  We consider the Cauchy problem for coupled systems of wave and Klein-Gordon
equations with quadratic nonlinearity in three space dimensions. We show global
existence of small amplitude solutions under certain condition including the
null condition on self-interactions between wave equations. Our condition is
much weaker than the strong null condition introduced by Georgiev for this kind
of coupled system. Consequently our result is applicable to certain physical
systems, such as the Dirac-Klein-Gordon equations, the Dirac-Proca equations,
and the Klein-Gordon-Zakharov equations.
