Global existence of spherically symmetric solutions to the coupled Einstein and nonlinear Klein-Gordon system

We prove the global unique existence of classical solutions to the Einstein equations coupled with the nonlinear Klein-Gordon equations for small init ial data under spherical symmetry. We also obtain the decay estimates of th e solutions. For its proof we reduce the system to a single first-order int egro-differential equation, and use the contraction mapping theorem in the appropriate function spaces.