Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 484185, 16 pages doi:10.1155/2009/484185 Research Article Bounds for Certain New 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 March 2009; Accepted 10 June 2009 Recommended by Victoria Otero-Espinar Our aim in this paper is to investigate some new integral inequalities on time scales, which provide explicit bounds on unknown functions Our results unify and extend some integral inequalities and their corresponding discrete analogues The inequalities given here can be used as handy tools to study the properties of certain dynamic equations on time scales Copyright q 2009 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 study of dynamic equations on time scales, which goes back to its founder Hilger , is an area of mathematics that has recently received a lot of attention For example, we refer the reader to literatures 2–8 and the references cited therein At the same time, some fundamental integral inequalities used in analysis on time scales have been extended by many authors 9–14 In this paper, we investigate some new nonlinear integral inequalities on time scales, which unify and extend some continuous inequalities and their corresponding discrete analogues Our results can be used as handy tools to study the properties of certain dynamic equations on time scales 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 2, Main Results In what follows, R denotes the set of real numbers, R 0, ∞ , Z denotes the set of integers, {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, Advances in Difference Equations Crd denotes the set of rd-continuous functions, R denotes the set of all regressive and rd{p ∈ R : μ t p t > 0, ∀t ∈ T} We use the usual conventions continuous functions, and R that empty sums and products are taken to be and 1, respectively Throughout this paper, we always assume that p ≥ 1, < q ≤ p, p, and q are real constants, and t ≥ t0 , t0 ∈ Tκ We firstly introduce the following lemmas, which are useful in our main results Lemma 2.1 15 Bernoulli’s inequality Let < α ≤ and x > −1 Then x α ≤1 αx Lemma 2.2 Let t0 ∈ Tκ and w : T × Tκ → R be continuous at t, t , where t ≥ t0 , t ∈ Tκ with 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, 2.1 where wΔ denotes the derivative of w with respect to the first variable, then t g t : w t, τ Δτ 2.2 t0 implies gΔ t t wΔ t, τ Δτ w σ t ,t 2.3 t0 Lemma 2.3 Comparison Theorem Suppose u, b ∈ Crd , a ∈ R Then uΔ t ≤ a t u t b t , t ≥ t0 , t ∈ Tκ , 2.4 implies u t ≤ u t0 ea t, t0 t ea t, σ τ b τ Δτ, t ≥ t0 , t ∈ Tκ 2.5 t0 Lemma 2.4 see 13 Let u, f, g ∈ Crd , u t , f t , and g t be nonnegative If f t is nondecreasing, then u t ≤f t t g τ u τ Δτ, t ∈ Tκ , 2.6 t0 implies u t ≤ f t eg t, t0 , t ∈ Tκ , 2.7 Advances in Difference Equations Next, we establish our main results Theorem 2.5 Assume that u, a, b, g, h ∈ Crd , a t > 0, u t , b t , g t and h t are nonnegative Then t up t ≤ a t b t h τ Δτ, g τ uq τ t ∈ Tκ , 2.8 t0 implies t 1/p−1 a t b t p u t ≤ a1/p t eB t, σ τ F τ Δτ, t ∈ Tκ , 2.9 t0 where F t g t aq/p t B t q q/p−1 a t b t g t , p h t , 2.10 t ∈ Tκ 2.11 Proof Define a function z t by t zt g τ uq τ h τ Δτ 2.12 t0 Then 2.8 can be restated as up t ≤ a t b t zt a t b t zt a t 2.13 Using Lemma 2.1, from the above inequality, we easily obtain u t ≤ a1/p t 1/p−1 a t b tzt , p 