Hindawi Publishing 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 Solutions of Integrodifferential Impulsive Periodic System on Banach 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 impulsive periodic systems on Banach space. Using impulsive periodic 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 impulsive periodic 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. Periodic system with applications on finite-dimensional spaces has been extensively studied. Particularly, impulsive periodic systems on finite-dimensional spaces are considered and some important results such as the existence and stability of periodic solution, the relationship between bounded solution and periodic solution, and robustness by perturbation are obtained see 1–4. Since the end of last century, many researchers pay great attention to impulsive systems on infinite-dimensional spaces. Particulary, Ahmed et al. investigated optimal control problems of system governed by impulsive system 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 on periodic solution for periodic system on infinite- dimensional spaces see 12, 21–23 and some results discussing integrodifferential system on finite Banach space and infinite Banach space see 11, 13. To our knowledge, inte- grodifferential impulsive periodic systems on infinite-dimensional spaces with unbounded operator have not been extensively investigated. Recently, we discuss the impulsive periodic system and integrodifferential impulsive system on infinite-dimensional spaces. Linear impulsive evolution operator is constructed and T 0 -periodic PC-mild solution is introduced. The existence of periodic solutions, alternative theorem criteria of Massera type, asymptotical stability, and robustness by perturbation is established see 24–26. For semilinear impulsive periodic system, a suitable Poincar ´ e operator is constructed which verifies its compactness and continuity. By virtue of a generalized Gronwall inequality with mixed integral operator and impulse given by us, the estimate of the 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 of periodic PC-mild solutions, respectively see 27, 28. For integrodifferential impulsive system, the existence of PC-mild solutions and optimal controls is presented see 15. Herein, we go on studying the following integrodifferential impulsive periodic system ˙xtAxtf  t, x,  t 0 gt, s, xds  ,t /  τ k , ΔxtB k xtc 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 kδ  τ k  T 0 ; Δxτ k xτ  k  − xτ − k , k ∈ Z  0 ; T 0 is a fi xed positive number; and δ ∈ N denoted t he number of impulsive points between 0 and T 0 . The operator A is the infinitesimal generator of a C 0 -semigroup {Tt,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 kδ  B k , c kδ  c k . This paper is mainly concerned with the existence of periodic solutions for integrodifferential impulsive periodic system on infinite- dimensional Banach space X. In this paper, we use Leray-Schauder fixed point theorem to obtain the existence of periodic solutions for integrodifferential impulsive periodic system 1.1. First, by virtue of impulsive evolution operator corresponding to linear homogeneous impulsive system, we construct a new Poincar ´ e operator P for integrodifferential impulsive periodic system 1.1, then we overcome some difficulties to show the compactness of Poincar ´ e operator P which is very important. By a new generalized Gronwall inequality 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 periodic system is shown. In order to obtain the existence of periodic 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 and the 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 of periodic solutions. In addition, the new generalized Gronwall inequality 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 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 inequality with impulse, mixed-type integral operator, and B-norm are established. In Section 4,theT 0 -periodic PC-mild solution for integrodifferential impulsive periodic system 1.1 is introduced. We construct the suitable Poincar ´ e operator P and give the relation between T 0 -periodic PC-mild solution and the fixed point of P. After showing the compactness of the Poincar ´ e operator P and obtaining the boundedness of the fixed point set {x  λPx, λ ∈ 0, 1} by virtue of the generalized Gronwall inequality, we can use Leray- Schauder fixed point theorem to establish the existence of T 0 -periodic PC-mild solutions for integrodifferential impulsive periodic system. At l ast, an example is given to demonstrate the applicability of our result. 