Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 4, 1-17; http://www.math.u-szeged.hu/ejqtde/

PERIODIC SOLUTIONS OF A CLASS OF INTEGRODIFFERENTIAL IMPULSIVE PERIODIC SYSTEMS

WITH TIME-VARYING GENERATING OPERATORS ON BANACH SPACE

JinRong Wang^{1}, X. Xiang^{1,2} and W. Wei^{2}

1. College of Computer Science and Technology of Guizhou University 2. College of Science of Guizhou University Guiyang, Guizhou 550025, P.R. China

ABSTRACT. This paper deals with a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space. Using impulsive periodic evolution operator given by us, the suitable T_{0}-periodic P C-mild solution is introduced and P oincaré operator is constructed. Showing the compactness of P oincaré 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 P C-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration. Keywords. Integrodifferential impulsive periodic systems, Time-varying generating operators, T_{0}- periodic P C-mild solution, Generalized Gronwall’s inequality, Existence.

1. Introduction

It is well known that impulsive periodic motion is a very important and special phenomena not only in natural science but also in social science such as climate, food supplement, insecticide population, sustainable development. Periodic system with applications on finite dimensional spaces have 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, robustness by perturbation) are obtained (See [7], [11], [12], [32]).

Since the end of last century, many researchers pay great attention on impulsive systems on infinite dimensional spaces. Particulary, Dr. Ahmed investigated optimal control problems of system governed by impulsive system (See [3], [4], [5], [6]). Many authors including us also gave a series of results for semilinear (integrodifferential, strongly nonlinear) impulsive systems and optimal control problems (See [8], [9], [10], [13], [14], [15], [16], [26], [27], [28], [29], [30], [31]).

This work is supported by Natural Science Foundation of Guizhou Province Education Department (No.2007008) and Key Projects of Science and Technology Research in the Ministry of Education (No.207104). This work is also supported by the undergraduate carve out project of Department of Guiyang Science and Technology([2008] No.15-2).

E-mail: wjr9668@126.com.

EJQTDE, 2009 No. 4, p. 1

Although, there are some papers on periodic solution for periodic systems on infinite dimensional spaces (See [1], [13], [24], [25]) and some results discussing integrodifferential system on finite Banach space and infinite Banach space (see [10], [14]), to our knowledge, integrodifferential impulsive periodic systems with time-varying generating operators on infinite dimensional spaces have not been extensively investigated. Recently, we discuss the impulsive periodic system and integrodifferential impulsive system on infinite dimensional spaces. For linear impulsive evolution operator is constructed and T_{0}-periodic P C-mild solution is introduced. Existence of periodic solutions and alternative theorem, criteria of Massera type, asymptotical stability and robustness by perturbation are established (See [17], [18], [19]). For semilinear impulsive periodic system, a suitable P oincaré operator is constructed and verify its compactness and continuity. By virtue of a generalized Gronwall inequality with mixed integral operator and impulse given by us, the estimate on the P C-mild solutions are derived. Some fixed point theorem such as Banach fixed point theorem, Horn’s fixed point theorem and Leray-Schauder fixed point theorem are applied to obtain the existence of periodic P C-mild solutions respectively (See [20], [21], [22]). For integrodifferential impulsive system, existence of P C-mild solutions and optimal controls are presented (See [26]).

Herein, we go on studying the following integrodifferential impulsive periodic system with timevarying generating operators:

(1.1)  \{ \begin{matrix}ẋ(t) = A(t)x(t) + f (t, x, ∫_{0}^{t} g(t, s, x)ds), t ≠ τ_{k}, \\ Δx(t) = B_{k}x(t) + c_{k},  t = τ_{k}. \end{matrix}

in the Banach X, where {A (t), t ∈ [0, T_{0}]} is a family of closed densely defined linear unbounded operators on X and the resolvent of the unbounded operator A(t) is compact. f is a T_{0}-periodic, with respect to t ∈ [0 + ∞), Carathéodory function, g is a continuous function from [0, ∞) × [0, ∞) × X to X and are T_{0}-periodic in t and s, and B_{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.

Here, we also use Leray-Schauder fixed point theorem to obtain the existence of periodic solutions for integrodifferential impulsive periodic system with time-varying generating operators (1.1). First, by virtue of impulsive evolution operators corresponding to linear homogeneous impulsive system with time-varying generating operators, we construct a new P oincaré operator P for integrodifferential impulsive periodic system with time-varying generating operators (1.1), then overcome some difficulties to show the compactness of P oincaré operator P which is very important. By a new generalized Gronwall inequality with impulse, mixed type integral operator and B-norm given by us, the estimate of fixed point set {x = λP x, λ ∈ [0, 1]} is established. Therefore, the existence of T_{0}-periodic P C-mild solutions for impulsive integrodifferential periodic system with time-varying generating operators is shown.

In order to obtain the existence of periodic solutions, many authors use Horn’s fixed point theorem or Banach fixed point theorem. However, the conditions for Horn’s fixed point theorem are not easy to be verified sometimes and the conditions for Banach’s fixed point theorem are too strong. Our methods is much different from other’s 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

EJQTDE, 2009 No. 4, p. 2

operators 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.

This paper is organized as follows. In section 2, some results of linear impulsive periodic system with time-varying generating operators and properties of impulsive periodic evolution operator corresponding to homogeneous linear impulsive periodic system with time-varying generating operators are recalled. In section 3, the new generalized Gronwall inequality with impulse, mixed type integral operator and B-norm is established. In section 4, the T_{0}-periodic P C-mild solution for integrodifferential impulsive periodic system with time-varying generating operators (1.1) is introduced. We construct the suitable P oincaré operator P and give the relation between T_{0}-periodic P C-mild solution and the fixed point of P. After showing the compactness of the P oincaré operator P and obtaining the boundedness of the fixed point set {x = λP x, 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 P C-mild solutions for integrodifferential impulsive periodic system with time-varying generating operators. At last, an example is given to demonstrate the applicability of our result.

2. Linear Impulsive Periodic System with time-varying generating operators

In order to study the integrodifferential impulsive periodic system with time-varying generating operators, we first recall some results about linear impulsive periodic system with time-varying generating operators 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 D̃={τ_{1}, ⋯, τ_{δ}} ⊂ [0, T_{0}], where δ ∈ ℕ denotes the number of impulsive points between [0, T_{0}]. We introduce P C([0, T_{0}]; X) ≡ {x : [0, T_{0}] → X | x is continuous at t ∈ [0, T_{0}]\D̃, x is continuous from left and has right hand limits at t ∈ D̃} and P C^{1}([0, T_{0}]; X) ≡ {x ∈ P C([0, T_{0}]; X) | ẋ ∈ P C([0, T_{0}]; X)}. Set

∥x∥_{P C} = max { sup_{t∈[0,T_{0}]} ∥x(t + 0)∥, sup_{t∈[0,T_{0}]} ∥x(t − 0)∥} and ∥x∥_{P C^{1}} = ∥x∥_{P C} + ∥ẋ∥_{P C}.

It can be seen that endowed with the norm ∥ · ∥_{P C} (∥ · ∥_{P C^{1}}), P C([0, T_{0}]; X) (P C^{1}([0, T_{0}]; X)) is a Banach space.

