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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 831817, 13 pages doi:10.1155/2008/831817 Research Article Bounds for Certain Delay Integral Inequalities on Time Scales Wei Nian Li1, 2 Department of Applied Mathematics, Shanghai Normal University, Shanghai 200234, China Department of Mathematics, Binzhou University, Shandong 256603, China Correspondence should be addressed to Wei Nian Li, wnli@263.net Received 31 August 2008; Revised 21 October 2008; Accepted 22 October 2008 Recommended by Martin J Bohner Our aim in this paper is to investigate some delay integral inequalities on time scales by using Gronwall’s inequality and comparison theorem Our results unify and extend some delay integral inequalities and their corresponding discrete analogues The inequalities given here can be used as handy tools in the qualitative theory of certain classes of delay dynamic equations on time scales Copyright q 2008 Wei Nian Li 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 The unification and extension of differential equations, difference equations, q-difference equations, and so on to the encompassing theory of dynamic equations on time scales was initiated by Hilger in his Ph.D thesis in 1988 During the last few years, some integral inequalities on time scales related to certain inequalities arising in the theory of dynamic equations had been established by many scholars For example, we refer the reader to literatures 2–8 and the references therein However, nobody studied the delay integral inequalities on time scales, as far as we know In this paper, we investigate some delay integral inequalities on time scales, which provide explicit bounds on unknown functions Our results extend some known results in Throughout this paper, a knowledge and understanding of time scales and time scale notation is assumed For an excellent introduction to the calculus on time scales, we refer the reader to monographes 10, 11 Main results 0, ∞ , Z denotes the set of integers, In what follows, R denotes the set of real numbers, R {0, 1, 2, } denotes the set of nonnegative integers, C M, S denotes the class of all N0 continuous functions defined on set M with range in the set S, T is an arbitrary time scale, Journal of Inequalities and Applications Crd denotes the set of rd-continuous functions, R denotes the set of all regressive and rd{p ∈ R : μ t p t > 0, for all t ∈ T} We use the usual continuous functions, and R conventions that empty sums and products are taken to be and 1, respectively Throughout t0 , ∞ T this paper, we always assume that t0 ∈ T, T0 The following lemmas are very useful in our main results Lemma 2.1 see Assume that p ≥ q ≥ 0, p / 0, and a ∈ R Then q k p aq/p ≤ q−p /p p − q q/p k , p a for any k > 2.1 Lemma 2.2 Gronwall’s inequality 10 Suppose u, b ∈ Crd , m ∈ R , m ≥ Then t u t ≤b t t0 m t u t Δt, t ∈ T0 , 2.2 implies t u t ≤b t t0 em t, σ s b s m s Δs, t ∈ T0 2.3 Lemma 2.3 comparison theorem 10 Suppose u, b ∈ Crd , a ∈ R Then uΔ t ≤ a t u t b t , t ∈ T0 , 2.4 implies