Internat. J. Math. & Math. Sci.
VOL. ii NO.
(1988) 115-120
115
NONLINEAR INTEGRAL INEQUALITY IN TWO INDEPENDENT VARIABLES
P.T. VAZ and S.G. DEO
Department of Mathematics
University Goa
Bambolim, Goa
403 001
INDIA
(Received May 27, 1986 and in revised form October 28, 1986)
ABSTRACT.
In this note, the authors obtain a generalization of the integral inequality
of Bihari [I] to a nonlinear inequality in two independent variables.
With the aid
of this inequality a bound for the solution of a nonlinear partial differential
equation is established.
KEYS WORDS AND PHRASES. Nonlinear integral inequality, submItiplicative function,
nonlinear partial differential equation.
1980 AMS SUBJECT CLASSIFICATION CODE. Primary 35R45, Secondary 34A40.
I.
INTRODUCTION.
In the qualitative analysis of differential equations integral inequalities play
a vital role
[2].
An inequality due to Gronwall continues to draw the attention of
mathematicians because of its usefulness.
The nonlinear generalization of this
inequality due to Bihari [I] is as follows:
LEMMA i.
K
o, M
u
o.
Let
Y(x), F(x)
o, further
W(u)
be positive continuous functions in
a
x
b
and
a non-negative non-decreaslng continuous function for
Then the inequality
x
Y(x)
K + M
f F(t)
W(Y(t))dt
(a
b)
x
a
implies the inequality
Y(x)
G-I(G(K)
x
+ M
f F(t) at)
(a
x
b
b)
a
where
u
G(u)
f
uo
(Uo > o, u
o).
W(t)
This inequality has been further generalized in several directions by Beesack
[3].
It has been recently established that the inequalities of this type in two and
more independent variables can be profitably employed in the analysis of partial
differential equations [4, and references listed therein].
by Wendroff given without proof in
[5] is as follows:
An interesting inequality
P.T. VAZ and S.G. DEO
I16
LEMMA 2.
{x
J
Let
X < } and
x
o
{y
K
Y < }.
y
o
be a scalar, non-negative continuous function defined on
U(x,y)
a, b and c, the integral inequality
any arbitrary non-negative constants
y
x
U(x,y)
U(x,y) --<
implies that
+
c
U(s,y)ds + b
f
a
o
f
o
U(x,t)dt
exp(ax + by + abxy).
c
The proof of this inequality can be constructed by observing that
P(x,y)
Further let
J x K, then for
U
--<
P, where
satisfies the equation
y
x
P(x,y)
+
c
I P(s,y)ds +
a
I P(x,t)dt.
b
o
o
P(x,o)
Observe that
exp(ax)
c
P(o,y)
and
c exp(by).
Now assuming
exp(ax + by)q(x,y), one can determine the inequality satisfied by
P(x,y)
q(x,y)
which results into the given conclusion.
Lemma 2, in view of Lemma I, suggests that it is possible to consider a nonlinear
generalization of the Wendroff’s inequality.
We do this in the present paper and
further show by an example that the generalization of this kind is truely beneficial
in the study of some nonlinear partial differential equations.
NOTATION.
2.
For any rectangle
denote the real line.
Let
J x K
we define the following
classes of functions:
C
(i)
+ (JxK)
the space of continuous functions
(ii)
CI(R+)
(iii)
C2(+)
+
JxK
u
]R
the space of non-decreasing, non-negative, submultiplicative,
continuous functions on
+.
the space of non-decreasing, non-negative, continuous functions on
+
and such that for
g2
C2 (+)
and for any real-valued
h(x,y), (x,y) e J x K,
function
g2 (u(x’y))
u(x,y)
g2 (h(x,y)
h(x,y)
h(x,y)
Further, we define
u
Gi(u
f
Uo
Let
3.
G.
1
-I
(u
1,2.
G i, i
be the inverse of
(2.2)
o, uo > o).
NAIN RESULT.
THEOREM 3. I.
u
o
Assume that
C+(J
(a)
u(x,y)
(b)
gl (u)
e C
(c)
g2(u)
C
2
x K),
(+),
> o such that
o
Then for any arbitrary non-negative constants
(d)
u
dt
gi (t)
the inequality
there exists a
u
-g’(u)
a, b
> o
and
and
g2(u)_
c, with
> o
c
>- I,
for
NONLINEAR INTEGRAL INEQUALITY IN TWO INDEPENDENT VARIABLES
y
x
u(x,y)
=<
+
c
f
a
f
ds + b
gl(u(s,y)
o
117
g2(u(x,t))dt,
o
x _-< X, o
(o
y
--< Y)
(3.1)
implies, on a nonempty rectangle, the inequality
u(x,y)
where
G .(u), i
l(G2(1)+by)]
[G 2
Gl(C)+axg
G
(G2(1)+bY)’
g2
1,2, are as defined in (2.2), (o
X’
x
(3.2)
X
and
o
Y’
y
Y).
1
PROOF.
