Existence of Almost Periodic Solutions for a Class of Abstract Impulsive Differential Equations

Abstract

We study the existence of piecewise almost periodic solutions for a class of abstract impulsive semilinear differential equations.

1. Introduction

Due to their numerous applications to model the dynamics of evolving processes arising in mechanics, electrical engineering, medicine, population dynamics, among many others important areas of technology and science, the theory of impulsive differential equations has attracted the attention of researchers. It is well known that there are many evolutionary processes in which the system is subject to abrupt changes of state at certain moments of time between intervals of continuous evolution. Such changes can be approximated as being instantaneous changes at states, or in the form of impulses. We refer the reader to [1–3] for numerous examples and applications.

Several of the systems mentioned in the preceding paragraph can be modeled by impulsive differential equations. The literature related to ordinary differential equations with impulses is very extensive and we refer the reader to [2, 4] and the references therein. Impulsive ordinary differential equations appear frequently in applications involving nonlinear systems. It is worth to point out here that the analysis of such systems with discontinuities is different from that for continuous systems (see for instance [5]). This has been a special motivation to attract the attention of authors to the theory of systems with impulses. For this reason the theory of impulsive differential equations has drawn the attention of many authors in recent years, see for instance [6–11].

On the other hand, the study of the existence of periodic solutions, as well as its numerous generalizations to almost periodic solutions, asymptotically almost periodic solutions, almost automorphic solutions, asymptotically almost automorphic solutions, pseudo almost periodic solutions, and so forth, is one of the most attracting topics in the qualitative theory of differential equations due both to its mathematical interest as to their applications in various areas of applied science. Likewise, the existence of periodic and almost periodic solutions of impulsive ordinary differential equations has been considered by many authors. The reader can see [4, 12–19]. Recently Stamov [16] have investigated the existence of almost periodic solutions for the Lasota-Wazewska model and the exponential stability of these solutions, while Ahmad and Stamov [14] have investigated the existence of almost periodic processes corresponding to competitive nonautonomous Lotka-Volterra systems described by integrodifferential equations with infinite delay. However, the study of almost periodic solutions of impulsive partial differential equations are not sufficiently considered in the literature. This fact is the main motivation of the present work.

In this work we are concerned with nonautonomous abstract differential equations. Let 𝕏 be a Banach space endowed with the norm ‖⋅‖_𝕏. Specifically, we study the existence of almost periodic mild solutions for a class of impulsive differential equation of the form𝑢^(𝑡)=𝐴(𝑡)𝑢(𝑡)+Δ𝐹𝑢(𝑢𝑡_𝑖(𝑡))+𝑓(𝑡),𝑡∈ℝ,𝑡≠𝑡_𝑖,𝑖∈ℤ,=𝐼_𝑖𝑢𝑡_𝑖,(1.1) where 𝑡_𝑖∈ℝ for 𝑖∈ℤ and 𝐴(𝑡)∶𝐷(𝐴(𝑡))⊆𝕏→𝕏, 𝑡∈ℝ, are closed linear operators that satisfy some properties that will be established later. Moreover, 𝑓∶ℝ→𝕏 and 𝐹,𝐼_𝑖∶𝕏→𝕏 for 𝑖∈ℤ are appropriate functions. In addition, the symbol Δ𝜉(𝑡) represents the jump of the function 𝜉 at 𝑡, which is defined by Δ𝜉(𝑡)=𝜉(𝑡⁺)−𝜉(𝑡⁻), where the notations 𝜉(𝑡⁺) and 𝜉(𝑡⁻) represent, respectively, the right hand side and the left hand side limits of the function 𝜉 at 𝑡.

Let (𝕏,‖⋅‖_𝕏) and (𝕐,‖⋅‖_𝕐) be Banach spaces, the symbol ℒ(𝕐,𝕏) stands for the Banach space formed by all bounded linear operators from 𝕐 into 𝕏 endowed with the uniform operator topology. In particular, we abbreviate this notation by ℒ(𝕏) when 𝕐=𝕏. Moreover, for 𝑟>0, we denote by 𝐵_𝑟(𝕏) the closed ball consisting of all 𝑥∈𝕏 such that ‖𝑥‖≤𝑟. For a linear operator 𝐴, we denote by 𝜌(𝐴) its resolvent set and, for 𝜆∈𝜌(𝐴), 𝑅(𝜆,𝐴)=(𝜆𝐼−𝐴)⁻¹ is the resolvent operator. In addition, we denote by AP(𝕏) the space of almost periodic functions from ℝ into 𝕏 in the Bohr sense. It is well known that AP(𝕏) endowed with the norm of uniform convergence is a Banach space. For details about the almost periodic functions we refer to [20].

We now give a brief summary of this work. In Section 2 we recall some definitions and properties that are needed to establish our results. In Section 3, we establish our results about the existence of almost periodic mild solutions to the impulsive Cauchy problem (1.1). Finally, in Section 4 we apply our abstract results to a concrete system with impulses.

2. Preliminaries

We begin by defining the countable subsets of ℝ where the impulsive effects are concentrated.

Let ℬ be the set consisting all real sequences 𝑇={𝑡_𝑖}_𝑖∈ℤ such that 𝛼=inf_𝑖∈ℤ(𝑡_𝑖+1−𝑡_𝑖)>0. It is immediate that this condition implies that lim_𝑖→∞𝑡_𝑖=∞ and lim_{𝑖→−∞}𝑡_𝑖=−∞.

Remark 2.1. If 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ and 𝑡_𝑖<𝑡≤𝑡_𝑖+1, then 𝑡−𝑡_𝑘=𝑡−𝑡_𝑖+𝑡_𝑖−𝑡_𝑘≥𝑡−𝑡_𝑖+(𝑖−𝑘)𝛼,(2.1) for 𝑘≤𝑖.

Definition 2.2. The sequence 𝑇={𝑡_𝑘}_𝑘∈ℤ∈ℬ is said to be almost periodic if for each 𝜖>0 there is a relatively dense set 𝒬_𝜖 in ℝ having the following property: for each 𝜏∈𝒬_𝜖 there is an integer 𝑞∈ℤ such that |𝑡_𝑖+𝑞−𝑡_𝑖−𝜏|<𝜖 for all 𝑖∈ℤ.

Given a sequence 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ, we consider 𝑡^𝑗𝑖=𝑡_𝑖+𝑗−𝑡_𝑖, for 𝑖,𝑗∈ℤ.

Definition 2.3 (see [4]). The sequence 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ is said to be uniformly almost periodic if for each 𝜖>0 there exists a relatively dense set 𝒬_𝜖 in ℤ such that ||𝑡^{𝑗𝑖+𝑞}−𝑡^𝑗𝑖||<𝜖,(2.2) for all 𝑖,𝑗∈ℤ and 𝑞∈𝒬_𝜖.

Remark 2.4. It has been established in [4, Lemma 27] that if 𝑇 is uniformly almost periodic, then it is almost periodic.

For 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ fixed, let 𝒫𝒞_𝑇(ℝ,𝕏) be the space consisting of all piecewise continuous functions 𝑥∶ℝ→𝕏 such that 𝑥(⋅) is continuous at 𝑡 for every 𝑡∉𝑇 and 𝑥(𝑡_𝑖)=𝑥(𝑡^−𝑖) for all 𝑖∈ℤ, where, as it is usual, we have denoted by 𝑥(𝑡⁻) the left hand limit of 𝑥(⋅) at 𝑡. Hence a function 𝑥∈𝒫𝒞_𝑇(ℝ,𝕏) if 𝑥 is left continuous on ℝ, right continuous on ℝ⧵𝑇 and it can have jump discontinuities on the right hand side at the points of 𝑇.

