In any case, the idea is that Tryambaka promulgated the 64 Bhairava Tantras which Kashmiri Shaivites focus on, whereas Amardaka and Srinatha promulgated the 28 traditional Shaiva Agamas which philosophical Shaivite sects like Shaiva Siddhanta are based on.
But my question is, what scriptures describe Durvasa's mind-born sons Tryambaka, Amardaka, and Srinatha? Do any Puranas or Agamas mention them?
The only putative sons of Durvasa I know of are Ilvala and Vatapi who were killed by the sage Agastya. Ilvala and Vatapi are mentioned in Hindu scripture, but their being sons of Durvasa may just be folklore.