Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 464976, 13 pages doi:10.1155/2010/464976 Research Article On Some Integral Inequalities on Time Scales and Their Applications Run Xu,1 Fanwei Meng,1 and Cuihua Song2 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165 Shandong, China School of Chemistry and Chemical Engineering, Qufu Normal University, Qufu, 273165 Shandong, China Correspondence should be addressed to Run Xu, xurun 2005@163.com Received 15 January 2010; Revised March 2010; Accepted 18 March 2010 Academic Editor: Martin Bohner Copyright q 2010 Run Xu 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 The purpose of this paper is to investigate some new dynamic inequalities on time scales We establish some new dynamic inequalities; the results unify and extend some continuous inequalities and their corresponding discrete analogues The inequalities given here can be used as tools in the qualitative theory of certain dynamic equations Some examples are given in the end of this paper Introduction The theory of time scales was introduced by Hilger in 1988 in order to contain both difference and differential calculus in a consistent way Recently, many authors have extended some fundamental integral inequalities used in the theory of differential and integral equations on time scales For example, we refer the reader to the papers 2–12 and the references cited there in In this paper, we investigate some nonlinear integral inequalities on time scales, which extend some inequalities established by Li and Sheng and Li The obtained inequalities can be used as important tools in the study of dynamic equations on time scales Throughout this paper, let us assume that we have already acquired the knowledge of time scales and time scales notation; for an excellent introduction to the calculus on time scales, we refer the reader to Bohner and Peterson for general overview Some Preliminaries on Time Scales In what follows, R denotes the set of real numbers, Z denotes the set of integers, N0 denotes the set of nonnegative integers, C denotes the set of complex numbers, and C M, S denotes Journal of Inequalities and Applications the class of all continuous functions defined on set M with range in the set S T is an arbitrary T − {m}; otherwise, time scale If T has a right-scattered maximum m, then the set Tk Tk T Crd denotes the set of rd-continuous functions; R denotes the set of all regressive and {p ∈ rd-continuous functions We define the set of all positively regressive functions by R R : μ t p t > 0, t ∈ T} Obviously, if p ∈ Crd and p t ≥ for t ∈ T, then p ∈ R For f : T → R and t ∈ Tk , we define f Δ t as follows provided it exists : fσ t − f s ; s→t σ t −s f Δ t : lim 2.1 we call f Δ t the delta derivative of f at t The following lemmas are very useful in our main results Lemma 2.1 see If p ∈ R and fix t0 ∈ T, then the exponential function ep ·, t0 is for the unique solution of the initial value problem xΔ p t x, x t0 on T 2.2 Lemma 2.2 see Let t0 ∈ Tk and w : T × Tk → R be continuous at t, t , where t ≥ t0 Assume that wΔ t, · is rd-continuous on t0 , σ t If for any ε > 0, there exists a neighborhood U of t, independent of τ ∈ t0 , σ t , such that w σ t , τ − w s, τ − wΔ t, τ σ t − s ≤ ε|σ t − s|, s ∈ U, 2.3 where