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volume 3, issue 4, article 65, 2002.

Received 8 April, 2001;

accepted 15 June, 2002.

Communicated by:D. Bainov

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Journal of Inequalities in Pure and Applied Mathematics

ON CERTAIN NEW INTEGRAL INEQUALITIES AND THEIR APPLICATIONS

S.S. DRAGOMIR AND YOUNG-HO KIM

School of Communications and Informatics Victoria University of Technology

PO Box 14428, Melbourne City MC 8001 Victoria, AUSTRALIA.

EMail:sever.dragomir@vu.edu.au

URL:http://rgmia.vu.edu.au/SSDragomirWeb.html Department of Applied Mathematics

Changwon National University Changwon 641-773, Korea.

EMail:yhkim@sarim.changwon.ac.kr

c

2000School of Communications and Informatics,Victoria University of Technology ISSN (electronic): 1443-5756

034-02

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On Certain New Integral Inequalities and their

Applications

S.S. DragomirandYoung-Ho Kim

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J. Ineq. Pure and Appl. Math. 3(4) Art. 65, 2002

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Abstract

The aim of the present paper is to establish some variants integral inequalities in two independent variables. These integral inequalities given here can be applied as tools in the boundedness and uniquness of certain partial differential equations.

2000 Mathematics Subject Classification:26D10, 26D15.

Key words: Integral inequality, Two independent variables, Partial differential equa- tions, Nondecreasing, Nonincreasing.

1. Introduction

The integral inequalities involving functions of one and more than one independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of differential equations (see [1]–[11]). In recent year, Pachpatte [11] discovered some new integral inequalities involving functions of two independent variables. These inequalities are applied to study the boundedness and uniqueness of the solutions of following terminal value problem for the hyperbolic partial differential equation (1.1) with the condition (1.2).

u_xy(x, y) =h(x, y, u(x, y)) +r(x, y), (1.1)

u(x,∞) =σ_∞(x), u(∞, y) = τ_∞(y), u(∞,∞) =d.

(1.2)

Our main objective here, motivated by Pachpatte’s inequalities [11], is to establish additional new integral inequalities involving functions of two independent variables which can be used in the analysis of certain classes of partial differential equations.

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2. Main Results

Throughout the paper, all the functions which appear in the inequalities are assumed to be real-valued and all the integrals are involved in existence on the domains of their definitions. We shall introduce some notation, Rdenotes the set of real numbers and R+ = [0,∞) is the given subset ofR.The first order partial derivatives of a functionsz(x, y)defined forx, y ∈ Rwith respect tox andyare denoted byzx(x, y)andzy(x, y)respectively.

We need the inequalities in the following Lemma2.1and Lemma2.2, which are given in the book [1].

Lemma 2.1. Letgbe a monotone continuous function in an intervalI,contain- ing a pointu₀,which vanishes in I.Letuandk be continuous functions in an intervalJ = [α, β]such thatu(J)⊂I,and suppose thatkis of fixed sign inJ.

Leta∈I.

(i) Assume thatg is nondecreasing andkis nonnegative. If u(t)≤a+

Z t

α

k(s)g(u(s))ds, t∈J, then

u(t)≤G⁻¹

G(a) + Z t

α

k(s)ds

, α≤t ≤β₁, whereG(u) = Ru

u0dx/g(x), u∈I,andβ₁ = min(v₁, v₂),with v₁ = sup

v ∈J :a+ Z t

α

k(s)g(u(s))ds∈I, α≤t ≤v

,

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v₂ = sup

v ∈J :G(a) + Z t

α

k(s)ds∈G(I), α≤t≤v

. (ii) Assume thatJ = (α, β].If

u(t)≤a+ Z β

t

k(s)g(u(s))ds, t∈J, then

u(t)≤G⁻¹

G(a) + Z β

t

k(s)ds

, α₁ < t≤β, whereα₁ = max(µ₁, µ₂),with

µ₁ = sup

µ₁ ∈J :a+ Z β

t

k(s)g(u(s))ds ∈I, µ≤t≤β

, µ₂ = sup

µ∈J :G(a) + Z β

t

k(s)ds∈G(I), µ≤t ≤β

. The proofs of the inequalities in (i), (ii) can be completed as in [1, p. 40–42].

Here we omit the details.

Letu(x, y), a(x, y), b(x, y)be nonnegative continuous functions defined for x, y ∈R⁺.

Lemma 2.2. (i) Assume thata(x, y)is nondecreasing inxand nonincreasing inyforx, y ∈R⁺.If

u(x, y)≤a(x, y) + Z x

0

Z ∞

y

b(s, t)u(s, t)dtds

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for allx, y ∈R+,then

u(x, y)≤a(x, y) exp Z x

0

Z ∞

y

b(s, t)dtds

.