Definition 2.5 (see [4]). Let 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ. The function 𝜑∈𝒫𝒞_𝑇(ℝ,𝕏) is said to be 𝑇-piecewise almost periodic if the following conditions are fulfilled.(i)The set 𝑇 is almost periodic. (ii)For every 𝜖>0 there exists 𝛿>0 such that ‖𝑥(𝑡)−𝑥(𝑡^)‖<𝜖 for all 𝑡,𝑡^∈(𝑡_𝑖,𝑡_𝑖+1) such that |𝑡−𝑡^|<𝛿 and some 𝑖∈ℤ. (iii)For each 𝜖>0 there exists a relatively dense set Ω_𝜖 in ℝ such that if 𝜏∈Ω_𝜖, then ‖𝜑(𝑡+𝜏)−𝜑(𝑡)‖_𝕏<𝜖,(2.3) for all 𝑡∈ℝ satisfying the condition |𝑡−𝑡_𝑖|>𝜖 for all 𝑖∈ℤ.

For a fixed sequence 𝑇={𝑡_𝑖}_𝑖∈ℤ as in the Definition 2.2, we denote by AP_𝑇(𝕏) the space formed for all 𝑇-piecewise almost periodic function. It is well known that the space AP_𝑇(𝕏) endowed with the norm of the uniform convergence ‖⋅‖_∞ is a Banach space.

For more details about the properties of piecewise almost periodic functions we refer the reader to [4]. We only mention here a pair of properties which are essentials for our developments.

Lemma 2.6. Let 𝑔∈AP_𝑇(𝕏). Then ℛ(𝑔) is a relatively compact subset of 𝕏.

Proof. Let 𝜖>0 be fixed. It follows from the Definition 2.5(ii) that there exists 𝛿>0 such that ‖𝑔(𝑡)−𝑔𝑡^‖<𝜖,(2.4) for all 𝑡,𝑡^∈(𝑡_𝑖,𝑡_𝑖+1) such that |𝑡−𝑡^|<𝛿 and 𝑖∈ℤ. We can assume that 𝛿<𝜖. Applying now the Definition 2.5(iii) we infer the existence of 𝐿>0 having the following property. In any interval of length greater or equal that 𝐿 there exists 𝜏 such that ‖𝑔(𝑡+𝜏)−𝑔(𝑡)‖_𝕏<𝛿 for all 𝑡∈ℝ satisfying the condition |𝑡−𝑡_𝑖|≥𝛿 for all 𝑖∈ℤ. Hence, if 𝑡∈(𝑡_𝑖,𝑡_𝑖+1, |𝑡−𝑡_𝑖|>𝛿 and |𝑡−𝑡_𝑖+1|>𝛿), we can select 𝜏∈[−𝑡,𝐿−𝑡] and 𝑔(𝑡)∈𝑔(𝑡+𝜏)+𝐵_𝛿⊆𝑔([0,𝐿])+𝐵_𝛿.(2.5) Since 𝑔 is piecewise continuous, the set 𝐾_𝐿=𝑔([0,𝐿]) is compact. Moreover, if 𝑡∈(𝑡_𝑖,𝑡_𝑖+𝛿), then ‖𝑔(𝑡)−𝑔(𝑡_𝑖+𝛿)‖<𝜖 which implies that 𝑔(𝑡)∈𝐾_𝐿+𝐵_𝜖. A similar property holds when 𝑡∈(𝑡_𝑖+1−𝛿,𝑡_𝑖+1). Consequently, for every 𝑡∈ℝ we have that 𝑔(𝑡)∈𝐾_𝐿+𝐵_𝜖. Since 𝜖 was arbitrarily chosen, this completes the proof.

Definition 2.7 (see [21, Definition 2]). The sequence of maps 𝐼_𝑖∶𝕏→𝕏, 𝑖∈ℤ, is said to be almost periodic if for each 𝑥∈𝕏 and 𝜖>0 there is a relatively dense set 𝑄_𝜖(𝑥) in ℤ such that ‖𝐼_𝑖+𝑞(𝑥)−𝐼_𝑖(𝑥)‖<𝜖 for all 𝑞∈𝑄_𝜖(𝑥) and 𝑖∈ℤ.

The following property follows easily from the Definition 2.7.

Lemma 2.8. Let 𝐼_𝑖∶𝕏→𝕏, 𝑖∈ℤ, be an almost periodic sequence of maps and let 𝐾⊂𝕏 a compact set. Assume that there is a positive constant 𝐿 such that ‖𝐼_𝑖(𝑥)−𝐼_𝑖(𝑦)‖≤𝐿‖𝑥−𝑦‖,(2.6) for all 𝑥∈𝕏 and 𝑖∈ℤ. Then the set of sequences {(𝐼_𝑖(𝑥))_𝑖∈ℤ∶𝑥∈𝐾} is uniformly almost periodic.

Combining Lemma 2.8 with Definition 2.5 we obtain the following property which will be essential for our developments.

Lemma 2.9. Let 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ and let 𝜑∈AP_𝑇(𝕏). Assume that the sequence of vector-valued functions {𝐼_𝑖}_𝑖∈ℤ is almost periodic. If there is a positive constant 𝐿 such that ‖𝐼_𝑖(𝑥)−𝐼_𝑖(𝑦)‖≤𝐿‖𝑥−𝑦‖,(2.7) for all 𝑥∈𝕏 and 𝑖∈ℤ, then {𝐼_𝑖(𝜑(𝑡_𝑖))}_𝑖∈ℤ is an almost periodic sequence.

Proof. To prove this lemma we are going to use the characterization of almost periodic sequences established in [4, Theorem 70]. Let (𝑞^𝑛)_𝑛 be a sequence of integer numbers. Since ℛ(𝜑) is relatively compact, it follows from the Lemma 2.8 that (𝐼_𝑖(𝑥))_𝑖∈ℤ is a uniformly almost periodic sequence for 𝑥∈ℛ(𝜑). Therefore, we can extract a subsequence (𝑞^𝑛)_𝑛 of (𝑞^𝑛)_𝑛 such that 𝐼_𝑖+𝑞^′𝑛(𝑥) is convergent as 𝑛→∞ uniformly for 𝑥∈ℛ(𝜑) and 𝑖∈ℤ. On the other hand, as 𝜑 is a 𝑇-piecewise almost periodic function and inf_𝑖∈ℤ(𝑡_𝑖+1−𝑡_𝑖)>0, it follows from [4, Lemma 37] that the sequence 𝜑(𝑡_𝑖) is almost periodic. Consequently, we can extract a subsequence (𝑞_𝑛)_𝑛 of (𝑞^𝑛)_𝑛 such that 𝜑(𝑡_{𝑖+𝑞𝑛})_𝑛 is convergent as 𝑛→∞ uniformly for 𝑖∈ℤ. This allows us to conclude that ‖‖𝐼≤_{𝑖+𝑞𝑚}‖‖𝜑𝐼𝑡_{𝑖+𝑞𝑚}−𝐼_{𝑖+𝑞𝑛}𝜑𝑡_{𝑖+𝑞𝑛}_{𝑖+𝑞𝑚}‖‖≤𝐿‖‖𝜑𝑡𝜑_{𝑖+𝑞𝑚}𝑡_{𝑖+𝑞𝑚}−𝐼_{𝑖+𝑞𝑚}𝜑𝑡_{𝑖+𝑞𝑛}‖‖+‖‖𝐼_{𝑖+𝑞𝑚}𝜑𝑡_{𝑖+𝑞𝑛}−𝐼_{𝑖+𝑞𝑛}𝜑𝑡_{𝑖+𝑞𝑛}‖‖≤𝐿‖‖𝜑𝑡_{𝑖+𝑞𝑚}−𝜑𝑡_{𝑖+𝑞𝑛}‖‖+‖‖𝐼_{𝑖+𝑞𝑚}𝜑𝑡_{𝑖+𝑞𝑛}−𝐼_{𝑖+𝑞𝑛}𝜑𝑡_{𝑖+𝑞𝑛}‖‖−𝜑𝑡_{𝑖+𝑞𝑛}‖‖+sup_{𝑥∈ℛ(𝜑)}‖‖𝐼_{𝑖+𝑞𝑚}(𝑥)−𝐼_{𝑖+𝑞𝑛}(𝑥)‖‖,(2.8) which implies that (𝐼_{𝑖+𝑞𝑛}(𝜑(𝑡_{𝑖+𝑞𝑛})))_𝑛 is a Cauchy sequence uniformly for 𝑖∈ℤ. This complete the proof.

