  More than a century ago, L. E. J. Brouwer proved a famous theorem, which says
that any orientation preserving homeomorphism of the plane having a periodic
point must have a fixed point. In recent years, there are still some authors
giving various proofs of this fixed point theorem. In \cite{Fa}, Fathi showed
that the condition``having a periodic point'' in this theorem can be weakened
to ``having a non-wandering point''. In this paper, we first give a new proof
of Brouwer's theorem, which is relatively more simpler and the statement is
more compact. Further, we propose a notion of BP-chain recurrent points, which
is a generalization of the concept of non-wandering points, and we prove that
if an orientation preserving homeomorphism of the plane has a BP-chain
recurrent point, then it has a fixed point. This further weakens the condition
in the Brouwer's fixed point theorem on plane.
