Some Nonlinear Weakly Singular Integral

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−s^β−1s^γ−1Fsusds, ut≤at

_t

0

t−s^β−1Fswusds,

1.1

and the estimates of solutions are given, respectively. In 4, Medve ˇd also generalized his results to an analogue of the Wendroffinequalities for functions in two variables. From then

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 Peˇcari´c9used 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

u x, y

≤a x, y

_x

0

_y

0

x^α−s^αβ−1s^γ−1

y^α−t^α_β−1 t^γ−1f

x, y, s, t

wus, 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−y²are satisfied.

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

0

t^α−s^αpβ−1s^pγ−1ds t^θ αB

p γ−1

1

α , p

β−1 1

, t∈R, 2.1

where Bξ, η ₁

0s^ξ−11 − s^η−1ds (Reξ > 0,Reη > 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;

2ifα, β, γ∈II,p2 14β/13β.

Then, fori1,2,

B pi

γ−1 1 α , p_i

β−1 1

∈0,∞, θ_ip_i

α β−1

γ−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,

u x, y

≤

W₁⁻¹

W₁ A₁

x, y B₁

x, y

x 0

_y

0

f

x, y, s, t_1/1−β ds dt

_1−β

2.3

for 0≤x≤Xand 0≤y≤Y, where

M₁ 1 αB

βγ−1 αβ ,2β−1

β

, A1

x, y

2^β/1−βa

x, y_1/1−β , B1

x, y

2^β/1−β M²₁

xy1/βαβ−1γ−11_β/1−β

,

2.4

W₁⁻¹is the inverse ofW₁,

W_{1 u}

u0

dξ w^1/1−β

ξ^1−β, u≥u₀>0, 2.5

andX, Y ∈Rare chosen such that

W₁ A₁

x, y B₁

x, y

x 0

_y

0

f

x, y, s, t_1/1−β

ds dt∈Dom W₁⁻¹

, 2.6

(2) forα, β, γ∈II,

u x, y

≤

W₂⁻¹

W2

A2

x, y B2

x, y

x 0

_y

0

f

x, y, s, t_14β/β ds dt

_β/14β

2.7

for 0≤x≤Xand 0≤y≤Y, where

M₂ 1 αB

γ 14β

−β α

13β , 4β² 13β

,

A₂ x, y

2^13β/βa

x, y_14β/β , B₂

x, y

2^13β/β

M²₂

xy14β/13βαβ−1γ−11_13β/β

,

2.8

W₂⁻¹is the inverse ofW2,

W2 _u

u0

dξ w^14β/β

ξ^β/14β, u≥u0>0, 2.9

andX, Y ∈Rare chosen such that

W₂ A₂

x, y B₂

x, y

x 0

_y

0

f

x, y, s, t_14β/β

ds dt∈Dom W₂⁻¹

. 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

u x, y

≤a x, y

_x

0

_y

0

x^α−s^αβ−1s^γ−1

y^α−t^α_β−1 t^γ−1f

x, y, s, t

wus, tds dt. 2.11

Next, for convenience, we introduce indicesp_i,q_i. 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 indicesp_i,q_ito2.11, we get

u x, y

≤a x, y

x 0

_y

0

x^α−s^αpiβ−1s^piγ−1

y^α−t^α_piβ−1

t^piγ−1ds dt 1/pi

× _x

0

_y

0

f

x, y, s, t^qi

wus, t^qids dt 1/qi

.

2.12

By

A1A2· · ·An^r ≤n^r−1

A^r₁A^r₂· · ·A^r_n

, Ai≥0, r≥1, 2.13

from2.12and Lemma2.2, we have u^qi

x, y

≤2^qi−1

a

x, yqi _x

0

_y

0

x^α−s^αpiβ−1s^piγ−1

y^α−t^αpiβ−1t^piγ−1ds dt qi/pi

× _x

0

_y

0

f

x, y, s, t^qi

wus, t^qids dt

2^qi−1a x, y_qi

2^qi−1 M_i²

xy_θiqi/pix

0

_y

0

f

x, y, s, t^qi

wus, t^qids dt

,

2.14

where

Mi 1 αB

p_i γ−1

1 α , pi

β−1 1

2.15

andθ_iis given in Lemma2.3fori1,2.