To study the impulsive system (1.1), we recall briefly some important properties of the classical nonautonomous abstract Cauchy problem. Let 𝐴(𝑡)∶𝐷(𝐴(𝑡))→𝕏 be closed linear operators. We begin by studying the homogeneous problem𝑢^(𝑡)=𝐴(𝑢𝑡)(𝑢𝑠)(𝑡=),𝑥∈𝑡≥𝕏𝑠.,𝑡,𝑠∈ℝ,(2.9) In what follows we introduce the concept of evolution family associated with the problem (2.9).

Definition 2.10. A family of bounded linear operators {𝑈(𝑡,𝑠),𝑡≥𝑠}𝑠𝑢𝑏𝑠𝑒𝑡ℒ(𝕏) is called a strongly continuous evolution family if the following conditions are fulfilled.(a)𝑈(𝑡,𝑡)=𝐼. (b)𝑈(𝑡,𝑠)𝑈(𝑠,𝑟)=𝑈(𝑡,𝑟) for every 𝑟≤𝑠≤𝑡. (c)For each 𝑥∈𝕏 the function (𝑡,𝑠)→𝑈(𝑡,𝑠)𝑥 is continuous.

If for each 𝑠≤𝑡 and each 𝑥∈𝐷(𝐴(𝑠)) we have that 𝑈(𝑡,𝑠)(𝐷(𝐴(𝑠)))⊆𝐷(𝐴(𝑡)), and the function 𝑡→𝑈(𝑡,𝑠)𝑥 is continuously differentiable with respect to 𝑡 and a solution of the problem (2.9), then the evolution family 𝑈(𝑡,𝑠) is said to be generated by {𝐴(𝑡)∶𝑡∈ℝ}.

The existence of an evolution family allows us to establish a variation of constants formula to solve the inhomogeneous problem𝑢^(𝑡)=𝐴(𝑡)𝑢𝑢(𝑡()𝑠+)=𝑓𝑥(𝑡)∈,𝕏𝑡,>𝑠,𝑡,𝑠∈ℝ,(2.10) where 𝑓 is a locally integrable function. We refer the reader to [22] for a discussion about this matter. We only recall here that the function𝑢(𝑡)=𝑈(𝑡,𝑠)𝑢₀+^𝑡𝑠𝑈(𝑡,𝜉)𝑓(𝜉)𝑑𝜉,𝑡≥𝑠,(2.11) is said to be the mild solution of the problem (2.10).

The problem of establishing the conditions under which {𝐴(𝑡)∶𝑡∈ℝ} generates an evolution family that has been studied by several authors. We refer to [23] for a discussion about this subject. In the rest of this work we use the terminology used in [23]. Specifically we consider the following conditions.(AT1)There are constants 𝐾≥0,  𝜔∈ℝ, and 𝜙∈(𝜋/2,𝜋) such that 𝜆∈𝜌(𝐴(𝑡)) and ‖𝑅(𝜆,𝐴(𝑡))‖≤𝐾1+||𝜆−𝜔||,(2.12) for 𝑡∈ℝ and 𝜆∈{𝑤}∪{𝜆∈ℂ∶|arg(𝜆−𝑤)|≤𝜙}. (AT2)There are constants 𝐿≥0 and 𝜇,𝜈∈(0,1] with 𝜇+𝜈>1 such that ||𝜆|^𝜈‖(𝜔𝐼−𝐴(𝑡))𝑅(𝜆+𝜔,𝐴(𝑡))(𝑅(𝜔,𝐴(𝑡))−𝑅(𝜔,𝐴(𝑡)))‖≤𝐿||𝑡−𝑠|^𝜇,(2.13) for 𝑡,𝑠∈ℝ and |arg𝜆|≤𝜙.

The concept of exponential dichotomy is a well-known technique used to study the asymptotic behavior of solutions of differential equations. In the next definition we precise this concept [23, 24].

Definition 2.11. An evolution family 𝑈(𝑡,𝑠), 𝑡,𝑠∈ℝ, 𝑠≤𝑡, on a Banach space 𝕏 has an exponential dichotomy if there are a uniformly bounded and strongly continuous map 𝑃∶ℝ→ℒ(𝕏) such that each 𝑃(𝑡) is a projection, and constants 𝑁,𝜇>0 such that for 𝑡≥𝑠 the following conditions hold.(ex1)𝑈(𝑡,𝑠)𝑃(𝑠)=𝑃(𝑡)𝑈(𝑡,𝑠). (ex2) Let 𝑄(𝑡)=𝐼−𝑃(𝑡). The restriction 𝑈_𝑄(𝑡,𝑠)∶𝑄(𝑠)𝕏→𝑄(𝑡)𝕏 of 𝑈(𝑡,𝑠) is invertible (we set 𝑈_𝑄(𝑠,𝑡)=𝑈_𝑄(𝑡,𝑠)⁻¹). (ex3)‖𝑈(𝑡,𝑠)𝑃(𝑠)‖≤𝑁𝑒^{−𝜇(𝑡−𝑠)}, and ‖𝑈_𝑄(𝑠,𝑡)𝑄(𝑡)‖≤𝑁𝑒^{−𝜇(𝑡−𝑠)}.

In this case, the operator-valued map Γ(𝑡,𝑠) given by Γ(𝑡,𝑠)=𝑈(𝑡,𝑠)𝑃(𝑠), for 𝑡≥𝑠 and Γ(𝑡,𝑠)=−𝑈_𝑄(𝑡,𝑠)𝑄(𝑠), for 𝑡<𝑠 is called Green's function corresponding to 𝑈(⋅,⋅) and 𝑃(⋅).

The existence, properties, and examples of Green's functions can be founded in [23]. We only mention here a few properties directly related with our objectives. If 𝑓 is a bounded function, it follows from (2.11) that the unique bounded mild solution of the equation𝑢^(𝑡)=𝐴(𝑡)𝑢(𝑡)+𝑓(𝑡),𝑡∈ℝ,(2.14) is given by𝑢(𝑡)=^∞−∞Γ(𝑡,𝜉)𝑓(𝜉)𝑑𝜉,𝑡∈ℝ.(2.15) Assuming that the conditions (AT1)-(AT2) hold, we introduce the following condition. (R)𝑅(𝜔,𝐴(⋅))∈AP(ℒ(𝕏)).

Lemma 2.12 (see [23, Proposition 5.8]). Assume that the conditions (AT1)-(AT2) and (R) hold. If 𝑈(⋅,⋅) has an exponential dichotomy with constants 𝑁,𝜇>0, then for each 𝜖>0 and ℎ>0 there is a relatively dense set Ω_𝜖,ℎ such that ‖Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠)‖≤𝜖𝑒^{−(𝜇/2)|𝑡−𝑠|},|𝑡−𝑠|>ℎ,𝑡,𝑠∈ℝ,𝜏∈Ω_𝜖,ℎ.(2.16)

We abbreviate this property by writing Γ∈AP(ℒ(𝕏)).

Remark 2.13. As was pointed out in [25], under the hypotheses of the Lemma 2.12, for all sequence of real numbers (𝑡^𝑛)_𝑛≥0 and ℎ>0 there is a subsequence (𝑡_𝑛)_𝑛≥0 such that Γ(𝑡+𝑡_𝑛,𝑠+𝑡_𝑛)𝑥 is uniformly convergent for all 𝑥∈𝑋 and 𝑡,𝑠∈ℝ with |𝑡−𝑠|>ℎ. Taking ℎ=1,1/2,…, and using the method of diagonal choice, it is possible to show that Γ(𝑡+𝑡_𝑛,𝑠+𝑡_𝑛)𝑥 is uniform convergent with respect to 𝑡,𝑠∈ℝ with 𝑡≠𝑠 for each 𝑥∈𝑋. Moreover, since 𝑃(⋅)∈AP(ℒ(𝕏)) and Γ(𝑡,𝑡)=𝑃(𝑡), then the preceding assertion holds for all 𝑡,𝑠∈ℝ.

