  We consider the coupled systems of nonlinear wave and Klein-Gordon equations
in two space dimensions with cubic nonlinearity. For this kind of systems, the
small data global existence is already known if the cubic nonlinearity
satisfies a certain condition related to the null condition. In this article,
our aim is to investigate the small data global existence under a condition
related to the weak null condition. We also make a remark on the asymptotic
behavior of global solutions.
