  Several observational works have shown the existence of Jupiter-mass planets
covering a wide range of semi-major axes around Sun-like stars. We aim to
analyse the planetary formation processes around Sun-like stars that host a
Jupiter-mass planet at intermediate distances ranging from $\sim$1 au to 2 au.
Our study focusses on the formation and evolution of terrestrial-like planets
and water delivery in the habitable zone (HZ) of the system. Our goal is also
to analyse the long-term dynamical stability of the resulting systems. A
semi-analytic model was used to define the properties of a protoplanetary disk
that produces a Jupiter-mass planet around the snow line, which is located at
$\sim$2.7 au for a solar-mass star. Then, it was used to describe the evolution
of embryos and planetesimals during the gaseous phase up to the formation of
the Jupiter-mass planet, and we used the results as the initial conditions to
carry out N-body simulations of planetary accretion. Our simulations produce
three different classes of planets in the HZ: 'water worlds', with masses
between 2.75 $M_{\oplus}$ and 3.57 $M_{\oplus}$ and water contents of 58% and
75% by mass, terrestrial-like planets, with masses ranging from 0.58
$M_{\oplus}$ to 3.8 $M_{\oplus}$ and water contents less than 1.2% by mass, and
'dry worlds', simulations of which show no water. A relevant result suggests
the efficient coexistence in the HZ of a Jupiter-mass planet and a
terrestrial-like planet with a percentage of water by mass comparable to the
Earth. Moreover, our study indicates that these planetary systems are
dynamically stable for at least 1 Gyr. Systems with a Jupiter-mass planet
located at 1.5 au - 2 au around solar-type stars are of astrobiological
interest. These systems are likely to harbour terrestrial-like planets in the
HZ with a wide diversity of water contents.
