Daf Yomi - Sukkah 46 - Minor and the First Days by Reb Jay

A person should not be makneh the Arba minim to a katan, as while a katan can be koneh, he cannot be makneh to others. What is defined as a katan? The Tur, the Rambam, and the Mechaber hold it is thirteen years old. The Mechaber brings a yesh omrim that says a katan is a child who does not cry out for his mother (six or seven years old, depending on the child). According to most opinions though, a katan is anyone under thirteen. This, however, leads to a problem on the first day of sukkos (and second day in chutz la'aretz), for many families (this is relevant to all children, not just one's own children; however, since in most cases we are dealing with a father and his children, we will use that case in our discussion here). Since there is a chiyuv to own the Arba minim on the first day (u'lkachtem lachem--and it shall be taken for you), and if the katan cannot be makneh the Arba minim back to his father, how can the gadol perform the mitzva the right way (if the katan did it first)? And even if the father has already fulfilled his chiyuv, the katan cannot be makneh the Arba minim to his brothers. And if the father does not give his son the Arba minim, then he is being deficient in his chiyuv of chinuch. In Eretz Yisroel this is not a problem, as the father can ensure that he performs the mitzva before his child, and then when he is makneh the Arba minim to his son, he has already been mikayim the mitzva, and even though the katan cannot give it back, nonetheless, the father can be yotzei the rest of sukkos as a borrower. But this only works if there is one son. In chutz la'aretz where there is a chiyuv of lachem--ownership-- on the first two days, and in Eretz Yisroel if the family has more than one child who is a katan, how can the mitzva be performed the right way?
This leads to the question of what is chinuch? Is chinuch teaching a child to be habitual in mitzvos (The Ra'avan), and therefore it is sufficient to lend him the Arba minim (and he does not need to be makneh the Arba minim to the child thus avoiding the problem that the child cannot return the Arba minim), and as long as the brocha is made and the na'anuim are done, the father has been mechanech the child in the mitzva of Arba minim. Or, is chinuch ensuring that the mitzva is done the right way in all aspects, which in the case of Arba minim includes owning the Arba minim, in which case the father is obligated to buy a set of Arba minim for all his children that are ketanim. The Igros Moshe writes that in Europe when the Arba minim were hard to come by and prohibitively expensive, people relied on the Ra'avan. Today, however, when they are not so expensive, if a father is able he should endeavor to purchase for each child a set of Arba minim.
The Bikkurei Yaakov gives an ingenious answer. He writes that the father can make a t'nai on each day of Yom Tov: "if today is really the first day of Yom Tov (as there is a Sefika D'yoma), then I am fulfilling my chiyuv, then I am being makneh the Arba minim to my son. If tomorrow is the first day, then I am only being lending the Arba minim to my son." A similar t'naiis made on the second day. However, this eitzah of the Bikkurei Yaakov only works if one has one child that is a katan, otherwise, we are still left with the problem of how to transmit the Arba minim to the other children.