View of On a nonlinear integral inequality arising in the theory of differential equations

REVUE D'ANALYSE NUT{ÉRIQUB ET DE ^THÉORIE DE L'APPROXII\{ATION Tome 25, N* 1-2, 1996, pp.173-183

ON A NONLINEAR INTEGRAL INEQUALITY ARISING IN THE THEORY OF DIFFERENTIAL EQUATIONS

B. G. PACHPATTE (Aurangabad)

l.INTRODUCTION

The following integral inequalify has played a very important role

in

the theory of differeritial equations.

THEOREM A. Let u and

f

^be real-valued nonnegative continuous funcfions

definedþr

t>-0.

If

,'Q)<c2 ⁺²

t

J

0

,f( ")u(

^s)ds,

for all

^{t > Q u,here c 2 0 ¿s} a constant, tlxen I

u(t)<c+l

0

f(

^s)ds,

forallt>0.

As far as we know, this inequality was first considered by L. Ou-lang [7] in

1957 while studying ^the boundedness of the solutions of certain second order dif- ferential equations, lnl979 C. M. Dafermos [3] ^{used the} following variant of the above inequality to establish a different connection between stabilify and second law of thennodynamics,

TueoRpv B.

Assnne

that

the nonnegatíve

functions u

^e

L*fO,sf

and

g e LllO,sf

satisfy the conditíon

t

y, (ù < M'y'

(0)

.

J lzoy2 (x)

+ 2Ng(x)y(')]e, r

e _{fo, s] ,}

0

Nonlinear Inteeral Inequality ¹⁷⁵

2 3

1',7 4 B.G. ^PacltPatte

yllere a, M, N are nonrtegative constanÍ;' ^Then

y

(") < Me*

^{v (o) + Àre* s( dir}

results to the literafure.

fot

^x,

^y

^{e R* and}

v) w)

^{0, v,here}

^k:

^R3*->

^R*

^{is a} conl.irtuous futxcli.on.

If

for

^x,

y

e R*, ^tlten

(2.2)

for

^x,

^y

^{e R*, where}

(2.3)

(2.4)

s=0 l=0

for

^{m, n e No, l.hen}

J

0

x)

(z r)

^u2

(r, y) <

"' * ^zji _1t,", ^r)

^{u (s, r) z (s,} t,u(s,L)) + ^g(s, t) u(s,r)] ^or

ci'

,

00

,(,, y)

< ^p (x, y) + q(x, _1,) ^exp

[{ i

t,",

^{r) k (s, t, p}

^{(s,'l) o'0"),}

Itts .f J'

p(r, y) ₌

^{c +}

lJ

^{s(", ,) a}

00

ry

q(x,

y) ₌

J J

f(",

^t)

L(s,t, p(s,r))dr

^cls ,

00

2. STÂTEN,IENT OF ^RESI]I,TS

hr this ^section we ^state our mainresults ^{to be} proved in this paper' In what follorvs we ^denote by R ^{the set} of real ttuntbers,

n* :

_[0, oo) and N(): {0, ^7, 2,

"'}

^'

Forênyfunctionz(x,y)definedforx,¡'eÃ*'weclenotethepartialderivatives

!

-

,(r,u), 3 _"(r,

^y),

* ^16,y)by

^z,çx.

^,,¡.

^zr(x,

^Ð.

^{zr(x, v)} respectivelv' ^For

ô¡ ' '" ' Ôl

^oYox

any fuirction zQn, n)defined ^{for m, n e No, we define the operaton Atz(ttr,}

n):

^z(m+

l,

n)

-

^{z(m, rt), L;z(rn' n) -- z(rn, ß-}

^{t) -}

^z(nt'

^n)'

^{and LrarzQn'}

n): ^L'l\'z(n'

^{n)]' (see'}

[g]). For all

m> ,,

^{,n,' n e'No and'any} functionP(ni defined ^for n e No we use the usual conventions

forx,leRn.

An interesting ^and useful discrete analogue of Theorem

l

is embodied in the

following theorem.

THEOREM 2. Let u(m,n),f(tn,n), ^{gQrt,rt) l:e} real-valued norutegativefunctions definedfor ^nx, n e No attd c a nonnegati.ve real conslanî' Let

H:Nfr

x

/ln +

À*

be a functíon ^v,hich salisf es the conditiott

(H) ^{0 <} H(m,n,v)- H(tn,n,v') < M(nt,n,v')(v -

^w),

for ^{m,n e}

^{No and}

^v ) w ) 0,

v,here

M

ís a real-valued nonnegaÍive fwtctí.otr definedfor ^tn,

n

^r- No,

w >

0.

