Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 518646, 9 pages doi:10.1155/2008/518646 ResearchArticleOnaGeneralizedRetardedIntegralInequalitywithTwo Variables Wu-Sheng Wang 1, 2 and Cai-Xia Shen 1 1 Department of Mathematics, Hechi College, Guangxi, Yizhou 546300, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Wu-Sheng Wang, wang4896@126.com Received 16 November 2007; Accepted 22 April 2008 Recommended by Wing-Sum Cheung This paper improves Pachpatte’s results on linear integral inequalities withtwo variables, and gives an estimation for a general form of nonlinear integralinequalitywithtwo variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for a integrodifferential equation. Copyright q 2008 W S. Wang and C X. Shen. 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. 1. Introduction Gronwall-Bellman inequality 1, 2 is an important tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of its generalizations in various cases from literature see, e.g., 1–12.In11, Pachpatte obtained an estimation for the integralinequality ux, y ≤ ax, y x 0 y 0 fs, t us, t s 0 t 0 gs, t, σ, τuσ, τdτdσ dtds. 1.1 His results were applied to a partial integrodifferential equation: u xy x, yF x, y,ux, y, x 0 y 0 h x, y,τ,σ, ux, y dτdσ , u x, y 0 αx,u x 0 ,y βy, 1.2 for boundedness and uniqueness of solutions. 2 Journal of Inequalities and Applications In this paper, we discuss a more general form of integral inequality: ψ ux, y ≤ ax, y bx bx 0 cy cy 0 fx, y,s, t ϕ 1 us, t s bx 0 t cy 0 gs, t, σ, τϕ 2 uσ, τ dτdσ dtds 1.3 for all x, y ∈ x 0 ,x 1 × y 0 ,y 1 . Obviously, u appears linearly in 1.1,butinour1.3 it is generalized to nonlinear terms: ϕ 1 us, t and ϕ 2 us, t. Our strategy is to monotonize functions ϕ i s with other two nondecreasing ones such that one has stronger monotonicity than the other. We apply our estimation to an integrodifferential equation, which looks similar to 1.2 but includes delays, and give boundedness and uniqueness of solutions. 2. Main result Throughout this paper, x 0 ,x 1 ,y 0 ,y 1 ∈ R are given numbers. Let R :0, ∞,I :x 0 ,x 1 ,J : y 0 ,y 1 , and Λ : I × J ⊂ R 2 . Consider inequality 1.3, where we suppose that ψ ∈ C 0 R , R is strictly increasing such that ψ∞∞, b ∈ C 1 I,I, and c ∈ C 1 J, J are nondecreasing, such that bx ≤ x and cy ≤ y, a ∈ C 1 Λ, R ,f∈ C 0 Λ 2 , R ,andgx, y,s, t ∈ C 0 Λ 2 , R are given, and ϕ i ∈ C 0 R , R i 1, 2 are functions satisfying ϕ i 00andϕ i u > 0 for all u>0. Define functions w 1 s : max τ∈0,s ϕ 1 τ , w 2 s : max τ∈0,s ϕ 2 τ/w 1 τ w 1 s, φs : w 2 s/w 1 s. 2.1 Obviously, w 1 ,w 2 ,andφ in 2.1 are all nondecreasing and nonnegative functions and satisfy w i s ≥ ϕ i s,i 1, 2. Let W 1 u u 1 ds w 1 ψ −1 s , 2.2 W 2 u u 1 ds w 2 ψ −1 s , 2.3 Φu u W 1 1 ds φ ψ −1 W −1 1 s . 