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ISSN 2229 – 5046
HYERS - ULAM STABILITY OF DIFFERENCE EQUATIONS OF SECOND ORDER
R. ARUL
1AND C. GAYATHRI*
21,2
Department of Mathematics,
Kandaswami Kandar’s College, Velur - 638 182, Tamilnadu, India.
(Received On: 19-05-18; Revised & Accepted On: 29-06-18)
ABSTRACT
I
n this paper, we investigate the Hyers - Ulam stability of second order difference equations of the form)
(
=
)
(
)
(
)
(
2n
r
n
qy
n
y
p
n
y
+
∆
+
∆
and
)
(
=
)
(
)
(
)
(
)
(
)
(
2n
r
n
y
n
q
n
y
n
p
n
y
−
∆
+
∆
where
p q
,
∈
R
and{
p
(
n
)},
{
q
(
n
)},
{
r
(
n
)}
are sequences of reals. Examples are provided to illustrate the main results.AMS Subject Classification: 39A45, 39B42
Keywords and Phrases: Hyers - Ulam stability, second order, difference equation.
1. INTRODUCTION
In recent years, there has been a great interest in investigating the Hyers - Ulam stability of various types of functional equations. This problem was first raised by Ulam [15] concerning the stability of group homomorphism, and the answer was given by Hyers [3], we refer the reader to [13] for the exact definition of Hyers - Ulam stability. Since then, the stability problems of various functional equations has been studied by many authors, see [13] and the references contained therein.
After that, Ulam stability problem for functional equations was replaced by stability of differential equations and difference equations. The differential equation
0
=
)
(
)
(
)
(
)
(
)
(
.
.
.
)
(
)
(
)
(
)
(
t
y
( )t
a
1t
y
( 1)t
a
1t
y
t
a
0t
y
t
h
t
a
n n+
n− n−+
+
′
+
+
has the Hyers - Ulam stability, if for given
ε
>
0,
I
be an open interval and for any functionf
satisfying the differential inequalityε
≤
+
+
′
+
+
+
−−
(
)
(
)
.
.
.
(
)
(
)
(
)
(
)
(
)
|
)
(
)
(
|
1 01) ( 1 )
(
t
h
t
y
t
a
t
y
t
a
t
y
t
a
t
y
t
a
n nn n
then there exists a solution
f
0(
t
)
of the above equation such that.
0,
=
)
(
lim
)
(
|
)
(
)
(
|
0
0
t
K
and
K
t
I
f
t
f
−
≤
∈
→
ε
ε
ε
In [9], the authors studied the Hyers - Ulam stability of second order differential equation of the form
0
=
y
y
y
′′
+
α
′
+
β
and
)
(
=
f
t
y
y
y
′′
+
α
′
+
β
where
α
,
β
∈
R
. The Hyers - Ulam stability of differential equations have been studied in many papers, see for example [4, 5, 6, 9, 10, 14], and the references cited therein. However only few results are reported in the literature regarding the Hyers - Ulam stability of difference equations, see [1, 2, 8, 11, 12, 14] and the references cited therein.Corresponding Author: C. Gayathri*
2,
The aim of this paper is to study the Hyers - Ulam stability of second order difference equations of the form
0,
=
)
(
)
(
)
(
2n
qy
n
y
p
n
y
+
∆
+
∆
(1.1)),
(
=
)
(
)
(
)
(
2n
r
n
qy
n
y
p
n
y
+
∆
+
∆
(1.2) and)
(
=
)
(
)
(
)
(
)
(
)
(
2n
r
n
y
n
q
n
y
n
p
n
y
−
∆
+
∆
(1.3) wherep
,
q
∈
R
and{
p
(
n
)},
{
q
(
n
)},
{
r
(
n
)}
are sequences of reals.Definition 1.1: The difference equation
0
=
)
(
)
(
)
(
)
(
)
(
.
.
.
)
(
)
(
)
(
)
(
n
y
n
a
1n
1y
n
a
1n
y
n
a
0n
y
n
b
n
a
k∆
k+
k−∆
k−+
+
∆
+
+
has the Hyers - Ulam stability, if for given
ε
>
0,
I
be an open interval and for any real functionf
(
n
)
satisfying the inequalityε
≤
+
+
∆
+
+
∆
+
∆
−−
(
)
(
)
.
.
.
(
)
(
)
(
)
(
)
(
)
|
)
(
)
(
|
a
kn
ky
n
a
k 1n
k 1y
n
a
1n
y
n
a
0n
y
n
b
n
then there exists a function
f
0(
n
)
of the above difference equation such that|
f
(
n
)
−
f
0(
n
)
|
≤
K
(
ε
)
and0
=
)
(
lim
ε→0K
ε
forn
∈
I
⊂
N
(0)
=
{0,
1,
2,
.
