Three-dimensional systems of
difference equations –
the asymptotic behavior of their
solutions
Josef Diblík
Irena Růžičková
(1)
ui (k )  ui (k  1)  ui (k ), i  1,2,3
k  N(a)  a, a  1,, a  N is fixed
f1 , f 2 , f 3 : N(a)  R  R
3
The solution of system (1) is defined as
an infinite sequence of number vectors
u(a), u(a  1), u(a  2),
with u ()  (u1 (), u2 (), u3 ()) such that
for any k  N(a) equalities (1) hold.
The existence and uniqueness of the
solution of initial problem (1),(2) with a
prescribed initial value
(u1 (a), u2 (a), u3 (a))  (u , u , u )  R (2)
a
1
is obvious.
a
2
a
3
3
If for every fixed k  N(a)
the right hand sides
f i (k , u1 , u2 , u3 ), i  1,2,3,
are continuous on (u1 , u2 , u3 )  R ,
then the solution of initial problem (1),(2)
depends continuously on the initial data.
3
Define a set   N(a)  R as
3

 ( k )
kN( a )
where
(k )  {( k , u1 , u2 , u3 ) : k  N(a ), ui  R,
bi (k )  ui  ci (k ), i  1,2,3}
Problem
Find sufficient conditions with respect to
the right-hand side of system (1) in order to
guarantee the existence of at least one
solution
u (k )  (u (k ), u (k ), u (k )), k  N(a),
*
satisfying (k , u (k ), u (k ), u3 (k ))  (k )
*
1
for every
*
2
*
1
k  N(a).
*
3
*
2
The boundary of the set  is
            
1
B
2
B
3
B
1
C
2
C
where
 Bj  {( k , u1 , u2 , u3 ) : k  N(a), u j  b j (k ),
bi (k )  ui  ci (k ), i  1,2,3, i  j}
Cj  {( k , u1 , u2 , u3 ) : k  N(a), u j  c j (k ),
bi (k )  ui  ci (k ), i  1,2,3, i  j}
3
C
c
1
A point M (k , u1 , u2 , u3 ) is called the first
consequent point of a point
M  (k , u , u , u )  N(a)  R
*
*
1
*
2
*
3
3
(and we write M  C[ M ] ) if
c
1
k  k 1
*
ui  u  f i (k , u , u , u ), i  1,2,3.
*
i
*
1
*
2
*
3
A point (k , u1 , u2 , u3 )   is a point of
the type of strict egress for the set 
with respect to system (1) if and only if
for some j {1,2,3}
u j  b j (k ) and
f j (k , u1 , u2 , u3 )  b j (k  1)  b j (k )
or
u j  c j (k ) and
f j (k , u1 , u2 , u3 )  c j (k  1)  c j (k )
The set  is of Liapunov type with respect
to the j-th variable with respect to system (1)
if for every (k , u )  
b j (k  1)  u j  f j (k , u1 , u2 , u3 )  c j (k  1)
Theorem 1
Let bi (k ), ci (k ), bi (k )  ci (k ), i  1,2,3 be real
functions defined on N(a) and let
f i : N(a)  R  R, i  1,2,3,
3
be functions that are continuous with respect
to their last three arguments.
Suppose that for a fixed j  {1,2,3} all the points
of the sets  ,  are points of strict egress.
j
B
j
C
Further suppose the set  is of the Liapunov
type with respect to the i - th variabl e for
i  1,2,3, i  j.
Then there exists a solution
u  (u1* (k ), u2* (k ), u3* (k ))
of system (1) satisfying the inequaliti es
bi (k )  u (k )  ci (k ), i  1,2,3,
*
i
for every k  N(a).
Proof
The proof of Theorem 1 will be performed
by contradiction.
We suppose that there exists no solution
satisfying inequalities (3) for every k  N(a)
In such situation we prove that there exists
a continuous mapping of a closed interval
onto its both endpoints.
Retract and retraction
If A  B are any two sets of a topologic al
space and π : B  A is a continuous mapping
from B onto A such that π( p)  p for every
p  A, then π is said to be a retraction of B
onto A.
If there exists a retraction of B onto A, A is
called a retract of B.
The general scheme of the proof
Each solution of system (1) is uniquely
determined by the chosen initial condition.
Fix values u2a and u3a with uia  (bi (a ), ci (a )).
Now the solution is given just by the choice
of u1a .
Define the closed interval
I  [b1 (a ), c1 (a )].
We show that ther e exists a retraction R
(as a compositio n of two auxiliary
mappings) of the set B  I onto the set
A  I  {b1 (a ), c1 (a )}.
This contradict ion will prove our result.
u2
~
M
~
R2 ( M 1 )
u1
a
k*
k* 1
~
M1
k