2.4 Obviously, W 1 ,W 2 ,andΦ are strictly increasing in u>0, and therefore the inverses W −1 1 ,W −1 2 , and Φ −1 are well defined, continuous, and increasing. We note that Φu u W 1 1 dx φ ψ −1 W −1 1 x u W 1 1 w 1 ψ −1 W −1 1 x dx w 2 ψ −1 W −1 1 x W −1 1 u 1 dx w 2 ψ −1 x W 2 W −1 1 u . 2.5 W S. Wang and C X. Shen 3 Furthermore, let fx, y,s, t : max τ∈x 0 ,x fτ,y,s,t, which is also nondecreasing in x for each fixed y, s,andt and satisfies fx, y,s, t ≥ fx, y, s, t ≥ 0. Theorem 2.1. If inequality 1.3 holds for the nonnegative function ux, y,then ux, y ≤ ψ −1 W −1 2 Ξx, y 2.6 for all x, y ∈ x 0 ,X 1 × y 0 ,Y 1 ,where Ξx, y : W 2 W −1 1 r 2 x, y bx bx 0 cy cy 0 fx, y,s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds, r 2 x, y : W 1 r 1 x, y bx bx 0 cy cy 0 fx, y,s, tdtds, r 1 x, y : a x 0 ,y x x 0 a x s, y ds, 2.7 and X 1 ,Y 1 ∈ Λ is arbitrarily given on the boundary of the planar region R : x, y ∈ Λ : Ξx, y ∈ Dom W −1 2 ,r 2 x, y ∈ Dom W −1 1 . 2.8 Here Dom denotes the domain of a function. Proof. By the definition of functions w i and f i ,from1.3 we get ψ ux, y ≤ ax, y bx bx 0 cy cy 0 fx, y,s, t w 1 us, t s bx 0 t cy 0 g s, t, σ, τ w 2 uσ, τ dτdσ dtds 2.9 for all x, y ∈ Λ. Firstly, we discuss the case that ax, y > 0 for all x, y ∈ Λ. It means that r 1 x, y > 0 for all x, y ∈ Λ. In such a circumstance, r 1 x, y is positive and nondecreasing on Λ and r 1 x, y ≥ a x 0 ,y x x 0 a x t, ydt. 2.10 Regarding 1.3, we consider the auxiliary inequality ψ ux, y ≤ r 1 x, y bx bx 0 cy cy 0 fX, y, s, t w 1 us, t s bx 0 t cy 0 gs, t, σ, τw 2 uσ, τ dτdσ dtds 2.11 4 Journal of Inequalities and Applications for all x, y ∈ x 0 ,X × J,wherex 0 ≤ X ≤ X 1 is chosen arbitrarily. We claim that ux, y ≤ ψ −1 W −1 2 W 2 W −1 1 W 1 r 1 x, y bx bx 0 cy cy 0 fX, y, s, tdtds bx bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds 2.12 for all x, y ∈ x 0 ,X × y 0 ,Y 1 ,whereY 1 is defined by 2.8. Let ηx, y denote the right-hand side of 2.11, which is a nonnegative and nondecreasing function on x 0 ,X × J. Then, 2.11 is equivalent to ux, y ≤ ψ −1 ηx, y ∀x, y ∈ x 0 ,Y × J. 2.13 By the fact that bx ≤ x for x ∈ x 0 ,X and the monotonicity of w i ,ψ,η,andbx,wehave ∂/∂xηx, y w 1 ψ −1 ηx, y ≤ ∂/∂xr 1 x, y w 1 ψ −1 r 1 x, y b x w 1 ψ −1 ηx, y × cy cy 0 f 1 X, y, bx,t w 1 u bx,t bx bx 0 t cy 0 g bx,t,τ,σ w 2 uτ,σ dτdσ dt ≤ ∂/∂xr 1 x, y w 1 ψ −1 r 1 x, y b x cy cy 0 f 1 X, y, bx,tdt b x cy cy 0 f 1 X, y, bx,t bx bx 0 t cy 0 g bx,t,τ,σ φ uτ,σ dτdσ dt 2.14 for all x, y ∈ x 0 ,X × J. Integrating the above from x 0 to x,weget W 1 ηx, y ≤ W 1 r 1 x, y bx bx 0 cy cy 0 f 1 X, y, s, tdtds bx bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σφ uτ,σ dτdσ dtds 2.15 for all x, y ∈ x 0 ,X × J.Let ψ ξx, y : W 1 ηx, y , r 2 x, y : W 1 r 1 x, y bx bx 0 cy cy 0 f 1 X, y, s, tdtds. 