Consider the following homogeneous linear impulsive periodic system with time-varying generating operators

(2.1)  \{ \begin{matrix}ẋ(t) = A(t)x(t),  t ≠ τ_{k}, \\ Δx(τ_{k}) = B_{k}x(τ_{k}),  t = τ_{k}. \end{matrix}

on Banach space X, where △x(τ_{k}) = x(τ_{k}^{+}) − x(τ_{k}^{−}), {A (t), t ≥ 0} is a family of closed densely defined linear unbounded operators on X satisfying the following assumption.

Assumption [A1]: (See [2], p.158) For t ∈ [0, T_{0}] one has (P_{1}) The domain D(A(t)) = D is independent of t and is dense in X. (P_{2}) For t ≥ 0, the resolvent R(λ, A(t)) = (λI − A(t))^{−1} exists for all λ with Reλ ≤ 0, and there is a constant M independent of λ and t such that

∥R (λ, A(t))∥ ≤ M(1 + |λ|)^{−1} for Reλ ≤ 0.

EJQTDE, 2009 No. 4, p. 3

(P_{3}) There exist constants L > 0 and 0 < α ≤ 1 such that

∥(A(t) − A(θ))A^{−1}(τ)∥ ≤ L|t − θ|^{α} for t, θ, τ ∈ [0, T_{0}].

Lemma 2.1: (See [2], p.159) Under the assumption [A1], the Cauchy problem

(2.2)  ẋ(t) + A(t)x(t) = 0, t ∈ (0, T_{0}] with x(0) = x̄

has a unique evolution system {U(t, θ) | 0 ≤ θ ≤ t ≤ T_{0}} in X satisfying the following properties:

(1) U(t, θ) ∈ £_{b}(X) for 0 ≤ θ ≤ t ≤ T_{0}.

(2) U(t, r)U(r, θ) = U(t, θ) for 0 ≤ θ ≤ r ≤ t ≤ T_{0}.

(3) U(·, ·)x ∈ C(Δ, X) for x ∈ X, Δ = {(t, θ) ∈ [0, T_{0}] × [0, T_{0}] | 0 ≤ θ ≤ t ≤ T_{0}}.

(4) For 0 ≤ θ < t ≤ T_{0}, U(t, θ): X ⟶ D and t ⟶ U(t, θ) is strongly differentiable in X. The derivative \frac{∂}{∂t}U(t, θ) ∈ £_{b}(X) and it is strongly continuous on 0 ≤ θ < t ≤ T_{0}. Moreover,

\frac{∂}{∂t}U(t, θ) = −A(t)U(t, θ) for 0 ≤ θ < t ≤ T_{0},
∥\frac{∂}{∂t}U(t, θ)∥_{£_{b}(X)} = ∥A(t)U(t, θ)∥_{£_{b}(X)} ≤ \frac{C}{t − θ} ,
∥A(t)U(t, θ)A(θ)^{−1}∥_{£_{b}(X)} ≤ C for 0 ≤ θ ≤ t ≤ T_{0}.

(5) For every v ∈ D and t ∈ (0, T_{0}], U(t, θ)v is differentiable with respect to θ on 0 ≤ θ ≤ t ≤ T_{0}

\frac{∂}{∂θ} U(t, θ)v = U(t, θ)A(θ)v.

And, for each x̄ ∈ X, the Cauchy problem (2.2) has a unique classical solution x ∈ C^{1}([0, T_{0}]; X) given by

x(t) = U(t, 0)x̄, t ∈ [0, T_{0}].

In addition to assumption [A1], we introduce the following assumptions.

Assumption [A2]: There exits T_{0} > 0 such that A(t + T_{0}) = A(t) for t ∈ [0, T_{0}].

Assumption [A3]: For t ≥ 0, the resolvent R(λ, A(t)) is compact.

Then we have

Lemma 2.2: Assumptions [A1], [A2] and [A3] hold. Then evolution system {U(t, θ) | 0 ≤ θ ≤ t ≤ T_{0}} in X also satisfying the following two properties:

(6) U(t + T_{0}, θ + T_{0}) = U(t, θ) for 0 ≤ θ ≤ t ≤ T_{0};

(7) U(t, θ) is compact operator for 0 ≤ θ < t ≤ T_{0}.

In order to introduce a impulsive evolution operator and give it’s properties, we need the following assumption.

Assumption [B]: For each k ∈ ℤ_{0}^{+}, B_{k} ∈ £_{b}(X), there exists δ ∈ ℕ such that τ_{k+δ} =

τ_{k} + T_{0} and B_{k+δ} = B_{k}.

EJQTDE, 2009 No. 4, p. 4

Consider the following Cauchy problem

(2.3)  \{ \begin{matrix}ẋ(t) = A(t)x(t),  t ∈ [0, T_{0}]\D̃, \\ Δx(τ_{k}) = B_{k}x(τ_{k}),  k = 1, 2, ⋯, δ, \\ x(0) = x̄. \end{matrix}

For every x̄ ∈ X, D is an invariant subspace of B_{k}, using Lemma 2.1, step by step, one can verify that the Cauchy problem (2.3) has a unique classical solution x ∈ P C^{1}([0, T_{0}]; X) represented by x(t) = 𝒮 (t, 0)x̄ where 𝒮 (·, ·) : Δ ⟶ £(X) given by

𝒮 (t, θ) = \{ \begin{matrix}U(t, θ), τ_{k−1} ≤ θ ≤ t ≤ τ_{k}, \\ U(t, τ_{k}^{+})(I + B_{k})U(τ_{k}, θ), τ_{k−1} ≤ θ < τ_{k} < t ≤ τ_{k+1}, \\ U(t, τ_{k}^{+})[ ∏_{θ<τ_{j}<t}(I + B_{j})U(τ_{j}, τ_{j−1}^{+})](I + B_{i})U(τ_{i}, θ), \\ τ_{i−1} ≤ θ < τ_{i} ≤ ⋯ < τ_{k} < t ≤ τ_{k+1}. \end{matrix} (2.4)

The operator {𝒮 (t, θ), (t, θ) ∈ Δ} is called impulsive evolution operator associated with {B_{k}; τ_{k}}_{k=1}^{∞}.

The following lemma on the properties of the impulsive evolution operator {𝒮 (t, θ), (t, θ) ∈ Δ} associated with {B_{k}; τ_{k}}_{k=1}^{∞} are widely used in this paper.

Lemma 2.3: (See Lemma 1 of [18]) Assumptions [A1], [A2], [A3] and [B] hold. 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), i.e., sup_{0≤θ≤t≤T_{0}} ∥𝒮 (t, θ)∥ ≤ M_{T_{0}} where M_{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 M ∈ Z_{0}^{+}, 𝒮 (MT_{0} + t, 0) = 𝒮 (t, 0) [𝒮 (T_{0}, 0)] ^{M}.

(5) 𝒮 (t, θ) is compact operator for 0 ≤ θ < t ≤ T_{0}.

Here, we note that system (2.1) has a T_{0}-periodic P C-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 P C-mild solutions for linear impulsive periodic system with time-varying generating operators to the existence of fixed points for an operator equation. This implies that we can build up the new framework to study the periodic P C-mild solutions for the integrodifferential impulsive periodic system with time-varying generating operators on Banach space.

Now we introduce the P C-mild solution of Cauchy problem (2.3) and T_{0}-periodic P C-mild solution of the system (2.1).