.
.}.
Definition 1.2: The difference equation (1.2) has the Hyers - Ulam stability if there exists a constant
K
>
0
with the property: for everyε
>
0,
y
(
n
),
r
(
n
)
defined forn
∈
(
a
,
b
+
1),
0
<
a
<
b
<
∞
,
ifε
≤
−
+
∆
+
∆
(
)
(
)
(
)
(
)
|
|
2y
n
p
y
n
qy
n
r
n
(1.4)then there exists some
z
(
n
),
n
∈
(
a
,
b
+
1)
satisfying)
(
=
)
(
)
(
)
(
2n
r
n
qz
n
z
p
n
z
+
∆
+
∆
such that
|
y
(
n
)
−
z
(
n
)
|
≤
K
ε
. We call suchK
as a Hyers - Ulam stability constant for equation (1.2).The results presented in this paper are new and complement to the results reported in the literature for difference equations.
2. STABILITY RESULTS
In this section, we study the Hyers - Ulam stability of equations (1.1), (1.2) and (1.3). We begin with the following theorem.
Theorem 2.1: If the characteristic equation
m
2+
(
p
−
2)
m
+
(
q
−
p
+
1)
=
0
have two different positive roots, then the equation (1.1) has the Hyers - Ulam stability.Proof: Let
ε
>
0
andy
(
n
),
n
∈
(
a
,
b
+
1)
be a solution of equation (1.1) satisfying the property.
|
)
(
)
(
)
(
|
∆
2y
n
+
p
∆
y
n
+
qy
n
≤
ε
Let
λ
andµ
be the two different positive roots of the characteristic equation. Forn
∈
(
a
,
b
+
1)
, define)
(
)
(
=
)
(
n
y
n
y
n
g
∆
−
λ
. Then)
(
)
(
=
)
(
n
2y
n
y
n
g
∆
−
∆
∆
λ
and hence
|
)
(
)
(
)
(
)
(
|=|
)
(
)
(
|
∆
g
n
−
µ
g
n
∆
2y
n
−
λ
∆
y
n
−
µ
∆
y
n
+
λµ
y
n
=|
∆
2y
(
n
)
+
p
∆
y
(
n
)
+
qy
(
n
)
|
≤
ε
.
Thus,
g
(
n
)
satisfies the relation.
)
(
)
(
µ
ε
ε
≤
∆
−
≤
−
g
n
g
n
(2.1)From (2.1), we have
1) ( 1)
( 1)
(
)
(1
)]
(
)
(1
1)
(
[
)
(1
)
(1
+
− +≤
+
− ++
−
+
≤
+
− +−
n n nn
g
n
g
µ
ε
µ
µ
µ
that is,
(
(1
)
(
)
)
(1
)
.
)
(1
+
−( +1)≤
∆
+
−≤
+
−( +1)−
n n nn
g
ε
µ
µ
µ
ε
(2.2)Summing (2.2) from
n
tob
, we obtain(
)
( 1)= =
1) (
=
)
(1
)
(
)
(1
)
(1
+
− +≤
∆
+
−≤
+
− +−
∑
∑
∑
jb
n j j
b
n j j b
n j
j
g
ε
µ
µ
µ
ε
which on simplification implies that
.
)
(1
)
(
)
(1
1)
(
)
(1
)
(1
( 1)µ
µ
ε
µ
µ
µ
µ
ε
b n nn
b
g
b
g
− −
+ −
−
+
≤
+
−
+
+
≤
+
−
Hence
1 1)
(
1
(1
µ
)
(
1)
(
)
ε
ε
≤
+
+
−
≤
−
− − +n
g
b
g
n b
where
µ
ε
ε
1=
. Letz
(
n
)
=
(1
+
µ
)
−(b−n+1)g
(
b
+
1)
. Then∆
z
(
n
)
−
µ
z
(
n
)
=
0
. Now|
g
(
n
)
−
z
(
n
)
|
≤
ε
1 impliesthat
1 1
(
)
λ
(
)
(
)
ε
ε
≤
∆
−
−
≤
−
y
n
y
n
z
n
and hence
(
(
1)
(1
)
(
)
(
)
)
(1
)
.
