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(1)

VOL. ii NO.

(1988)

115-120

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 integralinequality 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. Let Y(x),

F(x)

be positive continuous functions in a x b and K o, M o, further

W(u)

a non-negative non-decreaslng continuous function for u o. Then the inequality

x

Y(x)

K

+

M

f

F(t) W(Y(t))dt (a x b) a

implies the inequality

x

Y(x) G-I(G(K) +

M

f F(t)

at) (a x

b

b) a

where

u

G(u) f

(Uo > o, u o).

uo 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]. An interesting inequality by Wendroff given without proof in

[5]

is as follows:

(2)

I16

P.T. VAZ

and S.G. DEO

LEMMA 2. Let J {x o x X < } and K

{y

o y Y < }. Further let U(x,y) be a scalar, non-negative continuous function defined on J x K, then for any arbitrary non-negative constants a, b and c, the integral inequality

x y

U(x,y) c

+

a

f

U(s,y)ds

+

b

f

U(x,t)dt

o o

implies that U(x,y)

--<

c exp(ax

+

by

+ abxy).

The proof of this inequality can be constructed by observing that U

--<

P, where

P(x,y) satisfies the equation

x y

P(x,y) c

+

a

I

P(s,y)ds

+

b

I

P(x,t)dt.

o o

Observe that P(x,o) c exp(ax) and P(o,y) c exp(by). Now assuming

P(x,y) exp(ax +

by)q(x,y), one can determine the inequality satisfied by

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.

2. NOTATION.

Let denote the real line. For any rectangle J x K we define the following classes of functions:

(i) C

+

(JxK)

the space of continuous functions

u JxK ]R

+

(ii)

CI(R+)

the space of non-decreasing, non-negative, submultiplicative, continuous functions on

+.

C2(+)

the space of non-decreasing, non-negative, continuous functions on (iii)

+

and such that for

g2 C2 (+)

and for any real-valued function

h(x,y), (x,y)

e J x K,

g2 (u(x’y)) u(x,y)

h(x,y)

g2 (h(x,y)

h(x,y)

Further, we define

u

Gi(u f

dt (u o,

uo

> o).

(2.2)

Uo

gi (t)

Let G.

-I

be the inverse of G

i, i 1,2.

1

3. NAIN RESULT.

THEOREM 3.

I.

Assume that

(a)

u(x,y) C+(J

x K), (b)

gl (u)

e C

(c) g2(u)

C2

(+),

(d) there exists a uo > o such that

-g’(u)

> o and

g2(u)_

> o for

u uo Then for any arbitrary non-negative constants a, b and c, with c

>- I,

the inequality

(3)

x y

u(x,y) =<

c

+

a

f gl(u(s,y)

ds + b

f g2(u(x,t))dt,

(o x _-< X, o y

--<

Y) (3.1)

o o

implies, on a nonempty rectangle, the inequality

u(x,y) G

Gl(C)+axg [G

2

l(G2(1)+by)] g2 (G2(1)+bY)’

(3.2)

where G

.(u),

i 1,2, are as defined in (2.2), (o x

X’

X and o y

Y’

Y).

1

PROOF. We define

x

h(x,y) c

+

a

f gl(u(s’y))ds’

(x,y) J x K.

o

It is clear that

h(x,y)

is non-decreasing and h(x,y) on J x K.

Inequality (3.1) may be written as Y

u(x,y) h(x,y)

+

b

f g2(u(x’t))dt"

o

Dividing throughout by

h(x,y)

and using (2.1) of assumption

(c)

we have

(3.3)

(3.4)

Y

(u(x,t)

u(x,y)

__<

+

b

f g2 h(x,t)

dt

h(x,y)

o

For fixed x e J, an application of Lemma yields u(x,y) h(x,y).G

-I

2

(G2(1) +

by), (o

=<

x X, o

=<

y

--< Y’ --<

Y)

(3.5)

(3.6) Substituting (3.6) in

(3.3)

and employing submultiplicative property of

gl

we obtain

x

-I

h(x,y)

_-< c

+

a

f gl(h(s,y))g1(G2 (G2(1) + by))

ds. (3.7)

o

An application of Lemma again, to (3.7), yields a bound for

h(x,y)

on a nonempty rectangle. The desired inequality now follows by substituting the bound for h(x,y) in (3.6).