Definition 2.1: For every x̄ ∈ X, the function x ∈ P C([0, T_{0}]; X) given by x(t) = 𝒮 (t, 0)x̄ is said to be the P C-mild solution of the Cauchy problem (2.3) .

Definition 2.2: A function x ∈ P C([0, +∞); X) is said to be a T_{0}-periodic P C-mild solution of system (2.1) if it is a P C-mild solution of Cauchy problem (2.3) corresponding to some x̄ and x(t + T_{0}) = x(t) for t ≥ 0.

EJQTDE, 2009 No. 4, p. 5

Secondly, we recall the following nonhomogeneous linear impulsive periodic system with timevarying generating operators

(2.5)  \{ \begin{matrix}ẋ(t) = A(t)x(t) + f(t),  t ≠ τ_{k}, \\ Δx(τ_{k}) = B_{k}x(τ_{k}) + c_{k},  t = τ_{k}. \end{matrix}

where f ∈ L^{1} ([0, T_{0}]; X), f(t + T_{0}) = f(t) for t ≥ 0 and c_{k} satisfies the following assumption.

Assumption [C]: For each k ∈ ℤ_{0}^{+} and c_{k} ∈ X, there exists δ ∈ ℕ such that c_{k+δ} = c_{k}.

In order to study system (2.5), we need to consider the following Cauchy problem

(2.6)  \{ \begin{matrix}ẋ(t) = A(t)x(t) + f(t),  t ∈ [0, T_{0}]\D̃, \\ Δx(τ_{k}) = B_{k}x(τ_{k}) + c_{k},  k = 1, 2, ⋯, δ, \\ x(0) = x̄. \end{matrix}

and introduce the P C-mild solution of Cauchy problem (2.6) and T_{0}-periodic P C-mild solution of system (2.5).

Definition 2.3: A function x ∈ P C([0, T_{0}]; X), for finite interval [0, T_{0}], is said to be a P Cmild solution of the Cauchy problem (2.5) corresponding to the initial value x̄ ∈ X and input f ∈ L^{1} ([0, T_{0}]; X) if x is given by

x(t) = 𝒮 (t, 0)x̄ + ∫_{0}^{t} 𝒮 (t, θ)f(θ)dθ + ∑_{0≤τ_{k}<t} 𝒮 (t, τ_{k}^{+}) c_{k} for t ∈ [0, T_{0}].

Definition 2.4: A function x ∈ P C([0, +∞); X) is said to be a T_{0}-periodic P C-mild solution of system (2.5) if it is a P C-mild solution of Cauchy problem (2.6) to some x̄ and x(t+T_{0}) = x(t) for t ≥ 0.

3. The generalized Gronwall’s inequality

In order to use Leray-Schauder theorem to show the existence of periodic solutions, we need a new generalized Gronwall’s inequality with impulse, mixed type integral operator and B-norm which is much different from the classical Gronwall’s inequality and can be used in other problems (such as discussion on integrodifferential equation of mixed type, see [26]). It will play an essential role in the study of nonlinear problems on infinite dimensional spaces.

We first introduce the following generalized Gronwall’s inequality with impulse and B-norm.

Lemma 3.1: Let x ∈ P C([0, ∞); X) and satisfy the following inequality

(3.1)  ∥x(t)∥ ≤ a + b ∫_{0}^{t} ∥x(θ)∥^{λ_{1}}dθ + d ∫_{0}^{t} ∥x_{θ}∥_{B}^{λ_{3}}dθ,

where a, b, d ≥ 0, 0 ≤ λ_{1}, λ_{3} ≤ 1 are constants, and ∥x_{θ}∥_{B} = sup_{0≤ξ≤θ} ∥x(ξ)∥. Then

∥x(t)∥ ≤ (a + 1)e^{(b+c)t}.

Proof. See Lemma 3.1 of [23]. □

Using Gronwall’s inequality with impulse and B-norm, we can obtain the following new generalized Gronwall’s Lemma.

EJQTDE, 2009 No. 4, p. 6

Lemma 3.2: Let x ∈ P C([0, T_{0}]; X) satisfy the following inequality

∥x(t)∥ ≤ a+b ∫_{0}^{t} ∥x(θ)∥^{λ_{1}}dθ +c ∫_{0}^{T_{0}} ∥x(θ)∥^{λ_{2}}dθ +d ∫_{0}^{t} ∥x_{θ}∥_{B}^{λ_{3}}dθ +e ∫_{0}^{T_{0}} ∥x_{θ}∥_{B}^{λ_{4}}dθ for all t ∈ [0, T_{0}],

where λ_{1}, λ_{3} ∈ [0, 1], λ_{2}, λ_{4} ∈ [0, 1), a, b, c, d, e ≥ 0 are constants. Then exists a constant M^{∗} > 0 such that

∥x(t)∥ ≤ M^{∗}.

Proof. See Lemma 3.2 of [23]. □

4. Periodic solutions of Integrodifferential Impulsive Periodic System with time-varying generating operators

In this section, we consider the following integrodifferential impulsive periodic system with timevarying generating operators

(4.1)  \{ \begin{matrix}ẋ(t) = A(t)x(t) + f (t, x, ∫_{0}^{t} g(t, s, x)ds), t ≠ τ_{k}, \\ Δx(t) = B_{k}x(t) + c_{k},  t = τ_{k}. \end{matrix}

and the associated Cauchy problem

(4.2)  \{ \begin{matrix}ẋ(t) = A(t)x(t) + f (t, x, ∫_{0}^{t} g(t, s, x)ds),  t ∈ [0, T_{0}]\D̃, \\ Δx(τ_{k}) = B_{k}x(τ_{k}) + c_{k},  k = 1, 2, ⋯, δ, \\ x(0) = x̄. \end{matrix}

By virtue of the expression of the P C-mild solution of the Cauchy problem (2.6), we can introduce the P C-mild solution of the Cauchy problem (4.2).

Definition 4.1: A function x ∈ P C([0, T_{0}]; X) is said to be a P C-mild solution of the Cauchy problem (4.2) corresponding to the initial value x̄ ∈ X if x satisfies the following integral equation

x(t) = 𝒮 (t, 0)x̄+∫_{0}^{t} 𝒮 (t, θ)f (θ, x(θ), ∫_{0}^{θ} g(θ, s, x(s))ds) dθ + ∑_{0≤τ_{k}<t}𝒮 (t, τ_{k}^{+}) c_{k} for t ∈ [0, T_{0}].

Now, we introduce the T_{0}-periodic P C-mild solution of system (4.1).

Definition 4.2: A function x ∈ P C([0, +∞); X) is said to be a T_{0}-periodic P C-mild solution of system (4.1) if it is a P C-mild solution of Cauchy problem (4.2) corresponding to some x̄ and x(t + T_{0}) = x(t) for t ≥ 0.

We make the following assumptions.

Assumption [F]:

[F1]: f : [0, +∞) × X × X → X satisfies:

(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}∥).

EJQTDE, 2009 No. 4, p. 7

[F2]: There exists a positive constant M_{f} > 0 such that

∥f(t, x, y)∥ ≤ M_{f}(1 + ∥x∥ + ∥y∥) for all x, y ∈ X.

[F3]: f(t, x, y) is T_{0}-periodic in t, i.e., f(t + T_{0}, x, y) = f(t, x, y), t ≥ 0.

Assumption [G]:

[G1]: 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∥.

