Varying Cosmological Constant and the Machian Solution in the Generalized Scalar-Tensor Theory

The cosmological constant $(1/2)\lambda_{1}\phi_{, \mu}\phi ^{, \mu}/\phi ^{2}$ is introduced to the generalized scalar-tensor theory of gravitation with the coupling function $\omega (\varphi )=\eta \mathrm{/}(\xi -2)$ and the Machian cosmological solution satisfying $\varphi =O(\rho \mathrm{/}\omega )$ is discussed for the homogeneous and isotropic universe with a perfect fluid (with negative pressure). We require the closed model and the negative coupling function for the attractive gravitational force. The constraint $$ for $0\leqq \xi <2$ leads to $\eta >3$. If ${\lambda }_1<0$ and $0\leqq -\eta /\lambda_{1}<2$, the universe shows the slowly accelerating expansion. The coupling function diverges to $-\mathrm{\infty }$ and the scalar field $\varphi$ converges to $G_{\mathrm{\infty }}^{-1}$ when $\xi \to 2$ ($t\to +\mathrm{\infty }$). The cosmological constant decays in proportion to $t^{-2}$. Thus the Machian cosmological model approaches to the Friedmann universe in general relativity with $\ddot{a}=0$, $\lambda =0$, and $p=-\rho \mathrm{/}3$ as $t\to +\mathrm{\infty }$. General relativity is locally valid enough at present.