t u t ≤ u t0 ea t, t0 t0 ea t, σ τ b τ Δτ, t ∈ T0 2.5 Firstly, we study the delay integral inequality on time scales of the form xp t ≤ a t t t0 b s xp s Δs c t t t0 f s xq τ s g s xr s Δs, t ∈ T0 , E with the initial condition x t ϕ τ t ≤ a t ϕt , 1/p t ∈ α, t0 ∩ T, for t ∈ T0 with τ t ≤ t0 , where p, q, and r are constants, p / 0, p ≥ q ≥ 0, p ≥ r ≥ 0, τ : T0 → T, τ t ≤ t, − ∞ < α inf{τ t , t ∈ T0 } ≤ t0 , and ϕ t ∈ Crd α, t0 ∩ T, R I Wei Nian Li Theorem 2.4 Assume that x t , a t , b t , c t , f t , g t ∈ Crd T0 , R If a t and c t are nondecreasing for t ∈ T0 , then the inequality E with the initial condition I implies eb t, t0 x t ≤ a t c t t F t t0 1/p eG t, σ s F s G s Δs , 2.6 for any k > 0, t ∈ T0 , where F t t f s eb s, t0 q/p pk t0 k p−q r/p g s eb s, t0 qa s p−q /p pk k p−r s p−r /p Δs, 2.7 G t c t q/p qf t eb t, t0 rg t eb t, t0 p−q /p pk pk r/p p−r /p , t ∈ T0 2.8 Proof Define a function z t by zt t a t t0 b s xp s Δs t c t t0 f s xq τ s g s xr s Δs 1/p , t ∈ T0 2.9 It is easy to see that z t is a nonnegative and nondecreasing function, and x t ≤z t , t ∈ T0 2.10 Therefore, x τ t ≤z τ t ≤z t , for t ∈ T0 with τ t > t0 2.11 On the other hand, using the initial condition I , we have x τ t ϕ τ t ≤ a t 1/p ≤z t , for t ∈ T0 with τ t ≤ t0 2.12 Combining 2.11 and 2.12 , we obtain x τ t ≤z t , t ∈ T0 2.13 It follows from 2.9 , 2.10 , and 2.13 that zp t ≤ a t t t0 b s zp s Δs c t t t0 f s zq s g s zr s Δs, t ∈ T0 2.14 Define a function w t by w t a t c t u t , 2.15 Journal of Inequalities and Applications where u t t t0 f s zq s g s zr s Δs, t ∈ T0 2.16 Then 2.14 can be restated as t zp t ≤ w t t0 b s zp s Δs, t ∈ T0 2.17 Obviously, w ∈ Crd T0 , R , b t ≥ 0, b ∈ R Using Lemma 2.2, from 2.17 , we obtain t zp t ≤ w t t0 eb t, σ s w s b s Δs, t ∈ T0 2.18 Noting that w t is nondecreasing, from 2.18 , we have zp t ≤ w t w t t t0 eb t, σ s b s Δs t w t t0 eb t, σ s b s Δs , t ∈ T0 2.19 By 10, Theorems 2.39 and 2.36 i , we obtain t t0 eb t, σ s b s Δs eb t, t0 − eb t, t eb t, t0 − 1, t ∈ T0 2.20 It follows from 2.19 and 2.20 that zp t ≤ w t eb t, t0 eb t, t0 a t t ∈ T0 c t u t , 2.21 Using Lemma 2.1, from 2.21 , for any k > 0, we easily obtain zq t ≤ eb t, t0 q/p ≤ eb t, t0 q/p zr t ≤ eb t, t0 r/p ≤ eb t, t0 r/p a t c t u t k p−q pk a t qc t u t qa t p−q /p c t u t k p−r pk q/p t p−r /p , t ∈ T0 , rc t u t , pk p−r /p t ∈ T0 pk p−q /p 2.22 r/p 2.23 Wei Nian Li Combining 2.16 , 2.22 , and 2.23 , we have t u t ≤ t0 q/p f s eb s, t0 pk g s eb s, t0 t F t t0 k p−q qa s p−q /p k p−r r/p pk G s u s Δs, qc s u s pk s p−r /p p−q /p rc s u s pk p−r /p Δs 2.24 t ∈ T0 , where F t and G t are defined by 2.7 and 2.8 , respectively Using Lemma 2.2, from 2.24 , we have t u t ≤F t t0 eG t, σ s F s G s Δs, t ∈ T0 2.25 Therefore, the desired inequality 2.6 follows from 2.10 , 2.22 , and 2.25 This completes the proof Theorem 2.5 Suppose that all assumptions of Theorem 2.4 hold Then the inequality E with the initial condition I implies x t ≤ eb t, t0 a t c t F t eG t, t0 1/p , 2.26 for any k > 0, t ∈ T0 , where F t and G t are defined by 2.7 