Through context and also the definition of the word perpetuate, we may conclude that the best synonym would be: preserving.
Self-perpetuating is something that has the ability of continuing (preserving) its own existence without any external interference. It's something that creates the conditions for its own continuity. Another possible definition of self-perpetuating is something that prevents change and innovation by constantly creating the same old conditions - in other words, by preserving the same conditions.
In the excerpt, the cherished and self-perpetuating problem referring to the American culture was how women were taught to think, what they were told to talk about. It was a self-preserving problem since women themselves, for over fifteen years, did nothing or little to change such topics or expectations: children, house chores, happy husbands... The problem didn't need any external motivation, it was perpetuated from the inside.