Sinceqi≥0 andθi≥0i1,2, thenax, y ^qi andxy^θiqi/pi are also nondecreasing inxandy. Taking any arbitraryxandywithx≤X,y≤Y, we obtain

u^qi x, y

≤2^qi−1a x,y_qi

2^qi−1 M²_i

xy_θiqi/pix 0

_y

0

f

x,y, s, t ^qi

wus, t^qids dt 2.16

for 0≤x≤x, 0 ≤y≤y. Denote A_i

x,y

2^qi−1a

x,yqi, 2.17

and let

z_i x, y

A_i x,y

2^qi−1 M²_i

xyθi_qi/pix

0

_y

0

f

x,y, s, t ^qi

wus, t^qids dt

. 2.18

Then,u^qix, y≤zix, yorux, y≤z^1/q_i ⁱx, y. Meanwhile,zi0, y Aix, y, and zix, y is nondecreasing inxandy. Considering

D₁z_i x, y

2^qi−1 M²_i

xyθi_qi/piy

0

f

x,y, x, t ^qi

wux, t^qids dt

≤2^qi−1

M²_i

xy_θiqi/piy

0

f

x,y, x, t ^qi

wzix, t^1/qi_qi dt

,

2.19

we have

D₁z_i x, y w^qi

z^1/q_i ⁱ

x, y ≤2^qi−1 M_i²

xy_θiqi/piy 0

f

x,y, x, t ^qi dt

, 2.20

where we apply the fact thatw^qiz^1/q_i ⁱx, yis nondecreasing iny. Integrating both sides of the above inequality from 0 tox, we obtain

Wi

zi

x, y

≤Wi

zi

0, y

2^qi−1 M²_i

xy_θiqi/pi_x

0

_y

0

f

x,y, s, t ^qi ds dt

Wi

Ai

x, y

2^qi−1 M²_i

xy_θiqi/pix

0

_y

0

f

x,y, s, t ^qi ds dt

2.21

for 0≤x≤x, 0 ≤y≤y, where

Wiu _u

u0

dξ w^qi

ξ^1/qi, u≥u0>0. 2.22

From assumptionA2,Wiis strictly increasing so its inverseW_i⁻¹is continuous and increas- ing in its corresponding domain. Replacingxandybyxandy, we have

Wi

zi

x, y

≤Wi

Ai

x, y 2^qi−1

M²_i

xyθi_qi/pix

0

_y

0

f

x,y, s, t ^qi ds dt

.

2.23

Sincexandyare arbitrary, we replacexandybyxandy, respectively, and get

W_i z_i

x, y

≤W_i A_i

x, y

2^qi−1 M²_i

xyθi_qi/pix

0

_y

0

f

x, y, s, t^qi ds dt

. 2.24

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

z_i x, y

≤W_i⁻¹

W_i A_i

x, y

2^qi−1 M²_i

xyθi_qi/pix

0

_y

0

f

x, y, s, t^qi ds dt

. 2.25

Therefore, we have u

x, y

≤z^1/q_i ⁱ x, y

≤

W_i⁻¹

Wi

Ai

x, y 2^qi−1

M_i²

xyθi_qi/pix

0

_y

0

f

x, y, s, t^qi ds dt

1/qi

2.26

for 0≤x≤Xand 0≤y≤Y.

Finally, considering two situations fori1,2 and using parametersα,β, γto denote p_i,q_i,M_i, andθ_iin the above inequality, we can obtain the estimations, respectively. we omit the details here.

Remark 2.5. Medve ˇd4, Theorem 2.2investigated 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 takeα1 andwu u, then we can obtain the result of linear case4, Theorem 2.4.

Remark 2.6. Letu^px, y vx, y, thenux, y v^1/px, yoru^qx, y v^q/px, y. Therefore, if we takewv v^q/p, the formula2.6in10is the special case of inequality1.2, 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 with 0< k≤1. Suppose that

u x, y

≤a x, y

_x

0

_y

0

x^α−s^αβ−1s^γ−1

y^α−t^α_β−1 t^γ−1f

x, y, s, t

us, t^kds dt. 3.1

Then,

(1) forα, β, γ∈I, ifk1,

u x, y

≤2^βa x, y

exp 1−β

B₁ x, y

x 0

_y

0

f

x, y, s, t^1/1−β ds dt

, 3.2

if 0< k <1,

u x, y

≤

2^βa

x, y_1−k/1−β

1−kB1

x, y

x 0

_y

0

f

x, y, s, t^1/1−β ds dt

_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,

u x, y

≤2^13β/14βa x, y

exp β

14βB₂ x, y

x 0

_y

0

f

x, y, s, t14β/β

ds dt

, 3.4

if 0< k <1, u

x, y

≤ 2^13βa

x, y_14β1−k/β

1−kB2

x, y

x 0

_y

0

f

x, y, s, t^14β/β ds dt

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 u^k, we can get3.1.

iIfk1,

W_iu _u

u0

dξ ξ ln u

u0

, u≥u₀>0,

W_i⁻¹u u0e^u, Dom W_i⁻¹

0,∞, i1,2.

3.6

iiIf 0< k <1,

Wiu _u

u0

dξ ξ^k 1

1−k

u^1−k−u^1−k₀ ,

W_i⁻¹u

u^1−k₀ 1−ku_1/1−k

, Dom W_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

u x, y

≤a x, y

_x

0

_y

0

x^σ−s^σμ−1s^τ−1

y^σ−t^σ_μ−1 t^τ−1g

x, y, s, t

us, tds dt

_x

0

_y

0

x^α−s^αβ−1s^γ−1

y^α−t^α_β−1 t^γ−1f

x, y, s, t

wus, tds dt.