wΔ denotes the derivative of w with respect to the first variable, then t g t : w t, τ Δτ 2.4 t0 implies gΔ t : t wΔ t, τ Δτ w σ t ,t 2.5 t0 The following theorem is a foundational result in dynamic inequalities Lemma 2.3 Comparison Theorem Suppose u, b ∈ Crd , a ∈ R ; then uΔ t ≤ a t u t b t , t ≥ t0 , t ∈ Tk , 2.6 implies u t ≤ u t0 ea t, t0 t t0 b τ ea t, σ τ Δτ, t ≥ t0 , t ∈ Tk 2.7 Journal of Inequalities and Applications The following lemma is useful in our main results Lemma 2.4 see Let a ≥ 0, p ≥ q > 0, then aq/p ≤ q K p q−p /p a p − q q/p K , p K > 2.8 Main Results In this section, we study some integral inequalities on time scales We always assume that p, q, r, m are constants, p ≥ q > 0, p ≥ m > 0, p ≥ r > 0, and t ≥ t0 , t ∈ Tk Theorem 3.1 Assume that u, a, b, f, g, h ∈ Crd ; u t , a t , b t , f t , g t , and h t are nonnegative; then up t ≤ a t t f s uq s b t s g s ur s t0 h τ um τ Δτ Δs, t ∈ Tk , 3.1 t0 implies u t ≤ 1/p t a t B τ eA τ t, σ τ Δτ b t , K > 0, t ∈ Tk , 3.2 t0 where A t B t q K p q−p /p q K p f t t t0 q−p /p m K p r K p f t a t m−p /p a τ r−p /p m K p g t b t p − q q/p K p g t m−p /p t b τ h τ Δτ, t0 r K p r−p /p p − m m/p K h τ Δτ, p at p − r p/r K p 3.3 t ∈ Tk Proof Define z t by t zt f s uq s g s ur s t0 then z t0 s h τ um τ Δτ Δs, 3.4 t0 0, and 3.1 can be restated as up t ≤ a t b tzt 3.5 Journal of Inequalities and Applications Using Lemma 2.1, for any k > 0, we obtain uq t ≤ a t b tz t q/p ≤ q K p ur t ≤ a t b tzt r/p ≤ r K p um t ≤ a t m/p b t zt m K p ≤ q−p /p a t b t zt p − q q/p K , p r−p /p a t b t zt p − r r/p K , p a t b t zt p − m m/p K p m−p /p 3.6 It follows from 3.4 and 3.6 that zΔ t f t uq t g t ur t t h τ um τ Δτ t0 q K p ≤f t g t q−p /p r K p t t0 ≤B t r−p /p m K p h τ at p − q q/p K p b t zt a t m−p /p b t zt a τ p − r r/p K p 3.7 p − m m/p K Δτ p b τ zτ t ∈ Tk , At zt , where A t , and B t are defined as in 3.3 and A t is regressive obviously From Lemma 2.3 and 3.7 , noting z t0 0, we obtain zt ≤ t B τ eA τ t, σ τ Δτ 3.8 t0 Therefore, the desired inequality 3.2 follows from 3.5 and 3.8 Remark 3.2 Theorem 3.1 extends some known inequalities on time scales If q 1, r 0, h t 0, then Theorem 3.1 reduces to 7, Theorem 3.1 If q p, h t 0, then Theorem 3.1 reduces to 8, Theorem 3.2 Remark 3.3 The result of Theorem 3.1 holds for an arbitrary time scale If T R, then Theorem 3.1 becomes the Theorem established by Yuan et al 13 If T Z, we can have the following Corollary Corollary 3.4 Let T Z and assume that u t , a t , b t , f t , g t , and h t are nonnegative functions defined for t ∈ N0 Then the inequality up t ≤ a t t−1 b t s f s uq s g s ur s s−1 τ h τ um τ , t ∈ N0 , 3.9 Journal of Inequalities and Applications implies ut ≤ a t 1/p t−1 t−1 B s b t s A τ K > 0, t ∈ N0 , , 3.10 τ s where At B t q K p f t q−p /p q K p t−1 q−p /p m K p τ r K p f t m K p g t b t p − q q/p K p a t m−p /p r−p /p p − m m/p K h τ , p a τ b τ h τ , τ r K p g t t−1 m−p /p r−p /p p − r p/r K p a t 3.11 t ∈ N0 Corollary 3.5 Let T lZ ∩ 0, ∞ , where lZ {lk : k ∈ Z, l > 0} We assume that u t , a t , b t , f t , g t ,and h t are nonnegative functions defined for t ∈ T Then the inequality t/l−1 up t ≤ a t b t f ls uq ls g ls ur ls s/l−1 h lτ um lτ s , t∈T 3.12 τ implies u t ≤ a t b t 1/p t/l−1 t/l−1 B ls s A τ K > 