On the first day of Succot, one must fulfil the Misva of Lulav and Etrog with a set that belongs to him. This is derived from the Pasuk: "V’Lakachtem LA’CHEM B’yom Harishon. (And you shall take for yourself on the first day…)." "Lachem" (for yourself) implies "MiShelechem" (belonging to you). Therefore, a person may not use a borrowed set to fulfil his obligation. If someone doesn’t own a set, he must acquire it from his friend as a gift (with a stipulation that he return it). Accordingly, it is problematic to give a child a set to use on Yom Tob. The child has sufficient Da’at (Halachic competence) to receive the gift and become the owner as required, but he does not have the Da’at to perform the transaction of giving it back. Therefore, one should first fulfil the Misva himself and only then give it to a minor to use.
The following case happened in Kahal Kadosh Ateret Torah on the first day of Succot. A father fulfilled the Misva of Lulav and Etrog and then gave it to his son, as stated above. However, the question arose as to how he will fulfill his Misva the next day, the second day of Yom Tob, which is celebrated in the diaspora. The Etrog will remain the property of his son and many authorities hold that the requirement of owning the Arba’at Haminim applies to the second day as well.
While he was given the answer that there is nothing to be done, and he may not use his Etrog, Hacham Ovadia (Hazon Ovadia p. 399 in the footnote) found a way to solve the problem. He cites the minority opinion of the Ran that a child over six, who has reached the age referred to as "Peutot," has the Halachic competence to perform a transaction of giving. While this is not the accepted opinion, it is significant enough to create a Safek Sefeka (Double Doubt). Perhaps the second day is actually Hol Hamoed; and even if it is Yom Tob and ownership of the Etrog is required, perhaps the Halacha is in accordance with the Ran that a six year old child can make a transaction. Based on this, he permits the father to use his Etrog on the second day and make a Beracha. Moreover, an additional doubt can be added to this equation. Even if the second day is actually Yom Tob, perhaps the Halacha is in accordance with the opinions that the second day of Yom Tob does not require "MiShelechem," and one does not need to own the Etrog on the second day.
SUMMARY
One must own the set of Lulav and Etrog in order to fulfil the Misva on the first day of Succot. Therefore, he should first fulfil the Misva before letting his minor child do so. However, after giving it to his child on the first day, there is no problem fulfilling the Misva with that set on the second day of Yom Tob.
