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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 909156, 13 pages doi:10.1155/2008/909156 Research Article Some New Nonlinear Weakly Singular Integral Inequalities of Wendroff Type with Applications Wing-Sum Cheung,1 Qing-Hua Ma,2 and Shiojenn Tseng3 Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong Faculty of Information Science and Technology, Guangdong University of Foreign Studies, Guangzhou 510420, China Department of Mathematics, Tamkang University, Tamsui, Taipei 25137, Taiwan Correspondence should be addressed to Wing-Sum Cheung, wscheung@hku.hk Received 20 March 2008; Accepted 26 August 2008 Recommended by Sever Dragomir Some new weakly singular integral inequalities of Wendroff type are established, which generalized some known weakly singular inequalities for functions in two variables and can be used in the analysis of various problems in the theory of certain classes of integral equations and evolution equations Application examples are also given Copyright q 2008 Wing-Sum Cheung et al 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 Introduction In the study of differential and integral equations, one often deals with certain integral inequalities The Gronwall-Bellman inequality and its various linear and nonlinear generalizations are crucial in the discussion of existence, uniqueness, continuation, boundedness, oscillation, and stability properties of solutions The literature on such inequalities and their applications is vast; see 1–6 , and the references are given therein Usually, the integrals concerning such inequalities have regular or continuous kernels, but some problems arising from theoretical or practical phenomena require us to solve integral inequalities with singular kernels For example, Henry used this type of integral inequalities to prove global existence and exponential decay results for a parabolic Cauchy problem; Sano and Kunimatsu gave a sufficient condition for stabilization of semilinear parabolic distributed systems by making use of a modification of Henry-type inequalities; Ye et al proved a generalization of this type of inequalities and used it to study the dependence of the solution on the order and the initial condition of a fractional differential equation All such inequalities are proved by an iteration argument, and the estimation formulas are expressed by a complicated power series which are sometimes not very convenient for applications To avoid this shortcoming, ˘ Medved 10 presented a new method for studying Henry-type inequalities and established Journal of Inequalities and Applications explicit bounds with relatively simple formulas which are similar to the classic Gronwall˘ Bellman inequalities Very recently, Ma and Pe˘ ari´ 11 used a modification of Medved’s c c method to study certain class of nonlinear inequalities of Henry-type, which generalized some known results and were used as handy and effective tools in the study of the solutions’ boundedness of some fractional differential and integral equations ˘ In this paper, by applying Medved’s method of desingularization of weakly singular inequalities we establish some new singular version of the Wendroff inequality see 1, 12 for