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volume 7, issue 3, article 111, 2006.

Received 24 October, 2005;

accepted 24 May, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

ON SOME NEW RETARD INTEGRAL INEQUALITIES INn INDEPENDENT VARIABLES AND THEIR APPLICATIONS

XUEQIN ZHAO AND FANWEI MENG

Department of Mathematics Qufu Normal University Qufu 273165

People’s Republic of China.

EMail:xqzhao1972@126.com

c

2000Victoria University ISSN (electronic): 1443-5756 318-05

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On Some New Retard Integral Inequalities innIndependent

Variables and Their Applications Xueqin Zhao and Fanwei Meng

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J. Ineq. Pure and Appl. Math. 7(3) Art. 111, 2006

http://jipam.vu.edu.au

Abstract

In this paper, we established some retard integral inequalities innindependent variables and by means of examples we show the usefulness of our results.

2000 Mathematics Subject Classification:26D15, 26D10.

Key words: Retard integral inequalities; integral equation; Partial differential equa- tion.

1. Introduction

The study of integral inequalities involving functions of one or more independent variables is an important tool in the study of existence, uniqueness, bounds, stability, invariant manifolds and other qualitative properties of solutions of differential equations and integral equations. During the past few years, many new inequalities have been discovered (see [1, 3, 4, 7,8]). In the qualitative analy- sis of some classes of partial differential equations, the bounds provided by the earlier inequalities are inadequate and it is necessary to seek some new inequalities in order to achieve a diversity of desired goals. Our aim in this paper is to establish some new inequalities in n independent variables, meanwhile, some applications of our results are also given.

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2. Preliminaries and Lemmas

In this paper, we suppose R+ = [0,∞), is subset of real numbers R, e0 = (0, . . . ,0), α(t) = (αe 1(t1), . . . , αn(tn)) ∈ Rⁿ+, t = (t1, . . . , tn) ∈ Rⁿ+, s = (s₁, . . . , s_n) ∈ Rⁿ+, re = (r₁, . . . , r_n), re₀ = (r₁₀, . . . , r_n0), ze = (z₁, . . . , z_n), ze₀ = (z₁₀, . . . , z_n0), T = (T₁, . . . , T_n)∈Rⁿ+.

Iff :Rⁿ+→R+, we suppose

1. s≤t⇔s_i ≤t_i (i= 1,2, . . . , n);

2. Rα(t)_e

e0 f(s)ds=Rα1(t1)

0 · · ·Rαn(tn)

0 f(s₁, . . . , s_n)ds_n. . . ds₁; 3. D_i = _dt^d

i, i = 1,2,. . . , n.

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3. Main Results

In this part, we obtain our main results as follows:

Theorem 3.1. Letψ ∈ C(R+,R+)be a nondecreasing function withψ(u) >

0 on (0,∞), and let c be a nonnegative constant. Let αi ∈ C¹(R⁺,R⁺) be nondecreasing withα_i(t_i)≤t_i onR+(i= 1, . . . , n). Ifu, f ∈C(Rⁿ+,R+)and

(3.1) u(t)≤c+

Z α(t)_e e0

f(s)ψ(u(s))ds, fore0≤t < T, then

(3.2) u(t)≤G⁻¹

"

G(c) + Z α(t)_e

e0

f(s)ds

# ,

where

G(z) =e Z z_e

ze0

ds

ψ(s), ez ≥ze₀ >0.

G⁻¹ is the inverse ofG,T ∈Rⁿ+is chosen so that

(3.3) G(c) +

Z α(t)_e e0

f(s)ds∈Dom(G⁻¹), e0≤t < T.

