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 with Two 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 with two variables, and gives an estimation for a general form of nonlinear integral inequality with two 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.In11, Pachpatte obtained an estimation for the integral inequality ux, y ≤ ax, y  x 0  y 0 fs, t  us, t  s 0  t 0 gs, t, σ, τuσ, τdτdσ  dtds. 1.1 His results were applied to a partial integrodifferential equation: u xy x, yF  x, y,ux, y,  x 0  y 0 h  x, y,τ,σ, ux, y  dτdσ  , u  x, y 0   αx,u  x 0 ,y   βy, 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  ux, y  ≤ ax, y  bx bx 0   cy cy 0  fx, y,s, t  ϕ 1  us, t    s bx 0   t cy 0  gs, t, σ, τϕ 2  uσ, τ  dτdσ  dtds 1.3 for all x, y ∈ x 0 ,x 1  × y 0 ,y 1 . Obviously, u appears linearly in 1.1,butinour1.3 it is generalized to nonlinear terms: ϕ 1 us, t and ϕ 2 us, 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 bx ≤ x and cy ≤ y, a ∈ C 1 Λ, R  ,f∈ C 0 Λ 2 , R  ,andgx, 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 00andϕ 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, φs : 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 Φu  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 Φu  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  fx, y,s, t : max τ∈x 0 ,x fτ,y,s,t, which is also nondecreasing in x for each fixed y, s,andt and satisfies  fx, y,s, t ≥ fx, y, s, t ≥ 0. Theorem 2.1. If inequality 1.3 holds for the nonnegative function ux, y,then ux, y ≤ ψ −1  W −1 2  Ξx, y  2.6 for all x, y ∈ x 0 ,X 1  × y 0 ,Y 1 ,where Ξx, y : W 2  W −1 1  r 2 x, y    bx bx 0   cy cy 0   fx, y,s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds, r 2 x, y : W 1  r 1 x, y    bx bx 0   cy cy 0   fx, y,s, tdtds, 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 : Ξx, 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 ,from1.3 we get ψ  ux, y  ≤ ax, y  bx bx 0   cy cy 0   fx, y,s, t  w 1  us, t    s bx 0   t cy 0  g  s, t, σ, τ  w 2  uσ, τ  dτdσ  dtds 2.9 for all x, y ∈ Λ. Firstly, we discuss the case that ax, 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, ydt. 2.10 Regarding 1.3, we consider the auxiliary inequality ψ  ux, y  ≤ r 1 x, y  bx bx 0   cy cy 0   fX, y, s, t  w 1  us, t    s bx 0   t cy 0  gs, t, σ, τw 2  uσ, 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 ux, y ≤ ψ −1  W −1 2  W 2  W −1 1  W 1  r 1 x, y    bx bx 0   cy cy 0   fX, y, s, tdtds    bx bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds  2.12 for all x, y ∈ x 0 ,X × y 0 ,Y 1 ,whereY 1 is defined by 2.8. Let ηx, 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 ux, y ≤ ψ −1  ηx, y  ∀x, y ∈ x 0 ,Y × J. 2.13 By the fact that bx ≤ x for x ∈ x 0 ,X and the monotonicity of w i ,ψ,η,andbx,wehave ∂/∂xηx, y w 1  ψ −1  ηx, y  ≤ ∂/∂xr 1 x, y w 1  ψ −1  r 1 x, y   b  x w 1  ψ −1  ηx, y  ×  cy cy 0   f 1  X, y, bx,t   w 1  u  bx,t    bx bx 0   t cy 0  g  bx,t,τ,σ  w 2  uτ,σ  dτdσ  dt ≤ ∂/∂xr 1 x, y w 1  ψ −1  r 1 x, y   b  x  cy cy 0   f 1 X, y, bx,tdt  b  x  cy cy 0   f 1 X, y, bx,t   bx bx 0   t cy 0  g  bx,t,τ,σ  φ  uτ,σ  dτdσ  dt 2.14 for all x, y ∈ x 0 ,X × J. Integrating the above from x 0 to x,weget W 1  ηx, y  ≤ W 1  r 1 x, y    bx bx 0   cy cy 0   f 1 X, y, s, tdtds   bx bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σφ  uτ,σ  dτdσ  dtds 2.15 for all x, y ∈ x 0 ,X × J.Let ψ  ξx, y  : W 1  ηx, y  , r 2 x, y : W 1  r 1 x, y    bx bx 0   cy cy 0   f 1 X, y, s, tdtds. 2.16 W S. Wang and C X. Shen 5 From 2.15, 2.16,weobtain ψ  ξx, y  ≤ r 2 x, y  bx bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σφ  uτ,σ  dτdσ  dtds 2.17 for all x 0 ≤ x<X,y 0 ≤ y<y 1 .Letβx, 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  ξx, y  ≤ βx, y ∀x, y ∈ x 0 ,Y × J. 2.18 From 2.13, 2.16,and2.18,wehave ux, y ≤ ψ −1  ηx, y   ψ −1  W −1 1  ψ  ξx, y  ≤ ψ −1  W −1 1  βx, 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 , φψ −1 W −1 1 s is continuous and nondecreasing on 0, ∞ and satisfies φψ −1 W −1 1 s > 0 for s>0. Let hsψ −1 W −1 1 s. Since b  x ≥ 0andbx ≤ x for x ∈ x 0 ,X,from2.19 we have ∂/∂xβx, y φ  h  βx, y  ≤ ∂/∂xr 2 x, y φ  h  