2.14 uq t ≤ aq/p t q q/p−1 a t b tzt p 2.15 It follows from 2.12 and 2.15 that zΔ t g t uq t ≤g t F t h t aq/p t B t zt, q q/p−1 a t b t zt p t ∈ Tκ , h t 2.16 Advances in Difference Equations where F t and B t are defined as in 2.10 and 2.11 , respectively Using Lemma 2.3 and 0, from 2.16 we have noting z t0 z t ≤ t eB t, σ τ F τ Δτ, t ∈ Tκ 2.17 t0 Therefore, the desired inequality 2.9 follows from 2.14 and 2.17 This completes the proof of Theorem 2.5 Corollary 2.6 Assume that u, g ∈ Crd , u t , and g t are nonnegative If α > is a constant, then t up t ≤ α g τ uq τ Δτ, t ∈ Tκ , 2.18 t0 implies u t ≤ α1/p − e t, t0 q B q , t ∈ Tκ , 2.19 where q q/p−1 α g t , p B t Proof Letting a t α, b t F t 1, and h t αq/p g t , t ∈ Tκ 2.20 in Theorem 2.5, we obtain B t q q/p−1 α g t : B t, p t ∈ Tκ Therefore, u t ≤ α1/p α1/p 1/p−1 α p t 1/p−1 α p t t0 t0 eB t, σ τ F τ Δτ eB t, σ τ αq/p g τ Δτ 2.21 Advances in Difference Equations α1/p α1/p 1/p α q t 1/p α q t t0 t0 q q/p−1 α g τ Δτ p eB t, σ τ eB t, σ τ B τ Δτ α1/p 1/p α eB t, t0 − eB t, t q α1/p 1/p α eB t, t0 − α1/p q q α1/p − e t, t0 q B q t ∈ Tκ , 2.22 The proof of Corollary 2.6 is complete Remark 2.7 The result of Theorem 2.5 holds for an arbitrary time scale Therefore, using Theorem 2.5, we can obtain many results for some peculiar time scales For example, letting T R and T Z, respectively, we have the following two results Corollary 2.8 Let T Then the inequality R and assume that u t , a t , b t , g t , h t ∈ C R , R , and a t > t up t ≤ a t b t g s uq s h s ds, t∈R , 2.23 implies u t ≤ a1/p t 1/p−1 a t b t p t t F θ exp B s ds dθ, t∈R , 2.24 θ where F t and B t are defined as in Theorem 2.5 Corollary 2.9 Let T Z and assume that a t > 0, u t , b t , g t , and h t are nonnegative functions defined for t ∈ N0 Then the inequality up t ≤ a t t−1 b t g s uq s h s , t ∈ N0 , 2.25 s implies u t ≤ a1/p t 1/p−1 a t b t p t−1 t−1 F θ θ where F t and B t are defined as in Theorem 2.5 s θ B s , t ∈ N0 , 2.26 Advances in Difference Equations Investigating the proof procedure of Theorem 2.5 carefully, we easily obtain the following more general result Theorem 2.10 Assume that u, a, b, gi , hi ∈ Crd , a t > 0, u t , b t , gi t , and hi t are nonnegative, i 1, 2, , n If there exists a series of positive real numbers q1 , q2 , , qn such that p ≥ qi > 0, i 1, 2, , n, then n t i up t ≤ a t t0 b t gi τ uqi τ hi τ Δτ, t ∈ Tκ , 2.27 implies t 1/p−1 a t b t p u t ≤ a1/p t eB∗ t, σ τ F ∗ τ Δτ, t ∈ Tκ , 2.28 t0 where F∗ t n gi t aqi /p t B∗ t hi t , n b t i i qi qi /p−1 a t gi t p 2.29 Theorem 2.11 Assume that u, a, b, f, g, m ∈ Crd , a t > 0, u t , b t , f t , g t and m t are nonnegative If w t, s is defined as in Lemma 2.2 such that w t, s ≥ and wΔ t, s ≥ for t, s ∈ T with s ≤ t, then up t ≤ a t t b t w t, τ f τ up τ m τ Δτ, g τ uq τ t ∈ Tκ , 2.30 t0 implies t 1/p−1 a t b t p u t ≤ a1/p t eA t, σ τ G τ Δτ, t ∈ Tκ , 2.31 t0 where A t w σ t ,t b t t w Δ f t t, τ b τ q q/p−1 a tg t p f τ t0 G t w σ t ,t a t f t t t0 q q/p−1 a τ g τ p g t aq/p t wΔ t, τ a τ f τ 2.32 Δτ, m t 2.33 g τ aq/p τ m τ Δτ Advances in Difference Equations Proof Define a function z t by t zt w t, τ f τ up τ m τ Δτ, g τ uq τ t ∈ Tκ 2.34 t0 As in the proof of Theorem 2.5, we easily obtain 2.14 and 2.15 Using Then z t0 Lemma 2.2 and combining 2.34 and 2.15 , we have zΔ t w σ t , t f t up t t g