Define the nondecreasing positive functionz(t)and make (3.4) z(t) =c+ε+

Z α(t)_e e0

f(s)ψ(u(s))ds, e0≤t < T,

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whereεis an arbitrary small positive number. We know that

(3.5) u(t)≤z(t), D₁D₂· · ·D_nz(t) =f(α)ψ(u(e α))αe ⁰₁α⁰₂· · ·α⁰_n. Using (3.5), we have

(3.6) D₁D₂· · ·D_nz(t)

ψ(z(t)) ≤f(α)αe ⁰₁α⁰₂· · ·α⁰_n. For

(3.7) D_n

D₁D₂· · ·Dn−1z(t) ψ(z(t))

= D₁D₂· · ·D_nz(t)ψ(z(t))−D₁D₂· · ·D_n−1z(t)ψ⁰D_nz(t)

ψ²(z(t)) ,

usingD₁D₂· · ·Dn−1z(t)≥0,ψ⁰ ≥0,D_nz(t)≥0in (3.7), we get (3.8) D_n

D₁D₂· · ·Dn−1z(t) ψ(z(t))

≤ D₁D₂· · ·D_nz(t)

ψ(z(t)) ≤f(α)αe ⁰₁α⁰₂· · ·α⁰_n. Fixingt₁, . . . , tn−1, settingt_n =s_n, integrating fromt_nto∞, yields

(3.9) D₁D₂· · ·Dn−1z(t) ψ(z(t))

≤

Z αn(tn) 0

f(α1(t1), . . . , αn−1(tn−1), sn)α⁰₁α⁰₂· · ·α⁰_n−1dsn.

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Using the same method, we deduce that (3.10) D₁z(t)

ψ(z(t)) ≤

Z α2(t2) 0

· · ·

Z αn(tn) 0

f(t₁, s₂, . . . , s_n)α⁰₁ds_n. . . ds₂, and integration on[t1,∞)yields

(3.11) G(z(t))

≤G(c+ε) +

Z α1(t1) 0

· · ·

Z αn(tn) 0

f(s₁, s₂, . . . , s_n)ds_n. . . ds₁, t∈Rⁿ+. From the definition ofGand lettingε→0, we can obtain inequality (3.2).

Remark 1. If we letG(z)→ ∞,z → ∞, then condition (3.3) can be omitted.

Corollary 3.2. If we letψ = s^r,0 < r ≤ 1in Theorem 3.1, then fort ∈ Rⁿ+, we have

(3.12) u(t)≤









 h

c^1−r+ (1−r)Rα(t)_e

e0 f(s)dsi_1−r¹

, 0< r <1;

cexp Rα(t)_e

e0 f(s)ds

, r = 1.

Remark 2. If we let n = 1, r = 1, α(t) =e t, in Corollary3.2, we obtain the Mate-Nevai inequality.

Theorem 3.3. Letϕ∈C(R⁺,R⁺)be an increasing function withϕ(∞) =∞.

Let ψ ∈ C(R+,R+) be a nondecreasing function and let cbe a nonnegative

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constant. Letα_i ∈ C¹(R+,R+)be nondecreasing withα_i(t_i) ≤ t_i onR+(i = 1, . . . , n). Ifu, f ∈C(Rⁿ+,R⁺)and

(3.13) ϕ(u(t))≤c+

Z α(t)_e e0

f(s)ψ(u(s))ds, fore0≤t < T, then

(3.14) u(t)≤ϕ⁻¹

(

G⁻¹[G(c) + Z α(t)e

e0

f(s)ds]

) ,

where G(z) =e R_ez ze0

ds

ψ[ϕ⁻¹(s)],ez ≥ ze₀ > 0, ϕ⁻¹, G⁻¹ are respectively the inverse ofϕandG,T ∈Rⁿ+is chosen so that

(3.15) G(c) + Z α(t)_e

e0

f(s)ds∈Dom(G⁻¹), e0≤t < T.

Proof. From the definition of theϕ, we know (3.13) can be restated as

(3.16) ϕ(u(t))≤c+ Z α(t)_e

e0

f(s)ψ[ϕ⁻¹(ϕ(u(s)))]ds, t∈Rⁿ+. Now an application of Theorem3.1gives

(3.17) ϕ(u(t))≤G⁻¹

"

G(c) + Z α(t)_e

e0

f(s)ds

#

, e0≤t < T.

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So,

(3.18) u(t)≤ϕ⁻¹ (

G⁻¹[G(c) + Z α(t)_e

e0

f(s)ds]

)

, e0≤t < T.

