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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 430521, 22 pages doi:10.1155/2008/430521 ResearchArticleTheGeneralizedGronwallInequalityandItsApplicationtoPeriodicSolutionsofIntegrodifferentialImpulsivePeriodicSystemonBanach Space JinRong Wang, 1 X. Xiang, 1, 2 W. W ei, 2 and Qian Chen 2 1 College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China 2 College of Science, Guizhou University, Guiyang, Guizhou 550025, China Correspondence should be addressed to JinRong Wang, wjr9668@126.com Received 27 June 2008; Accepted 29 September 2008 Recommended by Ond ˇ rej Do ˇ sl ´ y This paper deals with a class of integrodifferential impulsiveperiodic systems onBanach space. Using impulsiveperiodic evolution operator given by us, the T 0 -periodic PC-mild solution is introduced and suitable Poincar ´ e operator is constructed. Showing the compactness of Poincar ´ e operator and using a new generalized Gronwall’s inequality with impulse, mixed type integral operators and B-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of T 0 -periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration. Copyright q 2008 JinRong Wang 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. 1. Introduction It is well known that impulsiveperiodic motion is a very important and special phenomenon not only in natural science, but also in social science such as climate, food supplement, insecticide population, and sustainable development. Periodicsystem with applications on finite-dimensional spaces has been extensively studied. Particularly, impulsiveperiodic systems on finite-dimensional spaces are considered and some important results such as the existence and stability ofperiodic solution, the relationship between bounded solution andperiodic solution, and robustness by perturbation are obtained see 1–4. Since the end of last century, many researchers pay great attention toimpulsive systems on infinite-dimensional spaces. Particulary, Ahmed et al. investigated optimal control problems ofsystem governed by impulsivesystem see 5–8. Many authors including us also gave a series of results for semilinear integrodifferential, strongly nonlinear impulsive systems and optimal control problems see 9–20. 2 Journal of Inequalities and Applications Although, there are some papers onperiodic solution for periodicsystemon infinite- dimensional spaces see 12, 21–23 and some results discussing integrodifferential systemon finite Banach space and infinite Banach space see 11, 13. To our knowledge, inte- grodifferential impulsiveperiodic systems on infinite-dimensional spaces with unbounded operator have not been extensively investigated. Recently, we discuss theimpulsiveperiodicsystemand integrodifferential impulsivesystemon infinite-dimensional spaces. Linear impulsive evolution operator is constructed and T 0 -periodic PC-mild solution is introduced. The existence ofperiodic solutions, alternative theorem criteria of Massera type, asymptotical stability, and robustness by perturbation is established see 24–26. For semilinear impulsiveperiodic system, a suitable Poincar ´ e operator is constructed which verifies its compactness and continuity. By virtue of a generalizedGronwallinequality with mixed integral operator and impulse given by us, the estimate ofthe PC-mild solutions is derived. Some fixed point theorems such as Banach fixed point theorem and Horn fixed point theorem are applied to obtain the existence ofperiodic PC-mild solutions, respectively see 27, 28. For integrodifferential impulsive system, the existence of PC-mild solutionsand optimal controls is presented see 15. Herein, we go on studying the following integrodifferential impulsiveperiodicsystem ˙xtAxtf t, x, t 0 gt, s, xds ,t / τ k , ΔxtB k xtc k ,t τ k . 