Toyota has Lexus, Honda Acura and Nissan has Infiniti. They are premium brands that were aimed at North America. Toyota has since taken Lexus around the world (started 1989; now in about 65 countries) Honda has hardly tried to with the Acura (started 1986; only in about 4 countries I know of) and Nissan is expanding Infiniti (started 1989; now in about 47 countries) around the world.
The fact is they all sell quite well in North America but they have all struggled elsewhere with market penetration. It seems many people around the world want European premium marques instead. Euro brands have been increasing their range of models too, which adds to the perception of their success with the extra sales that generates.
One problem Infiniti has had has been its cars are big, built with North American taste in mind. This will render limited sales in Europe for example. Now it plans to build a car for the 1-Series segment (see pic at bottom of article). It will be built by Nissan UK from 2015. This will mean moving a planned small car to another factory as the UK plant is almost at full capacity now. Another plan to improve all labelling of vehicles by calling all cars as 'Q' models and SUVs as 'QX' models.
Infiniti expects to sell about 170,000 cars in 2012, with at least 115,000 US sales. That is a modest figure for a brand with global ambitions. How it goes in wresting sales away from existing premium brands remains to be seen.