3. Existence of Solutions

In this section we keep the notations introduced in the Section 2. In addition, we will assume that 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ is an uniformly almost periodic sequence of moments, and we consider as standing hypotheses that conditions (AT1)-(AT2) hold and that 𝑈(𝑡,𝑠) is an evolution family generated by the operators 𝐴(𝑡) with an exponential dichotomy 𝑃(𝑡). Initially, we consider the abstract cauchy problem with impulses𝑢^(𝑡)=𝐴(𝑡)𝑢Δ(𝑢𝑡)+𝑡_𝑖𝑓(𝑡),𝑡∈ℝ,𝑡≠𝑡_𝑖,𝑖∈ℤ,=𝑥_𝑖,𝑖∈ℤ,(3.1) where 𝑓∈𝒫𝒞_𝑇(ℝ,𝕏).

Definition 3.1. A function 𝑢∈𝒫𝒞_𝑇(ℝ,𝕏) is said to be a piecewise continuous mild solution of the problem (3.1) if for all 𝑡∈ℝ, 𝑡_𝑖<𝑡≤𝑡_𝑖+1, we have 𝑢(𝑡)=𝑈𝑡,𝑡_𝑖𝑢𝑡_𝑖+𝑈𝑡,𝑡_𝑖𝑥_𝑖+^𝑡𝑡_𝑖𝑈(𝑡,𝑠)𝑓(𝑠)𝑑𝑠.(3.2)

If we assume that 𝑓 is bounded and that the set {‖𝑥_𝑖‖∶𝑖∈ℤ} is bounded, it follows from (2.15) that the unique bounded piecewise continuous mild solution of the problem (3.1) is given by𝑢(𝑡)=^∞−∞Γ(𝑡,𝜉)𝑓(𝜉)𝑑𝜉+_{𝑡𝑖<𝑡}Γ𝑡,𝑡_𝑖𝑥_𝑖,𝑡∈ℝ.(3.3)

Our results of existence of solutions for the problem (1.1) are based in several properties of this construction, which we will collect in the following lemma.

Lemma 3.2. Let 𝑇={𝑡_𝑖}_𝑖∈ℤ∈ℬ uniformly almost periodic. Assume that 𝑓∈AP_𝑇(𝕏), Γ∈AP(ℒ(𝕏)) and the sequence {𝑥_𝑖∶𝑖∈ℤ} is almost periodic. Then for each 𝜖>0 there are relatively dense sets Ω_𝜖 of ℝ and 𝒬_𝜖 of ℤ such that the following conditions hold. (i)‖𝑓(𝑡+𝜏)−𝑓(𝑡)‖<𝜖 for all 𝑡∈ℝ, |𝑡−𝑡_𝑖|>𝜖, 𝜏∈Ω_𝜖 and 𝑖∈ℤ. (ii)‖Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠)‖<𝜖 for all 𝑡,𝑠∈ℝ, |𝑡−𝑠|>0, |𝑠−𝑡_𝑖|>𝜖, |𝑡−𝑡_𝑖|>𝜖, 𝜏∈Ω_𝜖 and 𝑖∈ℤ. (iii)‖𝑥_𝑖+𝑞−𝑥_𝑖‖_𝕏<𝜖 for all 𝑞∈𝒬_𝜖 and 𝑖∈ℤ. (iv)|𝑡_𝑖+𝑞−𝑡_𝑖−𝜏|<𝜖 for all 𝑞∈𝒬_𝜖 and 𝜏∈Ω_𝜖 and 𝑖∈ℤ.

Remark 3.3. The proof of Lemma 3.2 is based on the technique of finding common almost periods. We refer the reader to [4] for a discussion about this method.

Lemma 3.4. Assume that the conditions of Lemma 2.12 hold. Let 𝑥∈AP_𝑇(𝕏) and let 𝑣(𝑡) be given by 𝑣(𝑡)=^∞−∞Γ(𝑡,𝜉)𝑥(𝜉)𝑑𝜉,𝑡∈ℝ.(3.4) Then the function 𝑣∈AP(𝕏).

Proof. For 𝜖>0, let Ω_𝜖 be a relatively dense set of ℝ formed by 𝜖-periods of Γ and 𝑥 such as in the Lemma 3.2. For 𝜏∈Ω_𝜖 we have 𝑣(𝑡+𝜏)−𝑣(𝑡)==^∞−∞(Γ(𝑡+𝜏,𝑠)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠_𝑡−∞(Γ(𝑡+𝜏,𝑠)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠+^∞𝑡(Γ(𝑡+𝜏,𝑠)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠.(3.5) The first term on the right hand side of (3.5) can be decomposed as ^𝑡−∞(Γ(𝑡+𝜏,𝑠)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠=+^𝑡−∞(Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠_𝑡−∞=𝐽₁Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))𝑑𝑠+𝐽₂.(3.6) Let ℎ>0. Applying the Lemma 2.12, we can estimate the first term on the right hand side of the above expression as ‖𝐽₁‖=≤‖‖‖‖𝑥‖^𝑡−∞_∞(Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠‖‖‖^{𝑡−ℎ−∞}≤𝜖‖𝑥‖_∞‖Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠)‖𝑑𝑠+^{𝑡𝑡−ℎ}‖(Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠))𝑓(𝑠)‖𝑑𝑠^{𝑡−ℎ−∞}≤2𝑒^{−(𝜇/2)(𝑡−𝑠)}𝑑𝑠+^{𝑡𝑡−ℎ}‖(Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠))𝑥(𝑠)‖𝑑𝑠𝜇‖𝑥‖^∞𝜖+2𝑁‖𝑥‖_∞ℎ.(3.7) In order to estimate the second term on the right hand side of (3.6), we assume that 𝑡_𝑖<𝑡≤𝑡_𝑖+1. For 0<ℎ<𝛼/2, then we can split the integral ∫^𝑡−∞‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠 appropriately to obtain ‖𝐽₂‖≤≤^𝑡−∞‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠_{𝑖−1𝑗=}_−∞^{𝑡𝑗+1𝑡−ℎ}_𝑗+ℎ+‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠_{𝑖−1𝑗=}_−∞^{𝑡𝑗𝑡+ℎ}_𝑗+‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠_{𝑖−1𝑗=}_−∞^𝑡𝑗+1_{𝑡𝑗+1−ℎ}+‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠_𝑡𝑡𝑖‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠.(3.8) Since 𝑥∈AP_𝑇(𝕏) we have that ‖𝑥(𝑠+𝜏)−𝑥(𝑠)‖<𝜖 for all 𝑠∈[𝑡_𝑗+ℎ,𝑡_𝑗+1−ℎ] and 𝑗∈ℤ, 𝑗≤𝑖. Moreover, in view of that 𝑡−𝑠≥𝑡−𝑡_𝑖+𝑡_𝑖−𝑠≥𝑡−𝑡_𝑖+𝑡_𝑖−𝑡_𝑗+1−ℎ≥𝑡−𝑡_𝑖+𝛼(𝑖−1−𝑗)+ℎ,(3.9) we can estimate ^𝑖−¹_𝑗=−∞^{𝑡𝑗+1𝑡−ℎ}_𝑗+ℎ≤𝜖_𝑖−1‖Γ(𝑡+𝜏,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠_𝑗=_−∞^{𝑡𝑗+1𝑡−ℎ}_𝑗+ℎ≤𝜖𝑁_𝑖−1‖Γ(𝑡+𝜏,𝑠+𝜏)‖𝑑𝑠_𝑗=_−∞^{𝑡𝑗+1𝑡−ℎ}_𝑗+ℎ≤𝜖𝑁𝑒^{−𝜇(𝑡−𝑠)}𝑑𝑠𝜇^𝑖−1_𝑗=−∞𝑒^{−𝜇(𝑡−𝑡𝑗+1+ℎ)}≤𝜖𝑁𝜇^𝑖−1_𝑗=−∞𝑒^{−𝜇𝛼(𝑖−𝑗−1)}≤𝜖𝑁𝜇(1−𝑒^−𝜇𝛼).(3.10) To estimate the second term on the right hand side of (3.8) we observe ^𝑖−¹_𝑗=−∞^{𝑡𝑗𝑡+ℎ}_𝑗≤2𝑁‖‖Γ𝑥‖(𝑡+𝜏_∞𝑖−1,𝑠+𝜏)(𝑥(𝑠+𝜏)−𝑥(𝑠))‖𝑑𝑠_𝑗=_−∞^{𝑡𝑗𝑡+ℎ}_𝑗≤2𝑁𝜖‖𝑥‖_∞𝑒^{−𝜇(𝑡−𝑠)}𝑑𝑠𝑒^{−𝜇ℎ𝑖−1}_𝑗=_−∞𝑒^{−𝜇(𝑡−𝑡𝑗)}≤2𝑁𝜖𝑒^{−𝜇(ℎ+𝑡−𝑡𝑖)}‖𝑥‖_{∞𝑖−1𝑗=}_−∞𝑒^{−𝜇𝛼(𝑖−𝑗)}≤2𝑁𝜖‖𝑥‖^∞1−𝑒^−𝜇𝛼.(3.11) We obtain similar estimates for the third and fourth term on the right hand side of (3.8). Combining these estimate, we get that ‖𝐽₁‖≤𝑂(𝜖) and ‖𝐽₂‖≤𝑂(𝜖) independent of 𝑡∈ℝ. Furthermore, proceeding in a similar way we can show that ‖‖‖^∞𝑡(Γ(𝑡+𝜏,𝑠)−Γ(𝑡,𝑠))𝑥(𝑠)𝑑𝑠‖‖‖=𝑂(𝜖),(3.12) which completes the proof that 𝑣(⋅) is an almost periodic function in the Bohr's sense.