REMARK 3.1. In particular, if b o and

g1(u)

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 y K, our estimate further reduces to

u(x,y)

_-< c exp(ax).

Thus,

Gronwall’s

estimate is included in

(3.2).

REMARK 3.2. In the case b o, the estimate in (3.2) reduces to u(x,y) G

-I [G l(c) +

ax

gl(1)]

for each y e K, o

-<-

x

X’ --<

X, which is a Bihari-like estimate.

Further, if in (3.1), a o, then for fixed x J, Y

u(x,y) --<

c

+

b

f g2(u(x’t))dt

o

(4)

118 P.T.

VAZ

and S.G. DEO impi ie s

Y

(u(x t))

u(x,Y)c

!

+

b of

g2 c’

dt y K.

An application of Lemma yields u(x,y)

-<

c G -I

2

[G2(1) +

by],

(x J is fixed, o g y g

Y’

g Y)

This estimate is the same as that obtained from (3.2) with a o.

REMARK 3.3. Let

gi(u)

u, i 1,2. Then

---,u u o, u o, i 1,2 G (u) io

i uo o

Clearly

G.-i(v) u exp(v), i 1,2.

i o

Hence the estimate (3.2) reduces to

u(x,y)

-<

u exp

[log __c +

ax u

exp(log

- +

bv)]- u exp(log

---1

+ by)

O U O O U

o o o

-<

c exp

lax

exp(by)

+

by]

c exp

lax +

by

+

abxy

+

higher order terms]

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

a

beUu

u u

+

a, b o,

Uxy

y x

(o

-<

x

-<

X, o

-<

y _< Y) satisfying the initial values

u(x,o) u(o,y) u(o,o)

c

->

1.

Under the condition (4.2) equation (4.1) can be reformulated in terms of the integral equation

ua+l(s,o)ds

b

Y

x

eU(’t)dt

u(x,y)

c

a+l o o

7

ua+l Y

eU(X,t)

+

(s,y)ds

+

b

f

dt.

o o

Therefore, using the initial data, we obtain the inequality

x y

lu(x,y) -<

k

+

t)

-

o

f lu(s,y) a+ld

s+b of e

lu(x’

dt,

where k

B+I +

be Y).

(4.1)

(4.2)

(4.3)

(5)

The present inequality is equivalent to (3.1). A direct application of Theorem 3.1 yields

-1 [G (k)

+

2

lu(x,y)l -<

G -1

--1

[G

(C,2(1) + by)]

a+l

G2-1(G2

(1)

+ by)

where

u

G1

(u) of

--ai

p dp

--

u uo

],

and

(;2(u)

o3"u

__1

ep dp e-u e o ].

Therefore,

-a

-1/a

-1 -u -1

-l(v)

u av and G

2 (v) log

e e o v

G1

o

Hence

]u(x y)

<

{k-a

ax a+l -I/a

-I -I

a+l

log(e- by)-

log(e by)

This provides a pointwise estimate for solutions of the given equation

(4.1).

REFERENCES

I. BIHARI,

I.

A

Generalization of a Lemma of Bellman and its Application to Unique- ness Problems of Differential Equations Acta Math. Acad. Sci.

Hungar.

7

(1956)

71-94.

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

Note on the Gronwall Inequality in Two Independent Variables, Journal of

Int.egral Equations 4(1982),

272-276.

5.

BECKEMBACK, E.F.

and

BELLMAN,

R.

Inequalities,

Springer-Verlag, Berlin, 1961.

(6)

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