I have three questions:
1) I have many relatives who live far from me. We have some family disputes that make us distant. Sometimes, they would call me but cause trouble, so my family ignored them. During Eid, however, we meet as a family and talk normally. Is this considered cutting the ties? Do I have to greet my family often or is being in a neutral state like this okay?
2) If I forsake someone out of personal anger and I greet them, does this greeting have to be the Salaam greeting, or will any other greeting be enough?
3) If I dislike a person due to personal reasons without hating him, so I avoid him although we are in a normal state and we do not hate each other nor are angry with each other, is this considered forsaking another Muslim?
Please reply, this is an important matter. Thank you .

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ʻalayhi wa sallam, is His slave and Messenger.

We shall answer your three questions in the three following points:

First, if these relatives are among the relatives with whom it is obligatory to uphold the ties of kinship, then it is an obligation to uphold the ties of kinship with them, and cutting those ties with them is regarded as a major sin. The relatives with whom it is obligatory to uphold ties of kinship are the Mahram relatives (permanently unmarriageable kin), according to the preponderant opinion of the scholars of Fiqh. Settling for meeting them during the 'Eed only is not sufficient to uphold the ties of kinship as required in the Shariah, as this is not customarily regarded by people as upholding ties of kinship.

If you cannot visit them, you are obliged to uphold the ties of kinship with them by any means available, such as phone calls or messages and the like. The Maaliki scholar Al-ʻAdawi said, “Upholding the ties of kinship can be done by visiting one's relatives in person, offering money for those in need, good words, and asking about them and their conditions ... visiting is required with the relatives who live close to the person; otherwise (if they live far), he upholds ties with them by writing to them or sending a messenger.” [Haashiyat Al-ʻAdawi]

It should be noted that maintaining ties of kinship with one's relatives and enduring their harm yield great rewards. Abu Hurayrah, may Allah be pleased with him, narrated that a man said to the Prophet, sallallahu ʻalayhi wa sallam, “I have relatives with whom I try to maintain good relations, but they sever the relations with me; whom I treat kindly, but they treat me badly; with whom I am gentle, but they are rough towards me.” The Prophet, sallallahu ʻalayhi wa sallam, said, “If you are as you say, it is as if you are feeding them hot ashes, and you will be supported by Allah against them as long as you continue to do so.” [Muslim]

A Muslim is not required to ask about his family members constantly; rather, he is required to do what upholds the ties with them by any of the customarily acceptable means highlighted above.

Second, if by forsaking someone for a personal reason, you mean that you avoid him for a worldly purpose, this is impermissible beyond three nights. You must put an end to this forsaking. It is sufficient to talk to him even if this person does not greet you, and this would put an end to the forsaking. If you greet this person, it is better to greet him with the greeting of Islam (with the Salaam). If he is away, it is sufficient to write to him. Al-Qaadhi ʻIyaadh said, “The least acceptable means of upholding ties of kinship is to refrain from forsaking one's relatives and talk to them even if by the Salaam.”

Third, if avoiding a person is not due to a dispute, it is not considered forsaking. Scholars stated that the prohibited forsaking is when two people are present in the same place and they turn away from one another and refrain from talking to each other. In Sharh Saheeh Al-Bukhaari, Ibn Battaal said, “Forsaking means that a person refrains from talking to his (Muslim) brother when they meet, and they turn away from each other and do not even greet one another with the Salaam. One of the due rights of a Muslim upon his fellow Muslim if they meet is that each of them greets the other. If they refrain from it due to discord, then they have committed what Allah has forbidden and deserve the due punishment, unless Allah chooses to pardon them.” [Sharh Saheeh Al-Bukhaari]

Allah knows best.