Lipschitz continuity and monotone decreasingness of the solution to the BCS gap equation for superconductivity

In the preceding work \cite{watanabe3}, it is shown that the solution to the BCS gap equation for superconductivity is continuous with respect to both the temperature and the energy under the restriction that the temperature is very small. Without this restriction, we show in this paper that the solution is continuous with respect to both the temperature and the energy, and that the solution is Lipschitz continuous and monotonically decreasing with respect to the temperature.