and 2.8 , respectively Proof As in the proof of Theorem 2.4, we obtain 2.25 It is easy to see that F t is nondecreasing for t ∈ T0 Therefore, by 10, Theorems 2.39 and 2.36 i , we have u t ≤ F t eG t, t0 , t ∈ T0 2.27 The desired inequality 2.26 follows from 2.10 , 2.22 , and 2.27 The proof is complete Remark 2.6 Let T R If b t 0, then Theorem 2.5 reduces to 9, Theorem 2.3 Letting T from Theorem 2.5, we easily establish the following result Z, Corollary 2.7 Assume that x n , a n , b n , c n , f n , g n are nonnegative functions defined for n ∈ N0 If a n and c n are nondecreasing in N0 , and x n satisfies the following delay discrete inequality: xp n ≤ a n n−1 s b s xp s c n n−1 s f s xq s − ρ g s xr s , n ∈ N0 , E1 Journal of Inequalities and Applications with the initial condition x n ϕn , n ∈ {−ρ, , −1, 0}, 1/p ϕ n−ρ ≤ a n I1 for n ∈ N0 with n − ρ ≤ 0, where p, q, r, and ρ are constants, p / 0, p ≥ q ≥ 0, p ≥ r ≥ 0, ρ ∈ N0 , ϕ n ∈ R , n ∈ {−ρ, , −1, 0}, then n−1 x n ≤ a n b s c n H n s 1/p n−1 , J s 2.28 s for any k > 0, n ∈ N0 , H n n−1 s−1 t f s q/p b t pk s s−1 t g s k p−q r/p b t k p−r pk J n c n n−1 b s p−q /p pk qf n qa s p−q /p p−r /p s q/p s rg n , n−1 b s p−r /p pk 2.29 s r/p , n ∈ N0 Next, using the Chain Rule, we consider a special case of the delay integral inequality E of the form xp t ≤ C t t0 t b s xp s Δs t0 f s xp−1 τ s Δs, t ∈ T0 , E with the initial condition x t ϕ τ t ϕt , t ∈ α, t0 ∩ T, I for t ∈ T0 with τ t ≤ t0 , ≤ C1/p where C and p ≥ are positive constants, τ t , α, and ϕ t are defined as in I Theorem 2.8 Assume that x t , b t , f t ∈ Crd T0 , R Then the inequality E with the initial condition I implies x t ≤ C1/p eb/p t, t0 p t t0 eb/p t, σ s f s Δs, t ∈ T0 2.30 Wei Nian Li Proof Define a function w t by wp t C t t0 t b s xp s Δs t0 f s xp−1 τ s Δs, t ∈ T0 2.31 Using a similar way in the proof of Theorem 2.4, we easily obtain that w t is a positive and nondecreasing function, and x t ≤w t , x τ t t ∈ T0 , ≤w t , 2.32 t ∈ T0 2.33 Differentiating 2.31 , we obtain pwp−1 θ wΔ t b t xp t f t xp−1 τ t , t ∈ T0 , 2.34 where θ ∈ t, σ t It follows from 2.32 – 2.34 that pwp−1 θ wΔ t ≤ b t wp t f t wp−1 t , t ∈ T0 2.35 Noting the fact that < w t ≤ w θ and wΔ t ≥ 0, from the above inequality, we have pwp−1 t wΔ t ≤ b t wp t f t wp−1 t , t ∈ T0 2.36 Therefore, wΔ t ≤ f t , p b t w t p t ∈ T0 2.37 By Lemma 2.3, from 2.37 , we have w t ≤ C1/p eb/p t, t0 t t0 eb/p t, σ s f s Δs, p t ∈ T0 2.38 Therefore, the desired inequality 2.30 follows from 2.32 and 2.38 This completes the proof of Theorem 2.8 Corollary 2.9 Assume that x t , b t , f t ∈ C R , R If x t satisfies the following delay integral inequality: xp t ≤ C t b s xp s ds t f s xp−1 ρ s ds, t∈R , E2 Journal of Inequalities and Applications with the initial condition x t φ ρ t φ t , ≤ C1/p t ∈ β, , I2 for t ∈ R with ρ t ≤ 0, where C and p ≥ are positive constants, ρ t ∈ C R , R , ρ t ≤ t, − ∞ < β 0, and φ t ∈ C β, , R , then t b s ds p x t ≤ C1/p exp p t f s exp t b τ dτ ds, s p inf{ρ t , t ∈ R } ≤ t∈R 2.39 Corollary 2.10 Assume that x n , b n , f n are