)
(1
)
(1
1 ( 1)1) ( 1)
( 1
+ − +
− +
−
≤
∆
+
+
−
+
−
≤
+
+
−
n n nn
z
n
y
n
y
λ
ε
λ
λ
λ
ε
Proceeding as above, one obtains
λ
λ
ε
λ
λ
λ
λ
λ
ε
j nb
n j n
b n
j
z
n
y
b
y
− +
− −
+ −
−
+
≤
+
−
+
−
+
+
≤
+
−
(1
)
(1
)
(
1)
(1
)
(
)
∑
(1
)
( 1)(
)
1(1
)
= 1)
( 1
or
.
)
(
)
(1
)
(1
)
(
1)
(
)
(1
( 1) 1= 1)
( 1
λ
ε
λ
λ
λ
λ
ε
≤
+
+
−
−
+
+
≤
−
− − +∑
− +j
z
n
y
b
y
jb
n j n n
b
Define
)
(
)
(1
1)
(
)
(1
=
)
(
( 1)= 1)
(
j
z
b
y
n
u
j nb
n j n
b− + − − +
−
+
−
+
+
λ
∑
λ
then
λµ
ε
λ
ε
=
|
)
(
)
(
|
−
≤
1n
y
n
u
. It is easy to see that∆
u
(
n
)
=
λ
u
(
n
)
+
z
(
n
)
and hence)
(
)
(
=
)
(
2n
z
n
u
n
u
∆
+
∆
∆
λ
=
λ
∆
u
(
n
)
+
µ
z
(
n
)
=
λ
∆
u
(
n
)
+
µ
(
∆
u
(
n
)
−
λ
u
(
n
))
=
(
λ
+
µ
)
∆
u
(
n
)
−
λµ
u
(
n
)
or
0.
=
)
(
)
(
)
(
2n
qu
n
u
p
n
u
+
∆
+
∆
Consequently, the equation (1.1) has the Hyers - Ulam stability with the stability constant
λµ
1
=
K
. This completesthe proof.
Theorem 2.2: Assume that the characteristic equation
m
2+
(
p
−
2)
m
+
(
q
−
p
+
1)
=
0
have two different positive roots. If condition (1.4) holds, then the equation (1.2) has the Hyers - Ulam stability.Proof: Proceeding as in the proof of Theorem 2.1, we obtain
|
)
(
)
(
)
(
)
(
)
(
|=|
)
(
)
(
)
(
|
∆
g
n
−
µ
g
n
−
r
n
∆
2y
n
−
λ
∆
y
n
−
µ
∆
y
n
+
λµ
y
n
−
r
n
=|
∆
2y
(
n
)
+
p
∆
y
(
n
)
+
qy
(
n
)
−
r
(
n
)
|
≤
ε
.
Hence,
g
(
n
)
satisfies the relation.
)
(
)
(
)
(
µ
ε
ε
≤
∆
−
−
≤
Similar to the proof of Theorem 2.1, we have
(
(1
)
(
)
)
(1
)
(
)
(1
)
,
)
(1
+
−( +1)≤
∆
+
−−
+
−( +1)≤
+
−( +1)−
n n n nn
r
n
g
µ
ε
µ
µ
µ
ε
and we let
(
)
=
(1
)
(
1)
(1
)
( 1)(
)
= 1) (j
r
b
g
n
z
b j nn j n
b− + − − +
−
+
−
+
+
µ
∑
µ
, thenz
(
n
)
satisfies the equation0,
=
)
(
)
(
)
(
n
z
n
r
n
z
−
−
∆
µ
and
|
g
(
n
)
−
z
(
n
)
|
≤
ε
1. Using the same type of argument as in Theorem 2.1, one can show that there exists)
(
)
(1
1)
(
)
(1
=
)
(
( 1)= 1) (
j
z
b
g
n
u
j nb
n j n
b− + − − +
−
+
−
+
+
λ
∑
λ
such that
λµ
ε
≤
−
(
)
|
)
(
|
u
n
y
n
andu
(
n
)
satisfies the equation0.
=
)
(
)
(
)
(
)
(
2n
r
n
qu
n
u
p
n
u
+
∆
+
−
∆
This completes the proof.
Next, we study the Hyers - Ulam stability of equation (1.3). For this, we need the following result.
Lemma 2.3: Assume that
0
≤
α
(
n
)
≤
α
<
∞
for everyn
∈
I
. Then forn
∈
(
a
,
b
+
1)
, the equation0
=
)
(
)
(
)
(
)
(
n
n
y
n
r
n
y
−
−
∆
α
(2.3)has the Hyers - Ulam stability.
