How is "every man be fully persuaded in his own mind" not contradictory to the "narrow is the way leading to life"? The following verses seem to imply that Paul was liberal when it came to the believers' practice of their faith in the Lord:


  One man esteemeth one day above another: another esteemeth every day
  alike. Let every man be fully persuaded in his own mind. He that
  regardeth the day, regardeth it unto the Lord; and he that regardeth
  not the day, to the Lord he doth not regard it. He that eateth, eateth
  to the Lord, for he giveth God thanks; and he that eateth not, to the
  Lord he eateth not, and giveth God thanks. (KJV, Romans 14:5-6)


However, the Lord's words on the way leading to life seem more strict:


  Enter ye in at the strait gate: for wide is the gate, and broad is the
  way, that leadeth to destruction, and many there be which go in
  thereat: Because strait is the gate, and narrow is the way, which
  leadeth unto life, and few there be that find it. (KJV, Matthew
  7:13-14)


So, how are Paul's words not contradictory to the Lord's words? I think there must be some explanation.
How is “every man be fully persuaded in his own mind” not contradictory to the “narrow is the way leading to life”?

The Romans verse in context is referring to disputes between Christians. 

The "narrow way" refers to those from the population at large who have become Christian.

Even the nation of Israel had the problem of the "broad way" that leads to destruction.


  Judges 17:6  In those days there was no king in Israel, but every man
  did that which was right in his own eyes.


The Romans verse is talking about those who have already entered in by the narrow gate by allowing Christians the elbow room to have their doctrine a little wrong.


  Romans 14:2  For one believeth that he may eat all things: another,
  who is weak, eateth herbs.