Corollary 3.4. If we letϕ = s^p, ψ =s^q, p, q are constants, andp ≥ q > 0in Theorem3.3, fore0≤t < T, then

(3.19) u(t)≤









 h

c¹⁻^q^p +

1− ^q_p Rα(t)_e

e0 f(s)dsi_p−q¹

, when p > q;

c¹^pexp

1 p

R_eα(t)

e0 f(s)ds

, when p=q.

Theorem 3.5. Let u, f and g be nonnegative continuous functions defined on Rⁿ+, let c be a nonnegative constant. Moreover, letw₁, w₂ ∈ C(R+,R+) be nondecreasing functions with w_i(u) > 0 (i = 1,2) on (0,∞). Let α_i ∈ C¹(R+,R+)be nondecreasing withα_i(t_i)≤t_i onR+(i= 1, . . . , n). If

(3.20) u(t)≤c+ Z α(t)_e

e0

f(s)w₁(u(s))ds+ Z t

e0

g(s)w₂(u(s))ds, fore0≤t < T, then

(i) for the casew₂(u)≤w₁(u), (3.21) u(t)≤G⁻¹₁

(

G₁(c) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds )

,

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(ii) for the casew₁(u)≤w₂(u), (3.22) u(t)≤G⁻¹₂

(

G₂(c) + Z α(t)e

e0

f(s)ds+ Z t

e0

g(s)ds )

,

where

(3.23) G_i(z) =e Z _ez

ze0

ds

w_i(s), ez ≥ze₀ >0, (i= 1,2) andG⁻¹_i (i= 1,2)is the inverse ofG_i, T ∈Rⁿ+is chosen so that (3.24) G_i(c) +

Z α(t)_e e0

f(s)ds +

Z t e0

g(s)ds ∈Dom(G⁻¹_i ), (i= 1,2), e0≤t < T.

Proof. Define the nonincreasing positive functionz(t)and make (3.25) z(t) =c+ε+

Z α(t)_e e0

f(s)w1(u(s))ds+ Z t

e0

g(s)w2(u(s))ds, whereεis an arbitrary small positive number. From inequality (3.20), we know

(3.26) u(t)≤z(t)

and

(3.27) D1D2· · ·Dnz(t) = [f(α)we 1(u(α))αe ⁰₁α⁰₂· · ·α⁰_n+g(t)w2(u(t))].

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The rest of the proof can be completed by following the proof of Theorem3.1 with suitable modifications.

Theorem 3.6. Let u, f and g be nonnegative continuous functions defined on Rⁿ+, and let ϕ ∈ C(R+,R+)be an increasing function withϕ(∞) = ∞ and letcbe a nonnegative constant. Moreover, letw₁, w₂ ∈ C(R⁺,R⁺)be nonde- creasing functions withw_i(u) > 0 (i = 1,2)on(0,∞), α_i ∈ C¹(R+,R+)be nondecreasing withα_i(t_i)≤t_ionR+(i= 1, . . . , n). If

(3.28) ϕ(u(t))≤c+ Z α(t)_e

e0

f(s)w₁(u(s))ds+ Z t

e0

g(s)w₂(u(s))ds, fore0≤t < T, then

(i) for the casew₂(u)≤w₁(u), (3.29) u(t)≤ϕ⁻¹

( G⁻¹₁

"

G1(c) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

#)

;

(ii) for the casew₁(u)≤w₂(u), (3.30) u(t)≤ϕ⁻¹

( G⁻¹₂

"

G₂(c) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

#) ,

where

G_i(ez) = Z _ez

ze0

ds

w_i(ϕ⁻¹(s)), z >e ze₀, (i= 1,2),

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andϕ⁻¹, G⁻¹_i (i = 1,2)are respectively the inverse of G_i, ϕ, T ∈ Rⁿ+is chosen so that

(3.31) Gi(c) + Z α(t)_e

e0

f(s)ds +

Z t e0

g(s)ds ∈Dom(G⁻¹_i ), (i= 1,2), e0≤t < T.

