Travel with maiden name in ppt and GC

[degu]

Hi,

I changed my maided name to my married name on the SSN card and Drivers lisence, but my maiden name is still there in my passport and Green card. Is it ok to Travel to India with the same maiden name even though I have changed it in my SSN card and Drivers lisence? Can I come back and then apply for a name change in PPT and GC? Will I have issues at Port of Entry?
Any one please let me know.

Thanks.