Abstract:
Consider the three-dimensional system of difference equations
xn+1 =
∏k
j=0 zn−3j
∏k
j=1 xn−(3j−1) (
an + bn
∏k
j=0 zn−3j
),
yn+1 =
∏k
j=0 xn−3j
∏k
j=1 yn−(3j−1) (
cn + dn
∏k
j=0 xn−3j
),
zn+1 =
∏k
j=0 yn−3j
∏k
j=1 zn−(3j−1) (
en + fn
∏k
j=0 yn−3j
), n ∈ N0,
where k ∈ N0, the sequences (an)n∈N0
, (bn)n∈N0
, (cn)n∈N0
, (dn)n∈N0
,
(en)n∈N0
, (fn)n∈N0
and the initial values x−3k, x−3k+1, . . . , x0, y−3k,
y−3k+1, . . . , y0, z−3k, z−3k+1, . . . , z0 are real numbers.
In this work, we give explicit formulas for the well defined solutions
of the above system. Also, the forbidden set of solution of the system is
found. For the constant case, a result on the existence of periodic solutions
is provided and the asymptotic behavior of the solutions is investigated in
detail.
AMS Mathematics Subject Classification : 39A10, 39A20, 39A23, 40A05.
Key words and phrases : Three-dimensional systems of difference equations, explicit formulas, periodicity, asymptotic behavior.