nonnegative functions defined for n ∈ N0 If x n satisfies the following delay discrete inequality: xp n ≤ C n−1 b s xp s s n−1 f s xp−1 s − ρ , n ∈ N0 , E3 s with the initial condition x n ϕn , ϕ n − ρ ≤ C1/p n ∈ {−ρ, , −1, 0}, I3 for n ∈ N0 with n − ρ ≤ 0, where C and p ≥ are positive constants, ρ and ϕ n are defined as in I1 , then x n ≤ C1/p n−1 s b s p n−1 f s ps n−1 i s b i p , n ∈ N0 2.40 Finally, we study the delay integral inequality on time scales of the form xp t ≤ a t c t t t0 f s xq s L s, x τ s Δs, t ∈ T0 , E with the initial condition I , where p ≥ 1, ≤ q ≤ p are constants, τ t is as defined in the inequality E , and L : T0 × R → R is a continuous function Theorem 2.11 Assume that x t , a t , c t , f t ∈ Crd T0 , R If a t and c t are nondecreasing for t ∈ T0 , and ≤ L t, x − L t, y ≤ K t, y x − y , 2.41 Wei Nian Li for x ≥ y ≥ 0, where K : T0 × R → R is a continuous function, then the inequality E initial condition I implies x t ≤ a t c t t H t t0 with the 1/p eJ t, σ s H s J s Δs , 2.42 Δs, 2.43 for any k > 0, t ∈ T0 , where t H t f s k p−q pk t0 qa s qc t f t J t pk L s, p−q /p K t, p−q /p p−1 p p−1 p a t p a s p c t p 2.44 Proof Define a function z t by zt t t0 f s xq s L s, x τ s Δs, t ∈ T0 2.45 We easily observe that z t is a nonnegative and nondecreasing function, and E restated as x t ≤ a t c t zt 1/p , t ∈ T0 can be 2.46 Using Lemma 2.1, from 2.46 , we have x t ≤ a t c t zt 1/p ≤ p−1 p a t p c t zt , p t ∈ T0 2.47 Therefore, for t ∈ T0 with τ t ≥ t0 , we obtain x τ t ≤ p−1 p a τ t p c τ t z τ t p ≤ p−1 p a t p c t zt , p 2.48 and for t ∈ T0 with τ t ≤ t0 , using the initial condition I and 2.47 , we get x τ t ϕ τ t ≤ a t 1/p ≤ p−1 p a t p c t zt p 2.49 10 Journal of Inequalities and Applications It follows from 2.48 and 2.49 that x τ t ≤ p−1 p a t p c t zt , p t ∈ T0 2.50 Combining 2.45 , 2.46 , and 2.50 , by Lemma 2.1, for any k > 0, we obtain zt ≤ ≤ t t0 t t0 f s a s f s t t0 ≤ t k p−q qa s p−1 p pk t0 t t0 H t pk t t0 p−q /p qa s K s, J s z s Δs, L s, p−q /p c s zs p L s, p−q /p qc s f s t0 pk a s p f s k p−q t Δs qc s z s p−q /p pk L s, q/p c s zs p−1 p p−1 p c s zs p Δs Δs − L s, p−1 p a s p a s p c s p p−1 p a s p a s p L s, p−1 p a s p Δs Δs z s Δs t ∈ T0 , 2.51 where H t and J t are defined by 2.43 and 2.44 , respectively By Lemma 2.2, from 2.51 , we have z t ≤H t t t0 eJ t, σ s H s J s Δs, t ∈ T0 Therefore, the desired inequality 2.42 follows from 2.46 and Theorem 2.11 is complete 2.52 2.52 The proof of Noting H t , defined by 2.43 , is nondecreasing for t ∈ T0 , we easily obtain the following result Theorem 2.12 Suppose that all assumptions of Theorem 2.11 hold Then the inequality E the initial condition I implies x t ≤ a t c t H t eJ t, t0 1/p , with 2.53 for any k > 0, t ∈ T0 , where H t and J t are defined by 2.46 and 2.47 , respectively Remark 2.13 If T R, then Theorem 2.12 reduces to 9, Theorem 2.8 Letting T Theorem 2.12, we can obtain the following corollary Z, from Wei Nian Li 11 Corollary 2.14 Assume that x n , a n , c n , f n are nonnegative functions defined for n ∈ N0 If a n and c n