1.1 on infinite-dimensional Banach space X, where 0 τ 0 <τ 1 <τ 2 < ···<τ k ···; lim k →∞ τ k ∞, τ kδ τ k T 0 ; Δxτ k xτ k − xτ − k , k ∈ Z 0 ; T 0 is a fi xed positive number; and δ ∈ N denoted t he number ofimpulsive points between 0 and T 0 . The operator A is the infinitesimal generator of a C 0 -semigroup {Tt,t≥ 0} on X; f is a T 0 -periodic, with respect to t ∈ 0 ∞, Carath ´ edory function; g is a continuous function from 0, ∞ × 0, ∞ × X to X and is T 0 -periodic in t and s;andB kδ B k , c kδ c k . This paper is mainly concerned with the existence ofperiodicsolutions for integrodifferential impulsiveperiodicsystemon infinite- dimensional Banach space X. In this paper, we use Leray-Schauder fixed point theorem to obtain the existence ofperiodicsolutions for integrodifferential impulsiveperiodicsystem 1.1. First, by virtue ofimpulsive evolution operator corresponding to linear homogeneous impulsive system, we construct a new Poincar ´ e operator P for integrodifferential impulsiveperiodicsystem 1.1, then we overcome some difficulties to show the compactness of Poincar ´ e operator P which is very important. By a new generalizedGronwallinequality with impulse, mixed- type integral operators, and B-norm given by us, the estimate of fixed point set {x λPx, λ ∈ 0, 1} is established. Therefore, the existence of T 0 -periodic PC-mild solutions for impulsive integrodifferential periodicsystem is shown. In order to obtain the existence ofperiodic solutions, many authors use Horn fixed point theorem or Banach fixed point theorem. However, the conditions for Horn fixed point theorem are not easy to be verified sometimes andthe conditions for Banach fixed point theorem are too strong. Our method is much different from others’, and we give a new way to show the existence ofperiodic solutions. In addition, the new generalizedGronwallinequality with impulse, mixed-type integral operator, and B-norm given by us, which can be used in other problems, have played an essential role in the study of nonlinear problems on infinite-dimensional spaces. JinRong Wang et al. 3 This paper is organized as follows. In Section 2, some results of linear impulsiveperiodicsystemand properties ofimpulsiveperiodic evolution operator corresponding to homogeneous linear impulsiveperiodicsystem are recalled. In Section 3, the new generalizedGronwallinequality with impulse, mixed-type integral operator, and B-norm are established. In Section 4,theT 0 -periodic PC-mild solution for integrodifferential impulsiveperiodicsystem 1.1 is introduced. We construct the suitable Poincar ´ e operator P and give the relation between T 0 -periodic PC-mild solution andthe fixed point of P. After showing the compactness ofthe Poincar ´ e operator P and obtaining the boundedness ofthe fixed point set {x λPx, λ ∈ 0, 1} by virtue ofthegeneralizedGronwall inequality, we can use Leray- Schauder fixed point theorem to establish the existence of T 0 -periodic PC-mild solutions for integrodifferential impulsiveperiodic system. At l ast, an example is given to demonstrate the applicability of our result. 