2.16 W S. Wang and C X. Shen 5 From 2.15, 2.16,weobtain ψ ξx, y ≤ r 2 x, y bx bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σφ uτ,σ dτdσ dtds 2.17 for all x 0 ≤ x<X,y 0 ≤ y<y 1 .Letβx, y denote the right-hand side of 2.17, which is a nonnegative and nondecreasing function on x 0 ,Y × J. Then, 2.17 is equivalent to ψ ξx, y ≤ βx, y ∀x, y ∈ x 0 ,Y × J. 2.18 From 2.13, 2.16,and2.18,wehave ux, y ≤ ψ −1 ηx, y ψ −1 W −1 1 ψ ξx, y ≤ ψ −1 W −1 1 βx, y 2.19 for all x 0 ≤ x<X, y 0 ≤ y<Y 1 ,whereY 1 is defined by 2.8. By the definitions of φ, ψ,andW 1 , φψ −1 W −1 1 s is continuous and nondecreasing on 0, ∞ and satisfies φψ −1 W −1 1 s > 0 for s>0. Let hsψ −1 W −1 1 s. Since b x ≥ 0andbx ≤ x for x ∈ x 0 ,X,from2.19 we have ∂/∂xβx, y φ h βx, y ≤ ∂/∂xr 2 x, y φ h r 2 x, y b x φ h βx, y cy cy 0 f 1 X, y, bx,t bx bx 0 t cy 0 g bx,t,τ,σ φ uτ,σ dτdσ dtds ≤ ∂/∂xr 2 x, y φ h r 2 x, y b x cy cy 0 f 1 X, y, bx,t bx bx 0 t cy 0 g bx,t,τ,σ dτdσ dtds 2.20 for all x, y ∈ x 0 ,X × y 0 ,Y 1 . Integrating the above from x 0 to x,by2.4 we get Φ βx, y ≤ Φ r 2 x, y bx bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds 2.21 for all x, y ∈ x 0 ,X × y 0 ,y 1 .By2.19 and the above inequality, we obtain ux, y ≤ ψ −1 W −1 1 Φ −1 Φ r 2 x, y bx bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds 2.22 6 Journal of Inequalities and Applications for all x, y ∈ x 0 ,X × y 0 ,Y 1 ,whereY 1 is defined by 2.8. It follows from 2.5 that ux, y ≤ ψ −1 W −1 2 W 2 W −1 1 W 1 r 1 x, y bx bx 0 cy cy 0 f 1 X, y, s, tdtds bx bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds , 2.23 which proves the claimed 2.12. We start from the original inequality 1.3 and see that ψ uX, y ≤ r 1 X, y bX bx 0 cy cy 0 fX, y, s, t ϕ 1 us, t s bx 0 t cy 0 gs, t, σ, τϕ 2 uσ, τ dτdσ dtds 2.24 for all y ∈ y 0 ,Y 1 ; n amely, the auxiliary inequality 2.11 holds for x X, y ∈ y 0 ,Y 1 .By 2.12,weget uX, y ≤ ψ −1 W −1 2 W 2 W −1 1 W 1 r 1 X, y bX bx 0 cy cy 0 f 1 X, y, s, tdtds bX bx 0 cy cy 0 f 1 X, y, s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds 2.25 for all x 0 ≤ X ≤ X 1 ,y 0 ≤ y ≤ Y 1 . This proves 2.6. The remainder case is that ax, y0 for some x, y ∈ Λ.Let r 1,ε x, y : r 1 x, yε, 2.26 where ε>0 is an arbitrary small number. Obviously, r 1,ε x, y > 0 for all x, y ∈ Λ. Using the same arguments as above, where r 1 x, y is replaced with r 1,ε x, y,weget ux, y ≤ ψ −1 W −1 2 W 2 W −1 1 W 1 r 1,ε x, y bx bx 0 cy cy 0 f 1 x, y, s, tdtds bx bx 0 cy cy 0 f 1 x, y, s, t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds 2.27 for all x 0 ≤ X ≤ X 1 ,y 0 ≤ y ≤ Y 1 . Letting ε → 0 ,weobtain2.6 because of continuity of r 1,ε in ε and continuity of ψ −1 ,W −1 1 ,W,W −1 2 ,andW 2 . This completes the proof. W S. Wang and C X. Shen 7 3. Applications In 11, the partial integrodifferential equation 1.2 was discussed for boundedness and uniqueness