(ii) Assume that a(x, y)is nonincreasing in each of the variablesx, y ∈ R+. If

u(x, y)≤a(x, y) + Z ∞

x

Z ∞

y

b(s, t)u(s, t)dtds for allx, y ∈R+,then

u(x, y)≤a(x, y) exp Z ∞

x

Z ∞

y

b(s, t)dtds

.

The proofs of the inequalities in (i), (ii) can be completed as in [1, p. 109- 110]. Here we omit the details.

To establish our results, we require the class of functionsSas defined in [2].

A functiong : [0,∞)→[0,∞)is said to belong to the classSif (i) g(u)is positive, nondecreasing and continuous foru≥0, (ii) (1/v)g(u)≤g(u/v), u >0, v ≥1.

Theorem 2.3. Let u(x, y), a(x, y), b(x, y), c(x, y), d(x, y)be nonnegative con- tinuous functions defined for x, y ∈ R+, let g ∈ S.Define a function z(x, y) by

z(x, y) = a(x, y) +c(x, y) Z x

0

Z ∞

y

d(s, t)u(s, t)dtds

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withz(x, y)is nondecreasing inxandz(x, y)≥1forx, y ∈R+.If (2.1) u(x, y)≤z(x, y) +

Z x

α

b(s, y)g u(s, y) ds, forα, x, y∈R⁺andα ≤x,then

(2.2) u(x, y)≤p(x, y)

a(x, y) +c(x, y)e(x, y)

× exp Z x

0

Z ∞

y

d(s, t)p(s, t)c(s, t)dtds

, forx, y ∈R+,where

p(x, y) = G⁻¹

G(1) + Z x

α

b(s, y)ds

, (2.3)

e(x, y) = Z x

0

Z ∞

y

d(s, t)p(s, t)a(s, t)dtds, (2.4)

G(u) = Z u

u0

ds

g(s), u≥u₀ >0, (2.5)

G⁻¹ is the inverse function ofG,and G(1) +

Z x

α

b(s, y)ds ∈Dom(G⁻¹).

Proof. Let z(x, y) is a nonnegative, continuous, nondecreasing and letg ∈ S.

Then (2.1) can be restated as

(2.6) u(x, y)

z(x, y) ≤1 + Z x

α

b(s, y) 1

z(x, y)g(u(s, y)) ds.

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Define a function w(x, y) by the right side of (2.6), then [u(x, y)/z(x, y)] ≤ w(x, y)and

(2.7) w(x, y)≤1 +

Z x

α

b(s, y)g(w(s, y))ds.

Treatingy, y∈R+fixed in (2.7) and using (i) of Lemma2.1to (2.7), we get

(2.8) w(x, y)≤G⁻¹

G(1) + Z x

α

b(s, y)ds

. Using (2.8) in[u(x, y)/z(x, y)]≤w(x, y),we obtain

u(x, y)≤z(x, y)p(x, y),

wherep(x, y)is defined by (2.3). From the definition ofz(x, y)we have (2.9) u(x, y)≤p(x, y) (a(x, y) +c(x, y)v(x, y)),

wherev(x, y)is defined by v(x, y) =

Z x

0

Z ∞

y

d(s, t)u(s, t)dtds.

From (2.9) we get v(x, y)≤

Z x

0

Z ∞

y

d(s, t)p(s, t) (a(s, t) +c(s, t)v(s, t))dtds

=e(x, y) + Z x

0

Z ∞

y

d(s, t)p(s, t)c(s, t)v(s, t)dtds,

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where e(x, y) is defined by (2.4). Clearly, e(x, y)is nonnegative, continuous, nondecreasing in x, x ∈ R⁺ and nonincreasing in y, y ∈ R⁺. Now, by (i) of Lemma2.2, we obtain

(2.10) v(x, y)≤e(x, y) exp Z x

0

Z ∞

y

. Using (2.10) in (2.9) we get the required inequality in (2.2).

Theorem 2.4. Let u(x, y), a(x, y), b(x, y), c(x, y), d(x, y)be nonnegative con- tinuous functions defined forx, y ∈R⁺and letg ∈S.Define a functionz(x, y) by

z(x, y) =a(x, y) +c(x, y) Z ∞

x

Z ∞

y

d(s, t)u(s, t)dtds withz(x, y)is nonincreasing inxandz(x, y)≥1forx, y ∈R⁺.If

u(x, y)≤z(x, y) + Z β

x

b(s, y)g(u(s, y))ds forβ, x, y ∈R⁺andβ ≥x,then

u(x, y)≤p(x, y)

a(x, y) +c(x, y)e(x, y)

×exp Z ∞

x

Z ∞

y

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forx, y ∈R+,where

p(x, y) =G⁻¹

G(1) + Z β

x

b(s, y)ds

, (2.11)

e(x, y) = Z ∞

x

Z ∞

y

d(s, t)p(s, t)a(s, t)dtds.