are nondecreasing in N0 , and x n satisfies the following delay discrete inequality: xp n ≤ a n c n n−1 f s xq s L s, x s − ρ , n ∈ N0 , E4 s where p, q, and ρ are constants, p ≥ 1, p ≥ q ≥ 0, ρ ∈ N0 , and L, K : N0 × R → R satisfying ≤ L n, x − L n, y ≤ K n, y x − y , 2.54 for x ≥ y ≥ 0, then the inequality E4 with the initial condition I1 implies x n ≤ a n c n H n 1/p n−1 , J s 2.55 s for any k > 0, n ∈ N0 , where n−1 H n f s k p−q pk s J n qa s p−q /p L s, p−1 p a s p , 2.56 qc n f n pk p−q /p p−1 K n, p a n p c n p Some applications In this section, we present some applications of our results Example 3.1 Consider the delay dynamic equation on time scales: xp t Δ M t, x t , x τ t , t ∈ T0 , 3.1 with the initial condition x t ψ τ t ψ t , C1/p t ∈ α, t0 ∩ T, for t ∈ T0 with τ t ≤ t0 , I where M : T0 × R2 → R is a continuous function, C xp t0 and p > are constants, α and τ t are as defined in the initial condition I , and ψ t ∈ Crd α, t0 ∩ T, R Theorem 3.2 Assume that M t, x t , x τ t ≤ f t xq τ t g t xr t , 3.2 12 Journal of Inequalities and Applications where f t , g t ∈ Crd T0 , R , q and r are constants, p ≥ q ≥ 0, p ≥ r ≥ If x t is a solution of 3.1 satisfying the initial condition I , then x t ≤ |C| t F t t0 1/p eG t, σ s F s G s Δs , 3.3 for any k > 0, t ∈ T0 , where F t t f s k p−q pk t0 q|C| g s k p−r p−q /p pk qf t G t pk p−q /p rg t pk p−r /p r|C| p−r /p Δs, 3.5 Proof Obviously, the solution x t of 3.1 with the initial condition I equivalent delay integral equation on time scales xp t t C t0 M s, x s , x τ s 3.4 Δs, t ∈ T0 , satisfies the 3.6 with the initial condition I Noting the assumption 3.2 , we have xp t ≤ |C| t t0 f s xq τ s g s xr s Δs, t ∈ T0 , 3.7 with the initial condition I Therefore, by Theorem 2.4, from 3.7 , we easily obtain the estimate 3.3 of solutions of 3.1 The proof of Theorem 3.2 is complete Using Theorem 2.5, we easily obtain the following result Theorem 3.3 Suppose that all assumptions of Theorem 3.2 hold If x t is a solution of 3.1 satisfying the initial condition I , then x t ≤ |C| F t e G t, t0 1/p , 3.8 for any k > 0, t ∈ T0 , where F t and G t are defined by 3.4 and 3.5 , respectively Remark 3.4 The right-hand sides of 3.3 and 3.8 give us the bounds on the solution x t of 3.1 satisfying the initial condition I in terms of the known functions for any k > 0, t ∈ T0 , respectively Example 3.5 Consider the delay discrete inequality as in E3 satisfying the initial condition 10−4 n, I3 with p 2, C 1/4, ρ 2, ϕ n 1/2, n ∈ {−2, −1, 0}, b n 10−3 n2 , f n n ∈ N0 , and we compute the values of x n from E3 and also we find the values of x n by using the result 2.40 In our computations, we use E3 and 2.40 as equations and as we see in Table the computation values as in E3 are less than the values of the result 2.40 Wei Nian Li 13 Table n 11 14 17 22 25 27 30 35 40 E3 5.000000000000000e–001 5.013992421214853e–001 5.240341550720497e–001 6.053588145272404e–001 7.428258989476674e–001 1.009578705314619e 000 2.189862704124656e 000 4.143517993238956e 000 6.839504919933415e 000 1.630102753510524e 001 