We study to what extent the phase boundary between the competing SC and SDW states is affected by the negative $t'$. The boundary is appreciably pushed up to the higher $\rho$ side with increase of the absolute value of the negative $t'$, although not largely.

We point out that the resultant SC region from $\rho \sim 0.76$ to $\rho \sim 0.86$ and the increased SC energy gain with increase of $\rho$ are in accord with experimental features of cuprate high-$T_c$ superconductors in the overdoped region. The observed energy gain in the SC state is found to be in fair agreement with the experimental condensation energy estimated from the critical magnetic field $H_c$ and the specific heat of YBa$_2$Cu$_3$O$_7$. On the other hand the $t$-$J$ model as an effective Hamiltonian of the Hubbard model is noticed to give an enormous overestimation of the condensation energy.

In the next section the model and the method are described. Results on the SC and SDW ground states are given in §3 and §4, respectively. In §5 the obtained results are compared with experiments and also with other computational works. The final section gives the summary. Short reports on the present results were given in 