We introduce a new condition on the evolution family 𝑈(⋅,⋅).(H) For each 𝑥∈𝕏, 𝑈(𝑡+ℎ,𝑡)𝑥→𝑥 as ℎ→0⁺ uniformly for 𝑡∈ℝ.

Lemma 3.5. Assume that the assumptions (R) and (H) hold. Let 𝐾 be a compact subset of 𝕏. Then ‖Γ(𝑡+ℎ,𝑠+ℎ^)𝑥−Γ(𝑡,𝑠)𝑥‖→0, as ℎ,ℎ^→0⁺ uniformly for 𝑠,𝑡∈ℝ and 𝑥∈𝐾.

Proof. Since 𝑃∈AP(ℒ(𝕏)) [23, Corollary 5.1] the set 𝑊={𝑃(𝑠)𝑥∶𝑥∈𝐾,𝑠∈ℝ} is relatively compact in 𝕏. Moreover, using (H), we have Γ(𝑠+ℎ,𝑠)𝑥−𝑃(𝑠)𝑥=[𝑈(𝑠+ℎ,𝑠)−𝐼]𝑃(𝑠)𝑥=[𝑈(𝑠+ℎ,𝑠)−𝐼]𝑦→0,ℎ→0,(3.13) uniformly for 𝑦∈𝑊.
We will prove separately that Γ(𝑡+ℎ,𝑠)𝑥−Γ(𝑡,𝑠)𝑥→0 and Γ(𝑡,𝑠+ℎ)𝑥−Γ(𝑡,𝑠)𝑥→0 as ℎ→0 uniformly for 𝑠,𝑡∈ℝ and 𝑥∈𝕏. Assume first that ℎ>0. We have Γ(𝑡+ℎ,𝑠)𝑥−Γ(𝑡,𝑠)𝑥==[[𝑈𝑈((𝑡𝑡++ℎℎ,,𝑡𝑡))−−𝐼𝐼,]ΓΓ((𝑡𝑡,,𝑠𝑠))−𝑥𝑃(𝑠)]𝑥+[𝑈(𝑡+ℎ,𝑡)−𝐼]𝑃(𝑠)𝑥.((33..1145)) Using (3.13), (3.15), and assumption (H) we deduce that there exists 𝛽>0 such that Γ(𝑡+ℎ,𝑠)𝑥−Γ(𝑡,𝑠)𝑥→0,ℎ→0,(3.16) uniformly for 𝑠,𝑡∈ℝ with 𝑡−𝑠≤𝛽 and 𝑥∈𝕏. Similarly, the assertion (3.16) is immediate when 𝑡−𝑠>0 is large enough, because ‖Γ(𝑡,𝑠)‖→0, as 𝑡−𝑠→∞. We consider now 0<𝛽<𝑎 and 𝛽≤𝑡−𝑠≤𝑎. It follows from [23, Theorem 5.9] that the function 𝑟↦Γ(𝑡+𝑟,𝑠+𝑟) is almost periodic. This implies that the set 𝑊^={Γ(𝑡,𝑠)𝑥∶𝑥∈𝐾,𝛽≤𝑡−𝑠≤𝑎} is relatively compact in 𝕏. Using again the assumption (H) we get that [𝑈(𝑡+ℎ,𝑡)−𝐼]𝑦→0, ℎ→0, uniformly for 𝑡∈ℝ and 𝑦∈𝑊^. Applying (3.14) we also obtain (3.16). A similar argument shows the assertion for ℎ<0.
Now we estimate Γ(𝑡,𝑠+ℎ)𝑥−Γ(𝑡,𝑠)𝑥 as ℎ→0. Assume now that ℎ>0. Using again the assumption (H) and the fact that 𝑃(⋅) is almost periodic, from the decomposition Γ(𝑡,𝑠+ℎ)𝑥−Γ(𝑡,𝑠)𝑥=𝑈(𝑡,𝑠+ℎ)[𝑃(𝑠+ℎ)𝑥−𝑈(𝑠+ℎ,𝑠)𝑃(𝑠)𝑥],(3.17) it follows easily that Γ(𝑡,𝑠+ℎ)𝑥−Γ(𝑡,𝑠)𝑥 as ℎ→0 uniformly for 𝑠,𝑡∈ℝ and 𝑥∈𝕏. Arguing in a similar way we obtain the property for ℎ<0.

Lemma 3.6. Assume that the assumptions (R) and (H) hold. Let (𝑥_𝑖)_𝑖∈ℤ be a discrete almost periodic sequence and let 𝑔(𝑡) be given by 𝑔(𝑡)=Γ𝑡,𝑡_𝑖𝑥_𝑖,𝑡_𝑖<𝑡≤𝑡_𝑖+1,𝑖∈ℤ.(3.18) Then the function 𝑔∈AP_𝑇(𝕏).