If

¡n-l n-l

(2.5)

u2Qn,n)

<

"2

* ztl lf(",r)ø(s, r)Ì/(s,í,u(s,r))

⁺ _s(s,

^t)u(s,t)),

Iz(")

ⁿ

⁼

^{o and}

[I ^{z(') =}

^t

s--tn

,l

s=nt

Our main result is given in the following ^theorern'

THEoREMLLetu(x,y),f(*,y),g(x,y)bereal-valuednowtegativecotttittuous functiohsdef.tredþrx,!cRnandcbeattonegativerealconstuttt,Let

L:R] -+

'R*

l:e a contiuous functíon v'ltich satisf ^{es the} condilion

(L)

⁰

^{< L(x,y,v)} - L(',y,r'r') ^<

^k('r:'

^y'u')(v -

^v') _'

(2.6) u(rn,,) <

^{a(rn,n) +}

b(m.,,,il [t

* ! ^{fin t)u(s,,,} ,("',))],

Nonlinear Integral Inequality r77 I /O

for

^{m, /? € No, v'l1ere}

(2.7)

B.G. ^{Pacirpatte 4 5}

for

^{m, /? € No.}

3. PROF OF THBORE]\{ ¹

We first ¿ìssurlle that c is positive and define ^a function z(x'y)by

(3.1)

^z(

Fronr (3.1) ^arrd

usi'g

the fact that u(*,

y)

= ,q"

^r'l

^'

^{it is easy to observe that}

(3.2) ^{,,,,(r, !) < z\t;Q, ù116, t)r'(r,}

^v,

^{G(,, ,)) n ,(", ,)]'}

Frour (3.2) ancl the facts that x,

y

e R.-, we ^{observe that}

z(x,1,)

> o, zr(x,y) > o, tr(r, ^y) > o

for

t,(x,

y)t

r(x, !)

Define

(3.s)

^u(*,

^{v) =} i i xy f(", t)L(s,t,.ft"¿)o'a"

00

Differentiating (3.5) and then using the fact that Ftr¡

=

< p(x,y)

+

v(x,y)

^{and the} þpothesis ^{(L) we} observe that

,,,y(x,

!) ⁼ f(r, ùL(*,y, F6, ^y))

^<

< f(r,y)L(r,y, ^p(x,y)

⁺

^v(x,y))

⁼

(3.6) ₌ f(r,t)ltçr,

^{y, p(x,}

^y)

^{+ v(x,}

^y)) - ^t(r,y, P(x,/))] *

+f

^(x,

y)L(r,y,

^p(x,

^y))

^<

< f(*,y)k(r,y, p(x,y))u(t' _v)

+

f(,, ^v)L(',

^{v, p(x,}

^v)),

By ^{keeping :r} fixetf in (3.6) ^{and seeting} y

:

f and integrating with ^respect to ^I frorn }'to

y

ånd'then

keepingyfixed in

thé resulting inequality ^and

setting.r:

^{s and}

integrating with respect to s from ^{0 to} -x we obtain

(3.7)

From (3.7) we observe that

(3.8)

wlrere

qr(r,y) ₌

^E

⁺ q(*,y) inwhiche>0

isanarbitrarysmallconstant' ^Siríoe

ør(x,l)

^{is positive and lnonotone} nondecreasing for x,

y

e R*, from (3'8) we ob- nt-t n-l

a(rn,n)

=

^c

+lf r(", ^r),

s=0 l=0 nt-l n-l

(2.8) ^t:(nt, n)

= I t

_j=0

^/(s,

^{r)H(s, t, a(s,}

^t)),

l=0

xy

x,

y) = ,' *

^/J

_{i lr(",}

^{t)u(s, t)L(s, t, u(s, r)) + s(ó-, r)a(s,} r)]aro"''' 00

(3,4) _F(ë, ^{y) <}

^{p(x,1,) +}

].e.