t uq t wΔ t, τ f τ up τ m t m τ Δτ g τ uq τ t0 ≤ w σ t ,t a t f t t g t aq/p t wΔ t, τ a τ f τ m t g τ aq/p τ b t q q/p−1 a t g t p f t zt mτ t0 b τ ≤ w σ t ,t b t t q q/p−1 a t g τ p f τ t0 w σ t ,t a t f t t zτ Δτ 2.35 q q/p−1 a t g t p f t wΔ t, τ b τ q q/p−1 a tg τ p f τ g t aq/p t wΔ t, τ a τ f τ Δτ z t mt g τ aq/p τ m τ Δτ t0 At zt Gt , t ∈ Tκ , where A t and G t are defined as in 2.32 and 2.33 , respectively Therefore, in the above inequality, using Lemma 2.3 and noting z t0 0, we get zt ≤ t eA t, σ τ G τ Δτ, t ∈ Tκ 2.36 t0 It is easy to see that the desired inequality 2.31 follows from 2.14 and 2.36 This completes the proof of Theorem 2.11 8 Advances in Difference Equations Corollary 2.12 Let T R and assume that u t , a t , b t , f t , g t , m t ∈ C R , R , a t > If w t, s and its partial derivative ∂w t, s /∂t are real–valued nonnegative continuous functions for t, s ∈ R with s ≤ t, then the inequality up t ≤ a t t b t w t, s f s up s g s uq s t∈R , m s ds, 2.37 implies t 1/p−1 a t b t p u t ≤ a1/p t t G s exp t∈R , A τ dτ ds, 2.38 s where At w t, t b t q q/p−1 a t g t p f t t ∂w t, s b s ∂t Gt w t, t a t f t q q/p−1 a s g s p f s q/p g t a t ∂w t, s a s f s ∂t t ds, 2.39 m t g s aq/p s m s ds Corollary 2.13 Let T Z and assume that a t > 0, u t , b t , f t , g t and m t are nonnegative functions defined for t ∈ N0 If w t, s and Δ1 w t, s are real-valued nonnegative functions for t, s ∈ N0 with s ≤ t, then the inequality up t ≤ a t t−1 b t w t, s f s up s g s uq s m s , t ∈ N0 , 2.40 s implies u t ≤ a1/p t 1/p−1 a t b t p t−1 t−1 G s s τ s A τ , t ∈ N0 , 2.41 Advances in Difference Equations where Δ1 w t, s 1, s − w t, s for t, s ∈ N0 with s ≤ t, w t A t w t t−1 1, t b t q q/p−1 a t g t p f t Δ1 w t, s b s q q/p−1 a sg s p f s s G t w t t−1 q/p 1, t a t f t g t a Δ1 w t, s a s f s t , 2.42 m t g s aq/p s m s s Corollary 2.14 Suppose that u t , a t , and w t, s are defined as in Theorem 2.11 If a t is nondecreasing for t ∈ Tκ , then up t ≤ a t t w t, τ uq τ Δτ, t ∈ Tκ , 2.43 e t, t0 , q A t ∈ Tκ , 2.44 t0 implies u t ≤ a1/p t − q where q p A t Proof Letting b t A t G t 1, f t q p t w σ t , t a1/p−1 t wΔ t, τ a1/p−1 τ Δτ 0, g t 1, and m t t w σ t , t a1/p−1 t in Theorem 2.11, we obtain wΔ t, τ a1/p−1 τ Δτ : A t, t0 w σ t , t aq/p t t wΔ t, τ aq/p τ Δτ t0 ≤a t 2.45 t0 w σ t , t a1/p−1 t 2.46 t t0 p a tA t , q t ∈ Tκ , wΔ t, τ a1/p−1 τ Δτ 10 Advances in Difference Equations where the inequality holds because a t is nondecreasing for t ∈ Tκ Therefore, using Theorem 2.11 and noting 2.46 , we easily have 1/p−1 a t p ≤ a1/p t t 1/p−1 a t p u t ≤ a1/p t t 1/p a t q ≤ a1/p t eA t, σ τ G τ Δ τ t0 e t0 t e t0 A t, σ τ A p a τ A τ Δτ q 2.47 t, σ τ A τ Δτ a1/p t 1 e t, t0 − e t, t A q A a1/p t − q e t, t0 , q A t ∈ Tκ The proof of Corollary 2.14 is complete By Theorem 2.11, we can establish the following more general result Theorem 2.15 Assume that u, a, b, f, gi , m ∈ Crd , a t > 0, u t , b t , f t , gi t , and m t are nonnegative, i 1, 2, , n, and there exists a series of positive real numbers q1 , q2 , , qn such that p ≥ qi > 0, i 1, 2, , n If w t, s is defined as in Lemma 2.2 such that w t, s ≥ and wΔ t, s ≥ for t, s ∈ T with s ≤ t, then up t ≤ a t t w t, τ b t n f τ up τ t0 gi τ uqi τ Δτ, mτ t ∈ Tκ , 2.48 i implies 1/p−1 a t b t p u t ≤ a1/p t t eA∗ t, σ τ G∗ τ Δ τ , t ∈ Tκ , 2.49 