We define
x
h(x,y)
c
+
f
a
h(x,y)
It is clear that
(x,y)
gl(u(s’y))ds’
o
is non-decreasing and
(3.3)
J x K.
h(x,y)
J x K.
on
Inequality (3.1) may be written as
Y
f
h(x,y) + b
u(x,y)
o
u(x,y) __<
h(x,y)
For fixed
and using (2.1) of assumption (c) we have
h(x,y)
Dividing throughout by
Y
f g2
+ b
o
(u(x,t)
h(x,t)
h(x,y).G 2
-I
(3.5)
dt
x e J, an application of Lemma
u(x,y)
(3.4)
g2(u(x’t))dt"
yields
+ by),
(G2(1)
(o
=<
x
X, o
=<
y
--< Y’ --< Y)
Substituting (3.6) in (3.3) and employing submultiplicative property of
x
h(x,y) _-< c + a
f
o
An application of Lemma
nonempty rectangle.
-I
gl(h(s,y))g1(G2 (G2(1)
+ by))
gl
(3.6)
we obtain
(3.7)
ds.
again, to (3.7), yields a bound for
h(x,y)
on a
The desired inequality now follows by substituting the bound for
h(x,y) in (3.6).
REMARK 3.1.
In particular, if
b
g1(u)
and
o
u, then the estimate in (3.2)
reduces to
u(x,y) _-< G -I [G I(c) + ax].
In view of (2.2), it is clear that for fixed
K, our estimate further reduces to
y
u(x,y) _-< c exp(ax).
Thus, Gronwall’s estimate is included in (3.2).
REMARK 3.2.
In the case
u(x,y)
for each
y e K, o
-<-
G
x
o, the estimate in (3.2) reduces to
b
-I [G
l(c) + ax
X’ --< X,
Further, if in (3.1), a
u(x,y) --< c + b
gl(1)]
which is a Bihari-like estimate.
o, then for fixed
Y
f
o
g2(u(x’t))dt
x
J,
P.T. VAZ and S.G. DEO
118
imp i ie s
u(x,Y)c
+
!
Y
f
b
g2
o
An application of Lemma
(u(xc’ t)) dt
K.
y
yields
u(x,y) -<
-I
c G
[G2(1)
2
(x
+ by],
Y’
is fixed, o g y g
J
g
Y)
This estimate is the same as that obtained from (3.2) with
REMARK 3.3.
gi(u)
Let
u
G (u)
io ---,
uo
i
1,2.
u, i
o,
u
u
Clearly
G.
-i
i
(v)
u
exp(v),
o
-<
u
O
exp [log __c + ax u
U
o
exp(log
O
o
o.
-
1,2
o, i
1,2.
i
Hence the estimate (3.2) reduces to
u(x,y) -<
a
Then
+ bv)]-
u
O
o
c exp
lax exp(by) + by]
c exp
lax + by + abxy + higher order terms]
exp(log
---1
U
+ by)
o
This estimate is obviously not as sharp as the one obtained in Lemma 2.
4.
AN APPLICATION.
Consider the characteristic initial value problem for the nonlinear partial
differential equation
Uxy
a
u u
y
+
beUux
a, b
(o -< x -< X,
o,
o
-< y
c
-> 1.
_<
(4.1)
Y)
satisfying the initial values
u(x,o)
u(o,o)
u(o,y)
(4.2)
Under the condition (4.2) equation (4.1) can be reformulated in terms of the
integral equation
u(x,y)
c
a+l
ua+l(s,o)ds
-
u
a+l
Y
(s,y)ds + b
k
B+I
+
f
eU(X,t) dt.
(4.3)
data, we obtain the inequality
Therefore, using the initial
where
eU(’t)dt
o
o
lu(x,y) -<
Y
o
o
7
+
x
b
k
be
+
Y).
x
f
o
lu(s,y)
a+ld s+b
y
f e lu(x’
o
t)
dt,
NONLINEAR INTF.GRAL INEQUALITY IN TWO INDEPENDENT VARIABLES
The present inequality is equivalent to (3.1).
I19
A direct application of Theorem
3.1 yields
lu(x,y)l -< G -1 [G (k) +
where
u
G1 (u)
f
o
dp
--ai
p
u
[G -1
2
u
Therefore,
G1 -l(v)
--1
-a
o
av
-1/a
G2-1(G2 (1) + by)
+ by)] a+l
(C,2(1)
u
(;2(u)
], and
u
o
o
and
G
-1
2
(v)
log
__1
3"
e
e
e
-u
p
dp
o
e
v
-u
e
o
].
-1
Hence
]u(x y)
<
{k-a
ax
a+l
log(e-
by)-
a+l
-I/a
log(e
-I
by)
-I
This provides a pointwise estimate for solutions of the given equation (4.1).
REFERENCES
I.
BIHARI, I.
2.
LAKSHMIKANTHAM, V. and LEELA, S. Differential and Integral Inequalities,Theory
and Applications, Vol. I, Academic Press, New York, 1969.
3.
BEESACK, P.R. On Integral Inequalities of Bihari Type, Acta Math. Acad. Sci.
Hungar. 28(1976), 81-88.
4.
CORDUNEANU, A.
A Generalization of a Lemma of Bellman and its Application to Uniqueness Problems of Differential Equations Acta Math. Acad. Sci. Hungar. 7
(1956) 71-94.
A Note on the Gronwall Inequality in Two Independent Variables,
Int.egral Equations 4(1982), 272-276.
BECKEMBACK, E.F. and BELLMAN, R. Inequalities, Springer-Verlag, Berlin, 1961.
Journal of
5.