functions in two variables An example is included to illustrate the usefulness of our results Main result 0, ∞ As usual, Ci M, S denotes In what follows, R denotes the set of real numbers, R the class of all i -times continuously differentiable functions defined on a set M with range in C M, S a set S i 1, 2, , and C0 M, S For convenience, before giving our main results, we first cite some useful lemmas and definitions here Lemma 2.1 see 13 Let a ≥ 0, p ≥ q ≥ and p / 0, then aq/p ≤ q K p q−p /p a p − q q/p K p 2.1 for any K > Definition 2.2 see 14 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 denoted by x, y, z ∈ I if conditions x ∈ 0, , y ∈ 1/2, and z ≥ 3/2 − y are satisfied; it is said to belong to the second-class distribution and denoted by x, y, z ∈ II if conditions x ∈ 0, , y ∈ 0, 1/2 , and z > − 2y2 / − y2 are satisfied Lemma 2.3 see 15, page 296 Let α, β, γ, and p be positive constants Then, t tα − sα p β−1 sp γ−1 ds tθ p γ − B α α ,p β − 1 , t∈R , 2.2 sξ−1 − s η−1 ds ξ, η ∈ C, Re ξ > 0, Re η > is the well-known beta function where B ξ, η and θ p α β − γ − 1 Lemma 2.4 see 14 Suppose that the positive constants α, β, γ, p1 , and p2 satisfy a if α, β, γ ∈ I, p1 1/β; 4β / 3β , then b if α, β, γ ∈ II, p2 B pi γ − α θi are valid for i 1, , pi β − pi α β − γ −1 ∈ 0, ∞ , 2.3 1≥0 Wing-Sum Cheung et al Lemma 2.5 see 6, page 329 Let u x, y , p x, y , q x, y , and k x, y continuous functions defined for x, y ∈ R If x y u x, y ≤ p x, y be nonnegative k s, t u s, t ds dt q x, y 2.4 for x, y ∈ R , then u x, y ≤ p x, y x y x y 0 k s, t p s, t ds dt exp q x, y 0 k s, t q s, t ds dt 2.5 for x, y ∈ R We also need the following well-known consequence of the Jensen inequality: A1 for Ai ≥ i A2 ··· An r ≤ nr−1 Ar Ar ··· Ar n JI 1, 2, , n and r ≥ Theorem 2.6 Let u x, y , a x, y , b x, y , and f x, y be nonnegative continuous functions for x, y ∈ D 0, T × 0, T < T ≤ ∞ Let p and q be constants with p ≥ q > If u x, y satisfies up x, y ≤ a x, y x y b x, y xα − sα β−1 γ−1 s y α − tα β−1 γ−1 t f s, t uq s, t ds dt, x, y ∈ D, 2.6 then for any K > one has the following i If α, β, γ ∈ I, u x, y ≤ x a x, y × exp y P1 x, y Q1 x, y x y 0 f 1/ 1−β s, t P1 s, t ds dt 1−β f 1/ 1−β s, t Q1 s, t ds dt 1/p 2.7 Journal of Inequalities and Applications for x, y ∈ D, where β γ − 2β − B , , α αβ β M1 q K p A x, y x P1 x, y Q1 x, y y A1 x, y 2β/ 1−β q−p /p 1−β p − q q/p K , p a x, y 2β/ β−1 M1 2β/ β−1 K q−p /p f 1/ 1−β s, t A1/ 1−β s, t ds dt, α β−1 xy A1 x, y b1/ 1−β x, y , 1/ 1−β q p 2β/ 1−β M1 γ / 1−β α β−1 xy γ / 1−β b1/ 1−β x, y 2.8 ii If α, β, γ ∈ II, u x, y ≤ x P2 x, y a x, y y Q2 x, y f x s, t P2 s, t ds dt β/ 4β y × exp 4β /β 2.9 f 4β /β 1/p s, t Q2 s, t ds dt , for x, y ∈ D, where γ 4β − β 4β2 B , , α α 3β 3β M2 x P2 x, y Q2 x, y 21 3β /β 21 K 3β /β y A2 x, y f 3β /β M2 q−p 4β /pβ 4β /β 3β /β q p 4β /β γ −β /β 4β α β−1 xy M2 s, t A 1 4β /β xy s, t ds dt, A2 x, y b 4β /β 4β α β−1 γ −β /β x, y , b1 4β /β x, y 2.10 Proof Define a function v x, y by x v x, y y b x, y xα − sα β−1 γ−1 s y α − tα β−1 γ−1 t f s, t uq s, t ds dt, x, y ∈ D, 2.11 Wing-Sum Cheung et al then up x, y ≤ a x, y v x, y , 2.12 or u x, y ≤ a x, y 1/p v x, y x, y ∈ D , 2.13 By Lemma 2.1 and inequality 2.13 , for any K > 0, we have uq x, y ≤ a x, y q/p v