  In this paper we analytically study the recurrence equations of an Ising
model with three competing interactions on a Cayley tree of order three. We
exactly describe paramagnetic and ferromagnetic phases of the Ising model. We
obtain some rigorous results: critical temperatures and curves, number of
phases, partition function. Ganikhodjaev et al. [J. Concrete and Applicable
Mathematics, 9 (1), 26-34 (2011)] have numerically studied the Ising model on a
second-order Cayley tree. We compare the numerical results to exact solutions
of mentioned model.