2. Linear impulsiveperiodicsystem In order to study the integrodifferential impulse periodic system, we first recall some results about linear impulse periodicsystem here. Let X be a Banach space. £X denotes the space of linear operators in X;£ b X denotes the space of bounded linear operators in X.£ b X is theBanach space with the usual supremum norm. Define D {τ 1 , ,τ δ }⊂0,T 0 , where δ ∈ N denotes the number ofimpulsive points between 0,T 0 . We introduce PC0,T 0 ; X ≡{x : 0,T 0 → X | xto be continuous at t ∈ 0,T 0 \ D; x is continuous from left and has right-hand limits at t ∈ D};andPC 1 0,T 0 ; X ≡{x ∈ PC0,T 0 ; X | ˙x ∈ PC0,T 0 ; X}. Set x PC max sup t∈0,T 0 xt 0 , sup t∈0,T 0 xt − 0 , x PC 1 x PC ˙x PC . 2.1 It can be seen that endowed with the norm · PC · PC 1 , PC0,T 0 ; XPC 1 0,T 0 ; X is a Banach space. Firstly, we consider homogeneous linear impulsiveperiodicsystem . x tAxt,t / τ k , ΔxtB k xt,t τ k . 2.2 We introduce the following assumption H1. H1.1 A is the infinitesimal generator of a C 0 -semigroup {Tt,t ≥ 0} on X with domain DA. H1.2 There exists δ such that τ kδ τ k T 0 . H1.3 For each k ∈ Z 0 ,B k ∈ £ b X and B kδ B k . In order to study system 2.2, we need to consider the associated Cauchy problem . x tAxt,t∈ 0,T 0 \ D, Δxτ k B k xτ k ,k 1, 2, ,δ, x0 x. 2.3 4 Journal of Inequalities and Applications If x ∈ DA and DA is an invariant subspace of B k , using Theorem 5.2.2, see 29, page 144, step by step, one can verify that the Cauchy problem 2.3 has a unique classical solution x ∈ PC 1 0,T 0 ; X represented by xtSt, 0x, where S·, · : Δ t, θ ∈ 0,T 0 × 0,T 0 | 0 ≤ θ ≤ t ≤ T 0 −→ £ b X2.4 given by St, θ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Tt − θ,τ k−1 ≤ θ ≤ t ≤ τ k , T t − τ k I B k T τ k − θ ,τ k−1 ≤ θ<τ k <t≤ τ k1 , T t − τ k θ<τ j <t I B j T τ j − τ j−1 I B i T τ i − θ , τ i−1 ≤ θ<τ i ≤···<τ k <t≤ τ k1 . 2.5 The operator {St, θ, t, θ ∈ Δ} is called impulsive evolution operator associated with {B k ; τ k } ∞ k1 . Now we introduce the PC-mild solution of Cauchy problem 2.3 and T 0 -periodic PC- mild solution ofthesystem 2.2. Definition 2.1. For every x ∈ X, the function x ∈ PC0,T 0 ; X given by xtSt, 0x is said to be the PC-mild solution ofthe Cauchy problem 2.3. Definition 2.2. A function x ∈ PC0, ∞; X is said to be a T 0 -periodic PC-mild solution ofsystem 2.2 if it is a PC-mild solution of Cauchy problem 2.3 corresponding to some x and xt T 0 xt for t ≥ 0. The following lemma gives the properties oftheimpulsive evolution operator {St, θ, t, θ ∈ Δ} associated with {B k ; τ k } ∞ k1 which are widely used in sequel. Lemma 2.3 see 24, Lemma 1. Impulsive evolution operator {St, θ, t, θ ∈ Δ} has the following properties. 1 For 0 ≤ θ ≤ t ≤ T 0 , St, θ ∈ £ b X, that is, sup 0≤θ≤t≤T 0 St, θ≤M T 0 ,whereM T 0 > 0. 2 For 0 ≤ θ<r<t≤ T 0 , r / τ k , St, θSt, rSr, θ. 3 For 0 ≤ θ ≤ t ≤ T 0 and N ∈ Z 0 , St NT 0 ,θ NT 0 St, θ. 4 For 0 ≤ t ≤ T 0 and N ∈ Z 0 , SNT 0 t, 0St, 0ST 0 , 0 N . 