This is the 344th most frequent Serbian word.
својих
"Својих" means "of one's own" in English.
In this sentence, 'својих' is the genitive plural form of 'свој' and refers to possessions or qualities belonging to the subject (his own memories).
Он је поносан на своје успомене из детињства.
He is proud of his own childhood memories.
Here, 'својих' also reflects ownership, in the dative plural form, indicating that the care is devoted to her own dogs.
Она брине о својим псима сваког дана.
She takes care of her own dogs every day.
In this context, 'својих' is the accusative plural form, indicating that the action (supporting) is directed towards their own children.