Proof: Let
ε
>
0
andy
(
n
),
n
∈
(
a
,
b
+
1)
be a solution of equation (2.3) satisfying the propertyε
α
−
≤
−
∆
(
)
(
)
(
)
(
)
|
|
y
n
n
y
n
r
n
or
.
)
(
)
(
))
(
(1
1)
(
α
ε
ε
≤
+
−
+
−
≤
−
y
n
n
y
n
r
n
Therefore
[
]
1= 1 =
))
(
(1
)
(
)
(
))
(
(1
1)
(
))
(
(1
− −
+
−
+
−
+
≤
+
−
∏
i
y
n
n
y
n
r
n
∏
i
n a i n a i
α
α
α
ε
1 =))
(
(1
−
+
≤
∏
i
n a i
α
ε
implies that
+
∆
≤
+
−
− −∏
∏
1 = 1 =))
(
(1
)
(
))
(
(1
i
y
n
i
n a i n a i
α
α
ε
.
))
(
(1
))
(
(1
)
(
1 = 1 = − −
+
≤
+
−
r
n
∏
i
∏
i
n a i n a i
α
ε
α
Summing the last inequality from
a
ton
−
1
, we have
+
∆
≤
+
−
− − − − −∏
∑
∏
∑
1 1= 1 = 1 = 1 =
))
(
(1
)
(
))
(
(1
i
y
j
i
j a i n a j j a i n a j
α
α
ε
1 = 1 = 1 = 1 =))
(
(1
))
(
(1
)
(
− − − −
+
≤
+
−
∑
r
j
∏
i
∑
∏
i
j a i n a j j a i n a j
α
ε
α
and hence)
(
))
(
(1
)
(
))
(
(1
1 1 = 1 = 1 =a
y
i
n
y
i
n a i j a i n a j−
+
≤
+
−
− − − −∏
∏
∑
α
α
ε
1 = 1 = 1 = 1 =))
(
(1
))
(
(1
)
(
− − − −
+
≤
+
−
∑
r
j
∏
i
∑
∏
i
where we have used the convention 1
(1
(
))
=
1.
=
i
a
a
i
+
α
∏
−Now
1
+
α
≥
1
, we have1 1
= =
1
1
1
(1
( ))
=
. . .
1
( )
(1
( ))(1
(
1))
(1
( )) . . . (1
(
1))
j n
j a i a
i
a
a
a
a
n
α
α
α
α
α
α
−
−
+
+
+ +
+
+
+
+
+
+
−
∑ ∏
1 1
=
))
(
(1
1
)
(1
− −
+
−
+
≤
∏
i
n
a i n
α
α
α
(1
)
1
(1
(
))
.
1 1=
− −
+
−
+
≤
∏
i
n
a i b
α
α
α
Using this in (??), we obtain
+
−
≤
−
+
−
(1
)
1
(
)
(
)
∏
−(1
(
))
1
=
i
a
y
n
y
n
a i b
α
α
α
ε
.
1
)
(1
))
(
(1
)
(
))
(
(1
1
= 1
= 1
=
α
α
ε
α
α
≤
+
−
+
+
−
− −
−
∏
∑
∏
j ba i n
a j n
a i
i
j
r
i
For
n
∈
(
a
,
b
+
1)
, if we define1
= 1
= 1
= 1
=
))
(
(1
)
(
))
(
(1
))
(
(1
)
(
=
)
(
− −
− −
+
+
+
∏
+
∏
∑
∏
i
j
r
i
i
a
y
n
z
j
a i n
a j n
a i n
a i
α
α
α
then it is easy to see that
.
1
)
(1
|
)
(
)
(
|
0
=
)
(
)
(
)
(
)
(
ε
α
α
α
−
−
≤
+
−
−
∆
z
n
n
z
n
r
n
and
y
n
z
n
bHence equation (2.3) has the Hyers - Ulam stability with stability constant
α
α
)
1
(1
=
+
−
b
K
. This completes theproof.
Theorem 2.4: Assume that
{
p
(
n
)}
and{
q
(
n
)}
are positive real sequences for everyn
. If{
c
(
n
)}
is a particular solution of0
=
)
(
)
(
)
(
)
(
1)
(
)
(
n
u
n
u
n
p
n
u
n
q
n
u
+
+
−
+
∆
(2.4)such that
0
<
c
(
n
)
≤
c
<
∞
and+
∆
≤
<
∞
)
(
)
(
)
(
=
)
(
d
n
c
n
c
n
q
n
d
forn
∈
(
a
,
b
+
1)
, then equation (1.3) has theHyers - Ulam stability.