9.500824036460114e 001 8.204195033362939e 002 2.43 5.000000000000000e–001 5.014006000000000e–001 5.242013057437409e–001 6.073138820474305e–001 7.507097542821271e–001 1.036912536372208e 000 2.391160696569409e 000 4.841349883598182e 000 8.504988064333858e 000 2.295320353713791e 001 1.816014350966817e 002 2.464464322608679e 003 From Table 1, we easily find that the numerical solution agrees with the analytical solution for some discrete inequalities The program is written in the programming Matlab 7.0 Acknowledgments The author thanks the referee very much for his careful comments and valuable suggestions on this paper This work is supported by the Natural Science Foundation of Shandong Province Y2007A08 , the National Natural Science Foundation of China 60674026, 10671127 , the Project of Science and Technology of the Education Department of Shandong Province J08LI52 , and the Doctoral Foundation of Binzhou University 2006Y01 References S Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol 18, no 1-2, pp 18–56, 1990 R Agarwal, M Bohner, and A Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities & Applications, vol 4, no 4, pp 535–557, 2001 E Akin–Bohner, M Bohner, and F Akin, “Pachpatte inequalities on time scales,” Journal of Inequalities in Pure and Applied Mathematics, vol 6, no 1, article 6, pp 1–23, 2005 W N Li, “Some new dynamic inequalities on time scales,” Journal of Mathematical Analysis and Applications, vol 319, no 2, pp 802–814, 2006 F.-H Wong, C.-C Yeh, and C.-H Hong, “Gronwall inequalities on time scales,” Mathematical Inequalities & Applications, vol 9, no 1, pp 75–86, 2006 W N Li and W Sheng, “Some nonlinear dynamic inequalities on time scales,” Proceedings of the Indian Academy of Sciences Mathematical Sciences, vol 117, no 4, pp 545–554, 2007 W N Li and W Sheng, “Some nonlinear integral inequalities on time scales,” Journal of Inequalities and Applications, vol 2007, Article ID 70465, 15 pages, 2007 D R Anderson, “Dynamic double integral inequalities in two independent variables on time scales,” Journal of Mathematical Inequalities, vol 2, no 2, pp 163–184, 2008 F Jiang and F Meng, “Explicit bounds on some new nonlinear integral inequalities with delay,” Journal of Computational and Applied Mathematics, vol 205, no 1, pp 479–486, 2007 10 M Bohner and A Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhă user, Boston, Mass, USA, 2001 a 11 M Bohner and A Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhă user, Boston, a Mass, USA, 2003  ... Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhă user, Boston, Mass, USA, 2001 a 11 M Bohner and A Peterson, Eds., Advances in Dynamic Equations on Time Scales,... F.-H Wong, C.-C Yeh, and C.-H Hong, “Gronwall inequalities on time scales,” Mathematical Inequalities & Applications, vol 9, no 1, pp 75–86, 2006 W N Li and W Sheng, “Some nonlinear dynamic inequalities. .. initial condition I equivalent delay integral equation on time scales xp t t C t0 M s, x s , x τ s 3.4 Δs, t ∈ T0 , satisfies the 3.6 with the initial condition I Noting the assumption 3.2 ,