0, t ∈ T, , 3.13 τ s/l where At B t q K p f t q−p /p q K p t/l−1 τ f t q−p /p m K p r K p a t m−p /p r−p /p g t b t p − q q/p K p a lτ g t m K p m−p /p r K p p − m m/p K h lτ , p t/l−1 b lτ h lτ , τ r−p /p a t t ∈ T p − r p/r K p 3.14 Journal of Inequalities and Applications Theorem 3.6 Assume that u, a, b, f, h are defined as in Theorem 3.1, L t, y , M t, y : Tk × R → R are continuous functions, and L t, y is nondecreasing about the second variable and satisfies ≤ L t, x − L t, y ≤ M t, y x−y 3.15 for t ∈ Tk and x ≥ y ≥ 0; then up t ≤ a t t f s uq s b t s L s, ur s t0 h τ um τ Δτ Δs, t ∈ Tk 3.16 t0 implies u t ≤ 1/p t a t B1 τ eA1 t, σ τ Δτ b t , K > 0, t ∈ Tk , 3.17 t0 where A1 t q K p q−p /p r K p B1 t r−p /p q K p f t f tb t r M t, K p q−p /p t h τ t0 r L t, K p m K p a t m K p r−p /p h τ b τ Δτ t0 p − r r/p K b t , p at p − q q/p K p m−p /p r−p /p t m−p /p aτ a t 3.18 p − m m/p K Δτ p p − r r/p K , p t ∈ Tk Proof Define z t by t zt f s uq s L s, ur s t0 then z t0 0, and 3.16 can be written as 3.5 s t0 h τ um τ Δτ Δs, 3.19 Journal of Inequalities and Applications Therefore, from 3.6 and 3.19 , we have zΔ t f t uq t t L t, ur t h τ um τ Δτ t0 q K p ≤f t q−p /p r L t, K p r − L t, K p t t0 r L t, K p q K p q K p r−p /p r−p /p m K p h τ ≤f t at q−p /p q−p /p r M t, K p r L t, K p at b t zt a t f t b t r−p /p r−p /p a τ at a t p − m m/p K Δτ p b τ zτ p − r r/p K p p − q q/p K p a t p − r r/p K p p − r r/p K p a t m−p /p r−p /p p − q q/p K p b t zt m K p m−p /p t h τ t0 t m K p m−p /p a τ p − m m/p K Δτ p h τ b τ Δτ z t t0 p − r r/p r K K p p p − r r/p K p r−p /p b t zt A1 t z t B1 t , t ∈ Tk , 3.20 where A1 t , and B1 t are defined as in 3.18 and A1 t is regressive obviously 0, we obtain From Lemma 2.3 and 3.20 , noting z t0 z t ≤ t B τ e A1 τ t, σ τ Δτ t0 Therefore, the desired inequality 3.17 follows from 3.5 and 3.21 3.21 Journal of Inequalities and Applications Remark 3.7 If T R, then Theorem 3.6 becomes 13, Theorem If T following Corollary Z, we can have the Corollary 3.8 Let T Z and assume that u t , a t , b t , f t , g t , and h t are nonnegative functions defined for t ∈ N0 L, M ∈ C R2 , R satisfy ≤ L t, x − L t, y ≤ M t, y x−y 3.22 for x ≥ y ≥ and L t, y is nondecreasing about the second variable Then the inequality t−1 up t ≤ a t f s uq s b t L s, ur s s s−1 t ∈ N0 3.23 K > 0, t ∈ N0 , 3.24 h τ um τ , τ implies u t ≤ a t b t 1/p t−1 t−1 B1 τ s A1 τ , τ s where A1 t q K p q−p /p r K p B1 t f t f t b t r−p /p q K p r M t, K p q−p /p r L t, K p m K p a t r−p /p a t m−p /p r−p /p t−1 h τ b τ τ a t p − r r/p K b t, p q − p q/p K p t−1 h τ τ p − r r/p K , p m K p m−p /p a τ p − m m/p K p t ∈ N0 3.25 Theorem 3.9 Assume that u t , a t , b t , f t , g t , and h t are defined as in Theorem 3.1, w t, s is defined as in Lemma 2.2 such that w σ t , t ≥ 0, wΔ t, s ≥ for t, s ∈ T with s ≤ t; then up t ≤ a t t b t t0 w t, s f s uq s g s ur s s h τ um τ Δτ Δs, t ∈ Tk , t0 3.26 Journal of Inequalities and Applications implies u t ≤ 1/p t a t B2 τ eA2 t, σ τ Δτ b t , K > 0, t ∈ Tk , 3.27 t0 where A2 t w σ t ,t t q K p q−p /p q K p wΔ t, s t0 m K p B2 t q K p w σ t ,t f t t q−p /p m−p /p t0 t0 g s q−p /p r−p /p m K p b t g s m−p /p t b τ h τ Δτ t0 b s b τ h τ Δτ Δs, h τ p − q q/p K p a t m−p /p q−p /p r K p s t0 s g t t0 q K p wΔ t, s