,i a\ ^L( +4''l ^. ,[rt". t)\,,

^y,

J,e, y))* ^{g(', y)].}

(3

3) urlf,+,,*) )- ^'L' \""'/-\

By keeping x fixed in (3.3) ^and

setting¡t: t

and iritegrating with ^resPect to ^I from }'to

y

änd"then keepirìg

y

nxeO

i1

the resulting inequality ^{arrcl setting}

¡ ^:s

^and

integrating with respect to s frotn 0 to -t we obtain

serve that

(3.e) ffi= ^{t. i} [tU,)k(s,t,p(",¿))ffi0'*'

Inequality (3.9) implies ^{the estimate (see,} l&'p.4921)

t'

u(r,

y) < q(x,ù . j'l ,rr,lk(s,t,p(s,

^{r))v(s, r)drds}

00

u(*,

y) 3

q"(x,

ù . Ii 16,lk(s,t,p(",

^{r))v(s, r) drds,}

00

,(*, y) t ^q,(x,r, *r[i

[rU, ^t)k(s,

^421,, r1)oro"),

4^Fø¡)

xl'

J J

r(', lL(s,t,.Et""l¡0,*

00

(3. 1 0)

1?8 B.G. Pachpatte ₆ 7 Nonlinear lutegral Inequality 179

By letting

r

-+ 0 in (3.10) we have By using

(2)

in (4.3) we have

(4,4)

^r^t(J;þ1,ù) =lf l,",lu(,n,,,,J;(*,ù)

⁺

^{g(,n,,ùf^}

Now keepingmfrxedin(4.4), set

¡¿:

^{/ and} sum over

l: 0,1.2,

..,, tt.-l and then keeping ¡z fixed in the resulting inequality setm:

s

and sum over^r: 0,7,2, ,.,

tn-|,

to obtain the following (3

u)

^t

(,, y) < q(x,r,*r[i

Ïr¡,lk(s,t,z(",

r))arc").

The desired inequality

in(2.2)

now follows by using (3.11) in (3.a) and tlie fact

r.---

tnat u(x,

y) < ,lt6.y)

^.

If c is nonnegative, we carÐ/ out the above procedure with c

*

ao instead of c.

rvhere so > 0 is an arbitrary srnall constant, and subsequently ^pass to the

limit

as eo -> ⁰ to obtain (2.2).The proof is complete.

nrl n-t

(4.s)

,þn,n) ^< a(m,r)* ILf(t,, ^)n(,,t, _^14',,))

¡'=0 l=0

Define

m-l n-l

4. PROOF OF THBOREI\{ 2

We assume that c is positive and defìne a function z(m,n) by nt-l n-l

(4,6) v(m, n)

= I I ^{¡9, )n(s,,, J;6:))}

s=0 l=0

z(m,n)

=

^{c2 +}

rI I lf(t,t)r(t,lH(s,t,u(s,l)

^{+ s(s, t)u(s,t)f '}

From (4.6) and using the fact that thesis

(fl

we observe that

lt(J",") . o(^,n) + v(m,n)

and tlre hypo- (4.1)

s=0 /=0

Fronr (4.1) and using the fact that u(m,

n) < _\EØ,,,t)

we obserue that

(4.2) Arlrz(nt,"¡ < z,{rç,41f{,n,n)n(,n,r, J;þ",,ù)

⁺

g(,r,,ùf"

By using tlre fact

ttut

^þ(*,r)>0,A,p(tn¡t¡>0, ,[tç,ru 4 ^< FØ', * t,,t * _ù,

(4,7)

A 2lrv(m, n) =

f

^{(tn, r) n(,n,} _",

$þnfl)

^<

< _.f

(*,fln(,n,n,a(m.n)

+ _v(nt.n))

- u(m,u,o(,n,r))f

+

+

f

þn, ⁿ⁾ n _þn, n, a(nt, n)) ^<

< _{-f (,n,} n)

u

_þn, n, a(tn, n))v (m, n) +

f

^{(tn, n) n (n r, n, a(m, n)) .}

z(tn,

n) <

_,,lzþn

+

l. n)

,

^{for tn, rz e N6,}

it

is easy to observe that

Now keeping m ftxed in (4.7), set n

:

_{/ and sum over /}

:

_{0, 1, 2, ...,}

_n-l

_{and then}

keeping n fixed in the resulting inequality set rz

:

_{s and stllt over.s}

:

_{0, l, 2, ...,}

_m-l

_to

obtain the following

Lrz(m,n) m-l n-l

z(tn + 7,n) + z(m,n)

(4.8) v(nt,n) < b(nt,r)* I I ^/(", lM(s,t,a(s,)þ(s,1.

s=0 t=0

ArAt(FQ,1,t1)) From (4.8) rve observed that

(4.3)

m-l n-l

(4.e)

vþn,n) < b,(rn,r) * I I ^{/(", lM(s,r,}

^{rz(s, r))v(s,}

^r)

^,

s=(l l=0

A,r\,p(tn,n)

wlrere br(m,n)

:

¿

I

b(m,n),

in

which e > _{0 is} an arbitrary small constant. Since br(m,n) is positive and monotone nondecreasing for tn,

n e

N,,, from (4.9) we observe that

z(tn

+7,n)

+ zQn, n)

180

(4.10) Define

Now keeping nt frxed in (a,13) and substituting n = t and then takirig the sum over

l:0,1,2,

.,.,

n-l

and using the fact that

Arw(m,0):

0, we have

Nonlinear Integral hequality 181

The desired inequality in (2.6) now follows by using (4.17) in (4.5) and then let-

ting

^s -+ 0 in the resulting inequality and using the fact that u(rn,

n)

^< zþn, n)

The proof of the case when c is nonnegative call be completed as mentioned in the proof of Theorem

l

^and hence the proof is complete.

