Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 485-495
485
ON SOME NEW N-INDEPENDENT-VARIABLE DISCRETE
INEQUALITIES OF THE GRONWALL TYPE
EN HAO YANG
Department of Mathematics
Jinan University
Guang Zhou, China
(Received June 26, 1985)
ABSTRACT.
In this paper we shall establish some new discrete inequalities of the
Gronwall type in N-independent variables.
They will have many applications for finite
several
difference
equations
analysis.
Their consequence for the case of
involving
independent
variables
and
for
numerical
N
3, generalizes all of the known
theorems obtained by Pachpatte and Singare in [I].
An example, to which those results
[I] are inapplicable,
established in
is given here to convey the usefulness of the
results obtained.
KEY WORDS AND PHRASES. Inequalities of the Gronwall type, difference equations.
1980 AMS SUBJECT CLASSIFICATION CODES. 26D05, 26DI0 and 26D15.
PRELIMINARIES.
1.
It is well-known that the discrete inequalities of the Gronwall
type played a vital role in the theory of finite difference equations and numerical
analysis (see [2-8] and [9-12] and the references therein).
Singare
[I]
variables.
have
established
new discrete
some
Recently, Pachpatte and
inequalities
in
three
independent
inequalities involving more than one independent variables are
Discrete
very useful in the study of many important problems concerning discrete versions of
some partial differential and integral equations in several variables.
The
aim
of
this
paper
is
to
obtain
several
N-independent-variable
inequalities which extend all results obtained in [I] for the case of
N
3.
discrete
In what
follows we shall make use of the following notations and definitions.
Let
be the infinite set consisting of the integers
o
use the convention of writing
N
b
j6Z
O,
a
if Z is the empty set.
x, and
by
xi, k
(Xl,X
c
jZ
j
0,1,2,..., and we shall
=_ I,
For simplicity, in the sequel we will denote (x ,x2,... ,x
,Xn),
(xi,xi+l,...,Xk)
and
xj+
2
respectively, here i,k are integers from 1,2,...,n with i
,xj), (xj,
the multlple-summatlon symbol
<
k.
by
xj, xj,
n)
N
and
Further, we denote
n
o
486
E.H. YANG
x.l
xj.-I
Xk-
x,k
by
yj=O yj+l=O
where
No’
xi’ Yl
j
<i<
< j <_ k <
k, and
Moreover, we define
n.
+I
A.()
and so on, where
".+,+)
Z. 3 (.
3
are numbers from
xj,xk,...
A X. (-)
N
and
j,k,..., are integers from
O
We write also here that
1,2,...,n.
=
/k(r)L(x)
/k L r
Xl2" "’’r
L(x)
for, any real-valued function
on
shall define a class of functions on
E:f:
f(x)
N
n
NOn’
<
r
<
n, x
N
n
O
In addition, we
by
O
/(r)f(x)
O,
here
O,r=l,...,n-l,and
_
(n)f()
O]
It is obvious that the following properties are true:
(1)
(2)
(3)
if f(x)
f(x), g(x)
if
>
K and c
is a real number, then
0
K then f(x) + g(x)
K.
K.
all functions of the form
.rl r2
x
xI x
are
<
cf(x)
the
in
<
i
i
2
class
rk- 1
x.
K,
here
< ik_ <_
<
(k
Xk_ I
2
r
h
_>
0 and
,,...,n)
ih(h=l,...,k-l)
are
integers
with
n.
2. LINEAR INEQUALITIES.
THEOREM I.
for
K.
x
N
n
O
Let u(x)
p(x)
and
.
-
and let
f(x)
be real-valued non-negatlve functions defined
be a real-valued positive and nondecreasing function in
Suppose further that the discrete inequality
u(x)
f’(:m)
+
p(y)u(y)
(2.1)
y,l
is satisfied for all
u(x) -z
x e
Who
O
f(O,x2)yl
O=
Then we have the inequality
1 +
G(Yl,
+
p(y)
x
n
@N O
(2.2)
DISCRETE INEQUALITIES OF THE GRONWALL TYPE
487
where
x,2
G(
Y2
(2.3)
and herein
A (k) :[’(k, 0 ,’k+ 2)
k k’ o, ;k+
<
for
<
k
(2.4)
N
n-l, x
n.
We define a function
PROOF.
U(x)
N
on
n
by the right member of (2.1), so that
o
by definition
U(Xk_ l,O,k+ I)
f(k_1,O,k+l)) O,
(2.5)
U(Xk_l,Xk+l Xk+l)
since
f(x)
U(x)>
O,
Xk6 N o
I_
x
k_
n.
is nondecreasing.