x, y ≤ q K p q−p /p a x, y p − q q/p K p v x, y 2.14 Substituting the last relation into 2.11 , we get x y v x, y ≤ b x, y β−1 γ−1 xα − sα s q K p × f s, t x y q−p /p β−1 γ−1 t a s, t p − q q/p K ds dt p v s, t b x, y q K p y α − tα q−p /p β−1 γ−1 xα − sα x y xα − sα b x, y β−1 γ−1 y α − tα s t β−1 γ−1 s f s, t A s, t ds dt y α − tα β−1 γ−1 t f s, t v s, t ds dt, 2.15 p − q /p K q/p where A x, y q/p K q−p /p a x, y 4β / 3β , q2 If α, β, γ ∈ I, let p1 1/β, q1 1/ − β ; if α, β, γ ∈ II, let p2 4β /β, then 1/pi 1/qi for i 1, By applying Holder’s inequality with indices pi , qi ¨ to 2.15 , we get x v x, y ≤ b x, y y × x pi β−1 xα − sα spi γ−1 y α − tα pi β−1 pi γ−1 t 1/pi ds dt 1/qi q K p × y f qi s, t Aqi s, t ds dt q−p /p x x y y b x, y xα − sα pi β−1 spi γ−1 y α − tα pi β−1 pi γ−1 t 1/pi ds dt 1/qi f qi s, t vqi s, t ds dt 2.16 Journal of Inequalities and Applications By Lemmas 2.3 and 2.4, the last inequality can be rewritten as v x, y ≤ Mi2 xy x y × θi 1/pi 1/qi Ai x, y b x, y K q−p /p q p Mi2 xy θi 1/pi b x, y 2.17 1/qi f qi s, t v qi s, t ds dt for x, y ∈ D, where pi γ − B α α Mi x Ai x, y y , pi β − 1 , 2.18 f qi qi s, t A s, t ds dt, and θi is given as in Lemma 2.4 for i 1, Applying inequality JI to 2.17 , we get vqi x, y ≤ 2qi −1 Mi2 xy 2qi −1 q p qi θi qi /pi K qi Ai x, y bqi x, y q−p /p Mi2 xy b x y θi qi /pi qi x, y f qi s, t vqi s, t ds dt 2.19 By Lemma 2.5 and the last inequality, we have x y vqi x, y ≤ P1i x, y Q1i x, y x y f qi s, t P1i s, t ds dt exp f qi s, t Q1i s, t ds dt , 2.20 where 2qi −1 Mi2 xy P1i x, y Q1i x, y qi −1 q p qi K θi qi /pi qi q−p /p Ai x, y bqi x, y , Mi2 xy θi qi /pi qi b 2.21 x, y Finally, substituting 2.20 into 2.13 , considering two situations for i 1, and using parameters α, β, and γ to denote pi , qi and θi in 2.20 , we can get the desired estimations 2.7 and 2.9 , respectively Remark 2.7 In 2.7 and 2.9 , we not only have given some bounds to a new class of nonlinear weakly singular integral inequalities of Wendroff type, but also note that function a x, y appearing in 2.7 and 2.9 is not required to satisfy the nondecreasing condition as some known results 16 Wing-Sum Cheung et al Corollary 2.8 Let functions u x, y , a x, y , b x, y , and f x, y be defined as in Theorem 2.6, and let q be a constant with < q ≤ Suppose that x y u x, y ≤ a x, y b x, y x−s β−1 γ−1 y−t s β−1 γ−1 t f s, t uq s, t ds dt 2.22 for x, y ∈ D, then one has the following i If β ∈ 1/2, , u x, y ≤ a x, y x P 11 x, y x f 1/ 1−β s, t P 11 s, t ds dt 2.23 1−β y × exp y Q11 x, y f 1/ 1−β s, t Q11 s, t ds dt for x, y ∈ D, where M11 x y 0 A11 x, y Q11 x, y 1/ 1−β f 1/ 1−β s, t A1 2β/ 1−β 2β/ β−1 M11 2β/ β−1 K − q Kq , qK q−1 a x, y A1 x, y P 11 x, y γ − 2β − , , β β β B q−1 / 1−β xy 2β γ−2 / 1−β 2β/ 1−β M11 2.24 s, t ds dt, A11 x, y b1/ 1−β x, y , q1/ 1−β xy 2β γ−2 / 1−β b1/ 1−β x, y ii If β ∈ 0, 1/2 , u x, y ≤ a x, y x P 12 x, y × exp Q12 x, y x 0 f 4β /β s, t P 12 s, t ds dt 2.25 β/ 4β y y f 4β /β s, t Q12 s, t ds dt , Journal of Inequalities and Applications where M12 x A12 x, y y Q12 x, y 21 21 3β /β 3β /β K 4β − β 4β2 , , 3β 3β 4β /β 4β /β f P 12 x, y γ B s, t A1 3β /β M12 q−1 4β /β xy 4β2 /β 4β γ−1 3β /β M12 q1 4β /β xy Proof Inequalities 2.23 and 2.25 follow by letting p and by simple computation