Can a capitalist economy based on selfishness have long term stability?
This depends to some extent on what you call "long term" and what constitutes "stability." However, it does seem that it is possible given that the United States and the United Kingdom (in particular) have been quite capitalistic for at least 200 years.
All capitalist societies are based on selfishness. Capitalism relies on all people pursuing their self interest. When people do this, the argument goes, all of society benefits. Therefore, any capitalist system in existence today is based on selfishness.
You can argue about how many countries have been capitalist for how long and about how stable they have been. However, it is true that no capitalist system has yet fallen and been replaced by anything else. This would seem to indicate that such systems can remain stable in the long term.