r 2 x, y   b  x φ  h  βx, y   cy cy 0   f 1 X, y, bx,t   bx bx 0   t cy 0  g  bx,t,τ,σ  φ  uτ,σ  dτdσ  dtds ≤ ∂/∂xr 2 x, y φ  h  r 2 x, y   b  x  cy cy 0   f 1 X, y, bx,t   bx bx 0   t cy 0  g  bx,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,by2.4 we get Φ  βx, y  ≤ Φ  r 2 x, y    bx bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds 2.21 for all x, y ∈ x 0 ,X × y 0 ,y 1 .By2.19 and the above inequality, we obtain ux, y ≤ ψ −1  W −1 1  Φ −1  Φ  r 2 x, y    bx bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  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 ux, y ≤ ψ −1  W −1 2  W 2  W −1 1  W 1  r 1 x, y    bx bx 0   cy cy 0   f 1 X, y, s, tdtds    bx bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds  , 2.23 which proves the claimed 2.12. We start from the original inequality 1.3 and see that ψ  uX, y  ≤ r 1 X, y  bX bx 0   cy cy 0   fX, y, s, t  ϕ 1  us, t    s bx 0   t cy 0  gs, t, σ, τϕ 2  uσ, 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 uX, y ≤ ψ −1  W −1 2  W 2  W −1 1  W 1  r 1 X, y    bX bx 0   cy cy 0   f 1 X, y, s, tdtds    bX bx 0   cy cy 0   f 1 X, y, s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds  2.25 for all x 0 ≤ X ≤ X 1 ,y 0 ≤ y ≤ Y 1 . This proves 2.6. The remainder case is that ax, y0 for some x, y ∈ Λ.Let r 1,ε x, y : r 1 x, yε, 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 ux, y ≤ ψ −1  W −1 2  W 2  W −1 1  W 1  r 1,ε x, y    bx bx 0   cy cy 0   f 1 x, y, s, tdtds    bx bx 0   cy cy 0   f 1 x, y, s, t   s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds  2.27 for all x 0 ≤ X ≤ X 1 ,y 0 ≤ y ≤ Y 1 . Letting ε → 0  ,weobtain2.6 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   Fx, y, u, v   ≤ fx, y  |u|  |v|  ,   h  x, y,s, t, us, t    ≤ gx, y,s, t   us, t   ,   F  x, y,u 1 ,v 1  − F  x, y,u 2 ,v 2    ≤ fx, y    u 1 − u 2      v 1 − v 2    ,   h  x, y,s, t, u 1  − h  x, y,s, t, u 2    ≤ gx, 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, yF  x, y,u  bx,cy  ,  bx bbx 0   cy ccy 0  h  bx,cy,τ,σ,uτ, σ  dτdσ  , u  x, y 0   αx,u  x 0 ,y   βy 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 α0β00. 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   Fx, y, u, v   ≤ fx, y  ϕ 1  |u|   |v|  ,   h  x, y,s, t, us, t    ≤ gx, y,s, t   ϕ 2  us, t    . 3.3 Corollary 3.1. If |αxβy| is nondecreasing in x and y and 3.3 holds, then every solution um, n of 3.2 satisfies ux, y ≤ W −1 2  Ξx, y  ∀x, y ∈  x 0 ,X 1  ×  y 0 ,Y 1  , 3.4 where Ξx, y : W 2  W −1 1  W 1    αxβy      bx bx 0   cy cy 0  f  b −1 s,c −1 t  b   b −1 s  c   c −1 t  dtds    bx bx 0   cy cy 0  f  b −1 s,c −1 t  b   b −1 s  c   c −1 t    s bx 0   t cy 0  gs, t, τ, σdτdσ  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   αxβy   <M,  bx bx 0   cy cy 0  f  b −1 s,c −1 t  b   b −1 s  c   c −1 t  dtds < M,  bx bx 0   cy cy 0  f  b −1 s,c −1 t  b   b −1 s  c   c −1 t    s bx 0   t cy 0  gs, t, τ, σdτdσ  dtds < M 3.6 on x 0 ,X 1  × y 0 ,Y 1 , then every solution ux, 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    ≤ fx, 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    ≤ gx, y,s, tϕ 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  bx bx 0   cy cy 0  f  b −1 s,c −1 t  b   b −1 s  c   c −1 t  dtds < M,  bx bx 0   cy cy 0  f  b −1 s,c −1 t  b   b −1 s  c   c −1 t    s bx 0   t cy 0  gs, t, τ, σdτdσ  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. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal,vol. 10, no. 4, pp. 643–647, 1943. 2 T. H. 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Pachpatte, “Bounds on certain integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 3, article 47, 10 pages, 2002. 12 W S. Wang, “A generalized sum-difference inequality and applications to partial difference equations,” Advances in Difference Equations, vol. 2008, Article ID 695495, 12 pages, 2008. . improves Pachpatte’s results on linear integral inequalities with two variables, and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not. “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005. 4 I. Bihari, A generalization of a lemma. 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