2. Linear impulsive periodic system In order to study the integrodifferential impulse periodic system, we first recall some results about linear impulse periodic system 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 the Banach space with the usual supremum norm. Define  D  {τ 1 , ,τ δ }⊂0,T 0 , where δ ∈ N denotes the number of impulsive points between 0,T 0 . We introduce PC0,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 ∈ PC0,T 0 ; X | ˙x ∈ PC0,T 0 ; X}. Set x PC  max  sup t∈0,T 0    xt  0   , sup t∈0,T 0    xt − 0    , x PC 1  x PC   ˙x PC . 2.1 It can be seen that endowed with the norm · PC · PC 1 , PC0,T 0 ; XPC 1 0,T 0 ; X is a Banach space. Firstly, we consider homogeneous linear impulsive periodic system . x tAxt,t /  τ k , ΔxtB k xt,t τ k . 2.2 We introduce the following assumption H1. H1.1 A is the infinitesimal generator of a C 0 -semigroup {Tt,t ≥ 0} on X with domain DA. H1.2 There exists δ such that τ kδ  τ k  T 0 . H1.3 For each k ∈ Z  0 ,B k ∈ £ b X and B kδ  B k . In order to study system 2.2, we need to consider the associated Cauchy problem . x tAxt,t∈ 0,T 0  \  D, Δxτ k B k xτ k ,k 1, 2, ,δ, x0 x. 2.3 4 Journal of Inequalities and Applications If x ∈ DA and DA 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 xtSt, 0x, where S·, · : Δ  t, θ ∈ 0,T 0  × 0,T 0  | 0 ≤ θ ≤ t ≤ T 0  −→ £ b X2.4 given by St, θ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Tt − 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 k1 , 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 k1 . 2.5 The operator {St, θ, t, θ ∈ Δ} is called impulsive evolution operator associated with {B k ; τ k } ∞ k1 . Now we introduce the PC-mild solution of Cauchy problem 2.3 and T 0 -periodic PC- mild solution of the system 2.2. Definition 2.1. For every x ∈ X, the function x ∈ PC0,T 0 ; X given by xtSt, 0x is said to be the PC-mild solution of the Cauchy problem 2.3. Definition 2.2. A function x ∈ PC0, ∞; X is said to be a T 0 -periodic PC-mild solution of system 2.2 if it is a PC-mild solution of Cauchy problem 2.3 corresponding to some x and xt  T 0 xt for t ≥ 0. The following lemma gives the properties of the impulsive evolution operator {St, θ, t, θ ∈ Δ} associated with {B k ; τ k } ∞ k1 which are widely used in sequel. Lemma 2.3 see 24, Lemma 1. Impulsive evolution operator {St, θ, t, θ ∈ Δ} has the following properties. 1 For 0 ≤ θ ≤ t ≤ T 0 , St, θ ∈ £ b X, that is, sup 0≤θ≤t≤T 0 St, θ≤M T 0 ,whereM T 0 > 0. 2 For 0 ≤ θ<r<t≤ T 0 , r /  τ k , St, θSt, rSr, θ. 3 For 0 ≤ θ ≤ t ≤ T 0 and N ∈ Z  0 , St  NT 0 ,θ NT 0 St, θ. 4 For 0 ≤ t ≤ T 0 and N ∈ Z  0 , SNT 0  t, 0St, 0ST 0 , 0 N . 5 If {Tt,t≥ 0} is a compact semigroup in X,thenSt, 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 ST 0 , 0 has a fixed point. The impulsive evolution operator {St, θ, t, θ ∈ Δ} can be used to reduce the existence of T 0 -periodic PC-mild solutions for linear impulsive periodic system to the 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 the periodic PC-mild solutions for integrodifferential impulsive periodic system on Banach space. Consider nonhomogeneous linear impulsive periodic system ˙xtAxtft,t /  τ k , ΔxtB k xtc k ,t τ k , 2.6 and the associated Cauchy problem ˙xtAxtft,t∈ 0,T 0  \  D, Δxτ k B k xτ k c k ,k 1, 2, ,δ, x0 x. 2.7 where f ∈ L 1 0,T 0 ; X, ft  T 0 ft and c kδ  c k . Now we introduce the PC-mild solution of Cauchy problem 2.7 and T 0 -periodic PC- mild solution of system 2.6. Definition 2.4. A function x ∈ PC0,T 0 ; X, for finite interval 0,T 0 ,issaidtobeaPC-mild solution of the Cauchy problem 2.6 corresponding to the initial value x ∈ X and input f ∈ L 1 0,T 0 ; X if x is given by xtSt, 0 x   t 0 St, θfθdθ   0≤τ k <t S  t, τ  k  c k . 2.8 Definition 2.5. A function x ∈ PC0, ∞; X is said to be a T 0 -periodic PC-mild solution of system 2.6 if it is a PC-mild solution of Cauchy problem 2.7 corresponding to some x and xt  T 0 xt 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, mixed-type integral operator, and B- norm which is much different from classical Gronwall inequality and 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 generalized Gronwall inequality with impulse and B-norm. Lemma 3.1. Let x ∈ PC0, ∞,X and satisfy the following inequality: xt≤a  b  t 0 xθ 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 xξ.Then   xt   ≤ a  1e bct . 