of the solutions under the assumptions that Fx, y, u, v ≤ fx, y |u| |v| , h x, y,s, t, us, t ≤ gx, y,s, t us, t , F x, y,u 1 ,v 1 − F x, y,u 2 ,v 2 ≤ fx, y u 1 − u 2 v 1 − v 2 , h x, y,s, t, u 1 − h x, y,s, t, u 2 ≤ gx, y,s, t u 1 − u 2 , 3.1 respectively. In this section, we further consider the nonlinear delay partial integrodifferential equation u xy x, yF x, y,u bx,cy , bx bbx 0 cy ccy 0 h bx,cy,τ,σ,uτ, σ dτdσ , u x, y 0 αx,u x 0 ,y βy 3.2 for all x, y ∈ Λ,whereb, c,andu are supposed to be as in Theorem 2.1; h : Λ 2 × R→R, F : Λ × R 2 →R, α : I→R,andβ : J→R are all continuous functions such that α0β00. Obviously, the estimation obtained in 11 cannot be applied to 3.2. We first give an estimation for solutions of 3.2 under the condition Fx, y, u, v ≤ fx, y ϕ 1 |u| |v| , h x, y,s, t, us, t ≤ gx, y,s, t ϕ 2 us, t . 3.3 Corollary 3.1. If |αxβy| is nondecreasing in x and y and 3.3 holds, then every solution um, n of 3.2 satisfies ux, y ≤ W −1 2 Ξx, y ∀x, y ∈ x 0 ,X 1 × y 0 ,Y 1 , 3.4 where Ξx, y : W 2 W −1 1 W 1 αxβy bx bx 0 cy cy 0 f b −1 s,c −1 t b b −1 s c c −1 t dtds bx bx 0 cy cy 0 f b −1 s,c −1 t b b −1 s c c −1 t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds, 3.5 and W 1 ,W −1 1 ,W 2 ,W −1 2 ,andX 1 ,Y 1 are defined as in Theorem 2.1 . 8 Journal of Inequalities and Applications Corollary 3.1 actually gives a condition of boundedness for solutions. Concretely, if there is a positive constant M such that αxβy <M, bx bx 0 cy cy 0 f b −1 s,c −1 t b b −1 s c c −1 t dtds < M, bx bx 0 cy cy 0 f b −1 s,c −1 t b b −1 s c c −1 t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds < M 3.6 on x 0 ,X 1 × y 0 ,Y 1 , then every solution ux, y of 3.2 is bounded on x 0 ,X 1 × y 0 ,Y 1 . Next, we give the condition of the uniqueness of solutions for 3.2. Corollary 3.2. Suppose F x, y,u 1 ,v 1 − F x, y,u 2 ,v 2 ≤ fx, y ϕ 1 u 1 − u 2 v 1 − v 2 , h x, y,s, t, u 1 − h x, y,s, t, u 2 ≤ gx, y,s, tϕ 2 u 1 − u 2 , 3.7 where f, g, ϕ 1 ,ϕ 2 are defined as in Theorem 2.1. There is a positive number M such that bx bx 0 cy cy 0 f b −1 s,c −1 t b b −1 s c c −1 t dtds < M, bx bx 0 cy cy 0 f b −1 s,c −1 t b b −1 s c c −1 t s bx 0 t cy 0 gs, t, τ, σdτdσ dtds < M 3.8 on x 0 ,X 1 × y 0 ,Y 1 .Then,3.2 has at most one solution on x 0 ,X 1 × y 0 ,Y 1 ,whereX 1 ,Y 1 are defined as in Theorem 2.1. Acknowledgments This work is supported by the Scientific Research Fund of Guangxi Provincial Education Department no. 200707MS112, the Natural Science Foundation no. 2006N001, and the Applied Mathematics Key Discipline Foundation of Hechi College of China. References 1 R. 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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 518646, 9 pages doi:10.1155/2008/518646 Research Article On a Generalized Retarded Integral Inequality