t0 where A∗ t n w σ t ,t b t f t i t wΔ t, τ b τ qi qi /p−1 a t gi t p n f τ t0 G∗ t i n w σ t ,t a t f t qi qi /p−1 a τ gi τ p gi t aqi /p t Δτ, 2.50 m t i t t0 wΔ t, τ n a τ f τ i gi τ aqi /p τ m τ Δτ Advances in Difference Equations 11 Theorem 2.16 Let u, a, r ∈ Crd , u t and r t be nonnegative, a t > 0, and a t be nondecreasing Assume that there exists a series of positive real numbers q1 , q2 , , qn such that p ≥ qi > 0, i 1, 2, , n If Si : Tκ × R → R is a continuous function such that ≤ Si t, xi − Si t, yi ≤ Hi t, yi for t ∈ Tκ and xi ≥ yi ≥ 0, i function, i 1, 2, , n, then xi − yi , 2.51 1, 2, , n, where Hi : Tκ × R → R is a nonnegative continuous t up t ≤ a t t0 n t i r τ up τ Δτ t0 Si τ, uqi τ Δτ, t ∈ Tκ , 2.52 implies u t ≤ R1/p t a1/p t 1/p−1 a t L t eJ t, t0 , p t ∈ Tκ , 2.53 where R t er t, t0 , n t i 2.54 t0 L t Si τ, Rqi /p τ aqi /p τ Δτ, 2.55 n qi Hi t, Rqi /p t aqi /p t Rqi /p t aqi /p−1 t p J t i 2.56 Si τ, uqi τ Δτ, 2.57 Proof Let n t i t0 v t z t a t v t , t ∈ Tκ Then 2.52 can be restated as up t ≤ z t t r τ up τ Δτ, t ∈ Tκ 2.58 t0 It is easy to see that z t ∈ Crd , z t > 0, and z t is nondecreasing Using Lemma 2.4, from 2.58 , we have up t ≤ R t z t , t ∈ Tκ , 2.59 where R t is defined as in 2.54 It follows from 2.57 and 2.59 that up t ≤ R t a t v t 2.60 12 Advances in Difference Equations Using Lemma 2.1 to the above inequality, we obtain u t ≤ R1/p t a t v t 2.61 1/p−1 a tv t , p ≤ R1/p t a1/p t uqi t ≤ Rqi /p t a t 1/p qi /p v t qi qi /p−1 a t v t , p ≤ Rqi /p t aqi /p t 2.62 t ∈ Tκ Noting the hypotheses on Si , from 2.62 , we get n t i t0 n t i t0 v t ≤ Si τ, Rqi /p τ Si τ, Rqi /p τ qi qi /p−1 a τ v τ p aqi /p τ aqi /p τ qi qi /p−1 a τ v τ p n t i n t i −Si τ, Rqi /p τ aqi /p τ ≤L t Δτ t0 Δτ t0 Hi τ, Rqi /p τ aqi /p τ Rqi /p τ Δτ Si τ, Rqi /p τ aqi /p τ qi qi /p−1 a τ v τ Δτ, p t ∈ Tκ , 2.63 where L t is defined by 2.55 Clearly, L t ≥ and L t are nondecreasing Therefore, for any ε > 0, from 2.63 , we obtain v t ≤1 L t ε n t i t0 Hi τ, Rqi /p τ aqi /p τ Rqi /p τ qi qi /p−1 v τ a Δτ, τ p Lτ ε t ∈ Tκ 2.64 Let ψ t v t , L t ε t ∈ Tκ , 2.65 and define k t by the right hand of 2.64 Then k t > 0, k t0 n kΔ t Hi t, Rqi /p t aqi /p t Rqi /p t qi qi /p−1 a t ψ t p Hi t, Rqi /p t aqi /p t Rqi /p t qi qi /p−1 a t k t p i ≤ n 1, ψ t ≤ k t , and i k t J t, t ∈ Tκ , 2.66 Advances in Difference Equations 13 where J t is defined by 2.56 Using Lemma 2.3 and noting k t0 k t ≤ eJ t, t0 , 1, from 2.66 , we have t ∈ Tκ 2.67 Therefore, v t ≤ Lt t ∈ Tκ ε eJ t, t0 , 2.68 It follows from 2.61 and 2.68 that u t ≤ R1/p t a1/p t 1/p−1 a t Lt p t ∈ Tκ ε eJ t, t0 , 2.69 Letting ε → in 2.69 , we immediately obtain the desired inequality 2.53 This completes the proof of Theorem 2.16 Corollary 2.17 Let T R, u, a, r ∈ C R , R , a t > 0, and a t be nondecreasing Assume that there exists a series of positive real numbers q1 , q2 , , qn such that p ≥ qi > 0, i 1, 2, , n If Si : R × R → R is a continuous function such that xi − yi , ≤ Si t, xi − Si t, yi ≤ Hi t, yi for t ∈ R and xi ≥ yi ≥ 0, i i 1, 2, , n, then 1, 2, , n, where Hi : R × R t up t ≤ a t n t i r τ up τ dτ 2.70 → R is a continuous function, Si τ, uqi τ dτ, t∈R , 2.71 implies 1/p u t ≤R t 1/p−1 a t L t exp p t a1/p t J s ds , t∈R , 2.72 where J t is defined as in 2.56 , t R t exp r s ds , n t L t Si τ, R i n J t i qi /p τ aqi /p τ 2.73 