Some Nonlinear Weakly Singular Integral Inequalities with Two Variables and Applications

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Volume 2010, Article ID 345701,12pages doi:10.1155/2010/345701

Research Article

Some Nonlinear Weakly Singular Integral

Inequalities with Two Variables and Applications

Hong Wang and Kelong Zheng

School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China

Correspondence should be addressed to Kelong Zheng,[email protected]

Received 23 October 2010; Accepted 22 December 2010

Academic Editor: Radu Precup

Copyrightq2010 H. Wang and K. Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some nonlinear weakly singular integral inequalities in two variables which generalize some known results are discussed. The results can be used as powerful tools in the analysis of certain classes of diﬀerential equations, integral equations, and evolution equations. An example is presented to show boundedness of solution of a diﬀerential equation here.

1. Introduction

Various singular integral inequalities play an important role in the development of the theory of differential equations, functional differential equations, and integral equations. For example, Henry1proposed a linear integral inequality with singular kernel to investigate some qualitative properties for a parabolic differential equation, and Sano and Kunimatsu

2gave a modified version of Henry type inequality. However, such results are expressed by a complicated power series which are sometimes inconvenient for their applications. To avoid the shortcoming of these results, Medve ˇd3presented a new method to discuss nonlinear singular integral inequalities of Henry type and their Bihari version as follows:

ut≤at

_t

0

t−sβ−1sγ−1Fsusds,

ut≤at

_t

0

t−sβ−1Fswusds,

1.1

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on, more attention has been paid to such inequalities with singular kernel see5–9. In particular, Ma and Yang 8 used a modification of Medve ˇd method to obtain pointwise explicit bounds on solutions of more general weakly singular integral inequalities of the Volterra type, and later Ma and Peˇcarić9used this method to study nonlinear inequalities of Henry type. Recently, Cheung et al. 10 applied the modified Medve ˇd method to investigate some new weakly singular integral inequalities of Wendrofftype and applications to fractional differential and integral equations.

In this paper, motivated mainly by the work of Ma et al.8,9and Cheung et al.10, we discuss more general form of nonlinear weakly singular integral inequality of Wendroﬀ type for functions in two variables

ux, y≤ax, y

x

0 y

0

xα−sαβ−1sγ−1yα−tαβ−1tγ−1fx, y, s, twus, tds dt. 1.2

Our results can generalize some known results and be used more eﬀectively to study the qualitative properties of the solutions of certain partial diﬀerential and integral equations. Moreover, an example is presented to show the usefulness of our results.

2. Main Result

In what follows,R denotes the set of real numbers, andR 0,∞.CX, Y denotes the

collection of continuous functions from the setXto the setY.D1zx, yandD2zx, ydenote

the first-order partial derivatives ofzx, ywith respect toxandy, respectively. Before giving our result, we cite the following definition and lemmas.

Definition 2.1 see 8. Let x, y, z be an ordered parameter group of nonnegative real

numbers. The group is said to belong to the first-class distribution and is denoted by

x, y, z ∈ I if conditionsx ∈ 0,1,y ∈ 1/2,1, andz ≥ 3/2−yare satisfied; it is said to belong to the second-class distribution and is denoted byx, y, z∈IIif conditionsx∈0,1,

y∈0,1/2andz >1−2y2_/₁₋_y2_{are satisfied.}

Lemma 2.2see8. Letα, β, γ, andpbe positive constants. Then,

_t

0

tα−sαpβ−1spγ−1ds t

θ

αB

pγ−11

α , p

β−11

, t∈R, 2.1

where Bξ, η ₀1sξ−1₁ ₋ _sη−1_ds ₍_Re_{ξ >} ₀_,_Re_{η >} ₀_{) is well-known} _B_{-function and}

θpαβ−1 γ−1 1.

Lemma 2.3see8. Suppose that the positive constantsα, β, γ,p1, andp2 satisfy the following

conditions:

1ifα, β, γ∈I,p11/β;

(3)

Then, fori1,2,

B

pi

γ−11

α , pi

β−11

∈0,∞,

θipi

αβ−1γ−1 1≥0

2.2

are valid.