Proof. From the definition ofϕ, we know (3.28) can be restated as

(3.32) ϕ(u(t))≤c+ Z α(t)e

e0

f(s)w₁[ϕ⁻¹(ϕ(u(s)))]ds +

Z t e0

g(s)w₂[ϕ⁻¹(ϕ(u(s)))]ds, t∈Rⁿ+. Now an application of Theorem3.5gives

ϕ(u(t))≤G⁻¹_i (

G_i(c) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds )

, e0≤t < T, where T satisfies (3.31). We can obtain the desired inequalities (3.29) and (3.30).

Theorem 3.7. Let u, f and g be nonnegative continuous functions defined on Rⁿ+ and let c be a nonnegative constant. Moreover, let ϕ ∈ C(R+,R+) be an increasing function with ϕ(∞) = ∞,ψ ∈ C(R⁺,R⁺)be a nondecreasing

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function withψ(u)>0on(0,∞)andα_i ∈C¹(R+,R+)be nondecreasing with αi(ti)≤ti onR⁺(i= 1, . . . , n). If

(3.33) ϕ(u(t))≤c+ Z α(t)_e

e0

[f(s)u(s)ψ(u(s)) +g(s)u(s)]ds, fore0≤t < T, then

(3.34) u(t)

≤ϕ⁻¹ (

Ω⁻¹

"

G⁻¹ G[Ω(c) + Z α(t)_e

e0

g(s)ds] + Z α(t)_e

e0

f(s)ds

!#) , where

Ω(er) = Z _er

re0

ds

ϕ⁻¹(s), er ≥re0 >0, G(z) =e

Z _ez

ze0

ds

ψ{ϕ⁻¹[Ω⁻¹(s)]}, ez ≥ze0 >0,

Ω⁻¹, ϕ⁻¹, G⁻¹are respectively the inverse ofΩ, ϕ, G. AndT ∈R+is chosen so that

G

"

Ω(c) + Z α(t)_e

e0

g(s)ds

# +

Z α(t)_e e0

f(s)ds ∈Dom(G⁻¹), e0≤t < T, and

G⁻¹ (

G

"

Ω(c) + Z α(t)_e

e0

g(s)ds

# +

Z α(t)_e e0

f(s)ds )

∈Dom(Ω⁻¹), e0≤t < T.

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Proof. Let us first assume that c > 0. Defining the nondecreasing positive function z(t)by the right-hand side of (3.33)

z(t) =c+ Z α(t)_e

e0

[f(s)u(s)ψ(u(s)) +g(s)u(s)]ds, we know

(3.35) u(t)≤ϕ⁻¹[z(t)]

and

(3.36) D1D2· · ·Dnz(t) = [f(α)u(e α)ψ(u(e α)) +e g(α)u(e α)]αe ₁⁰α⁰₂· · ·α⁰_n. Using (3.35), we have

(3.37) D₁D₂· · ·D_nz(t)

ϕ⁻¹(z(t)) ≤[f(α)ψ(ϕe ⁻¹(z(α))) +g(α)]αe ⁰₁α⁰₂· · ·α⁰_n. For

(3.38) D_n

D₁D₂· · ·Dn−1z(t) ϕ⁻¹(z(t))

= D₁D₂· · ·D_nz(t)ϕ⁻¹(z(t))−D₁D₂· · ·Dn−1z(t)(ϕ⁻¹(z(t)))⁰D_nz(t)

(ϕ⁻¹(z(t)))² ,

usingD₁D₂· · ·Dn−1z(t)≥0,D_nz(t)≥0,(ϕ⁻¹(z(t)))⁰ ≥0in (3.38), we get D_n

D₁D₂· · ·Dn−1z(t) ϕ⁻¹(z(t))

≤ D₁D₂· · ·D_nz(t) ϕ⁻¹(z(t)) (3.39)

≤[f(α)ψ(ϕe ⁻¹z(α)) +e g(α)]αe ₁⁰α⁰₂· · ·α⁰_n.