[G2]: There exists a positive constant M_{g} > 0 such that

∥g(t, s, x)∥ ≤ M_{g}(1 + ∥x∥) for all x ∈ X.

[G3]: g(t, s, x) are T_{0}-periodic in t and s, i.e., g(t + T_{0}, s + T_{0}, x) = g(t, s, x), t ≥ s ≥ 0 and

∫_{0}^{T_{0}} g(t, s, x)ds = 0, t ≥ s ≥ 0.

Lemma 4.1: Under assumptions [G1] and [G2], one has the following properties: (1) ∫_{0}^{·} g (·, s, x(s)) ds : P C ([0, T_{0}]; X) → P C ([0, T_{0}]; X).

(2) For all x_{1}, x_{2} ∈ P C ([0, T_{0}]; X) and ∥x_{1}∥_{P C([0,T_{0}];X)} , ∥x_{2}∥_{P C([0,T_{0}];X)} ≤ ρ,

∥∫_{0}^{t} g (t, s, x_{1}(s)) ds − ∫_{0}^{t} g (t, s, x_{2}(s)) ds∥ ≤ L_{g} (ρ) T_{0} ∥(x_{1})_{t} − (x_{2})_{t}∥_{B} .

(3) For x ∈ P C ([0, T_{0}]; X) ,

∥∫_{0}^{t} g (t, s, x(s)) ds∥ ≤ M_{g}T_{0} (1 + ∥x_{t}∥_{B}) .

Proof. See Lemma 4.3 of [23]. □

Now we present the existence of P C-mild solution for system (4.2).

Theorem 4.1: Assumptions [A1], [F1], [F2], [G1] and [G2] hold. Then system (4.2) has a unique P C-mild solution given by the following integral equation

x(t, x̄) = 𝒮 (t, 0)x̄ + ∫_{0}^{t} 𝒮 (t, θ)f (θ, x(θ), ∫_{0}^{θ} g(θ, s, x(s))ds) dθ + ∑_{0≤τ_{k}<t}𝒮 (t, τ_{k}^{+}) c_{k}.

Proof. In order to make the process clear we divide it into three steps.

Step 1, we consider the following general integro-differential equation without impulse

(4.3)  \{ \begin{matrix}x^{.} (t) = A(t)x(t) + f (t, x, ∫_{0}^{t} g(t, s, x)ds) , t ∈ [s, τ], \\ x(s) = x̄ ∈ X. \end{matrix}

In order to obtain the local existence of mild solution for system (4.3), we only need to set up the framework for use of the contraction mapping theorem. Consider the ball given by

𝔅 = {x ∈ C ([s, t_{1}]; X) | ∥x(t) − x̄∥ ≤ 1, s ≤ t ≤ t_{1}}

EJQTDE, 2009 No. 4, p. 8

where t_{1} would be chosen, and ∥x (t)∥ ≤ 1 + \left∥{ \atop x̄}\right∥ = ρ̄, s ≤ t ≤ t_{1}. 𝔅 ⊆ C ([s, t_{1}], X) is a closed convex set. Define a map Q on 𝔅 given by

(Qx)(t) = U(t, 0)x̄ + ∫_{s}^{t} U (t, θ) f (θ, x (θ) , ∫_{0}^{θ} g(θ, s, x(s))ds) dθ.

Under the assumptions [A1], [F1], [F2], [G1], [G2] and Lemma 3.1, one can verify that map Q is a contraction map on 𝔅 with chosen t_{1} > 0. This means that system (4.3) has a unique mild solution x ∈ C([s, t_{1}]; X) given by

x(t) = U(t, 0)x̄ + ∫_{s}^{t} U (t, θ) f (θ, x(θ), ∫_{0}^{θ} g(θ, s, x(s))ds) dθ on [s, t_{1}].

Again, using the Lemma 3.1 and (3) of Lemma 4.1, we can obtain the a prior estimate of the mild solutions for system (4.3) and present the global existence of mild solutions.

Step 2, for t ∈ (τ_{k}, τ_{k+1}], consider Cauchy problem

\{ \begin{matrix}x^{.} (t) = A(t)x(t) + f (t, x, ∫_{0}^{t} g(t, s, x)ds), t ∈ (τ_{k}, τ_{k+1}], \\ x(τ_{k}) = x_{k}. \end{matrix} (4.4)

where x_{k} ≡ (I + B_{k})x(τ_{k}) + c_{k} ∈ X.

By Step 1, Cauchy problem (4.4) also has a unique P C-mild solution

x(t) = U(t, τ_{k})x_{k} + ∫_{τ_{k}}^{t} U(t, θ)f (θ, x(θ), ∫_{0}^{θ} g(θ, s, x(s))ds) dθ.

Step 3, combining the all of solutions on (τ_{k}, τ_{k+1}] (k = 1, ⋯, δ), one can obtain the P C-mild solution of the Cauchy problem (4.2) given by

x(t, x̄) = 𝒮 (t, 0)x̄ + ∫_{0}^{t} 𝒮 (t, θ)f (θ, x(θ), ∫_{0}^{θ} g(θ, s, x(s))ds) dθ + ∑_{0≤τ_{k}<t}𝒮 (t, τ_{k}^{+}) c_{k}.

This completes the proof. □

To establish the periodic solutions for the system (4.1), we define a P oincaré operator from X to X as following

P (x̄) = x(T_{0}, x̄)
(4.5) = 𝒮 (T_{0}, 0)x̄ + ∫_{0}^{T_{0}} 𝒮 (T_{0}, θ)f (θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds) dθ + ∑_{0≤τ_{k}<T_{0}} 𝒮 (T_{0}, τ_{k}^{+}) c_{k}

where x(·, x̄) denote the P C-mild solution of the Cauchy problem (4.2) corresponding to the initial value x(0) = x̄, then, examine whether P has a fixed point.

We first note that a fixed point of P gives rise to a periodic solution.

Lemma 4.2: System (4.1) has a T_{0}-periodic P C-mild solution if and only if P has a fixed point.

Proof. Suppose x(·) = x(· + T_{0}), then x(0) = x(T_{0}) = P (x(0)). This implies that x(0) is a fixed point of P. On the other hand, if P x_{0} = x_{0}, x_{0} ∈ X, then for the P C-mild solution x(·, x_{0}) of the Cauchy problem (4.2) corresponding to the initial value x(0) = x_{0}, we can define y(·) = x(·+T_{0}, x_{0}),

EJQTDE, 2009 No. 4, p. 9

then y(0) = x(T_{0}, x_{0}) = P x_{0} = x_{0}. Now, for t > 0, we can use the (2), (3) and (4) of Lemma 2.3 and assumptions [A2], [B], [C], [F3], [G3] to arrive at

y(t) = x(t + T_{0}, x_{0})
= 𝒮 (t + T_{0}, T_{0})𝒮 (T_{0}, 0)x_{0}

+ ∫_{0}^{T_{0}} 𝒮 (t + T_{0}, T_{0})𝒮 (T_{0}, θ)f (θ, x(θ, x_{0}), ∫_{0}^{θ} g(θ, s, x(s, x_{0}))ds) dθ

+ ∑_{0≤τ_{k}<T_{0}} 𝒮 (t + T_{0}, T_{0})𝒮 (T_{0}, τ_{k}^{+}) c_{k}

+ ∫_{T_{0}}^{t+T_{0}} 𝒮 (t + T_{0}, θ)f (θ, x(θ, x_{0}), ∫_{0}^{θ} g(θ, s, x(s, x_{0}))ds) dθ