Proof. It is clear that 𝑔 is 𝑇-piecewise continuous. Let 𝜖,𝛽>0 be fixed. Let 𝑀>0 be a constant such that ‖𝑥_𝑗‖≤𝑀 for all 𝑗∈ℤ. Moreover, applying the Lemma 3.5 we can choose 𝛿>0 such that ‖‖Γ(𝑡,𝑠)𝑥_𝑗−Γ𝑡^,𝑠^𝑥_𝑗‖‖≤𝜖3,(3.19) for |𝑡−𝑡^|≤𝛿, |𝑠−𝑠^|≤𝛿, and for all 𝑗∈ℤ.
We consider 0<𝜖^=min{𝜖/3𝑀,𝜖/3𝑁,𝛽,𝛿}.
Applying the Lemma 3.2 we infer that there are relatively dense sets Ω of ℝ and 𝒬 of ℤ such that the following conditions hold.(i)‖Γ(𝑡+𝜏,𝑠+𝜏)−Γ(𝑡,𝑠)‖<𝜖^ for all 𝑡,𝑠∈ℝ, |𝑡−𝑠|>0, |𝑠−𝑡_𝑗|>𝜖^, |𝑡−𝑡_𝑗|>𝜖^, 𝜏∈Ω and 𝑗∈ℤ.(ii)‖𝑥_𝑗+𝑞−𝑥_𝑗‖_𝕏<𝜖^ for all 𝑞∈𝒬 and 𝑗∈ℤ.(iii)|𝑡_𝑗+𝑞−𝑡_𝑗−𝜏|<𝜖^ for all 𝑞∈𝒬, 𝜏∈Ω and 𝑖∈ℤ.
We consider 𝑡∈ℝ and 𝑖∈ℤ such that 𝑡_𝑖<𝑡<𝑡_𝑖+1, |𝑡−𝑡_𝑖|>𝛽 and |𝑡−𝑡_𝑖+1|>𝛽. By using (iii) we select 𝑞∈𝒬 such that ||𝑡_𝑗+𝑞−𝑡_𝑗−𝜏||<𝜖^,(3.20) for all j∈ℤ. This implies that 𝑡_𝑖+𝑞−𝑡_𝑖−𝜏<𝜖^<𝛽,(3.21) which in turn implies that 𝑡+𝜏>𝑡_𝑖+𝛽+𝜏>𝑡_𝑖+𝑞.(3.22) Arguing in a similar way, we have 𝑡_𝑖+𝑞+1−𝑡_𝑖+1−𝜏>−𝜖^>−𝛽,(3.23) from which follows that 𝑡_𝑖+𝑞+1>𝑡_𝑖+1−𝛽+𝜏>𝑡+𝜏.(3.24) Next we abbreviate 𝑘=𝑖+𝑞. Since 𝑡_𝑘<𝑡+𝜏<𝑡_𝑘+1 we can write 𝑔(𝑡+𝜏)−𝑔(𝑡)=Γ𝑡+𝜏,𝑡_𝑘𝑥_𝑘−Γ𝑡,𝑡_𝑖𝑥_𝑖=+ΓΓ𝑡𝑡++𝜏𝜏,,𝑡𝑡_𝑖𝑘+𝜏−Γ𝑡,𝑡_𝑖𝑥_𝑘+Γ𝑡,𝑡_𝑖𝑥_𝑘−𝑥_𝑖𝑥_𝑘−Γ𝑡+𝜏,𝑡_𝑖+𝜏𝑥_𝑘.(3.25) Using (i) we deduce that ‖‖Γ𝑡+𝜏,𝑡_𝑖+𝜏−Γ𝑡,𝑡_𝑖𝑥_𝑘‖‖≤𝜖3.(3.26) Similarly, using (ii) we infer that ‖Γ𝑡,𝑡_𝑖𝑥_𝑘−𝑥_𝑖‖≤𝜖3.(3.27) Using again that |𝑡_𝑖+𝑞−𝑡_𝑖−𝜏|<𝜖^<𝛿 and (3.19), we have that ‖Γ𝑡+𝜏,𝑡_𝑘𝑥_𝑘−Γ𝑡+𝜏,𝑡_𝑖+𝜏𝑥_𝑘‖≤𝜖3.(3.28) Collecting these estimate, it follows from (3.25) that ‖𝑔(𝑡+𝜏)−𝑔(𝑡)‖≤𝜖 for all 𝜏∈Ω.
Next we will prove that 𝑔 is uniformly continuous on the ∪_𝑖∈ℤ(𝑡_𝑖,𝑡_𝑖+1). Let 𝑡,ℎ∈ℝ, ℎ>0 such that 𝑡_𝑖<𝑡,𝑡+ℎ<𝑡_𝑖+1. We have 𝑔(𝑡+ℎ)−𝑔(𝑡)=Γ𝑡+ℎ,𝑡_𝑖𝑥_𝑖−Γ𝑡,𝑡_𝑖𝑥_𝑖=[Γ(𝑡+ℎ,𝑡)−Γ(𝑡,𝑡)]Γ𝑡,𝑡_𝑖𝑥_𝑖.(3.29) We choose appropriately 0<𝛽<𝑎 and we analyze Γ(𝑡,𝑡_𝑖)𝑥_𝑖 in the cases 𝑡−𝑡_𝑖<𝛽,  𝛽≤𝑡−𝑡_𝑖≤𝑎 and 𝑎<𝑡−𝑡_𝑖. Arguing as in the proof of the Lemma 3.5 we conclude that 𝑔(𝑡+ℎ)−𝑔(𝑡) converges to zero as ℎ→0 independent of 𝑡 and 𝑖.

The next theorem is the main result of this section. In this statement we denote𝐶=2𝑁𝜇‖𝑓‖^∞+‖𝐹(0)‖+𝑁1−𝑒^−𝜇𝛼sup_𝑗∈ℤ‖𝐼_𝑗(0)‖.(3.30)

Theorem 3.7. Assume that the conditions (R) and (H) hold. Let 𝑓∈AP_𝑇(𝑋) and let 𝐹,𝐼_𝑖∶𝕏→𝕏, 𝑖∈ℤ be maps that satisfy the following conditions. (a)The family {𝐼_𝑖∶𝑖∈ℤ} is almost periodic and there exists a constant 𝐿₁≥0 such that ‖𝐼_𝑖(𝑥)−𝐼_𝑖(𝑦)‖≤𝐿₁‖𝑥−𝑦‖,(3.31) for all 𝑥,𝑦∈𝕏 and 𝑖∈ℤ. (b)There exists a nondecreasing function 𝐿₂∶[0,∞)→[0,∞) such that ‖𝐹(𝑥)−𝐹(𝑦)‖≤𝐿₂(𝑟)‖𝑥−𝑦‖,(3.32) for all 𝑥,𝑦∈𝕏, ‖𝑥−𝑦‖≤𝑟. If there exists 𝑟>0 such that 𝐶𝑟+𝑁1−𝐿𝑒^1−𝜇𝛼+2𝐿²(𝑟)𝜇<1,(3.33) then the problem (1.1) has a unique 𝑇-piecewise continuous almost periodic mild solution 𝑢(⋅) which satisfies 𝑢(𝑡)=^∞−∞Γ(𝑡,𝑠)𝑓(𝑠)𝑑𝑠+^∞−∞Γ(𝑡,𝑠)𝐹(𝑢(𝑠))𝑑𝑠+_{𝑡𝑖<𝑡}Γ𝑡,𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖.(3.34)