Assume that

A1ax, y∈CR2, Randfx, y, s, t∈CR4, R;

A2wu∈CR, Ris nondecreasing andw0 0.

Letax, y max0≤τ≤x,0≤η≤yaτ, ηandfx, y, s, t max0≤τ≤x,0≤η≤y fτ, η, s, t.

Theorem 2.4. Under assumptions (A1) and (A2), ifum, n∈CR2, Rsatisfies1.2, then

(1) forα, β, γ∈I,

ux, y≤

W₁−1

W1

A1

x, yB1

x, y

x

0 _y

0

fx, y, s, t1/1−βds dt

1−β

2.3

for0≤x≤Xand0≤y≤Y, where

M1

1

αB

_β_γ₋₁

αβ ,

2β−1

β

,

A1

x, y2β/1−βax, y1/1−β,

B1

x, y2β/1−βM2₁xy1/βαβ−1γ−11β/1−β,

2.4

W₁−1is the inverse ofW1,

W1 u

u0

dξ

w1/1−β_ξ1−β, u≥u0>0, 2.5

andX, Y ∈Rare chosen such that

W1

A1

x, yB1

x, y

x

0 _y

0

fx, y, s, t1/1−βds dt∈DomW₁−1, 2.6

(2) forα, β, γ∈II,

ux, y≤

W₂−1

W2

A2

x, yB2

x, y

x

0 _y

0

fx, y, s, t14β/βds dt

β/14β

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for0≤x≤Xand0≤y≤Y, where

M2

1

αB

γ14β−β α13β ,

4β2

13β

,

A2

x, y213β/βax, y14β/β,

B2

x, y213β/βM2₂xy14β/13βαβ−1γ−1113β/β,

2.8

W₂−1is the inverse ofW2,

W2 _u

u0

dξ

w14β/β_ξβ/14β, u≥u0>0, 2.9

andX, Y ∈Rare chosen such that

W2

A2

x, yB2

x, y

x

0 _y

0

fx, y, s, t14β/βds dt∈DomW₂−1. 2.10

Proof. With the definition of ax, y and fx, y, s, t, clearly, ax, y and fx, y, s, t are nonnegative and nondecreasing inxandy. Furthermore,ax, y≥ax, yandfx, y, s, t≥ fx, y, s, t. From1.2, we have

ux, y≤ax, y

x

0 y

0

xα−sαβ−1sγ−1yα−tαβ−1tγ−1fx, y, s, twus, tds dt. 2.11

Next, for convenience, we introduce indicespi,qi. Denote that ifα, β, γ∈I, then letp11/β

and q1 1/1−β; ifα, β, γ ∈ II, then let p2 14β/13βand q2 14β/β.

Then 1/pi1/qi1 holds fori1,2.

Using the H ¨older inequality with indicespi,qito2.11, we get

ux, y≤ax, y

x

0 y

0

xα−sαpiβ−1spiγ−1yα−tαpiβ−1tpiγ−1ds dt

1/pi

× x

0 y

0

fx, y, s, tqiwus, tqids dt

1/qi

.

2.12

By

A1A2· · ·Anr ≤nr−1

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from2.12and Lemma2.2, we have

uqix, y

≤2qi−1

ax, yqi

x

0 _y

0

xα−sαpiβ−1spiγ−1yα−tαpiβ−1tpiγ−1ds dt

qi/pi

× x

0 y

0

2qi−1ax, yqi2qi−1M_i2xyθiqi/pi

x

0 _y

0

,

2.14

where

Mi

1

αB

pi

γ−11

α , pi

β−11

2.15

andθiis given in Lemma2.3fori1,2.