Details are omitted here s, t ds dt, A12 x, y b 4β γ−1 4β /β 4β2 /β b1 x, y , 4β /β x, y 2.26 and < q ≤ in Theorem 2.6 α Remark 2.9 When b x, y ≡ 1, the inequality 2.22 has been studied in 16 , but here we not only have given some new estimates for u x, y which are unfortunately incomparable with the results in 16 , but also eliminated the nondecreasing condition for function a x, y Let p 2, q α 1, we get the following interesting Henry-Ou-Iang type singular integral inequality For a more detailed account of Ou-Iang type inequalities and their applications, one is referred to and references cited therein Corollary 2.10 Let functions u x, y , a x, y , b x, y , and f x, y be defined as in Theorem 2.6 Suppose that x y u2 x, y ≤ a x, y b x, y x−s β−1 γ−1 s y−t β−1 γ−1 t f s, t u s, t ds dt 2.27 for x, y ∈ D, then for any K > 0, one has the following i If β ∈ 1/2, , u2 x, y ≤ a x, y x P 21 x, y f 1/ 1−β s, t P 21 s, t ds dt 1−β y × exp x y Q21 x, y f 1/ 1−β s, t Q21 s, t ds dt for x, y ∈ D, where M11 A2 x, y B β γ − 2β − , , β β −1/2 K a x, y 1/2 K , 2.28 Wing-Sum Cheung et al x y 0 A21 x, y 2β/ 1−β 2β/ β−1 M11 P 21 x, y 2β Q21 x, y / β−1 1/ 1−β f 1/ 1−β s, t A2 2β γ−2 / 1−β xy 2β/ 1−β K −1/2 1−β M11 s, t ds dt, A21 x, y b1/ 1−β x, y , 2β γ−2 / 1−β xy b1/ 1−β x, y 2.29 ii If β ∈ 0, 1/2 , x u2 x, y ≤ a x, y P 22 x, y Q22 x, y x f 4β /β s, t P 22 s, t ds dt 2.30 β/ 4β y × exp y f 4β /β s, t Q22 s, t ds dt for x, y ∈ D, where M12 B x A22 x, y y f P 22 x, y 21 Q22 x, y 3β /β 2−1 K γ 4β − β 4β2 , , 3β 3β 4β /β 3β /β M12 q−1 4β /β 4β /β s, t A2 s, t ds dt, 2.31 xy 4β γ−1 3β /β M12 xy 4β2 /β A22 x, y b 4β γ−1 Proof Inequalities 2.28 and 2.30 follow by letting p simple computation Details are omitted 4β2 /β 2, q 4β /β b1 α 4β /β x, y , x, y in Theorem 2.6 and by Theorem 2.11 Let u x, y , a x, y , b x, y , and f x, y be defined as in Theorem 2.6, let p ≥ be a constant, and let L : D × R →R be a continuous function which satisfies the condition ≤ L x, y, v − L x, y, w ≤ N x, y, w v − w C for x, y ∈ D and v ≥ w ≥ 0, where N : D × R →R is a continuous function If u x, y satisfies that x y up x, y ≤ a x, y b x, y xα − sα β−1 γ−1 s y α − tα β−1 γ−1 t f s, t L s, t, u s, t ds dt 2.32 for x, y ∈ D, then for any K > one has the following 10 Journal of Inequalities and Applications i If α, β, γ ∈ I, u x, y ≤ ∗ P1 x, y a x, y x x y × exp y ∗ Q1 x, y ∗ f 1/ 1−β s, t P1 s, t ds dt s, t, a s, t p f 1/ 1−β s, t N 1/ 1−β 1−β ∗ × Q1 s, t ds dt p−1 p 2.33 1/p for x, y ∈ D, where β γ − 2β − B , , α αβ β M1 x y L1 x, y 2.34 2β/ 1−β ∗ P1 x, y 2β/ β−1 M1 α β−1 xy 2β/ 1−β ∗ Q1 x, y p−1 ds dt, p s, t, a s, t p f 1/ 1−β s, t L1/ 1−β 2β/ β−1 M1 xy γ / 1−β α β−1 L1 x, y b1/ 1−β x, y , b x, y p γ / 1−β 1/ 1−β ii If α, β, γ ∈ II, u x, y ≤ ∗ P2 x, y a x, y × exp x x y f y ∗ Q2 x, y f 4β /β 4β /β s, t N ∗ s, t P2 s, t ds dt 4β /β β/ 4β ∗ × Q2 s, t ds dt s, t, a s, t p p−1 p 2.35 1/p for x, y ∈ D, where M2 x L2 x, y f ∗ P2 x, y ∗ Q2 x, y y 3β /β 21 γ 4β − β 4β2 B , , α α 3β 3β 4β /β s, t L 3β /β M2 3β /β xy 3β /β M2 4β /β s, t, a s, t p 4β α β−1 xy γ −β /β 4β α β−1 p−1 ds dt, p L2 x, y b γ −β /β b x, y p 4β /β 2.36 x, y , 4β /β Wing-Sum Cheung et al 11 Proof Define a function v x, y by x v x, y y β−1 γ−1 xα − sα b x, y s y α − tα β−1 γ−1 t