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Fixingt₁, . . . , tn−1, settingt_n =s_n, integrating from0tot_nwith respect tos_n yields

(3.40) D₁D₂· · ·D_n−1z(t) ϕ⁻¹(z(t))

≤

Z αn(tn) 0

[f(α₁(t₁), . . . , αn−1(tn−1), s_n)ψ(ϕ⁻¹(z(α₁(t₁), . . . , αn−1(tn−1), s_n))) +g(α₁(t₁), . . . , α_n−1(t_n−1), s_n)]α⁰₁α⁰₂· · ·α⁰_n−1ds_n. Using the same method, we deduce that

(3.41) D1z(t) ϕ⁻¹(z(t))

≤

Z α2(t2) 0

· · ·

Z αn(tn) 0

[f(α₁(t₁), s₂, . . . , s_n)ψ(ϕ(z(α₁(t₁), s₂, . . . , s_n))) +g(α₁(t₁), s₂, . . . , s_n)]α⁰₁ds_n. . . ds₂. Settingt1 =s1, and integrating it from0tot1 with respect tos1 yields (3.42) Ω(z(t))≤Ω(c) +

Z α(t)_e e0

f(s)ψ(ϕ⁻¹(z(s))ds+ Z α(t)_e

e0

g(s)ds,

LetT₁ ≤T be arbitrary, we denotep(T₁) = Ω(c) +Rα(Te 1)

0 g(s)ds, from (3.42), we deduce that

Ω(z(t))≤p(T₁) + Z α(t)_e

0

f(s)ψ[ϕ⁻¹z(s)]ds, e0≤t≤T₁ ≤T.

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Now an application of Theorem3.3gives z(t)≤Ω⁻¹

( G⁻¹

"

G(p(T₁)) + Z α(t)_e

e0

f(s)ds

#)

, e0≤t≤T₁ ≤T, so

u(t)≤ϕ⁻¹ (

Ω⁻¹

"

G⁻¹ G(p(T₁)) + Z α(t)e

e0

f(s)ds

!#)

, e0≤t≤T₁ ≤T.

Taking t = T₁ in the above inequality, sinceT₁ is arbitrary, we can prove the desired inequality (3.34).

If c = 0 we carry out the above procedure with ε > 0 instead of c and subsequently letε→0.

Settingf(t) = 0,n = 1, we can obtain a retarded Ou-Iang inequality.

Letu, f andgbe nonnegative continuous functions defined onRⁿ+and letc be a nonnegative constant. Moreover, let ψ ∈ C(R+,R+)be a nondecreasing function withψ(u)>0on(0,∞)andα_i ∈C¹(R+,R+)be nondecreasing with αi(ti)≤ti onR⁺(i= 1, . . . , n). If

u²(t)≤c²+ Z α(t)_e

e0

[f(s)u(s)ψ(u(s)) +g(s)u(s)]ds, fore0≤t < T, then

u(t)≤Ω⁻¹

"

Ω c+1 2

Z α(t)_e e0

g(s)ds

! +1

2 Z α(t)_e

e0

f(s)ds

# ,

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where

Ω(z) =e Z _ez

ze0

ds

ψ(s) z >e ze₀, Ω⁻¹ is the inverse ofΩ, andT ∈Rⁿ+is chosen so that

Ω c+1 2

Z α(t)_e e0

g(s)ds

! +1

2 Z α(t)_e

e0

f(s)ds∈Dom(Ω⁻¹), e0≤t < T.

Corollary 3.8. Let u, f andg be nonnegative continuous functions defined on Rⁿ+and letcbe a nonnegative constant. Moreover, letp, qbe positive constants withp≥q, p 6= 1. Letα_i ∈C¹(R+,R+)be nondecreasing withα_i(t_i)≤t_ion R+(i= 1, . . . , n). If

u^p(t)≤c+ Z α(t)_e

e0

[f(s)u^q(s) +g(s)u(s)]ds, t≥0 fore0≤t < T then

u(t)≤











c⁽¹⁻¹^p⁾+ ^p−1_p Rα(t)e

e0 g(s)ds_p−1^p exph

1 p

Rα(t)e

e0 f(s)dsi

, whenp=q;

c⁽¹⁻^p¹⁾+^p−1_p Rα(t)_e

e0 g(s)ds^p−q_p−1

+ ^p−q_p Rα(t)_e e0 f(s)ds

_p−q¹

,whenp > q.