+ ∑_{T_{0}≤τ_{k+δ}<t+T_{0}} 𝒮 (t + T_{0}, τ_{k+δ}^{+}) c_{k+δ}

= 𝒮 (t, 0){𝒮 (T_{0}, 0)x_{0} + ∫_{0}^{T_{0}} 𝒮 (T_{0}, θ)f (θ, x(θ, x_{0}), ∫_{0}^{θ} g(θ, s, x(s, x_{0}))ds) dθ

+ ∑_{0≤τ_{k}<T_{0}} 𝒮 (T_{0}, τ_{k}^{+}) c_{k+δ}}

+ ∫_{0}^{t} 𝒮 (t + T_{0}, θ + T_{0})f (θ + T_{0}, x(θ + T_{0}, x_{0}), ∫_{0}^{θ+T_{0}} g(θ + T_{0}, s, x(s, x_{0}))ds) dθ

+ ∑_{T_{0}≤τ_{k+δ}<t+T_{0}} 𝒮 (t + T_{0}, τ_{k+δ}^{+}) c_{k+δ}

= 𝒮 (t, 0)x(T_{0})

+ ∫_{0}^{t} 𝒮 (t + T_{0}, θ + T_{0})f (θ + T_{0}, x(θ + T_{0}, x_{0}), ∫_{T_{0}}^{θ+T_{0}} g(θ + T_{0}, s, x(s, x_{0}))ds) dθ

+ ∑_{T_{0}≤τ_{k+δ}<t+T_{0}} 𝒮 (t + T_{0}, τ_{k+δ}^{+}) c_{k+δ}

= 𝒮 (t, 0)x(T_{0}) + ∫_{0}^{t} 𝒮 (t, θ)f (θ, x(θ + T_{0}, x_{0}), ∫_{0}^{θ} g(θ + T_{0}, s + T_{0}, x(s + T_{0}, x_{0}))ds) dθ

+ ∑_{T_{0}≤τ_{k+δ}<t+T_{0}} 𝒮 (t + T_{0}, τ_{k+δ}^{+}) c_{k+δ}

= 𝒮 (t, 0)x(T_{0}) + ∫_{0}^{t} 𝒮 (t, θ)f(θ, y(θ, y(0)), ∫_{0}^{θ} g(θ, s, y(s, y(0)))ds)dθ + ∑_{0≤τ_{k}<t} 𝒮 (t, τ_{k}^{+}) c_{k}.
= 𝒮 (t, 0)y(0) + ∫_{0}^{t} 𝒮 (t, θ)f(θ, y(θ, y(0), ∫_{0}^{θ} g(θ, s, y(s, y(0)))ds)dθ + ∑_{0≤τ_{k}<t} 𝒮 (t, τ_{k}^{+}) c_{k}.

This implies that y(·, y(0)) is a P C-mild solution of Cauchy problem (4.2) with initial value y(0) = x_{0}. Thus the uniqueness implies that x(·, x_{0}) = y(·, y(0)) = x(· + T_{0}, x_{0}), so that x(·, x_{0}) is a T_{0}- periodic. □

Next, we show that P defined by (4.5) is a continuous and compact operator.

Lemma 4.3: The operator P is a continuous and compact operator.

EJQTDE, 2009 No. 4, p. 10

Proof. (1) Show that P is a continuous operator on X.

Let x̄, ȳ ∈ Ξ ⊂ X, where Ξ is a bounded subset of X. Suppose x(·, x̄) and x(·, ȳ) are the P C-mild solutions of Cauchy problem (4.2) corresponding to the initial value x̄ and ȳ ∈ X respectively given by

x(t, x̄) = 𝒮 (t, 0)x̄ + ∫_{0}^{t} 𝒮 (t, θ)f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)dθ + ∑_{0≤τ_{k}<t}𝒮 (T_{0}, τ_{k}^{+}) c_{k};
x(t, ȳ) = 𝒮 (t, 0)ȳ + ∫_{0}^{t} 𝒮 (t, θ)f(θ, x(θ, ȳ), ∫_{0}^{θ} g(θ, s, x(s, ȳ))ds)dθ + ∑_{0≤τ_{k}<t}𝒮 (T_{0}, τ_{k}^{+}) c_{k}.

Thus, we obtain

∥x(t, x̄)∥ ≤ M_{T_{0}}∥x̄∥ + (1 + M_{g}T_{0})M_{T_{0}}M_{f}T_{0} + M_{T_{0}} ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥ + M_{T_{0}}M_{f} ∫_{0}^{t} ∥x(θ, x̄)∥dθ
+M_{T_{0}}M_{f}M_{g}T_{0} ∫_{0}^{t} ∥x(s, x̄)∥ds
≤ a_{0} + M_{T_{0}}M_{f} ∫_{0}^{t} ∥x(θ, x̄)∥dθ + M_{T_{0}}M_{f}M_{g}T_{0} ∫_{0}^{t} ∥x_{s,x̄}∥_{B}ds,

and

∥x(t, ȳ)∥ ≤ M_{T_{0}}∥ȳ∥ + (1 + M_{g}T_{0})M_{T_{0}}M_{f}T_{0} + M_{T_{0}} ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥ + M_{T_{0}}M_{f} ∫_{0}^{t} ∥x(θ, ȳ)∥dθ
+M_{T_{0}}M_{f}M_{g}T_{0} ∫_{0}^{t} ∥x(s, ȳ)∥ds
≤ b_{0} + M_{T_{0}}M_{f} ∫_{0}^{t} ∥x(θ, ȳ)∥dθ + M_{T_{0}}M_{f}M_{g}T_{0} ∫_{0}^{t} ∥x_{s,ȳ}∥_{B}ds

where ∥x_{s,x̄}∥_{B} = sup_{0≤ξ≤s} ∥x(ξ, x̄)∥ and ∥x_{s,ȳ}∥_{B} = sup_{0≤ξ≤s} ∥x(ξ, ȳ)∥.

By Lemma 3.1, one can verify that there exist constants M_{1}^{∗} and M_{2}^{∗} > 0 such that

∥x(t, x̄)∥ ≤ M_{1}^{∗} and ∥x(t, ȳ)∥ ≤ M_{2}^{∗}.

Let ρ = max{M_{1}^{∗}, M_{2}^{∗}} > 0, then ∥x(·, x̄)∥, ∥x(·, ȳ)∥ ≤ ρ which imply that they are locally bounded.

By assumptions [F1], [F2], [G1], [G2] and (2) of Lemma 4.1, we obtain

∥x(t, x̄) − x(t, ȳ)∥

≤ ∥𝒮 (t, 0)∥∥x̄ − ȳ∥

+ ∫_{0}^{t} ∥𝒮 (t, θ)∥ ∥f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds) − f(θ, x(θ, ȳ), ∫_{0}^{θ} g(θ, s, x(s, ȳ))ds)∥ dθ

≤ M_{T_{0}}∥x̄ − ȳ∥ + M_{T_{0}}L_{f}(ρ) ∫_{0}^{t} ∥x(θ, x̄) − x(θ, ȳ)∥dθ + M_{T_{0}}L_{f}(ρ)L_{g}(ρ)T_{0} ∫_{0}^{t} ∥x_{s,x̄} − x_{s,ȳ}∥_{B}ds.