Further, we can obtain from the definition of
A(’)(,,)
A(n)u()
U(x)
that,
p(,,),()
+
(2.6)
_Z p(x)U (m_l, Xn+l)
since
(n)
f(x)
is valid for
(2.6)
<
0, p(x)
<
k
<
0,
(2.1), and (2.5).
n-i and x e
Zl (-)u (,_ ’",:+
u(,_,,,
Keeping
>
Xn_
fixed
obtain the inequality
in
N
n
O
A(’-
Xn
-
By applying (2.5) and (2.7), we derive from
+)
(2.6’) set
In view of the fact that
Yn
p(x).
and sum over
Yn
(2.6’1
0,1,2,...,
Xn-I
to
(2.8)
488
where
E.H. YANG
gn-1
(2.4).
is given by
We may rewrite the last inequality as
,(n-2)U(n_2Xn_l+l
x
/ (n-2) U(x)
n
U(Xn-2’Xn-1 +l’xn
&
en_l(Sn_l,O)
(2.5) and (2.7).
since
and sum over
Keeping
_---_
+
now
(2.9)
P(Xn_l’Yn)’
and
Xn_ 2
fixed
Xn
in
(2.9),
set
x
Yn-I
n-l
to get the inequality
Yn_l 0,1,2,..., Xn_l-1
x,n-1
g’n- (Xn-2’ Yn-1,0)
2 ,,,n-2’
t,(x)
y, n-I
(2.10)
+
here
gn-2
is given by
(’’’ Yn-1 )’
P
(2.4).
If
>
n-2
I, then by using a similar argument as used
above for (2.8) to (2.10), we can obtain
x,n-2
_A(n-3)U.(x)_ _z gn_3(n_3,0,n_l)
U(x)
gn_2(Xn_3, Yn_2,0 Xn
+
y,n-2
x,n-I
gn-l(Xn-3’Tn-2,n-l’O)
+
+
y,n-2P(Sn-3’n-2)"
Continuing in this way then we obtain
,/(I)U(x)__
G(x)
+
y,
u()
G(x)
where
is defined by
U(Xl+l’x2)
(2.3).
p(x 1,’2)
Obviously, the last inequality can be rewritten as
x,n
Z.
1 +
G(x)
u(,,)
+
y,2
y
x2 fixed in (2.11), set x
successively in (2.11), we then get
and then substitute
Keeping
xl-I
P(xl’Y2)"
(2.11)
y
I), 1,2,...,
Xl-I
1 +
Yl=O
Thus the desired bound for
immediately.
G(Yl,X2)
p(y)
+
)
u(x)
(2.12)
2
in (2.2) follows from (2.1),
(2.5) and (2.12)
DISCRETE INEQUALITIES OF THE GRONWALL TYPE
EXAMPLE I.
489
Suppose that the discrete inequality
x,3
).. z_
v(x,x2,x3
functions
+
+x2x 3
N3o’
(Xl’X2’X3)
holds for
non-negative
a
xix 2
where
defined
+
>
a
N
on
y,l
0
3.o
Q(y,y2,Y3)v(yI,Y2,Y3
(*)
is a constant, v and Q are real-valued
Then,
by
Theorem
here
have
we
the
nondecreasing function
f(xt,x2,x3)
a +
x2x 3
+
4
XlX2(>O)
K,
since the following conditions are satisfied
:(1)f(x 1,x2’x 3)
+6x21 +2Xl)
x2(1 +lCXl
(2)(x,x2,x3)
+
x
+6x z
x_ O,
+x > o,
for
(xl,x2,x3)
so that,
()(,,,o,,, 3)
(,o,,, 3)
I +4x
f(xl,x2,0
v
,
(*)
from
Q(yl,y2,y 3
(*)
lle note here that the above inequality
I
a +
,3
+
o,
(,,,o,,, 3)
](2)f(xl,x2,O
Hence we derive the desired bound on
-
N3o
+6x21
+
4Xl3
XlX 2
such that
)t
for
(xl,x2,x3)
(:N
3.o
can not be treated by means of the known
results established in [1].
THEORE 2.
LeC
u(x), f(x), p(x)
be he same as in Theorem 1, and let
a real-valued non-negative function defined for
x
N
II
o
3 ,I
O
Jl
(y)f(O,Y2)
+
q(x) be
Suppose ChaC Che inequaliCy
Then we also have
x,n
u(x)- F(x)
N
z,l
y,l
is satisfied for all
x
x,n
I
z =0
1
+G(Zl,Y2)
p(z)+q(z)
+
z,2
t2.14)
490
E.H. YANG
for all
x e N
F()=
n
o
here
f(O,,)
x,n-2
G
)f(y,o,,,))+
+
y,l
-_
+
x,n-I
(n-2)f(n-2’O’Xn) >_
V(x)
We define two functions
PROOF.