Theorem 3.9. Letu, f and g be nonnegative continuous functions defined on Rⁿ+, and let ϕ ∈ C(R+,R+)be an increasing function withϕ(∞) = ∞ and

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letcbe a nonnegative constant. Moreover, letw₁, w₂ ∈ C(R+,R+)be nonde- creasing functions withwi(u) >0 (i= 1,2)on(0,∞), andαi ∈C¹(R⁺,R⁺) be nondecreasing withα_i(t_i)≤t_i (i= 1, . . . , n). If

(3.43) ϕ(u(t))≤c+ Z α(t)_e

e0

f(s)u(s)w₁(u(s))ds+ Z t

e0

g(s)u(s)w₂(u(s))ds, fore0≤t < T, then

(i) for the casew₂(u)≤w₁(u), (3.44) u(t)

≤ϕ⁻¹ (

Ω⁻¹

"

G⁻¹₁ G₁(Ω(c)) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

!#) ,

(ii) for the casew₁(u)≤w₂(u), (3.45) u(t)

≤ϕ⁻¹ (

Ω⁻¹

"

G⁻¹₂ G₂(Ω(c)) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

!#) , where

Ω(r) =e Z _er

re0

ds

ϕ⁻¹(s), er≥re0 >0, G_i(ez) =

Z _ez

ze0

ds

w_i{ϕ⁻¹[Ω⁻¹(s)]}, ez ≥ze₀ >0 (i= 1,2)

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Ω⁻¹, ϕ⁻¹, G⁻¹ are respectively the inverse of Ω, ϕ, G, and T ∈ R+ is chosen so that

G_i Ω(c) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

!

∈Dom(G⁻¹_i ), e0≤t≤T, and

G⁻¹_i

"

G_i Ω(c) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

!#

∈Dom(Ω⁻¹), e0≤t ≤T.

Proof. Letc >0and define the nonincreasing positive functionz(t)and make (3.46) z(t) =c+

Z _eα(t) e0

f(s)u(s)w₁(u(s))ds+ Z t

e0

g(s)u(s)w₂(u(s))ds.

From inequality (3.43), we know

(3.47) u(t)≤ϕ⁻¹[z(t)],

and

(3.48) D1D2· · ·Dnz(t)

= [f(α)u(e α)we ₁(u(α))αe ⁰₁α⁰₂· · ·α⁰_n+g(t)u(t)w₂(u(t))].

Using (3.47), we have (3.49) D₁D₂· · ·D_nz(t)

ϕ⁻¹(z(t)) ≤f(α)we ₁(u(α))αe ⁰₁α⁰₂· · ·α⁰_n+g(t)w₂(u(t)).

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For

(3.50) D_n

D₁D₂· · ·Dn−1z(t) ϕ⁻¹(z(t))

= D₁D₂· · ·D_nz(t)ϕ⁻¹(z(t))−D₁D₂· · ·Dn−1z(t)(ϕ⁻¹(z(t)))⁰D_nz(t)

(ϕ⁻¹(z(t)))² ,

usingD₁D₂· · ·Dn−1z(t)≥0,(ϕ⁻¹(z(t)))⁰ ≥0,D_nz(t)≥0in (3.50), we get D_n

D₁D₂· · ·Dn−1z(t) ϕ⁻¹(z(t))

≤ D₁D₂· · ·D_nz(t) ϕ⁻¹(z(t))

≤f(α)αe ₁⁰α⁰₂· · ·α⁰_nw₁(ϕ⁻¹(α(t))) +e g(t)w₂(ϕ⁻¹(t)).

