According to your sentence a "Takeshima¡± is a name that is one and only
 internationally established Name, but  internationally is is accepted both name of
 " Takeshima", and " Dokdo" about the name of the island between Korea and 
Japan. But there is only used "Takeshima" in your website. 
Japan insists that the United Nations has officially approved of the name
 "Takeshima." from UN.  Only the  Takeshima does not represent the opinion of all members of the UN. 
We  have different opinions about the name of the island that the two countries share, 
but I think it is best  for now to use both names, " Dokdo ", and " Takeshima" in consideration of international issue.