F(x) is given by
)f (y 2’ o )
y,l
+
yI
I, and
is the same as in above Theorem
and
W(x)
(n-l)f(n_l,O).
on
N
n
o
(2.15)
by the right number of
(2.13) and the following equality
(x)
V(x)
q(y)V(y)
+
(2.16)
yl
respectively, so that by the definitions: (f(x) is nondecreasing)
w(o
,o, J+l )=
v(-J-l’ o
V(Xj_l,Xj+l,j+l)
w(~j-l’
where
xj
e N
n
+t,
j+l
j=l,2,...,n.
o’
/(r)v(x)=
-
x_ V(x)
w()
j+l
)= f( j- t,o, j+l )>o
> O,
> o,
(2.17)
(2.t8)
(2.9)
In addition, we obtain here
r)f(x)+
y,
and
r+--- P(Xr,Yr+ 1) u(xr,v r+ l
-
x,r
+
y,n
(2.20)
z,l z,r+l
x,n
(r)w(x)
/(r)V(x)
+
y,r+l
q(Xr’gr+l)V(Xr’r+l)"
for
Letting
r
n
in the above
(2.20) and using
z()v(,,)
since
p(x), q(x)
r
A(n)f(.x)
<
x
N
_z p(,,),(),
are non-negative and
u(x)
<
z_ n,x
V(x)
<
0
N
(2.21)
n
o
we then derive that
n
o
(2.22)
W(x).
Now by (2.21) we obtain
for x
Nn
o
DISCRETE INEQUALITIES OF THE GRONWALL TYPE
>
and
V(x)
Zh()w( o
r+ 2
q(x)
since
0
w(
_< r _<
where
<
W(x).
It is clear from (2.17) that,
(r)v(}r, O,r+2)
o, r+ 2
gr(xr,O,xr+
2 ),
v(Xr,O, r+ 2
N
n-l, x
491
n
gr
and
o’ j=l,2,...,n,
is given by
(2.24)
(2.4).
Now by following the same argument as used in the proof of Theorem I, and using
(2.17), (2.24), we get from (2.23)
Xl-I
W(X)
z_
f(0,x2
I +
G(Yl,2)
Substituting this bound for
W(x)
in
Keeping
for x
Nn
O
rewrite it as follows
h(),
(2.26)
is defined by
fixed in the above (2.26), set
Xn_
to get the
h(x)
(2.25)
y,2
(2.22), and then
(’-)v(_,./) Z%(-)v()
where the function
_
(p(y)+ q(y)
+
yl= O
estimate
/,,(-)v(,) A(-)v(_l,o)
Xn
Yn
and sum over
Yn
0,1,2,...,
Xn-1
n(E,,_,>n
+
")n
.A(n-1)f(n_l,O
h(Xn_l,Yn)"
+
)n
Keeping
Yn-I
now
Xn_2,
0,I,2,...,
fixed
Xn
Xn_1-1
the
in
last inequality, set
Xn_l= Yn-l
and sum over
to get the inequality
(n-2)V(x-z (n-2)f(xn
)7,
x,n-I
2,0
x
n
+
n
+
t(x
y, rl-I
n- 2
Yn- i )
y,n-I
(n-I)f( x
c,.,n- 2
Yn- I’ %..]
492
E.H. YANG
Continuing in this way then we, obtain
z()v(,)
-
v(+,’)
_=
Z()(,o,’3)
+
y,2
Z()f( I,Y2,O,x 4)
x,n-I
x,n
Z (-) f(,g, _, o)+
y,2
x
fixed in (2.27), set
2
successively in (2.27)to derive the bound for
Keeping
Yl
Xl
(2.27)
(,y).
y,2
and
substitute
V(x)
such that
0,I,2,...,
Yl
xl-I
(2.28)
h(y),
F(x)+
V(x)
+... +
y,l
V(0,x 2)
f(0,x 2), where F(x) is given by (2.15). Hence the desired bound
(2.14) follows from (2.13), (2.28), and the definitions of V(x) and h(x) immedi-
since
in
Q.E.D.
REMARK i.
ately.
f(x)
al(x I)
(0, ), Aaj(z) _> 0
the Theorems 1,2 of [I] respectively.
for all
3.
3 and
n
Letting
a.3: No
1,2, where
+
a2(x2) + a3(x3) in
No, j 1,2,3,
z
above Theorems
then we derive
A NONLINEAR GENERALIZATION.
THEOREM 3.
let
W(z)
u(x)
>
u(x), p(x),
Let
f(x)
and
be satisfied for all
u
N
x
0
be the same as in above Theorem I, and
n
where
0
u
is a positive number.