Fixingt1, . . . , tn−1, settingtn =sn, integrating fromtnto∞, yields D₁D₂· · ·Dn−1z(t)

ϕ⁻¹(z(t)) ≤

Z αn(tn) 0

f(α₁(t₁), . . . , αn−1(tn−1), s_n)

×w₁(ϕ⁻¹(α₁(t₁), . . . , αn−1(tn−1), s_n))α⁰₁α⁰₂· · ·α⁰_n−1ds_n +

Z tn

0

g(t1, . . . , tn−1, sn)w2(ϕ⁻¹(t1, . . . , tn−1, sn))dsn. Deductively

(3.51) D₁z(t) ϕ⁻¹(z(t)) ≤

Z α2(t2) 0

· · ·

Z αn(tn) 0

f(α₁(t₁), s₂, . . . , s_n)

×w₁(ϕ⁻¹(α₁(t₁), s₂, . . . , s_n))α₁⁰ds_n. . . ds₂ +

Z t2

0

· · · Z tn

0

g(t₁, s₂, . . . , s_n)w₂(ϕ⁻¹(t₁, s₂, . . . , s_n))ds_n. . . ds₂.

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Fixing t₂, . . . , t_n, setting t₁ = s₁, integrating from 0 to t₁ with respect to s₁ yields

(3.52) Ω(z(t))≤Ω(c) + Z α(t)_e

e0

f(s₁, . . . , s_n)w₁(ϕ⁻¹(z(s)) +

Z t e0

g(s)w₂(ϕ⁻¹(z(s))ds, t∈Rⁿ+. From Theorem3.6, we obtain

z(t)≤Ω⁻¹

"

G⁻¹₁ G₁(Ω(c)) + Z α(t)_e

e0

f(s)ds+ Z t

e0

g(s)ds

!#

,

using (3.47), we get the inequality (3.44).

If c = 0 we carry out the above procedure with ε > 0 instead of c and subsequently letε→0.

(ii) whenw₁(u)≤w₂(u).

The proof can be completed with suitable changes.

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4. Some Applications

Example 4.1. Consider the integral equation:

(4.1) u^p(t₁, . . . , t_n)

=f(t₁, . . . , t_n)+

Z α(t)e e0

K(s₁, . . . , s_n)g(s₁, . . . , s_n, u(s₁, . . . , s_n))ds₁. . . ds_n, where f, K : Rⁿ+ → R, g : Rⁿ+×R → Rare continuous functions andp > 0 and p 6= 1is constant, αe_i(t) ∈ C¹(R+,R+)is nondecreasing with α_i(t) ≤ t_i onR⁺ (i = 1, . . . , n). In [8] B.G. Pachpatte studied the problem whenα(t) = t, n = 1. Here we assume that every solution under discussion exists on an intervalRⁿ+. We suppose that the functionsf,K,gin (4.1) satisfy the following conditions

|f(t₁, . . . , t_n)| ≤c₁, |K(t₁, . . . , t_n)| ≤c₂, (4.2)

|g(t₁, . . . , t_n, u)| ≤r(t₁, . . . , t_n)|u|^q+h(t₁, . . . , t_n)|u|,

where c₁, c₂,are nonnegative constants, and p ≥ q > 0, andr : Rⁿ+ → R+, h:Rⁿ+ →R⁺are continuous functions. From (4.1) and using (4.2), we get (4.3) |u^p(t₁, . . . , t_n)|

≤c₁+ Z α(t)_e

e0

[c₂r(s₁, . . . , s_n)|u|^q+c₂h(s₁, . . . , s_n)|u|ds₁. . . ds_n.

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Now an application of Corollary3.8yields

|u(t)| ≤











c⁽¹⁻

1 p)

1 + ^c²^(p−1)_p Rα(t)_e

e0 h(s)ds1. . . dsn

^p−1_p

×exph

c2

p

Rα(t)e

e0 r(s)ds₁. . . ds_ni when p=q,

"

c⁽¹⁻

1 p)

1 +^c²^(p−1)_p Rα(t)_e

e0 h(s)ds₁. . . ds_n ^p−q_p−1

+^c²^(p−q)_p Rα(t)_e

e0 r(s)ds₁. . . ds_n

#_p−q¹

when p > q.

If the integrals of r(s), h(s) are bounded, then we can have the bound of the solutionu(t)of (4.1). Similarly, we can obtain many other kinds of estimates.