3.8

Then,

iifα, β, γ,σ, μ, τ∈I,

u x, y

≤ W₁⁻¹

W1

A1

x, y Ω1

x, y_1/1−β

Ω1

x, y_1/1−β B1

x, y

x 0

_y

0

f

x, y, s, t_1/1−β ds dt

_1−β 3.9

for 0≤x≤X1and 0≤y≤Y1, where

Ω1

x, y

2^μexp 1−μ

B₁ x, y

x 0

_y

0

g

x, y, s, t_1/1−μ ds dt

, 3.10

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

W1

A1

x, y Ω1

x, y_1/1−β Ω1

x, y_1/1−β B₁

x, y

x 0

_y

0

f

x, y, s, t_1/1−β

ds dt∈Dom W₁⁻¹

,

3.11

iiifα, β, γ,σ, μ, τ∈II,

u x, y

≤

W₂⁻¹

W2

A2

x, y Ω2

x, y_14β/β

Ω2

x, y_14β/β B₂

x, y

x 0

_y

0

f

x, y, s, t_14β/β ds dt

_β/14β 3.12

for 0≤x≤X₂and 0≤y≤Y₂, where

Ω2

x, y

2^13μ/14μexp

μ 14μB2

x, y

x 0

_y

0

g

x, y, s, t_14μ/μ ds dt

, 3.13

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

W2

A2

x, y Ω2

x, y_14β/β Ω2

x, y_14β/β B2

x, y

x 0

_y

0

f

x, y, s, t_14β/β ds dt

∈Dom W₂⁻¹

.

3.14

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

u x, y

≤P_i x, y

_x

0

_y

0

x^σ−s^σμ−1s^τ−1

y^σ−t^σ_μ−1 t^τ−1g

x, y, s, t

us, tds dt, 3.15

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

Pi

x, y a

x, y

M²_i

xyθi1/pi_x

0

_y

0

f^qi

x, y, s, t

wus, t^qids dt 1/qi

. 3.16

iForα, β, γ,σ, μ, τ∈I,

applying Corollary3.1to3.15, we have

u x, y

≤2^μP1

x, y exp

1−μ B1

x, y

x 0

_y

0

g

x, y, s, t_1/1−μ ds dt

. 3.17

Letting

Ω1

x, y

2^μexp 1−μ

B1

x, y

x 0

_y

0

g

x, y, s, t_1/1−μ ds dt

, 3.18

we get

u x, y

≤P1

x, y Ω1

x, y a

x, y Ω1

x, y Ω1

x, y M²₁

xy_θ11/p1x 0

_y

0

f^q1

x, y, s, t

wus, t^q1ds 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:

u x, y

≤ 1 2

_x

0

_y

0

x−s^−1/3s^−1/6

y−t_−1/3

t^−1/6e^−s−2t

us, tds dt 4.1

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

a x, y

1

2, α1, β 2

3, γ 5 6, f

x, y, s, t

e^−s−2t, wu

us, t.

4.2

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

a

x, y 1

2, f

x, y, s, t

e^−s−2t, A₁

x, y 2²

1 2

₃ 1

2, M₁B 3

4,1 2

,

B₁ x, y

2²

B 3

4,1 2

₂

xy_1/42

4

B 3

4,1 2

₄ xy,

W1u _u

u0

dξ

ξ 2√ u−√

u0

,

W₁⁻¹u √ u₀u

2 ₂

, Dom W₁⁻¹

0,∞.

4.3

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

u x, y

≤

W₁⁻¹

W1

A1

x, y B1

x, y

x 0

_y

0

e^−s−2t₃

ds 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 xy

1−e^−3x

1−e^−6y1/3

√

2 2 1

9

B 3

4,1 2

₄ xy

1−e^−3x

1−e^−6y2/3

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

1 D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.

2 H. Sano and N. Kunimatsu, “Modified Gronwall’s inequality and its application to stabilization problem for semilinear parabolic systems,” Systems & Control Letters, vol. 22, no. 2, pp. 145–156, 1994.

3 M. Medve ˇd, “A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,” Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 349–366, 1997.

4 M. Medve ˇd, “Nonlinear singular integral inequalities for functions in two and n independent variables,” Journal of Inequalities and Applications, vol. 5, no. 3, pp. 287–308, 2000.

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.

8 Q. H. Ma and E. H. Yang, “Estimates on solutions of some weakly singular Volterra integral inequalities,” Acta Mathematicae Applicatae Sinica, vol. 25, no. 3, pp. 505–515, 2002.

9 Q.-H. Ma and J. Peˇcari´c, “Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 894–905, 2008.

10 W.-S. Cheung, Q.-H. Ma, and S. Tseng, “Some new nonlinear weakly singular integral inequalities of Wendrofftype with applications,” Journal of Inequalities and Applications, vol. 2008, Article ID 909156, 13 pages, 2008.