Sinceqi≥0 andθi≥0i1,2, thenax, yqi andxyθiqi/pi are also nondecreasing

inxandy. Taking any arbitraryxandywithx≤X,y≤Y, we obtain

uqix, y≤2qi−1ax, yqi2qi−1M2_ixyθiqi/pi

x

0 y

0

fx, y, s, t qiwus, tqids dt

2.16

for 0≤x≤x, 0≤y≤y. Denote

Ai

x,y2qi−1ax, yqi, 2.17

and let

zi

x, yAi

x,y2qi−1_M2 i

xyθiqi/pi

x

0 _y

0

fx, y, s, t qiwus, tqids dt

.

2.18

Then,uqi_{x, y}_≤_z

ix, yorux, y≤z1_i/qix, y. Meanwhile,zi0, y Aix, y, andzix, y

is nondecreasing inxandy. Considering

D1zi

x, y2qi−1_M2 i

xyθiqi/pi

y

0

fx, y, x, t qiwux, tqids dt

≤2qi−1

M2_ixyθi

qi/piy

0

fx, y, x, t qi

wzix, t1/qi qi

dt

,

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we have

D1zi

x, y wqi_z1/qi

i

x, y ≤

2qi−1M_i2xyθiqi/pi

y

0

fx, y, x, t qidt

, 2.20

where we apply the fact thatwqi_z1/qi

i x, yis nondecreasing iny. Integrating both sides of

the above inequality from 0 tox, we obtain

Wi

zi

x, y≤Wi

zi

0, y2qi−1M_i2xyθiqi/pi

x 0 _y 0

fx, y, s, t qids dt

Wi Ai

x,y2qi−1M2_ixyθiqi/pi

x 0 y 0

2.21

for 0≤x≤x, 0≤y≤y, where

Wiu u

u0

dξ

wqi_ξ1/qi, u≥u0>0. 2.22

From assumptionA2,Wiis strictly increasing so its inverseW_i−1is continuous and

increas-ing in its correspondincreas-ing domain. Replacincreas-ingxandybyxandy, we have

Wi

zi

x,y≤Wi

Ai

x,y2qi−1

M2_ixyθi

qi/pix

0 _y

0

.

2.23

Sincexandyare arbitrary, we replacexandybyxandy, respectively, and get

Wi

zi

x, y≤Wi

Ai

x, y2qi−1M2_ixyθiqi/pi

x 0 _y 0

fx, y, s, tqids dt

. 2.24

for 0≤x≤Xand 0≤y≤Y. The above inequality can be rewritten as

zi

x, y≤W_i−1

Wi

Ai

x, y2qi−1_M2 i

xyθiqi/pi

x 0 _y 0

.

2.25

Therefore, we have

ux, y≤z1_i/qix, y

≤

W_i−1

Wi

Ai

x, y2qi−1

M_i2xyθi

qi/pix

0 _y

0

1/qi

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for 0≤x≤Xand 0≤y≤Y.

Finally, considering two situations fori1,2 and using parametersα,β, γto denote

pi,qi,Mi, andθiin the above inequality, we can obtain the estimations, respectively. we omit

the details here.

Remark 2.5. Medve ˇd4,Theorem2.2investigated the special caseαγ1,fx, y, s, t Fs, t

of inequality1.2under the assumption that “wusatisfies the conditionq.” However, in our

result, theqcondition is eliminated. If we takeα1andwu u, then we can obtain the result of linear case4,Theorem2.4.

Remark 2.6. Letup_{x, y} _v_{x, y}_{, then}_u_{x, y} _v1/p_{x, y}_or_uq_{x, y} _vq/p_{x, y}_{. Therefore,}

if we takewv vq/p_{, the formula}_2.6_in₁₀_{is the special case of inequality}_1.2_,_{and we can}

obtain more concise results than2.7and2.9in10. Moreover, here the conditionp ≥qalso can be eliminated.

Remark 2.7. Whenα, β, γdoes not belong toIorII, there are some technical problems which we do not discuss here.