Proof: Let
ε
>
0
andy
(
n
),
n
∈
(
a
,
b
+
1)
be a solution of equation (1.3) satisfying the property.
|
)
(
)
(
)
(
)
(
)
(
)
(
|
∆
2y
n
−
p
n
∆
y
n
+
q
n
y
n
−
r
n
≤
ε
For
n
∈
(
a
,
b
+
1)
, definev
(
n
)
=
∆
y
(
n
)
−
c
(
n
)
y
(
n
)
. Then2
|
∆
v n
( )
−
d n v n
( ) ( )
−
r n
( ) |=|
∆
y n
( ) ( (
−
c n
+ +
1 )
d n
( ))
∆
y n
( ) ( ( ) ( )
+
c n d n
− ∆
c n y n
( )) ( )
−
r n
( ) |
ε
≤
−
+
∆
−
∆
(
)
(
)
(
)
(
)
(
)
(
)
|
=|
2y
n
p
n
y
n
q
n
y
n
r
n
where we have used the fact that
{
c
(
n
)}
is a particular solution of equation (2.4). From Lemma 2.3, it follows that0
=
)
(
)
(
)
(
)
(
n
d
n
v
n
r
n
v
−
−
∆
has the Hyers - Ulam stability with the property that
d
n
w
n
v
(
)
−
(
)
|
≤
ε
|
, where.
))
(
(1
)
(
))
(
(1
))
(
(1
)
(
=
)
(
1
= 1
= 1
= 1
=
− −
− −
+
+
+
+
∏
∑
∏
∏
d
i
d
i
r
j
d
i
a
v
n
w
j
a i n
a i n
a i n
Using
v
(
n
)
ind
n
w
n
v
(
)
−
(
)
|
≤
ε
|
, we haved
n
w
n
y
n
c
n
y
−
−
≤
ε
∆
(
)
(
)
(
)
(
)
|
|
for
n
∈
(
a
,
b
+
1)
. Again using Lemma 2.3, we see that0
=
)
(
)
(
)
(
)
(
n
c
n
y
n
w
n
y
−
−
∆
has the Hyers - Ulam stability with the property that
cd
n
z
n
y
(
)
−
(
)
|
≤
ε
|
, where.
))
(
(1
)
(
))
(
(1
))
(
(1
)
(
=
)
(
1
= 1
= 1
= 1
=
− −
− −
+
+
+
+
∏
∑
∏
∏
c
i
c
i
w
j
c
i
a
y
n
z
j
a i n
a j n
a i n
a i
Hence, equation (1.3) has the Hyers - Ulam stability with the stability constant
cd
K
=
1
. The proof is now complete.3. EXAMPLE
In this section we provide two examples to illustrate the main results.
Example 3.1: Consider the second order difference equation
1.
0,
=
)
(
3
1
)
(
6
7
)
(
2
+
∆
+
≥
∆
y
n
y
n
y
n
n
(3.1)
The characteristic equation is
=
0
3
1
6
7
2
−
+
m
m
and hence the characteristic roots are2
1
and
.
3
1
Therefore by
Theorem 2.1 the equation (3.1) has Hyers-Ulam stability with stability constant
K
=
6.
Example 3.2: Consider the second order difference equation
2)
1)(
(
2
=
)
(
2)
1)(
(
1
)
(
2)
(
1
)
(
2+
+
+
+
+
∆
+
−
∆
n
n
n
n
y
n
n
n
y
n
n
y
(3.2)on the interval
I
=
(1,
∞
)
.Let
n
I
n
n
c
∈
+
1
,
1
=
)
(
be a particular solution of equation0.
=
2)
1)(
(
2
)
(
2
2
)
(
1)
(
)
(
+
+
+
+
−
+
+
∆
n
n
n
u
n
n
u
n
u
n
u
By Theorem 2.4,
3
2
=
))
(
(1
1,
<
2
1
=
)
(
11 =
+
+
≤
+
∏
−
n
i
d
d
n
n
d
ni and
2
1
=
))
(
(1
1 1 =
+
+
∏
−n
i
c
n
i . It is easy to
verify that
3
2
)
(
n
≥
n
+
w
andn
I
n
n
n
z
≥
+
,
∈
3
1
)
(
. Indeed,n
n
y
(
)
=
1
is a solution of equation (3.2) andcd
n
z
n
y
(
)
−
(
)
|
≤
ε
|
. Hence equation (3.2) has the Hyers - Ulam stability.We conclude this paper with the following remark.
Remark 3.3: In this paper we investigated the Hyers - Ulam stability of different types of second order difference
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Source of support: Nil, Conflict of interest: None Declared.