Example 4.2. Consider the partial delay differential equation:

(4.4) ∂²u^p(x, y)

∂x₁∂x₂ =f(x, y, u(x, y), u(x−h₁(x), y−h₂(y))), u^p(x,0) =a₁(x) u^p(0, y) =a₂(y) (4.5)

a₁(0) =a₂(0) = 0, |a₁(x) +a₂(y)| ≤c, (4.6)

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wheref ∈ C(R+×R+×R²,R), a₁ ∈ C¹(R+,R), a₂ ∈C¹(R+,R), candp are nonnegative constants. h1 ∈C¹(R⁺,R⁺), h2 ∈C¹(R⁺,R⁺), such that

x−h₁(x)≥0, y−h₂(y)≥0, h⁰₁(x)<1, h⁰₂(y)<1.

Suppose that

(4.7) |f(x, y, u, v)| ≤a(x, y)|v|^q+b(x, y)|v|, wherea, b∈C(R+×R+,R)and let

(4.8) M₁ = max

x∈R+

1

1−h⁰₁(x), M₂ = max

y∈R+

1 1−h⁰₂(y). Ifu(x, y)is any solution of (4.4) – (4.7), then

(i) ifp=q, we have (4.9) |u(x, y)|

≤ c⁽¹⁻¹^p⁾+M1M2(p−1) p

Z φ1(x) 0

Z φ1(y) 0

eb(σ, τ)dσdτ

!_p−1^p

×exp

"

M₁M₂ p

Z φ1(x) 0

Z φ1(y) 0

ea(σ, τ)dσdτ

#

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(ii) ifp > q, we have (4.10) |u(x, y)|

≤



 c⁽¹⁻¹^p⁾+M1M2(p−1) p

Z φ1(x) 0

Z φ1(y) 0

eb(σ, τ)dσdτ

!^p−q_p−1

+M₁M₂(p−q) p

Z φ1(x) 0

Z φ1(y)

0 ea(σ, τ)dσdτ

#_p−q¹ . In whichφ₁(x) =x−h₁(x), x∈Rⁿ+, φ₂(y) = y−h₂(y), y ∈Rⁿ+ and

eb(σ, τ) = b(σ+h1(s), τ +h2(t)),ea(σ, τ) =a(σ+h1(s), τ +h2(t)), forσ, s, τ, t∈Rⁿ+.

In fact, ifu(x, y)is a solution of (4.4) – (4.7), then it satisfies the equivalent integral equation:

(4.11) [u(x, y)]^p

=a₁(x) +a₂(y) + Z x

0

Z y 0

f(s, t, u(s, t), u(s−h₁(s), t−h₂(t)))dtds.

forx, y ∈(Rⁿ+×Rⁿ+,R).

Using (4.5), (4.7) in (4.11) and making the change of variables, we have (4.12) |u(x, y)|^p

≤c+M₁M₂

Z φ1(x) 0

Z φ1(y) 0

ea(σ, τ)|u(σ, τ)|^q+eb(σ, τ)|u(σ, τ)|dσdτ.

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Now a suitable application of the inequality in Corollary 3.8 to (4.12) yields (4.9) and (4.10).

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References

[1] B.G. PACHPATTE, On some new inequalities related to certain inequalities in the theory of differential equations, J. Math. Anal. Appl., 189 (1995), 128–144.

[2] B.G. PACHPATTE, Explicit bounds on certain integral inequalities, J. Math.

Anal. Appl., 267 (2002), 48–61.

[3] B.G. PACHPATTE, On some new inequalities related to a certain inequality arising in the theory of differential equations, J. Math. Anal. Appl., 251 (2000), 736–751.

[4] B.G. PACHPATTE, On a certain inequality arising in the theory of differen- tial equations, J. Math. Anal. Appl., 182 (1994), 143–157.

[5] I. BIHARI, 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.

[6] M. MEDVED, Nonlinear singular integral inequalities for functions in two and n independent variables, J. Inequalities and Appl., 5 (2000), 287–308.

[7] O. LIPOVAN, A retarded Gronwall-like inequality and its applications, J.

Math. Anal. Appl., 252 (2000), 389–401.

[8] O. LIPOVAN, A retarded integral inequality and its applications, J. Math.

Anal. Appl., 285 (2003), 436–443.