5. SOX{E APPLICATIONS

In this section, we present some applications of our results to obtain bounds on the solutions of certain differential and sum-difference equations. These appli- cations are given as examples,

Exatnple

L

As a first application we obtain a bound on the solution of the following partial differential equati orl

(5.1)

^{(z(x, y)2,(x,}

^y)), = '(r, )lr(r, ^!,

^z(x,

^{y)) * s(*,y)],}

with the given boundary conditions

(s,2)

z(x,o)

= @("), ,(0,y) ₌

Y(_y) ,

forx,yeR*,where

g:Rl -+ R, F:4 _" ^R I 4,0, _V:Àn -)

Rarecontinuous functions and O(0)

=

^{V(0) . It is easy to} observe that proble¡r (5 . 1)-(5.2) is equiva- lent to the integral equation

(s,3)

wlrere

d(r,y) = O2(r)+V'(y) -O'(0) .If z(x,)isasolutionof

(5.1)-(5.2), then clearly it is also a solutiorr of the integral equation (5.3). We assume that

(s.4) _la(,, y)l <

"2,

(s

5)

_lr(.*,, ^.r,.

^,(r.t,))l

^<

fG, ùL(r,t,lz(x,

^{y)l) ,}

wheref(x,

y) is

^a nonnegative real-valued continuous

function

defined

for

x,y € À*,cisanonnegativerealconstant

and

L:Rl ^{x R*} I R* is acon- tinuous function

satisfying the condition (L)

in

Theorem.

l

From (5.3), (5.4) ancl (5.5) we observe that

B.G. Pachpatte

, _L

n-l

fþ",1M(m,t,a(m,l),

t=0

I

₉

94=, ^* f Ï ^r(", lM(s,t,a(s,t))-v(s'

¿) .

b,(m,n\ - ' L^

_.Ltr

\''

^{- /-'-}

\''

^{-' --\-) -} '/ '/ å"(s, r) -u\"-)',/ _S=0r=0

,)

=

¹⁺

Ï"i fg,t)u(s,t,a(s,))#

s=0 l=0

Fonn ^(4,1

l)

^and using (4,10) we observe that

(4.12) L2Ly(m,n)< f(m,n)M(m,n,a(m,n))wþn,n).

From the definition s¡y,(n,n) we observe that w(m,n)

<

u,(m,n

+ l),

for m, n e No. Using this fact in (4.12) we observe that

(4.13) -ttt:'þn'y). f(,n,n)tø(rn,r,a(nt,n)).

u,(nt, n)

^i

mrn)

(4.15)

^u'(tn

^*

^1,

') t

w(tn, n r

* I fþn,t)M(tn,t,a(m,r))

(4

t4)

_v'(m,n)

Frorn (4.14) we observe that

(4.16)

llsing ⁽⁴ .16) in (4. I 0) we get

w(m,n)

< fl

^I

^* I f$,t)M(s,t,a(s,t))

n-l

t=0

xy

,'(', ^y) =

^d(-r,

ù * zlJ _4", r¡lr'(" ,t,

^z(s,r)) + s(s, r)]aras ^, 00

. ,2 n ^tr),

,(r,v)' ' _"' ^{. rl} I[r(",t) ,(r,)t(r,t, z(s,r))+

g(s,

t) z(s,r)]ara'

00 Now keepingrtfrxed

in

(4.15) and letting

m:

s and substituting

s:0,1,2,

¡rz-1 successively, we have

m-l n-1

¡=0 /=0

(4,t1)

^v(m,n)

< b"(tn,n\ll

I n

I f(s,t)u(s,t,a(s,t))

n-l n-l

.r=0 /=O

Nonlinear Integral Inequality r83

10 11

B.G. PachPatte 182

Now ^an application of Theorem ¹ yields

where

(5 .7)

s=0 /=0

wlrere ^{å, g:} N'o

-,

^R,

(s .B)

n-1

n-l

b1þn,

n) = > t ^/(s,

^{r)H(s, t, a1(s,}

^t)),

s=0 l=0

for m, n e No. Inequality (5.10) gives the bound ou the solutionz(nl, ⁿ⁾ of equation (5.7) in tenns of ^the known functious'