Proof. Let Λ∶AP_𝑇(𝕏)→𝒫𝒞_𝑇(ℝ,𝕏) be the operator defined by Λ𝑢(𝑡)=^∞−∞Γ(𝑡,𝑠)𝑓(𝑠)𝑑𝑠+^∞−∞Γ(𝑡,𝑠)𝐹(𝑢(𝑠))𝑑𝑠+_{𝑡𝑖<𝑡}Γ𝑡,𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖.(3.35) Since the function 𝑡↦𝐹(𝑢(𝑡)) belongs to the space AP_𝑇(𝕏), it follows from Lemma 3.4 that the function 𝑡↦∫^∞−∞Γ(𝑡,𝑠)𝑓(𝑠)𝑑𝑠+∫^∞−∞Γ(𝑡,𝑠)𝐹(𝑢(𝑠))𝑑𝑠 also belongs to the space AP_𝑇(𝕏).
Let 𝑥_𝑖=𝐼_𝑖(𝑢(𝑡_𝑖)) for 𝑖∈ℤ. We define 𝑔_𝑘∶ℝ→𝕏 by 𝑔_𝑘(𝑡)=Γ(𝑡,𝑡_𝑖−𝑘)𝑥_𝑖−𝑘 for 𝑡_𝑖<𝑡≤𝑡_𝑖+1. It follows from Lemma 3.6 that 𝑔_𝑘∈AP_𝑇(𝕏). Moreover, ‖𝑔_𝑘(𝑡)‖=‖Γ𝑡,𝑡_𝑖−𝑘𝑥_𝑖−𝑘‖≤𝑁sup_𝑖∈ℤ‖𝑥_𝑖‖𝑒^{−𝜇(𝑡−𝑡𝑖−𝑘)}≤𝑁sup_𝑖∈ℤ‖𝑥_𝑖‖𝑒^{−𝜇(𝑡−𝑡𝑖)}𝑒^{−𝜇𝛼𝑘}.(3.36) Therefore, the series ∑^∞𝑘=0𝑔_𝑘 is uniformly convergent on ℝ. Consequently, ∑^∞𝑘=0𝑔_𝑘∈AP_𝑇(𝕏). Since ∑_{𝑡𝑖<𝑡}Γ(𝑡,𝑡_𝑖)𝐼_𝑖(𝑢(𝑡_𝑖))=∑^∞𝑘=0𝑔_𝑘, and combining this property with the previous assertion, it follows that Λ𝑢∈AP_𝑇(𝕏).
This shows that Λ∶AP_𝑇(𝕏)→AP_𝑇(𝕏). Let 𝑢∈AP_𝑇(𝕏) such that ‖𝑢‖_∞≤𝑟. It follows from (3.35) that ‖Λ𝑢(𝑡)‖≤+^𝑡−∞‖Γ(𝑡,𝑠)𝑓(𝑠)‖𝑑𝑠+^∞𝑡‖Γ(𝑡,𝑠)𝑓(𝑠)‖𝑑𝑠_𝑡−∞+‖Γ(𝑡,𝑠)(𝐹(𝑢(𝑠))−𝐹(0))‖𝑑𝑠+^𝑡−∞‖Γ(𝑡,𝑠)𝐹(0)‖𝑑𝑠_∞𝑡+‖Γ(𝑡,𝑠)(𝐹(𝑢(𝑠))−𝐹(0))‖𝑑𝑠+^∞𝑡‖Γ(𝑡,𝑠)𝐹(0)‖𝑑𝑠_{𝑡𝑖<𝑡}≤𝐶+2‖𝑁Γ𝑡,𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖−𝐼_𝑖(0)‖+_{𝑡𝑖<𝑡}‖Γ𝑡,𝑡_𝑖𝐼_𝑖(0)‖𝜇𝐿²(𝑟)𝑟+𝑁1−𝑒^−𝜇𝛼≤𝑟,𝐿₁𝑟(3.37) which implies that Λ(𝐵_𝑟(AP_𝑇(𝕏)))⊆𝐵_𝑟(AP_𝑇(𝕏)).
Finally, we verify that Λ is a contraction, let 𝑢₁,𝑢₂ be arbitrary elements of 𝐵_𝑟(AP_𝑇(𝕏)) and 𝑡_𝑖<𝑡≤𝑡_𝑖+1. Since 𝑡−𝑡_𝑗≥𝑡−𝑡_𝑖+(𝑖−𝑗)𝛼 for 𝑗≤𝑖, we get ‖Λ𝑢₁(𝑡)−Λ𝑢₂(𝑡)‖≤+_{−∞<𝑡𝑗<𝑡}‖Γ𝑡,𝑡_𝑗‖_ℒ(𝕏)‖𝐼_𝑗𝑢₁𝑡_𝑗−𝐼_𝑗𝑢₂𝑡_𝑗_∞−∞‖≤‖Γ(𝑡,𝑠)‖_ℒ(𝕏)‖𝐹𝑢₁(𝑠)−𝐹𝑢₂(𝑠)‖𝑑𝑠_𝑖𝑗=_−∞+𝑁𝑒^{−𝜇(𝑡−𝑡𝑗)}𝐿₁‖𝑢₁−𝑢₂‖_∞+^𝑡−∞𝑁𝑒^{−𝜇(𝑡−𝑠)}𝐿₂(𝑟)‖𝑢₁−𝑢₂‖_∞𝑑𝑠_∞𝑡≤𝑁𝐿₁𝑁𝑒^{−𝜇(𝑠−𝑡)}𝐿₂(𝑟)‖𝑢₁−𝑢₂‖_∞𝑑𝑠𝑒^{−𝜇(𝑡−𝑡𝑖)𝑖}_𝑗=_−∞𝑒^{−𝜇𝛼(𝑖−𝑗)}‖𝑢₁−𝑢₂‖_∞+2𝑁𝐿²(𝑟)𝜇‖𝑢¹−𝑢₂‖_∞≤𝑁𝐿¹1−𝑒^−𝜇𝛼+2𝐿²(𝑟)𝜇‖𝑢¹−𝑢₂‖_∞.(3.38) It follows from (3.3) that the fixed point 𝑢 of Λ is the 𝑇-piecewise continuous almost periodic mild solution of the problem (1.1).

In the case 𝐹 satisfies a uniform Lipschitz condition we obtain the following immediate consequence.

Corollary 3.8. Assume that the conditions (R) and (H) hold. Let 𝑓∈AP_𝑇(𝑋) and let 𝐹,𝐼_𝑖∶𝕏→𝕏, 𝑖∈ℤ be maps that satisfy the following conditions. (a)The family {𝐼_𝑖∶𝑖∈ℤ} is almost periodic and there exists a constant 𝐿₁≥0 such that ‖𝐼_𝑖(𝑥)−𝐼_𝑖(𝑦)‖≤𝐿₁‖𝑥−𝑦‖,(3.39) for all 𝑥,𝑦∈𝕏 and 𝑖∈ℤ. (b)There exists a constant 𝐿₂≥0 such that ‖𝐹(𝑥)−𝐹(𝑦)‖≤𝐿₂‖𝑥−𝑦‖,(3.40) for all 𝑥,𝑦∈𝕏. If 𝑁𝐿¹1−𝑒^−𝜇𝛼+2𝐿²𝜇<1,(3.41) then the problem (1.1) has a unique 𝑇-piecewise continuous almost periodic mild solution.

4. Applications

As an application we consider a system described by the partial differential equation with impulses𝜕𝜕𝑡𝑧(𝑡,𝜉)=𝜕𝜕𝜉²²Δ𝑧𝑡_𝑖𝑧(𝑡𝑧,𝜉()𝑡,+0)𝑏=(𝑡)𝑧𝑧(0(𝑡,,𝜋𝜉))+=𝑤0,(𝑡,𝑡𝜉∈),ℝ𝑡∈ℝ,𝜉∈[0,𝜋],,𝜉=^𝜋0𝑝𝑡_𝑖,𝜂,𝜉𝑧𝑡_𝑖,𝜂𝑑𝜂,𝜉∈[0,𝜋],𝑖∈ℤ.(4.1) In this equation, we assume that 𝑏∶ℝ→ℝ, 𝑤∶ℝ×[0,𝜋]→ℝ and 𝑝∶ℝ×[0,𝜋]×[0,𝜋]→ℝ are functions that satisfy some properties to be specified later. Moreover, 𝑇={𝑡_𝑖}_𝑖∈ℤ is a uniformly almost periodic sequence of moments. Such systems arises, among other diffusion systems, in the temperature control of a heated metal bar with insulated ends. To model this system we consider 𝕏=𝐿²([0,𝜋]). Let 𝐴₀ be the operator defined by𝐴₀𝜑(𝜉)=𝑑²𝜑(𝜉)𝑑𝜉²(4.2) with domain𝐷𝐴₀=𝜑∈𝕏∶𝜑∈𝐻²([0,𝜋]),𝜑(0)=𝜑(𝜋)=0.(4.3) The spectrum of 𝐴₀ consists of the eigenvalues −𝑛² for 𝑛∈𝐈𝐍, with associated eigenvectors𝜑_𝑛(𝜉)=2𝜋^1/2sin(𝑛𝜉).(4.4) Furthermore, the set {𝜑_𝑛∶𝑛∈𝐈𝐍} is an orthonormal basis of 𝕏. In particular,𝐴₀𝜑=^∞_𝑛=1−𝑛²⟨𝜑,𝜑_𝑛⟩𝜑_𝑛,(4.5) for 𝜑∈𝐷(𝐴₀). Using the above expression, one easily verifies that 𝐴₀ is the infinitesimal generator of a strongly continuous semigroup 𝑆(𝑡) given by𝑆(𝑡)𝜑=^∞_𝑛=1𝑒^−𝑛2𝑡⟨𝜑,𝜑_𝑛⟩𝜑_𝑛,𝑡≥0.(4.6)

Let 𝐴(𝑡)=𝐴₀+𝑏(𝑡)𝐼 for 𝑡∈ℝ defined on 𝐷(𝐴(𝑡))=𝐷(𝐴₀). We consider the following conditions.(a)The function 𝑏(⋅) is almost periodic. (b)The function 𝑝(⋅) is continuous. We denote 𝐿₁=^𝜋0𝑝²𝑡_𝑖,𝜂,𝜉𝑑𝜂𝑑𝜉^1/2.(4.7)(c)The function 𝑤 verifies the Carathéodory conditions: (i)𝑤(𝑡,⋅)∶[0,𝜋]→ℝ is measurable;(ii)𝑤(⋅,𝜉)∶ℝ→ℝ is continuous a.e.; (iii)there exists a function 𝑔∈𝐿²([0,𝜋]) such that |𝑤(𝑡,𝜉)|≤𝑔(𝜉) for all 𝑡∈ℝ and 𝜉∈[0,𝜋].