5 If {Tt,t≥ 0} is a compact semigroup in X,thenSt, θ is a compact operator for 0 ≤ θ< t ≤ T 0 . Here, we note that system 2.2 has a T 0 -periodic PC-mild solution x if and only if ST 0 , 0 has a fixed point. Theimpulsive evolution operator {St, θ, t, θ ∈ Δ} can be used to reduce the existence of T 0 -periodic PC-mild solutions for linear impulsiveperiodicsystemtothe existence of fixed points for an operator equation. This implies that we can build up JinRong Wang et al. 5 the new framework to study theperiodic PC-mild solutions for integrodifferential impulsiveperiodicsystemonBanach space. Consider nonhomogeneous linear impulsiveperiodicsystem ˙xtAxtft,t / τ k , ΔxtB k xtc k ,t τ k , 2.6 andthe associated Cauchy problem ˙xtAxtft,t∈ 0,T 0 \ D, Δxτ k B k xτ k c k ,k 1, 2, ,δ, x0 x. 2.7 where f ∈ L 1 0,T 0 ; X, ft T 0 ft and c kδ c k . Now we introduce the PC-mild solution of Cauchy problem 2.7 and T 0 -periodic PC- mild solution ofsystem 2.6. Definition 2.4. A function x ∈ PC0,T 0 ; X, for finite interval 0,T 0 ,issaidtobeaPC-mild solution ofthe Cauchy problem 2.6 corresponding tothe initial value x ∈ X and input f ∈ L 1 0,T 0 ; X if x is given by xtSt, 0 x t 0 St, θfθdθ 0≤τ k <t S t, τ k c k . 2.8 Definition 2.5. A function x ∈ PC0, ∞; X is said to be a T 0 -periodic PC-mild solution ofsystem 2.6 if it is a PC-mild solution of Cauchy problem 2.7 corresponding to some x and xt T 0 xt for t ≥ 0. 3. ThegeneralizedGronwallinequality In order to use Leray-Schauder theorem to show the existence ofperiodic solutions, we need a new generalizedGronwallinequality with impulse, mixed-type integral operator, and B- norm which is much different from classical Gronwallinequalityand can be used in other problems such as discussion on integrodifferential equation of mixed type, see 15. It will play an essential role in the study of nonlinear problems on infinite-dimensional spaces. We first introduce the following generalizedGronwallinequality with impulse and B-norm. Lemma 3.1. Let x ∈ PC0, ∞,X and satisfy the following inequality: xt≤a b t 0 xθ λ 1 dθ d t 0 x θ λ 3 B dθ, 3.1 6 Journal of Inequalities and Applications where a, b, d ≥ 0, 0 ≤ λ 1 ,λ 3 ≤ 1 are c onstants, and x θ B sup 0≤ξ≤θ xξ.Then xt ≤ a 1e bct . 3.2 Proof. i For 0 ≤ λ 1 , λ 3 < 1, let λ max{λ 1 ,λ 3 }∈0, 1 and yt ⎧ ⎨ ⎩ 1, xt≤1, xt, xt > 1. 3.3 Then xt ≤ yt ≤ a 1b t 0 yθ λ dθ d t 0 y θ λ B dθ ∀t ∈ 0,T 0 . 3.4 Using 3.4,weobtain y t λ B ≤ a 1b d t 0 y θ λ B dθ. 3.5 Define uta 1b d t 0 y θ λ B dθ, 3.6 we get ˙utb d y t λ B ,t / τ k , u0a 1,u τ k 0 u τ k . 3.7 Since y t λ B ≤ ut, we then have ˙ut ≤ b dut,t / τ k , u0a 1,u τ k 0 u τ k . 3.8 For t ∈ τ k ,τ k1 ,by3.8,weobtain ut ≤ u τ k 0 e bdt−τ k u τ k e bdt−τ k , 3.9 JinRong Wang et al. 7 further, ut ≤ a 1e bdt , 3.10 thus, xt ≤ y t B ≤ a 1e bdt . 3.11 ii For λ 1 λ 3 1, we only need to define u 1 ta b d t 0 x θ B dθ, 3.12 Similar tothe proof in i, one can obtain xt ≤ x t B ≤ ae bdt . 3.13 Combining i and ii, one can complete the proof. Using Gronwall’s inequality with impulse and B-norm, we can obtain the following new generalizedGronwall Lemma. Lemma 3.2. Let x ∈ PC0,T 0 ; X satisfy the following inequality: xt ≤ a b t 0 xθ λ 1 dθ c T 0 0 xθ λ 2 dθ d t 0 x θ λ 3 B dθ e T 0 0 x θ λ 4 B dθ ∀t ∈ 0,T 0 , 3.14 where λ 1 ,λ 3 ∈ 0, 1, λ 2 ,λ 4 ∈ 0, 1, a, b, c, d, e ≥ 0 are constants. Then there exists a constant M ∗ > 0 such that xt ≤ M ∗ . 3.15 Proof. By Lemma 3.1,weobtainthat xt ≤ yt ≤ y t B ≤ e bdt a 1c T 0 0 yθ λ dθ e T 0 0 y θ λ B dθ , 3.16 8 Journal of Inequalities and Applications where yt ⎧ ⎨ ⎩ 1, xt ≤ 1, xt, xt > 1, λ ⎧ ⎨ ⎩ max λ 1 ,λ 2 ,λ 3 ,λ 4 ∈ 0, 1, if λ 1 ,λ 2 ,λ 3 ,λ 4 ∈ 0, 1, max λ 2 ,λ 4 ∈ 0, 1, if λ 1 λ 3 1,λ 2 ,λ 4 ∈ 0, 1. 3.17 Define qt ≡ e bdT 0 a 1c t 0 yθ λ dθ c T 0 0 yθ λ dθ e t 0 y θ λ B dθ e T 0 0 y θ λ B dθ , 3.18 then q is a monotone increasing function and ˙qt e bdT 0 c yt λ e y t λ B ≤ c ee bdT 0 yt λ y t λ B ≤ 2c ee bdT 0 q λ t. 3.19 Consider d dt q 1−λ t1 − λq −λ t ˙qt ≤ 2c ee bdT 0 1 − λ. 3.20 Integrating from 0 to t,weobtain q 1−λ t − q 1−λ 0 ≤ 2c ee bdT 0 1 − λt, 3.21 that is, qt ≤ q 1−λ 02c ee bdT 0 1 − λt 1/1−λ . 