dτ, qi qi /p Hi t, R t aqi /p t p qi /p R t aqi /p−1 t 14 Advances in Difference Equations Corollary 2.18 Let T Z, a t > 0, a t be nondecreasing, u t and r t be nonnegative functions defined for t ∈ N0 Assume that there exists a series of positive real numbers q1 , q2 , , qn such that p ≥ qi > 0, i 1, 2, , n If Si : N0 × R → R such that ≤ Si t, xi − Si t, yi ≤ Hi t, yi for t ∈ N0 and xi ≥ yi ≥ 0, i xi − yi , 1, 2, , n, where Hi : N0 × R → R , i up t ≤ a t t−1 r τ up τ τ n t−1 Si τ, uqi τ , 2.74 1, 2, , n, then t ∈ N0 , 2.75 i τ implies u t ≤ R1/p t a1/p t 1/p−1 a t L t p t−1 J s , t ∈ N0 , 2.76 s where J t is defined as in 2.56 , t−1 Rt r s , s n t−1 Lt Si τ, Rqi /p τ aqi /p τ , 2.77 i τ n J t i qi Hi t, Rqi /p t aqi /p t Rqi /p t aqi /p−1 t p Remark 2.19 Using our main results, we can obtain many dynamic inequalities for some peculiar time scales Due to limited space, their statements are omitted here Some Applications In this section, we present two applications of our main results Example 3.1 Consider the inequality as in 2.25 with a t α t ,b t α t2 , g t −6 t, h t 0, p 2, q 1, α 10 , and we compute the values of u t from 2.25 and also we find the values of u t by using the result 2.26 In our computations we use 2.25 and 2.26 as equations and as we see in Table the computation values as in 2.25 are less than the values of the result 2.26 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 Advances in Difference Equations 15 Table t 10 12 14 17 20 25 2.25 1.414213562373095e − 003 2.661293464584210e − 003 5.486250637546570e − 002 2.670738191264154e − 001 1.527219045903506e 000 3.720520602864323e 000 7.856747926470754e 000 1.997586843703775e 001 4.331228422296512e 001 1.241251179017371e 002 2.26 1.414213562373095e − 003 2.910562109546456e − 003 1.103460932829943e − 001 5.410171853718061e − 001 3.137697944498020e 000 8.436559692675310e 000 2.187361362745254e 001 9.900992670086097e 001 5.854191578762491e 002 2.937887676184530e 004 Example 3.2 Consider the following initial value problem on time scales: up t Δ M t, u t , u t0 t ∈ Tκ , β, 3.1 where p ≥ and β / are constants, and M : Tκ × R → R is a continuous function Assume that |M t, u t | ≤ g t |uq t |, 3.2 where g t is defined as in Corollary 2.6, < q ≤ p is a constant If u t is a solution of IVP 3.1 , then 1− |u t | ≤ β q eV t, t0 q , t ∈ Tκ , 3.3 where q β p V t q−p g t , t ∈ Tκ 3.4 In fact, the solution u t of IVP 3.1 satisfies the following equation: up t t βp M τ, u τ Δτ, t ∈ Tκ 3.5 t0 Using the assumption 3.2 , from 3.5 , we have |u t |p ≤ β p t g τ |u τ |q Δτ, t ∈ Tκ t0 Now a suitable application of Corollary 2.6 to 3.6 yields 3.2 3.6 16 Advances in Difference Equations Acknowledgments This work is supported by the Natural Science Foundation of Shandong Province Y2007A08 , the National Natural Science Foundation of China 60674026, 10671127 , China Postdoctoral Science Foundation Funded Project 20080440633 , Shanghai Postdoctoral Scientific Program 09R21415200 , 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 M Bohner and A Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhă user, Boston, Mass, USA, 2001 a M Bohner and A Peterson, Advances in Dynamic Equations on Time Scales, Birkhă user, Boston, Mass, a USA, 2003 M Bohner, L Erbe, and A Peterson, “Oscillation 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