  The mixed spin-(1/2, S) Ising model on the union jack (centered square)
lattice is investigated by establishing the mapping relationship with its
corresponding eight-vertex model. An interplay between the nearest-neighbour
interaction, the competing next-nearest-neighbour interaction and the
single-ion anisotropy gives rise to a rather complex critical behaviour
displayed in the reentrant phase transitions, the weak universal critical
behaviour, as well as, a presence of first- and second-order phase transitions.
The most interesting finding to emerge from the present study relates to a
variation of the weak-universal critical exponents along the line of bicritical
points, which is being twice as large for the mixed spin-(1/2, S) systems with
the integer spin-S atoms as for the ones with the half-odd-integer spin-S
atoms.
