  Phase transitions of the $J_1$-$J_2$ Ising model on a square lattice are
studied using the higher-order tensor renormalization group(HOTRG) method. This
system involves a competition between the ferromagnetic interaction $J_1$ and
antiferromagnetic interaction $J_2$. Furthermore, weak first-order and
second-order transitions are observed near the ratio $g=J_2/|J_1|=1/2$. Our
results (based on HOTRG calculations for significantly larger sizes) indicate
that the region of the first-order transition is marginally narrower than that
in previous studies. Moreover, the universality class of the second-order
transition connected to the transition line is not necessarily fully consistent
with the Ashkin--Teller class considered earlier.