By Lemma 3.1 again, one can verify that there exists constant M_{3}^{∗} > 0 such that

∥x(t, x̄) − x(t, ȳ)∥ ≤ M_{3}^{∗}M_{T_{0}}∥x̄ − ȳ∥ ≡ L∥x̄ − ȳ∥, for all t ∈ [0, T_{0}],

which implies that

∥P (x̄) − P (ȳ)∥ = ∥x(T_{0}, x̄) − x(T_{0}, ȳ)∥ ≤ L∥x̄ − ȳ∥.

Hence, P is a continuous operator on X.

EJQTDE, 2009 No. 4, p. 11

(2) Verify that P takes a bounded set into a precompact set in X. Let Γ is a bounded subset of X. Define K = P Γ = {P (x̄) ∈ X | x̄ ∈ Γ}. For 0 < ε ≤ T_{0}, define

K_{ε} = P_{ε}Γ = 𝒮 (T_{0}, T_{0} − ε){x(T_{0} − ε, x̄) | x̄ ∈ Γ}.

Next, we show that K_{ε} is precompact in X. In fact, for x̄ ∈ Γ fixed, we have ∥x(T_{0} − ε, x̄)∥

≤ ∥𝒮 (T_{0} − ε, 0)x̄∥ + ∫_{0}^{T_{0}−ε} ∥𝒮 (T_{0} − ε, θ)f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)∥ dθ

+ ∑_{0≤τ_{k}<T_{0}−ε} ∥𝒮 (T_{0} − ε, τ_{k}^{+}) c_{k}∥

≤ M_{T_{0}}∥x̄∥ + M_{T_{0}}M_{f}T_{0}(1 + M_{g}T_{0}) + M_{T_{0}}M_{f} ∫_{0}^{T_{0}} ∥x(θ, x̄)∥dθ

+M_{T_{0}} ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥ + M_{T_{0}}M_{f}M_{g}T_{0} ∫_{0}^{T_{0}} ∥x_{s,x̄}∥_{B}ds

≤ M_{T_{0}}∥x̄∥ + M_{T_{0}}M_{f}T_{0}(1 + M_{g}T_{0}) + (1 + M_{g}T_{0})M_{T_{0}}M_{f}T_{0}ρ + M_{T_{0}} ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥.

This implies that the set {x(T_{0} − ε, x̄) | x̄ ∈ Γ} is totally bounded.

By virtue of (5) of Lemma 2.3, 𝒮 (T_{0}, T_{0} − ε) is a compact operator. Thus, K_{ε} is precompact in X.

On the other hand, for arbitrary x̄ ∈ Γ,

P_{ε}(x̄) = 𝒮 (T_{0}, 0)x̄ + ∫_{0}^{T_{0}−ε} 𝒮 (T_{0}, θ)f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)dθ + ∑_{0≤τ_{k}<T_{0}−ε} 𝒮 (T_{0}, τ_{k}^{+}) c_{k}.

Thus, combined with (4.5), we have

∥P_{ε}(x̄) − P (x̄)∥ ≤ ∥ ∫_{0}^{T_{0}−ε} 𝒮 (T_{0}, θ)f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)dθ

− ∫_{0}^{T_{0}} 𝒮 (T_{0}, θ)f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)dθ∥

+ ∥ ∑_{0≤τ_{k}<T_{0}−ε} 𝒮 (T_{0}, τ_{k}^{+}) c_{k} − ∑_{0≤τ_{k}<T_{0}} 𝒮 (T_{0}, τ_{k}^{+}) c_{k}∥

≤ ∫_{T_{0}−ε}^{T_{0}} ∥𝒮 (T_{0}, θ)∥ ∥f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)∥ dθ

+M_{T_{0}} ∑_{T_{0}−ε≤τ_{k}<T_{0}} ∥c_{k}∥ for τ_{k} ∈ [T_{0} − ε, T_{0})

≤ M_{T_{0}}M_{f}(1 + M_{g}T_{0})(1 + ρ)ε + M_{T_{0}} ∑_{T_{0}−ε≤τ_{k}<T_{0}} ∥c_{k}∥ for τ_{k} ∈ [T_{0} − ε, T_{0}).

If there are no impulsive points between [T_{0} − ε, T_{0}], it is clear that ∑_{T_{0}−ε≤τ_{k}<T_{0}} ∥c_{k}∥ being zero.

As a result, it is showing that the set K can be approximated to an arbitrary degree of accuracy by a precompact set K_{ε}. Hence K itself is precompact set in X. That is, P takes a bounded set into a precompact set in X. As a result, P is a compact operator. □

EJQTDE, 2009 No. 4, p. 12

In order to use Leray-Schauder fixed pointed theorem to examine the operator P has a fixed point, we have to make the assumptions [F2] and [G2] a little stronger as following.

[F2′]: There exist constant N_{f} > 0 and 0 < λ < 1 such that

∥f(t, x, y)∥ ≤ N_{f} (1 + ∥x∥^{λ} + ∥y∥^{λ}) for all x, y ∈ X.

[G2′]: There exists a positive constant N_{g} > 0 and 0 < λ < 1 such that

∥g(t, s, x)∥ ≤ N_{g}(1 + ∥x∥^{λ}) for all x ∈ X.

Now, we can give the main results in this paper.

Theorem 4.2: Assumptions [A1], [A2], [A3], [B], [C], [F1], [F2′], [F3], [G1], [G2′], [G3] hold. Then system (4.1) has a T_{0}-periodic P C-mild solution on [0, +∞).

Proof. By virtue of (5) of Lemma 2.3, 𝒮 (T_{0}, 0) is a compact operator on infinite dimensional space X. Thus, 𝒮 (T_{0}, 0) ≠ αI, α ∈ ℝ. Then, there exists β > 0 such that ∥[σ𝒮 (T_{0}, 0) − I]x̄∥ ≥ β∥x̄∥ for σ ∈ [0, 1]. In fact, define Π_{σ} = I−σ𝒮 (T_{0}, 0), σ ∈ [0, 1], and Π_{σ} : [0, 1] → £_{b}(X) and h(σ) = ∥Π_{σ}∥ : [0, 1] → ℝ^{+}. It is obvious that h ∈ C([0, 1]; ℝ^{+}). Thus, there exist σ_{∗} ∈ [0, 1] and β > 0 such that

h(σ_{∗}) = min{h(σ) | σ ∈ [0, 1]} ≥ β > 0.

If not, there exits σ̄ ∈ [0, 1] such that h(σ̄) = 0. We can assert that σ̄ ≠ 0 unless h(σ̄) = 1. Thus, for σ̄ ∈ (0, 1],

𝒮 (T_{0}, 0) = \frac{1}{σ̄} I where \frac{1}{σ̄} ≥ 1,

which is a contradiction with 𝒮 (T_{0}, 0) ≠ αI, α ∈ ℝ.

By Theorem 4.1, for fixed x̄ ∈ X, the Cauchy problem (4.2) corresponding to the initial value x(0) = x̄ has the P C-mild solution x(·, x̄). By Lemma 4.3, the operator P defined by (4.5), is compact.