In

conclutling this paper, we note that the inequalities established

in

this paper can be extenãed very easily to more ^than two independent variables. ^The

pråu..

fonnulations of thése inequalities is very ^close to that of given

in

Theo- l."rrr, 1 and 2 with suitable rnodificatious and heuce we ^do not discuss it here.

ry

qr(*,

y) = I

^J

f(', t)L(s,t,p,(s,

^r))arcs,

00

for x,

y e R*.

lnequality

(5'6)

gives.us the bound on the solution

z(x'y) of

fS

¡-(S

^{Zl iritenns of the} known functions'

Example2'Asasecolrdapplication,weslrallobtainaboundontlresolution of the following sum-difTereuce equation

,2

(,n,

^rr)

=

^h(nt,

ù . |Ëiz(s, ^r)lr(s,

^{t' z(s'}

^{t)) +}

^{g("' r)]} '

ar(m,n)

=

^c

+ tlls(",r)1,

m-1n-l

r=0 /=0

REFERENC ^{E S}

('t'

(s.6) _lr(r,y)l 3

^pt(x,v)

⁺ q{x,v)'*p[ I

_J

fi'' t)k(s't'p'(s'

^r))ords

\o

^o

xy

pír,y) ₌ c* JI

₀₀

"("

dtds,

)l

F:Nfr x

R

->

^{-R are functions such that}

lh(,n,,r)l

<

"2 ^,

1, V. Barbu, Dffirential Equatiotts (in Romanian), Ed Junime4 Iaçi' 1985'

2. H. Brêzis, Opérateure ,rl*ín ou, nlotlotones el sénigroupes tle conlracliotts ¿ans les é'tpaces de Hílb ert, North-Holand, Amsterdarn, ^{1 973}

3

^{C' M. Dafennos, The seco'ntJ law of} Therntotb'nantics and slabilíry' Arch Rat Ìr4ech Anal' 70 (19',79), 167-1',19.

4. S. S Dragomir, The Gronwall Type Lemmas anil Applicatiorrs, Motiografii Matenratice, univ Timiçoara 29, 1981 ^.

5. A, Haraux, Nonlinear Evolution Equations: Global behavior of solutions,Lectore Notes ^irl Math- etnatics, No. 841, Springer-Verlag, Berlin, New York' 1981'

6. S. N. Olekhnik, Bound"l.n"ri and unbounclerJness of solutiotts ol some Ð'stettls of ordinarv dilferen- tial equatíorts, Vestnik Moskov Univ Mat' 27 (19'72)'34-44'

7 . L. Ovlang, The boundedness of solutiorts of linear differential equatiotts y" ⁺ A(l)l'

:

0' shuxue Jinzhan 3(1951), ^409''415 '

g. B. G. Pacbpatte, on ^son'rc integrotlffiretttial inequalities of tl'te wendorf _{4'pe, J.} Math Anal"

Appl. 73 (1980), 491-500.

g. B. G. Pachpatte, Discrete inequalities ín two t,ariables ancl their applicatiotts, Radovi lr4atematicki 6 (1990), 235-24',7 ^.

10. M. Tsutsumi and L Fukuncla, On solutiotts of the derh,atites ttonlittear Schrö¿inget'equation.

Exístenceonduniquenesslheoretn'FurikcialajEkvacioj,23(1980)'259.277

Received 1.04.1994 57. ^Shri ltrìketan Colon.Y

Aurangabad ⁴³¹⁰⁰¹ (Ilaharashtra) Indio

(s.e) lr(,r,n,r(nt,l)l' f('',')a('',",1"(*,")l)'

wheref(m,n)isareal-r,aluednomegativefunctiondefinedforn4,?€N0,cisa

nonnegative real ^constan

t

^{and H'.Nfr}

^x

^tR*

+

^{Ãn is a} function satisfyi'g ^{the cou-} dition (H) in Theorem 2, From (5'7), (5'8) ^and (5'9) we observe that

l,(,,,,,)12

<

"2 r-rjî[/(', ^{r)l'1'' r)lø(s'}

t'lz(s'r)l) +

ls("' t)l'("'

^t)l]

s=0 f =0

Now ^an application of ^{Theoretn 2} yields

^-1f n-r 'l

(s,10) lr(,n,n)l<

^a1(rn,n)

\ ⁺ ð,(rrr.,')flll * ';äL t=o t f(s,t)M(s'r'a1(s'r))l'

^J

for m, n € Na, where