Gis defined in (2.5),G⁻¹ is the inverse function ofG,and G(1) +

Z β

x

b(s, y)ds∈Dom(G⁻¹).

The details of the proof of Theorem2.4 follows by an argument similar to that in the proofs of Theorem2.3with suitable changes. We omit the details.

Theorem 2.5. Let u(x, y), a(x, y), b(x, y), c(x, y) be nonnegative continuous functions defined for x, y ∈ R⁺ andF : R³+ → R⁺ be a continuous function which satisfies the condition

(2.12) 0≤F(x, y, u)−F(x, y, v)≤K(x, y, v)(u−v)

foru ≥ v ≥ 0,whereK(x, y, v)is a nonnegative continuous function defined forx, y, v ∈R+.And letg ∈S.Define a functionz(x, y)by

z(x, y) = a(x, y) +c(x, y) Z x

0

Z ∞

y

F(s, t, u(s, t))dtds with nondecreasing inxandz(x, y)≥1forx, y ∈R⁺.If

(2.13) u(x, y)≤z(x, y) + Z x

α

b(s, y)g u(s, y) ds

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forα, x, y∈R+andα ≤x,then (2.14) u(x, y)≤p(x, y)

a(x, y) +c(x, y)A(x, y)

×exp Z x

0

Z ∞

y

K(s, t, p(s, t)a(s, t))p(s, t)c(s, t)dtds

forx, y ∈R+,wherep(x, y)is defined by (2.3) and

(2.15) A(x, y) =

Z x

0

Z ∞

y

F (s, t, p(s, t)a(s, t))dtds, G(u) = Ru

u0(ds/g(s)), u≥u₀ >0, G⁻¹ is the inverse function ofG,and G(1) +

Z x

α

b(s, y)ds ∈Dom(G⁻¹).

Proof. The proof of this theorem follows by argument similar to that given in the proof of Theorem2.3. Letz(x, y)is a nonnegative, continuous, nondecreasing and letg ∈S,then, we observe that

u(x, y)≤z(x, y)p(x, y),

wherep(x, y)is defined by (2.3). From the definition ofz(x, y)we have (2.16) u(x, y)≤p(x, y) a(x, y) +c(x, y)w(x, y)

, wherew(x, y)is defined by

w(x, y) = Z x

0

Z ∞

y

F(s, t, u(s, t))dtds.

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From (2.12) and (2.16) we get w(x, y)≤

Z x

0

Z ∞

y

F (s, t, p(s, t) (a(s, t) +c(s, t)w(s, t))) +F (s, t, p(s, t)a(s, t))−F (s, t, p(s, t)a(s, t))

dtds

≤A(x, y) + Z x

0

Z ∞

y

K((s, t, p(s, t)a(s, t))p(s, t)c(s, t)w(s, t)dtds, whereA(x, y)is defined by (2.15). Clearly,A(x, y)is nonnegative, continuous, nondecreasing in x, x ∈ R+ and nonincreasing in y, y ∈ R+. Now, by (i) of Lemma2.2, we obtain

(2.17) w(x, y)≤A(x, y)

×exp Z x

0

Z ∞

y

K((s, t, p(s, t)a(s, t))p(s, t)c(s, t)dtds

. Using (2.16) in (2.17) we get the required inequality in (2.14).

Theorem 2.6. Let the assumptions of Theorem2.5be fulfilled. Define a function z(x, y)by

z(x, y) =a(x, y) +c(x, y) Z ∞

x

Z ∞

y

F(s, t, u(s, t))dtds, with nonincreasing inxandz(x, y)≥1forx, y ∈R+.If

u(x, y)≤z(x, y) + Z β

x

b(s, y)g(u(s, y))ds

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forβ, x, y ∈R+andβ ≥x,then u(x, y)≤p(x, y)

a(x, y) +c(x, y)A(x, y)

×exp Z ∞

x

Z ∞

y

K(s, t, p(s, t)a(s, t))p(s, t)c(s, t)dtds

forx, y ∈R+,wherep(x, y)is defined by (2.11) and A(x, y) =

Z ∞

x

Z ∞

y

F(s, t, p(s, t)a(s, t)) dtds.

Gis defined in (2.5),G⁻¹ is the inverse function ofG,and G(1) +

Z β

x

b(s, y)ds∈Dom(G⁻¹).