3.22 Onthe other hand, 2q02e bdT 0 a 1c T 0 0 yθ λ dθ e T 0 0 y θ λ B dθ ; qT 0 e bdT 0 a 12c T 0 0 yθ λ dθ 2e T 0 0 y θ λ B dθ . 3.23 JinRong Wang et al. 9 Now, we observe that 2q0 − e bdT 0 a 1q T 0 ≤ q 1−λ 02c ee bdT 0 T 0 1 − λ 1/1−λ . 3.24 As a result, we get 2q0 − e bdT 0 a 1 1−λ − q 1−λ 0 ≤ 2c ee bdT 0 T 0 1 − λ. 3.25 Letting Υz 2z − e bdT 0 a 1 1−λ − z 1−λ − 2c ee bdT 0 T 0 1 − λ, 3.26 we have Υ ∈ Ce bdT 0 a 1/2, ∞; R and Υe bdT 0 a 1/2 < 0. Moreover, lim z → ∞ Υz z 1−λ 2 1−λ − 1 > 0. 3.27 Hence, there exists enough large z 0 >e bdT 0 a 1/2 > 0 such that Υz > 0 for arbitrary z ≥ z 0 . Meanwhile, Υq0 ≤ 0. Thus, q0 ≤ z 0 . As a result, we obtain xt ≤ yt ≤ q T 0 2q0 − e bdT 0 a 1 ≤ 2z 0 − e bdT 0 a 1 ≡ M ∗ > 0 ∀t ∈ 0,T 0 . 3.28 4. Periodicsolutionsofintegrodifferentialimpulsiveperiodicsystem In this section, we consider the following integrodifferential impulsiveperiodic system: ˙xtAxtf t, x, t 0 gt, s, xds ,t / τ k , ΔxtB k xtc k ,t τ k . 4.1 andthe associated Cauchy problem ˙xtAxtf t, x, t 0 gt, s, xds ,t∈ 0,T 0 \ D, Δx τ k B k x τ k c k ,k 1, 2, ,δ, x0 x. 4.2 10 Journal of Inequalities and Applications By virtue ofthe expression ofthe PC-mild solution ofthe Cauchy problem 2.7,we can introduce the PC-mild solution ofthe Cauchy problem 4.2. Definition 4.1. A function x ∈ PC0,T 0 ; X is said to be a PC-mild solution ofthe Cauchy problem 4.2 corresponding tothe initial value x ∈ X if x satisfies the following integral equation: xtSt, 0 x t 0 St, θf θ, xθ, θ 0 g θ, s, xs ds dθ 0≤τ k <t S t, τ k c k for t ∈ 0,T 0 . 4.3 Now, we introduce the T 0 -periodic PC-mild solution ofsystem 4.1. Definition 4.2. A function x ∈ PC0, ∞; X is said to be a T 0 -periodic PC-mild solution ofsystem 4.1 if it is a PC-mild solution of Cauchy problem 4.2 corresponding to some x and xt T 0 xt for t ≥ 0. Assumption H2 includes the following. H2.1 f : 0, ∞ × X × X → X satisfies the following. i For each x, y ∈ X × X, t → ft, x, y is measurable. ii For each ρ>0, there exists L f ρ > 0 such that, for almost all t ∈ 0, ∞ and all x 1 , x 2 , y 1 ,y 2 ∈ X, x 1 , x 2 , y 1 , y 2 ≤ρ, we have f t, x 1 ,y 1 − f t, x 2 ,y 2 ≤ L f ρ x 1 − x 2 y 1 − y 2 . 4.4 H2.2 There exists a positive constant M f such that ft, x, y ≤ M f 1 x y ∀x, y ∈ X. 4.5 H2.3 ft, x, y is T 0 -periodic in t,thatis,ft T 0 ,x,yft, x, y,t≥ 0. H2.4 Let D {t, s ∈ 0 ∞ × 0 ∞;0 ≤ s ≤ t}. The function g : D × X → X is continuous for each ρ>0, there exists L g ρ > 0 such that, for each t, s ∈ D and each x, y ∈ X with x, y≤ρ, we have gt, s, x − gt, s, y ≤ L g ρx − y. 4.6 H2.5 There exists a positive constant M g such that gt, s, x ≤ M g 1 x ∀x, y ∈ X. 4.7 [...]... 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Corporation Journal of Inequalities and Applications Volume 2008, Article ID 430521, 22 pages doi:10.1155/2008/430521 Research Article The Generalized Gronwall Inequality and Its Application to Periodic. of linear impulsive periodic system and properties of impulsive periodic evolution operator corresponding to homogeneous linear impulsive periodic system are recalled. In Section 3, the new generalized Gronwall. T 0 xt for t ≥ 0. 3. The generalized Gronwall inequality In order to use Leray-Schauder theorem to show the existence of periodic solutions, we need a new generalized Gronwall inequality with impulse,