According to Leray-Schauder fixed point theory, it suffices to show that the set {x̄ ∈ X | x̄ = σP x̄, σ ∈ [0, 1]} is a bounded subset of X. In fact, let x̄ ∈ {x̄ ∈ X | x̄ = σP x̄, σ ∈ [0, 1]}, we have

β \left∥{ \atop x̄}\right∥ ≤ ∥[σ𝒮 (T_{0}, 0) − I]x̄∥
= σ ∫_{0}^{T_{0}} ∥𝒮 (T_{0}, θ)∥ ∥f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)∥ dθ + σ ∑_{0≤τ_{k}<T_{0}} ∥𝒮 (T_{0}, τ_{k}^{+})∥ ∥c_{k}∥.

By assumptions [F2′] and [G2′],

\left∥{ \atop x̄}\right∥ ≤ \frac{σ}{β} ∫_{0}^{T_{0}} ∥𝒮 (T_{0}, θ)∥ ∥f(θ, x(θ, x̄), ∫_{0}^{θ} g(θ, s, x(s, x̄))ds)∥ dθ + \frac{σ}{β} ∑_{0≤τ_{k}<T_{0}} ∥𝒮 (T_{0}, τ_{k}^{+})∥ ∥c_{k}∥

≤ \frac{σ}{β} M_{T_{0}} ((N_{f} + N_{g}T_{0})T_{0} + N_{f} ∫_{0}^{T_{0}} ∥x(θ, x̄)∥^{λ}dθ + N_{f}N_{g}T_{0} ∫_{0}^{T_{0}} ∥x_{s,x̄}∥_{B}^{λ}dθ + ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥) .

For t ∈ [0, T_{0}], we obtain

∥x(t, x̄)∥

≤ M_{T_{0}}∥x̄∥ + M_{T_{0}} ((N_{f} + N_{g}T_{0})T_{0} + N_{f} ∫_{0}^{t} ∥x(θ, x̄)∥^{λ}dθ + N_{f}N_{g}T_{0} ∫_{0}^{t} ∥x_{s,x̄}∥_{B}^{λ}dθ + ∑_{0≤τ_{k}<t} ∥c_{k}∥)

≤ \frac{σ}{β} M_{T_{0}}^{2} ((N_{f} + N_{g}T_{0})T_{0} + N_{f} ∫_{0}^{T_{0}} ∥x(θ, x̄)∥^{λ}dθ + N_{f}N_{g}T_{0} ∫_{0}^{T_{0}} ∥x_{s,x̄}∥_{B}^{λ}dθ + ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥)

+M_{T_{0}} ((N_{f} + N_{g}T_{0})T_{0} + N_{f} ∫_{0}^{t} ∥x(θ, x̄)∥^{λ}dθ + N_{f}N_{g}T_{0} ∫_{0}^{t} ∥x_{s,x̄}∥_{B}^{λ}dθ + ∑_{0≤τ_{k}<t} ∥c_{k}∥)

≤ (\frac{σ}{β} M_{T_{0}} + 1) M_{T_{0}} ((N_{f} + N_{g}T_{0})T_{0} + ∑_{0≤τ_{k}<T_{0}} ∥c_{k}∥) + M_{T_{0}}N_{f} ∫_{0}^{t} ∥x(θ, x̄)∥^{λ}dθ

+\frac{σ}{β} M_{T_{0}}^{2}N_{f} ∫_{0}^{T_{0}} ∥x(θ, x̄)∥^{λ}dθ + M_{T_{0}}N_{f}N_{g}T_{0} ∫_{0}^{t} ∥x_{s,x̄}∥_{B}^{λ}dθ + \frac{σ}{β} M_{T_{0}}^{2}N_{f}N_{g}T_{0} ∫_{0}^{T_{0}} ∥x_{s,x̄}∥_{B}^{λ}dθ.

By Lemma 3.2, there exists M^{∗} > 0 such that

∥x(t, x̄)∥ ≤ M^{∗} for t ∈ [0, T_{0}].

This implies that ∥x(0, x̄)∥ = ∥x̄∥ ≤ M^{∗} for all x̄ ∈ {x̄ ∈ X | x̄ = σP x̄, σ ∈ [0, 1]}.

Thus, by Leray-Schauder fixed pointed theory, there exits x_{0} ∈ X such that P x_{0} = x_{0}. By Lemma 4.2, we know that the P C-mild solution x(·, x_{0}) of Cauchy problem (4.2) corresponding to the initial value x(0) = x_{0}, is just T_{0}-periodic. Therefore x(·, x_{0}) is a T_{0}-periodic P C-mild solution of system (4.1). □

5. An Example

Consider the following non-autonomous integrodifferential periodic population evolution equation with periodic impulsive perturbations

(5.1) \{ \begin{matrix}\frac{∂}{∂t}x(r, t) + sin t\frac{∂}{∂r} x(r, t) = −0.2 sin tx(r, t) + x^{\frac{2}{3}} (r, t) \\ + ∫_{0}^{t} ψ(s)(1 + sin(t − s))\sqrt{3x^{\frac{2}{3}} (r, s) + 2}ds, \\ r ∈ Ω = (0, r_{m}), t > s and t, s ∈ (0, 2π] \ \{\frac{1}{2}π, π, \frac{3}{2}π\}, \\ Δx(r, τ_{i}) = x(r, τ_{i}^{+}) − x(r, τ_{i}^{−}) = \{ \begin{matrix}0.05Ix(r, τ_{i}), i = 1, \\ −0.05Ix(r, τ_{i}), i = 2, \\ 0.05Ix(r, τ_{i}), i = 3, \end{matrix} r ∈ Ω, τ_{i} = \frac{i}{2}π, i = 1, 2, 3, \\ x(r, 0) = x_{0}(r), r ∈ Ω, \\ x(0, t) = φ_{0}(t), t ∈ (0, 2π) \ \{\frac{1}{2}π, π, \frac{3}{2}π\}, \\ x(r, 0) = x(r, 2π). \end{matrix}

where t denotes time, r denotes age, r_{m} is the highest age ever attained by individuals of the population, x(r, t) is called age density function, x_{0}(r) is an initial age density of the people, φ_{0}(t) is the absolute infant fertility rate of population.

EJQTDE, 2009 No. 4, p. 14

Set X = L^{2}(0, r_{m}). Let A(t) be defined by

(A(t)φ)(r) = sin t( − \frac{d}{dr} φ(r) − 0.2φ(r)), for arbiartry φ ∈ D(A(t))

where

D(A(t)) = {φ | φ, A(t)φ ∈ L^{2}(0, r_{m}); φ(0) = φ_{0}}.

It is well known that operator A ≡ −\frac{d}{dr} φ(r) − 0.2φ(r) with t > r_{m} is the infinitesimal generator of a compact semigroup {T (t) , t ≥ 0} on X with domain D(A) = {φ | φ, Aφ ∈ L^{2}(0, r_{m}); φ(0) = φ_{0}}. Thus, one can immediately obtain that {A(t), t > r_{m}} can determine a compact, 2π-periodic evolutionary process {U (t, θ), t ≥ θ ≥ 0}.