f s r−p /p r K p f s m K p h τ t r K p f t m K p r K p r−p /p p − r p/r K p a t p − m m/p K Δτ p a τ p − q q/p K p a s r−p /p g t p − r p/r K p as m−p /p a τ p − m m/p K Δτ Δs, p t ∈ Tk 3.28 Proof Define z t by t zt w t, s f s uq s g s ur s t0 then z t0 0, and 3.26 can be written as 3.5 s t0 h τ um τ Δτ Δs, 3.29 10 Journal of Inequalities and Applications Therefore, from 3.6 and 3.29 we have zΔ t w σ t , t f t uq t t g t ur t h τ um τ Δτ t0 t wΔ t, s f s uq s s g s ur s t0 q K p ≤ w σ t ,t f t t t0 q K p wΔ t, s f s t0 s h τ t0 a t m K p p − r r/p K p b t zt a τ a s r−p /p p − q q/p K p b t zt m−p /p q−p /p r K p g s a t r−p /p m K p h τ ≤ B2 t q−p /p r K p g t t h τ um τ Δτ Δs t0 b sz s as m−p /p p − m m/p K p b τ zτ 3.30 p − q q/p K p b s zs a τ Δτ p − r r/p K p p − m m/p K Δτ Δs p b τ zτ t ∈ Tk , A2 t z t , where A2 t , and B2 t are defined by 3.28 and A2 t is regressive obviously 0, we obtain From Lemma 2.3 and 3.30 , noting z t0 z t ≤ t B τ e A2 τ t, σ τ Δτ 3.31 t0 Therefore, the desired inequality 3.27 follows from 3.5 and 3.31 Remark 3.10 If q p, h t 0, then Theorem 3.9 reduces to 8, Theorem 3.8 Using our results, we can also obtain many dynamic inequalities for some peculiar time scales; here, we omit them Some Applications In this section, we present some applications of Theorem 3.9 to investigate certain properties of solution u t of the following dynamic equation: up t Δ t F t, U t, u t , t0 H s, u s Δs , up t0 C, t ∈ Tk , 4.1 Journal of Inequalities and Applications 11 where C is a constant, F : Tk × R × R → R is a continuous function, and U : Tk × R → R, H : Tk × R → R are also continuous functions Example 4.1 Assume that |F t, U, V | ≤ |U| |U t, u | ≤ f t |u|q |V |, g t |u|r , |H t, u | ≤ h t |u|m , 4.2 t ∈ Tk , where p, q, r, and m are constants, p ≥ q > 0, and p ≥ m > 0, p ≥ r > f, g, h ∈ Crd , f t , g t and h t are nonnegative Then every solution u t of 4.1 satisfies |u t | ≤ t |C| 1/p B τ eA τ t, σ τ Δτ K > 0, t ∈ Tk , , 4.3 t0 where A, B are defined as in 3.3 with a t |C|, b t Indeed, the solution u t of 4.1 satisfies the following equivalent equation up t t τ H s, u s Δs Δτ, F τ, U τ, u τ , C t0 t ∈ Tk 4.4 t0 It follows from 4.2 and 4.4 that |up t | ≤ |C| t τ F τ, U τ, u τ , t0 ≤ |C| t H s, u s Δs Δτ t0 f τ |u τ | q τ r g τ |u τ | t0 4.5 m h s |u s | Δs Δτ t0 Using Theorem 3.1, the inequality 4.3 is obtained from 4.5 Example 4.2 Assume that |F t, U1 , V1 − F t, U2 , V2 | ≤ |U1 − U2 | p |V1 − V2 |, p |U t, u1 − U t, u2 | ≤ f t u1 − u2 , p p |H t, u1 − H t, u2 | ≤ h t u1 − u2 , 4.6 t ∈ Tk , p, f, and h are defined as in Example 4.1 If p m/n m, n ∈ N and m is odd, then 4.1 has at most one solution; otherwise, the two solutions u1 t , and u2 t of 4.1 have the relation p p u2 t u1 t 12 Journal of Inequalities and Applications Proof Let u1 t , and u2 t be two solutions of 4.1 Then we have p t p u1 t − u2 t τ F τ, U τ, u1 τ , t0 H s, u1 s Δs t0 τ −F τ, U τ, u2 τ , 4.7 H s, u2 s Δs Δτ, t ∈ Tk t0 It follows from 4.6 and 4.7 that p p u1 t − u2 t t ≤ p p f τ u1 τ − u τ t0 p τ t0 p p h s u1 s − u2 s Δs Δτ, t ∈ Tk 4.8 p By Theorem 3.1, we have u1 t − u2 t ≡ 0, t ∈ Tk The results are obtained Example 4.3 Consider the equation up t a t b t s F t, s, U s, u , t0 H τ, u Δτ Δs, t ∈ Tk 4.9 t0 If |F t, s, U, V | ≤ w t, s |U| |U t, u | ≤ f t |u|q |V | , g t |u|r , |H t, u | ≤ h t |u|m , 4.10 t ∈ Tk , where p, q, r, m are constants, p ≥ q > 0, p ≥ m > 0, p ≥ r > a, b, f, g, h ∈ Crd , a t , b t , f t , g t and h t are nonnegative, w t, s is defined as in Lemma 2.2 such that w σ t , t ≥ 0, wΔ t, s ≥ for t, s ∈ T with s ≤ t Then we have the estimate of the solution u t of 4.9 that |u t | ≤ t a t b t 1/p B2 τ eA2 t, σ τ Δτ , K > 0, t ∈ Tk , 4.11 t0 where A2 , B2 are defined as in 3.28 Proof From 4.10 and 4.9 , we have |u t |p ≤ a t t b t w t, s f s |u s |q g s |u s |r t0 s h τ |u τ |m Δτ Δs, t ∈ Tk t0 4.12 By Theorem 3.9 and 4.12 , we have that 4.11 holds Journal of Inequalities and Applications 13 Acknowledgments The authors thank the referees very much for their careful comments and valuable suggestions on this paper This research is supported by the National Natural Science Foundation of China 10771118 , the Natural Science Foundation of Shandong Province ZR2009AM011 , and the Science Foundation of the Education Department of Shandong Province, China J07yh05 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 Applied Mathematics, vol 6, no 1, article 6, 23 pages, 2005 M Bohner and A Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, An introduction with application, Birkhă auser, Boston, Mass, USA, 2001 a M Bohner and A Peterson, Advances in Dynamic Equations on Time Scales, Birkhă user, Boston, Mass, a USA, 2003 D B Pachpatte, “Explicit estimates on integral inequalities with time scale,” Journal of Inequalities in Pure and Applied Mathematics, vol 7, no 4, article 143, p 8, 2006 W N Li, “Some new dynamic inequalities on time scales,” Journal of Mathematical Analysis and Applications, vol 319, no 2, pp 802–814, 2006 W N Li and W Sheng, “Some nonlinear integral inequalities on time scales,” Journal of Inequalities and Applications, Article ID 70465, 15 pages, 2007 W N Li, “Some Pachpatte type inequalities on time scales,” Computers & Mathematics with Applications, vol 57, no 2, pp 275–282, 2009 10 W N Li, “Some delay integral inequalities on time scales,” Computers and Mathematics with Applications, vol 59, no 6, pp 1929–1936, 2010 11 W N Li, “Bounds for certain new integral inequalities on time scales,” Advances in Difference Equations, vol 2009, Article ID 484185, 16 pages, 2009 12 D R Anderson, “Nonlinear dynamic integral inequalities in two independent variables on time scale Pairs,” Advances in Dynamical Systems and Applications, vol 3, no 1, pp 1–13, 2008 13 Z Yuan, X Yuan, F Meng, and H Zhang, “Some new delay integral inequalities and their applications,” Applied Mathematics and Computation, vol 208, no 1, pp 231–237, 2009  ... Mathematical Analysis and Applications, vol 319, no 2, pp 802–814, 2006 W N Li and W Sheng, ? ?Some nonlinear integral inequalities on time scales, ” Journal of Inequalities and Applications, Article ID 70465,... estimates on integral inequalities with time scale,” Journal of Inequalities in Pure and Applied Mathematics, vol 7, no 4, article 143, p 8, 2006 W N Li, ? ?Some new dynamic inequalities on time scales, ”... N Li, ? ?Some Pachpatte type inequalities on time scales, ” Computers & Mathematics with Applications, vol 57, no 2, pp 275–282, 2009 10 W N Li, ? ?Some delay integral inequalities on time scales, ”