3. Some Corollaries

Corollary 3.1. Let functionsux, y,ax, y,fx, y, s, tbe defined as in Theorem2.4, and letkbe a constant with0< k≤1. Suppose that

ux, y≤ax, y

_x

0 _y

0

xα₋_sαβ−1_sγ−1_yα₋_tαβ−1_tγ−1_f_{x, y, s, t}_u_{s, t}k_{ds dt.} _3.1

Then,

(1) forα, β, γ∈I, ifk1,

ux, y≤2βax, yexp1−βB1

x, y

x

0 _y

0

, 3.2

if0< k <1,

ux, y≤

2βax, y1−k/1−β 1−kB1

x, y

x

0 _y

0

1−β/1−k

3.3

forx≥0,y≥0, whereax, y,fx, y, s, t,B1x, yare defined as in Theorem2.4,

(2) forα, β, γ∈II, ifk1,

ux, y≤213β/14βax, yexp

_β

14βB2

x, y

x

0 _y

0

(8)

if0< k <1,

ux, y

≤213βax, y14β1−k/β 1−kB2

x, y

x

0 y

0

β/14β1−k

,

3.5

forx≥0,y≥0, whereax, y,fx, y, s, t,B2x, yare defined as in Theorem2.4.

Proof. Clearly, inequality3.1is the special case of1.2. Takingwu uk_{, we can get}_3.1_.

iIfk1,

Wiu _u

u0

dξ ξ ln

u u0

, u≥u0>0,

W_i−1u u0eu, Dom

W_i−1 0,∞, i1,2.

3.6

iiIf 0< k <1,

Wiu u

u0

dξ ξk

1 1−k

u1−k−u1−₀ k,

W_i−1u u1−₀ k 1−ku1/1−k, DomW_i−1 0,∞, i1,2.

3.7

Therefore, the positive numbersXandY in2.6and2.10can be taken as∞, and the results can be obtained by simple computation. We omit the details.

Corollary 3.2. Let functionsux, y,ax, y,fx, y, s, tbe defined as in Theorem2.4. Suppose that

gx, y, s, t∈CR4

, Randux, ysatisfies

ux, y≤ax, y

_x

0 _y

0

xσ−sσμ−1sτ−1yσ−tσμ−1tτ−1gx, y, s, tus, tds dt

x

0 y

0

xα−sαβ−1sγ−1yα−tαβ−1tγ−1fx, y, s, twus, tds dt.

3.8

Then,

iifα, β, γ,σ, μ, τ∈I,

ux, y≤

W₁−1

W1

A1

x, yΩ1

x, y1/1−β

Ω1

x, y1/1−βB1

x, y

x

0 _y

0

(9)

for0≤x≤X1and0≤y≤Y1, where

Ω1

x, y2μ_exp₁₋_μ_B 1

x, y

x

0 _y

0

gx, y, s, t1/1−μds dt

, 3.10

W1,W₁−1,A1x, y,B1x, yare defined as in Theorem2.4, andX1, Y1 ∈Rare chosen such that

W1

A1

x, yΩ1

x, y1/1−β

Ω1

x, y1/1−βB1

x, y

x

0 _y

0

fx, y, s, t1/1−βds dt∈DomW₁−1,

3.11

iiifα, β, γ,σ, μ, τ∈II,

ux, y≤

W₂−1

W2

A2

x, yΩ2

x, y14β/β

Ω2

x, y14β/βB2

x, y

x

0 _y

0

β/14β 3.12

for0≤x≤X2and0≤y≤Y2, where

Ω2

x, y213μ/14μexp

_μ

14μB2

x, y

x

0 y

0

gx, y, s, t14μ/μds dt

, 3.13

W2,W₂−1,A2x, y,B2x, yare defined as in Theorem2.4, andX2, Y2 ∈Rare chosen such that

W2

A2

x, yΩ2

x, y14β/β

Ω2

x, y14β/βB2

x, y

x

0 y

0

∈DomW₂−1.