Define x(·)(r) = x(r, ·), sin(·)(r) = sin(r, ·), f (·, x(·), ∫_{0}^{·} g(·, s, x)ds) (r) = x^{\frac{2}{3}} (·)(r) + ∫_{0}^{t} ψ(s)(1 + sin(· − s))\sqrt{3x^{\frac{2}{3}} (·) + 2}ds(r) where ψ(· + 2π) = ψ(·) ∈ L_{loc}^{1}([0, +∞); X), ∫_{0}^{2π} ψ(s)(1 + sin(t − s))\sqrt{3x^{\frac{2}{3}} (t) + 2}ds = 0, and

B_{i} = \{ \begin{matrix}0.05I, i = 1, \\ −0.05I, i = 2, \\ 0.05I, i = 3. \end{matrix}

Thus we formulate (5.1) as the following abstract integrodifferential impulsive periodic system

(5.2)  \{ \begin{matrix}ẋ(t) = A(t)x(t) + f (t, x, ∫_{0}^{t} g(t, s, x)ds) , t ∈ (0, 2π] \ \{\frac{1}{2}π, π, \frac{3}{2}π\}, \\ Δx (\frac{i}{2}π) = B_{i}x (\frac{i}{2}π),  i = 1, 2, 3, \\ x(0) = x(2π). \end{matrix}

It satisfies all the assumptions given in Theorem 4.2, our results can be used to system (5.1). That is, system (5.1) has a 2π-periodic P C-mild solution x_{2π}(r, ·) ∈ P C_{2π} ([0 + ∞); L^{2}(0, r_{m})), where

P C_{2π}([0, +∞); L^{2}(0, r_{m})) ≡ {x ∈ P C ([0, +∞); L^{2}(0, r_{m})) | x(t) = x(t + 2π), t ≥ 0};

References

[1] H. Amann, Periodic solutions of semilinear parabolic equations, Nonlinear Anal.: A collection of papers in Honour of Erich Rothe, Academic Press, New York 1978, 1–29.

[2] N. U. Ahmed, Semigroup theory with applications to system and control, Longman Scientific Technical, New York, 1991.

[3] N. U. Ahmed, Optimal impulsive control for impulsive systems in Banach space, Int. J. Differential Equations Appl., Vol. 1, pp. 37-52, 2000.

[4] N. U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach space, Mathematical Anal., Vol. 8, pp. 261-274, 2001.

[5] N. U Ahmed, K. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Anal., Vol. 54, pp. 907-925, 2003.

[6] N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach space, SIAM J. Control Optimal, Vol. 42, pp. 669-685, 2003.

[7] D. D. Bainov, P. S. Simeonov, Impulsive differential equations: periodic solutions and applications, New York, Longman Scientific and Technical Group. Limited, 1993.

[8] D. Bahuguna and J. Dabas, Existence and uniqueness of a solution to a partial integro-differential

equation by the method of lines, E. J. Qualitative Theory of Diff. Equ., No. 4, pp. 1-12, 2008.

EJQTDE, 2009 No. 4, p. 15

[9] S. H. Hong and Z. Qiu, Existence of solutions of nth order impulsive integro-differential equations in Banach spaces, E. J. Qualitative Theory of Diff. Equ., No. 22, pp. 1-11, 2008.

[10] D. Guo and X. Liu, Extremal solutions of nonlinear impulsive integro-differential equations in Banach space, J. Math. Anal. Appl., Vol. 177, pp. 538-552, 1993.

[11] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore-London, 1989.

[12] X. Liu, Impulsive stabilization and applications to population growth models, Rocky Mountain J. Math., Vol. 25, pp. 381-395, 1995.

[13] J. Liu, Bounded and periodic solutions of differential equations in Banach space, J. Appl. Math. Comput., Vol. 65, pp. 141-150, 1994.

[14] J. Y. Park, Y.C. Kwunm J.M. Jeong, Existence of periodic solutions for delay evolution integrodifferential equations, J. Math. and Comput. Model., Vol. 65, pp. 597-603, 2004.

[15] I. K. Purnaras, On the existence of solutions to some nonlinear integrodifferential equations with delays, E. J. Qualitative Theory of Diff. Equ., No. 22, pp. 1-21, 2007.

[16] Y. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, E. J. Qualitative Theory of Diff. Equ., No. 16, pp. 1-10, 2007.

[17] J. R. Wang, Linear impulsive periodic system on Banach space, in Proceedings of the 4th International Conference on Impulsive and Hybrid Dynamical Systems (ICIDSA’07), Vol. 5, pp. 20-25, Nanning, China, July 2007.

[18] J. R. Wang, X. Xiang, and W. Wei, Linear impulsive periodic system with time-varying generating operators on Banach space, Advances in Difference Equations, Vol. 3, Article ID 26196, 16 pages, 2007.

[19] J. R. Wang, X. Xiang, W. Wei, and Q. Chen, Existence and global asymptotical stability of periodic solution for the T -periodic logistic system with time-varying generating operators and T_{0}-periodic impulsive perturbations on Banach spaces, Discrete Dynamics in Nature and Society, Vol. 2008, Article ID 524945, 16 pages, 2008.

[20] J. R. Wang, X. Xiang, W. Wei, and Q. Chen, Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces, Fixed Point Theory and Applications, Vol. 2008, Article ID 401947, 15 pages, 2008.

[21] J. R. Wang, X. Xiang, and W. Wei, Existence of periodic solution for semilinear periodic logistic systems with periodic impulsive perturbations on Banach space, in Proceedings of the 6th conference of Biomathematics, Advance in BioMathematics, Vol. 1, pp. 288-291, Tai’An, China, July 2008.

[22] J. R. Wang, X. Xiang, and W. Wei, Periodic solutions of semilinear impulsive periodic system with time-varying generating operators on Banach space, Mathematical Problems in Engineering, Vol. 2008, Article ID 183489, 15 pages, 2008.

[23] J. R. Wang, X. Xiang, W. Wei, and Q. Chen, The generalized gronwall inequality and its application to periodic solutions of integrodifferential impulsive periodic system on Banach space, Journal of Inequalities and Applications, Vol. 2008, Article ID 430521, 22 pages, 2008.

[24] P. Sattayatham, S. Tangmanee and Wei Wei, On periodic solution of nonlinear evolution equations in Banach spaces, J. Math. Anal. Appl., Vol. 276, pp. 98-108, 2002.

[25] X. Xiang and N. U. Ahmed, Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal., Vol. 18, pp. 1063-1070, 1992.

[26] W. Wei, X. Xiang and Y. Peng, Nonlinear impulsive integro-differential equation of mixed type and optimal controls, Optimization, Vol. 55, pp. 141-156, 2006.

[27] X. Xiang and W. Wei, Mild solution for a class of nonlinear impulsive evolution inclusion on Banach space, Southeast Asian Bulletin of Mathematics, Vol. 30, pp. 367-376, 2006.

[28] X. Xiang, W. Wei and Y. Jiang, Strongly nonlinear impulsive system and necessary conditions of optimality, Dynamics of Continuous, Discrete and Impulsive Systems, A: Mathematical Analysis, Vol. 12, pp. 811-824, 2005.

[29] X. L. Yu, X. Xiang and W. Wei, Solution bundle for class of impulsive differential inclusions on Banach spaces, J. Math. Anal. Appl. Vol. 327, pp. 220-232, 2007.

[30] Y. Peng, X. Xiang and W. Wei, Nonlinear impulsive integro-differential equations of mixed type with time-varying generating operators and optimal controls, Dynamic Systems and Applications, Vol. 6, pp. 481-496, 2007.

[31] Y. Peng and X. Xiang, Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls, Journal of Industrial and Management Optimization, Vol. 4, pp. 17-32, 2008.

[32] T. Yang, Impulsive control theory, Springer-Verlag Berlin Heidelberg, 2001.

(Received October 4, 2008)