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.
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