The details of the proof of Theorem2.6 follows by an argument similar to that in the proofs of Theorem2.5with suitable changes. We omit the details.

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3. Some Applications

In this section we present some immediate applications of Theorem2.3to study certain properties of solutions of the following terminal value problem for the hyperbolic partial differential equation

u_xy(x, y) = h(x, y, u(x, y)) +r(x, y), (3.1)

u(x,∞) = σ∞(x), u(0, y) =τ(y), u(0,∞) =k, (3.2)

where h : R²+ ×R → R, r : R²+ → R, σ∞, τ(y) : R+ → R are continuous functions andkis a real constant.

The following example deals with the estimate on the solution of the partial differential equation (3.1) with the conditions (3.2).

Example 3.1. Letc(x, y)continuous, nonnegative, nondecreasing inxand nonincreasing inyforx, y ∈R+,and let

(3.3) |h(x, y, u)| ≤c(x, y)d(x, y)|u|, (3.4)

σ_∞(x) +τ(y)−k− Z x

0

Z ∞

y

r(s, t)dtds

≤a(x, y) + Z x

α

b(s, y)g(|u|)ds, where a(x, y), b(x, y), d(x, y), g are as defined in Theorem2.3. If u(x, y) is a solution of (3.1) with the conditions (3.2), then it can be written as (see [1, p.

80])

(3.5) u(x, y) = σ∞(x) +τ(y)−k− Z x

0

Z ∞

y

(h(s, t, u(s, t)) +r(s, t)) dtds

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forx, y ∈R.From (3.3), (3.4), (3.5) we get (3.6) |u(x, y)| ≤a(x, y) +

Z x

α

b(s, y)g(|u|)ds +c(x, y)

Z x

0

Z ∞

y

d(s, t)|u|dtds.

Now, a suitable application of Theorem2.3to (3.6) yields the required estimate following

|u(x, y)| ≤p(x, y)

a(x, y) +c(x, y)e(x, y)

× exp Z x

0

Z ∞

y

forx, y ∈R+,wheree(x, y), p(x, y)are define in Theorem2.3.

Our next result deals with the uniqueness of the solution of the partial differential equation (3.1) with the conditions (3.2).

Example 3.2. Suppose that the functionhin (3.1) satisfies the condition (3.7) |h(x, y, u)−h(x, y, v)| ≤c(x, y)d(x, y)|u−v|,

wherec(x, y), d(x, y)is as defined in Theorem2.3withc(x, y)is nonincreasing iny.Letu(x, y), v(x, y)be two solutions of equation (3.1) with the conditions (3.2). From (3.5), (3.7) we have

(3.8) |u(x, y)−v(x, y)| ≤c(x, y) Z x

0

Z ∞

y

d(s, t)|u(s, t)−v(s, t)|dtds.

Now a suitable application of Theorem 2.3 yields u(x, y) = v(x, y), that is, there is at most one solution to the problem (3.1) with the conditions (3.2).

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References

[1] D. BAINOVAND P. SIMEONOV, Integral Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, 1992.

[2] P.R. BEESACK, Gronwall inequalities, Carleton University Mathematical Lecture Notes, No. 11, 1987.

[3] S.S. DRAGOMIR AND N.M. IONESCU, On nonlinear integral inequal- ities in two independent variables, Studia Univ. Babe¸s-Bolyai, Math., 34 (1989), 11–17.

[4] A. MATEANDP. NEVAL, Sublinear perturbations of the differential equation y⁽ⁿ⁾ = 0 and of the analogous difference equation, J. Differential Equations, 52 (1984), 234–257.

[5] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and their Integrals and Derivatives, Kluwer Academic Pub- lishers, Dordrecht, Boston, London, 1991.

[6] B.G. PACHPATTE, Inequalities for Differential and Integral Equations, Academic Press, New York, 1998.

[7] B.G. PACHPATTE, On some new discrete inequalities and their applica- tions, Proc. Nat. Acad. Sci., India, 46(A) (1976), 255–262.

[8] B.G. PACHPATTE, On certain new finite difference inequalities, Indian J.

Pure Appl. Math., 24 (1993), 373–384.

[9] B.G. PACHPATTE, Some new finite difference inequalities, Computer Math. Appl., 28 (1994), 227–241.

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[10] B.G. PACHPATTE, On some new discrete inequalities useful in the the- ory of partial finite difference equations, Ann. Differential Equations, 12 (1996), 1–12.

[11] B.G. PACHPATTE, On some fundamental integral inequalities and their discrete analogues, J. Ineq. Pure Appl. Math., 2(2) (2001), Article 15.

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