3.2 Proof. i For 0 ≤ λ 1 , λ 3 < 1, let λ  max{λ 1 ,λ 3 }∈0, 1 and yt ⎧ ⎨ ⎩ 1, xt≤1, xt, xt > 1. 3.3 Then   xt   ≤   yt   ≤ a  1b  t 0   yθ   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  1b  d  t 0   y θ   λ B dθ. 3.5 Define uta  1b  d  t 0   y θ   λ B dθ, 3.6 we get ˙utb  d   y t   λ B ,t /  τ k , u0a  1,u  τ k  0   u  τ k  . 3.7 Since y t  λ B ≤ ut, we then have ˙ut ≤ b  dut,t /  τ k , u0a  1,u  τ k  0   u  τ k  . 3.8 For t ∈ τ k ,τ k1 ,by3.8,weobtain ut ≤ u  τ k  0  e bdt−τ k   u  τ k  e bdt−τ k  , 3.9 JinRong Wang et al. 7 further, ut ≤ a  1e bdt , 3.10 thus,   xt   ≤   y t   B ≤ a  1e bdt . 3.11 ii For λ 1  λ 3  1, we only need to define u 1 ta b  d  t 0   x θ   B dθ, 3.12 Similar to the proof in i, one can obtain   xt   ≤   x t   B ≤ ae bdt . 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 generalized Gronwall Lemma. Lemma 3.2. Let x ∈ PC0,T 0 ; X satisfy the following inequality:   xt   ≤ a  b  t 0   xθ   λ 1 dθ  c  T 0 0   xθ   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   xt   ≤ M ∗ . 3.15 Proof. By Lemma 3.1,weobtainthat   xt   ≤   yt   ≤   y t   B ≤ e bdt  a  1c  T 0 0   yθ   λ dθ  e  T 0 0   y θ   λ B dθ  , 3.16 8 Journal of Inequalities and Applications where yt ⎧ ⎨ ⎩ 1,   xt   ≤ 1, xt,   xt   > 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 qt ≡ e bdT 0  a  1c  t 0   yθ   λ dθ  c  T 0 0   yθ   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 ˙qt  e bdT 0  c   yt   λ  e   y t   λ B  ≤ c  ee bdT 0    yt   λ    y t   λ B  ≤ 2c  ee bdT 0 q λ t. 3.19 Consider d dt q 1−λ t1 − λq −λ t ˙qt ≤ 2c  ee bdT 0 1 − λ. 3.20 Integrating from 0 to t,weobtain q 1−λ t − q 1−λ 0 ≤ 2c  ee bdT 0 1 − λt, 3.21 that is, qt ≤  q 1−λ 02c  ee bdT 0 1 − λt  1/1−λ . 3.22 On the other hand, 2q02e bdT 0  a  1c  T 0 0   yθ   λ dθ  e  T 0 0   y θ   λ B dθ  ; qT 0 e bdT 0  a  12c  T 0 0   yθ   λ dθ  2e  T 0 0   y θ   λ B dθ  . 3.23 JinRong Wang et al. 9 Now, we observe that 2q0 − e bdT 0 a  1q  T 0  ≤  q 1−λ 02c  ee bdT 0 T 0 1 − λ  1/1−λ . 3.24 As a result, we get  2q0 − e bdT 0 a  1  1−λ − q 1−λ 0 ≤ 2c  ee bdT 0 T 0 1 − λ. 3.25 Letting Υz  2z − e bdT 0 a  1  1−λ − z 1−λ − 2c  ee bdT 0 T 0 1 − λ, 3.26 we have Υ ∈ Ce bdT 0 a  1/2, ∞; R and Υe bdT 0 a  1/2 < 0. Moreover, lim z → ∞ Υz z 1−λ  2 1−λ − 1 > 0. 3.27 Hence, there exists enough large z 0 >e bdT 0 a  1/2 > 0 such that Υz > 0 for arbitrary z ≥ z 0 . Meanwhile, Υq0 ≤ 0. Thus, q0 ≤ z 0 . As a result, we obtain   xt   ≤   yt   ≤ q  T 0   2q0 − e bdT 0 a  1 ≤ 2z 0 − e bdT 0 a  1 ≡ M ∗ > 0 ∀t ∈  0,T 0  . 3.28 4. Periodic solutions of integrodifferential impulsive periodic system In this section, we consider the following integrodifferential impulsive periodic system: ˙xtAxtf  t, x,  t 0 gt, s, xds  ,t /  τ k , ΔxtB k xtc k ,t τ k . 4.1 and the associated Cauchy problem ˙xtAxtf  t, x,  t 0 gt, s, xds  ,t∈ 0,T 0  \  D, Δx  τ k   B k x  τ k   c k ,k 1, 2, ,δ, x0 x. 4.2 10 Journal of Inequalities and Applications By virtue of the expression of the PC-mild solution of the Cauchy problem 2.7,we can introduce the PC-mild solution of the Cauchy problem 4.2. Definition 4.1. A function x ∈ PC0,T 0 ; X is said to be a PC-mild solution of the Cauchy problem 4.2 corresponding to the initial value x ∈ X if x satisfies the following integral equation: xtSt, 0 x   t 0 St, θf  θ, xθ,  θ 0 g  θ, s, xs  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 of system 4.1. Definition 4.2. A function x ∈ PC0, ∞; X is said to be a T 0 -periodic PC-mild solution of system 4.1 if it is a PC-mild solution of Cauchy problem 4.2 corresponding to some x and xt  T 0 xt 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 → ft, 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   ft, x, y   ≤ M f  1  x  y  ∀x, y ∈ X. 4.5 H2.3 ft, x, y is T 0 -periodic in t,thatis,ft  T 0 ,x,yft, 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   gt, s, x − gt, s, y   ≤ L g ρx − y. 4.6 H2.5 There exists a positive constant M g such that   gt, 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 xt 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,