3.14

Proof. By the two mentioned lemmas, it follows from3.8that

ux, y≤Pi

x, y

_x

0 _y

0

xσ−sσμ−1sτ−1yσ−tσμ−1tτ−1gx, y, s, tus, tds dt, 3.15

wheregx, y, s, t max0≤τ≤x,0≤η≤ygτ, η, s, tand

Pi

x, yax, yM2_ixyθi1/pi

x

0 _y

0

fqix, y, s, twus, tqids dt

1/qi

. 3.16

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applying Corollary3.1to3.15, we have

ux, y≤2μP1

x, yexp1−μB1

x, y

x

0 _y

0

. 3.17

Letting

Ω1

x, y2μexp1−μB1

x, y

x

0 _y

0

, 3.18

we get

ux, y≤P1

x, yΩ1

x, y

ax, yΩ1

x, y

Ω1

x, yM2₁xyθ11/p1

x

0 y

0

fq1_{x, y, s, t}_w_u_{s, t}q1_{ds dt}

1/q1

.

3.19

Since inequality3.19is similar to2.12, we can repeat the procedure of proof in Theorem2.4

and get3.9.

iiAs for the case thatα, β, γ,σ, μ, τ∈II, the proof is similar to the argument in the proof of caseiwith suitable modification. We omit the details.

Remark 3.3. Whenα, β, γ∈I,σ, μ, τ∈IIorα, β, γ∈II,σ, μ, τ∈I, we can get the results which are similar to that in Corollary3.2and omit them here.

4. Application

In this section, we will apply our result to discuss the boundedness of certain partial integral equation with weakly singular kernel.

Suppose thatux, y∈CR2, Rsatisfies the inequality as follow:

ux, y≤ 1

2

_x

0 _y

0

x−s−1/3s−1/6y−t−1/3t−1/6e−s−2t

us, tds dt 4.1

forx≥0,y≥0. Then,4.1is the special case of inequality1.2that is,

ax, y 1

2, α1, β 2

3, γ 5 6,

fx, y, s, te−s−2t, wu

us, t.

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Obviously,α, β, γ 1,2/3,5/6∈I. Lettingp13/2,q1 3, we have

ax, y1

2, f

x, y, s, te−s−2t,

A1

x, y22

1 2

3

1

2, M1B

3 4, 1 2 , B1

x, y22

B 3 4, 1 2 2

xy1/4

2 4 B 3 4, 1 2 4 xy,

W1u u

u0

dξ

ξ 2

√

u−√u0

,

W₁−1u √u0

u

2

, DomW₁−1 0,∞.

4.3

Applying2.3in Theorem2.4, we get forx≥0,y≥0

ux, y≤

W₁−1

W1

A1

x, yB1 x, y x 0 _y 0

e−s−2t3ds dt

1/3

W₁−1

W1 1 2 4 B 3 4, 1 2 4 xy _x 0 _y 0

e−3e−6tds dt

1/3

W₁−1

√

2−2√u02

9 B 3 4, 1 2 4

xy1−e−3x1−e−6y

1/3

√ 2 2 1 9 B 3 4, 1 2 4

xy1−e−3x₁₋_e−6y 2/3

4.4

which implies thatux, yin4.1is bounded.

Acknowledgments

This work is supported by Scientific Research Fund of Sichuan Provincial Education Depart-mentno. 09ZC109. The authors are very grateful to the referees for their helpful comments and valuable suggestions.

References

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5 J. P. Dauer and N. I. Mahmudov, “Integral inequalities and mild solutions of semilinear neutral evolution equations,”Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 189–202, 2004.

6 K. M. Furati and N. Tatar, “Power-type estimates for a nonlinear fractional diﬀerential equation,” Non-linear Analysis: Theory, Methods & Applications, vol. 62, no. 6, pp. 1025–1036, 2005.

7 M. Medve ˇd, “Integral inequalities and global solutions of semilinear evolution equations,”Journal of Mathematical Analysis and Applications, vol. 267, no. 2, pp. 643–650, 2002.

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