f s, t L s, t, u s, t ds dt, x, y ∈ D, 2.37 then up x, y ≤ a x, y v x, y 2.38 By Lemma 2.1, we have u x, y ≤ a x, y v x, y 1/p ≤ a x, y p v x, y p−1 , p x, y ∈ D 2.39 Substituting the last inequality into 2.37 and using condition C , we get x v x, y ≤ b x, y y xα − sα x s y a s, t p xα − sα β−1 γ−1 t p−1 ds dt p v s, t β−1 γ−1 s y α − tα β−1 γ−1 t × f s, t L s, t, a s, t p b x, y p y α − tα × f s, t L s, t, ≤ b x, y β−1 γ−1 x y xα − sα 2.40 p−1 ds dt p β−1 γ−1 s y α − tα β−1 γ−1 t × f s, t M s, t, a s, t p p−1 v s, t ds dt p Applying similar procedures used from 2.15 to the end of the proof of Theorem 2.6 to the last inequality, we get the desired inequalities 2.33 and 2.35 Applications In this section, we will indicate the usefulness of our main results in the study of the boundedness of certain partial integral equations with weakly singular kernel Consider the partial integral equation: zp x, y x l x, y y h x, y 0 xα − sα β−1 γ−1 s y α − tα β−1 γ−1 t F s, t, z s, t ds dt 3.1 12 Journal of Inequalities and Applications for x, y ∈ D, where l x, y and h x, y ∈ C D, R , F ∈ C D × R, R satisfies F x, y, u ≤ b x, y |u|q 3.2 for some b ∈ C D, R , and p ≥ q > are constants Plugging 3.2 into 3.1 and by applying Theorem 2.6, we obtain a bound on the solutions z x, y of 3.1 Remark 3.1 i Obviously, the boundedness of the solutions of 3.1 - 3.2 cannot be derived by the known results in 16 ii By our results and under some suitable conditions, other basic properties’ solutions of 3.1 such as the uniqueness and the continuous dependence can also be derived here, but in order to save space, the details are omitted Acknowledgments The first author’s research was supported in part by the Research Grants Council of the Hong Kong SAR, China Project no HKU7016/07P The second author’s research was supported by NSF of Guangdong Province, China Project no 8151042001000005 References E F Beckenbach and R Bellman, Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, Germany, 1961 V Lakshmikantham and S Leela, Differential and Integral Inequalities: Theory and Applications Vol I: Ordinary Dfferential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1969 V Lakshmikantham and S Leela, Differential and Integral Inequalities: Theory and Applications Vol II: Functional, Partial, Abstract, and Complex Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1969 D Ba˘nov and P Simeonov, Integral Inequalities and Applications, vol 57 of Mathematics and Its ı Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992 R P Agarwal, Difference Equations and 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Applications, vol 5, no 3, pp 287–308, 2000  ... boundedness of some fractional differential and integral equations ˘ In this paper, by applying Medved’s method of desingularization of weakly singular inequalities we establish some new singular. .. Remark 2.7 In 2.7 and 2.9 , we not only have given some bounds to a new class of nonlinear weakly singular integral inequalities of Wendroff type, but also note that function a x, y appearing... modification of Medved’s c c method to study certain class of nonlinear inequalities of Henry -type, which generalized some known results and were used as handy and effective tools in the study of the