0
Let
be a real-valued continuous, positive and strictly increasing function defined on
(u
the interval
o
).
u(x)
holds for
x
N
n
o
-
Suppose further that the inequality
Then for 0
f(x)
p(y)W(u(y)),
+
(3.1)
y,l
<
x
<
X (this is
0
<
x
< Xi,
i
1,2,
i
n)
we also
have the inequality
u(x) z-
K
-I
K
f(O,2)
+
y,l
where
K
-I
denotes the inverse of
K
(YI’2
+
P(YI’ Z2)
(3.2)
),
(3.3)
z,2
and
r
()
and
f(xj
the function
0,
xj+ 2)
by
G*(x)
to_
ds
(
is obtained from
W[f(xj,
0,
xj+2)]
for r
G(x)
r
_x
o
u
o
by replacing all of its denominators
respectively, here
1,2,...,n-|.
Here x
N
n
o
DISCRETE INEQUALITIES OF THE GRONWALL TYPE
is chosen so that the expression contained in the brackets
-I
the domain of
PROOF.
as long as
0
i,O
o+l
<
x
R(x)
Define a function
R(O
for j
K
<
{...} in (3.2) belongs to
X.
N
on
)= f(x~j-l’ O
n
o
by the right member of (3.1), so that
u( j-I ,O
o+l
o+l
)A
u
(3.4)
o
1,2,...n, and
y,r+l
1
-
r-
n-l,x
/(n)R(.)= ,/(")f(x)+ p(x)W[u(x)] p(x)-wER()]
-Z
A
since
(n)
f(x)
<
0, p(x)
>
0, u(x)
<
R(x), and W(z)
from (3.6) that,
(n-l)R(n_l
.(n-I )R (x)
Xn*l
[ (_,/)3
since
493
(3.5),
keeping
Xn_
R(xj_l xj+l, xj+ I) > Uo,
fixed in
(3.7),
set
W(z)
and
-
is
n
o
x
increasing.
n
No
(3.6)
We can observe
p(x),
(3.7)
is increasing.
and sum over
Xn Yn
N
Yn
Using (3.4) and
0,1,2,... ,x
n
-i
to
derive the inequality
A (’-)(n-Z ’)
A (’-z)(,<)
p(x
, [(_,o)]
,[()]
n-I
,n
yn)
(3.8)
Replacing now the left member of (3.8) by the smaller term
(n-2)R(n_2,Xn_l+l,x n)
[(-, -* , n)]
and
Yn-i
then
keeping
Xn_ 2
0,l,2,...,Xn_l-I
and
Xn
fixed
,(n-2)R(x)
’[()J
in
(3.8),
set
Xn-I
Yn-I
and
sum
over
to get the estimate
A (n- S" (,.,_ , 0, <,_,)
[" (r,_ O. ,.)]
.
y,n-I
P (i,n_2
x,n.l
>,n-I
Yn_ 1).
A
’’- l
(._2, y._ ,o)
,,
,’[r (.n-2’ y n-I ,o)]
494
E.H. YANG
Proceeding in this way we then obtain
/k(1)R(x)
G-(x)
3
+
2
P(Xl
G*(x) is obtained from
(3.9) implies the following
G(x)
where
x
N
n
o
(3.9)
by the method as described in above.
o
Z-2
where
(Xl+l x2). Keeping now
Yl O,1,2,...Xl-1 to
x
and su over
x
2
Obviously,
fixed in the last inequality, set
Xl
Yl
obtain the inequality
n
";’
y)l
and hence we have for
R(X) K-
K
-I
0
<
x
<
K[f(O,x("2))
R(x)
xe N
o
X
immediately.
+
G’)(-(yl,
+
Hence the desired upper bound on
definition of
Z,2
P(YI’ Z2
u(x)
in
P(YI’ Z2)
(3.10)
(3.2) follows from (3.1), (3.10), and the
The choice of
X
N
n
O
is obvious.
REMARK 2. By applying the same argument as used in the proofs of Theorems 2 and
3, we can easily establish an extension of the Theorem 4 of [1], which yields an
upper bound for the solutions of the following inequality
Here
x
N
n
O
and
u(x), f(x), p(x), and W(z)
are the same as defined in Theorem 3.
Because the proof of this result is not difficult, so we leave it here to the reader.
DISCRETE INEQUALITIES OF THE GRONWALL TYPE
495
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BELLMAN, R. and COOKE, K.L.
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Academic
Euations,
Press,
New York, 1963.
3.
4.
BOPAEV, K.B. On Some Discrete Inequalities (Russian), Differencial’nye Uravnenja,
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JONES, G.S.
7.
MCKEE, S. Generalized
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PACHPATTE, B.G.
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POPENDA, J. and WERBOWSKI, J.
II (1979), 143-154.
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Lemmas,
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12.
YANG, E.H.
Difference
Inequalities
Their
On Some New Discrete Inequalities
Nonlinear AnaI.:TMA 7 (1983), 1237-1246.
Applications
of
the
to
Stability
Bellman-Bihari
Type,