養子 = adopted son, sometimes foster son or sponsored son
義子 = adopted son, foster son, sponsored son, godson (formal or informal), or even stepson, son-in-law, and possibly others, with the precise meaning dependent on the culture of the speaker and the context of the conversation
義子 = adopted son, foster son, sponsored son, godson (formal or informal), or even stepson, son-in-law, and possibly others, with the precise meaning dependent on the culture of the speaker and the context of the conversation