In this case, the operators 𝐴(𝑡) satisfy the conditions (AT1)-(AT2). Moreover, the evolution family 𝑈(𝑡,𝑠) generated by 𝐴(𝑡) is given by𝑈(𝑡,𝑠)=𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑆(𝑡−𝑠),𝑠≤𝑡.(4.8) It is immediate that this evolution family satisfies the assumption (H). It follows from (c) that the function 𝑓∶ℝ→𝕏, 𝑓(𝑡)=𝑤(𝑡,⋅) is measurable. Moreover, it follows from (b) that the map 𝐼_𝑖∶𝕏→𝕏 given by𝐼_𝑖(𝜑)(𝜉)=^𝜋0𝑝𝑡_𝑖,𝜂,𝜉𝜑(𝜂)𝑑𝜂,(4.9) is a bounded linear map with ‖𝐼_𝑖‖≤𝐿₁ for all 𝑖∈ℤ. Consequently, if we define 𝑢(𝑡)=𝑧(𝑡,⋅), the problem (4.1) can be modeled by the abstract system (1.1). In the rest of this section we assume that conditions (a), (b), and (c) are fulfilled. We assume further that 𝑤 satisfies the following condition. (d) The function 𝑤 is uniformly 𝑇-piecewise almost periodic. This means that (i)for the set {𝑤(⋅,𝜉)∶0≤𝜉≤𝜋}⊆AP_𝑇(ℝ) and for each 𝜖>0, there exists a relatively dense set Ω_𝜖 in ℝ such that ||𝑤(𝑡+𝜏,𝜉)−𝑤(𝑡,𝜉)||<𝜖,(4.10) for all 𝜏∈Ω_𝜖, 𝜉∈[0,𝜋] and 𝑡∈ℝ satisfying the condition |𝑡−𝑡_𝑖|>𝜖 for all 𝑖∈ℤ. (ii)for every 𝜖>0 there exists 𝛿>0 such that |𝑤(𝑡,𝜉)−𝑤(𝑡^,𝜉)‖<𝜖 for all 𝑖∈ℤ, 𝜉∈[0,𝜋],  𝑡,𝑡^∈(𝑡_𝑖,𝑡_𝑖+1) such that |𝑡−𝑡^|<𝛿,

It is not difficult to show that if 𝑤 satisfies conditions (c) and (d), then the function 𝑓∈AP_𝑇(𝕏).

On the other hand, we consider the following condition. (e) There are 𝑛∈𝐈𝐍 and 𝜇>0 such that (𝑛−1)²+𝜇≤𝑏(𝑡)≤𝑛²−𝜇 for all 𝑡∈ℝ.

Assuming that this condition holds, we define 𝑃(𝑡)∶𝕏→𝕏 by𝑃(𝑡)𝜑=^∞_𝑘=𝑛⟨𝜑,𝜑_𝑘⟩𝜑_𝑘.(4.11) It is immediate to see that 𝑈(𝑡,𝑠)𝑃(𝑠)𝜑=^∞_𝑘=𝑛⟨𝜑,𝜑_𝑘⟩𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑆(𝑡−𝑠)𝜑_𝑘=^∞_𝑘=𝑛𝑒^{−𝑘2∫(𝑡−𝑠)+𝑡𝑠𝑏(𝜏)𝑑𝜏}⟨𝜑,𝜑_𝑘⟩𝜑_𝑘,(4.12) which implies that‖𝑈(𝑡,𝑠)𝑃(𝑠)𝜑‖≤𝑒^{−𝑛2∫(𝑡−𝑠)+𝑡𝑠𝑏(𝜏)𝑑𝜏}‖𝜑‖.(4.13) Since^𝑡𝑠𝑏(𝜏)𝑑𝜏−𝑛²(𝑡−𝑠)≤−𝜇(𝑡−𝑠),(4.14) we infer that‖𝑈(𝑡,𝑠)𝑃(𝑠)‖≤𝑒^{−𝜇(𝑡−𝑠)},𝑠≤𝑡.(4.15) Arguing in a similar way, if 𝑄(𝑡)=𝐼−𝑃(𝑡), then𝑈(𝑡,𝑠)𝑄(𝑠)𝜑=^𝑛−1_𝑘=1𝑒^{−𝑘2∫(𝑡−𝑠)+𝑡𝑠𝑏(𝜏)𝑑𝜏}⟨𝜑,𝜑_𝑘⟩𝜑_𝑘,(4.16) which implies that𝑈(𝑡,𝑠)𝑄(𝑠)𝜑=^𝑛−1_𝑘=1𝑈_𝑄𝑒^{−𝑘2∫(𝑡−𝑠)+𝑡𝑠𝑏(𝜏)𝑑𝜏}⟨𝜑,𝜑_𝑘⟩𝜑_𝑘,(𝑠,𝑡)𝜑=(𝑈(𝑡,𝑠)𝑄(𝑠))⁻¹𝜑=^𝑛−1_𝑘=1𝑒^{𝑘2∫(𝑡−𝑠)−𝑡𝑠𝑏(𝜏)𝑑𝜏}⟨𝜑,𝜑_𝑘⟩𝜑_𝑘.(4.17) Hence‖𝑈_𝑄(𝑠,𝑡)𝜑‖≤𝑒^{(𝑛−1)2∫(𝑡−𝑠)−𝑡𝑠𝑏(𝜏)𝑑𝜏}‖𝜑‖.(4.18) In view of that^𝑡𝑠𝑏(𝜏)𝑑𝜏−(𝑛−1)²(𝑡−𝑠)≥𝜇(𝑡−𝑠),(4.19) we infer that‖𝑈_𝑄(𝑠,𝑡)‖≤𝑒^{−𝜇(𝑡−𝑠)},𝑠≤𝑡.(4.20) As a consequence we can affirm that the evolution family 𝑈(⋅,⋅) has an exponential dichotomy and the associated Green's function Γ satisfies the estimate ‖Γ(𝑡,𝑠)‖≤𝑒^{−𝜇|𝑡−𝑠|} for 𝑠,𝑡∈ℝ.

Finally, we note that for Re(𝜆) enough small, the resolvent operator 𝑅(𝜆,𝐴(𝑡))=𝜆𝐼−𝐴₀−𝑏(𝑡)𝐼⁻¹==_∞𝐼−𝑏(𝑡)𝑅𝜆,𝐴₀⁻¹𝑅𝜆,𝐴₀_𝑘=0𝑏(𝑡)^𝑘𝑅𝜆,𝐴₀^𝑘+1,(4.21) where the series on the right hand side converges in the space ℒ(𝕏) uniformly for 𝑡∈ℝ. Therefore, the operator-valued function 𝑅(𝜆,𝐴(⋅))∈AP(ℒ(𝕏)), which in turn shows that the evolution family 𝑈(⋅,⋅) satisfies the condition (R).

The next result is an immediate consequence of the Theorem 3.7.

Theorem 4.1. Let us assume the conditions (a)-(e) are fulfilled. If 𝐿₁/1−𝑒^−𝜇𝛼<1, then the problem (4.1) has a unique 𝑇-piecewise almost periodic mild solution.

This result can be generalized to include the system 𝜕𝜕𝑡𝑧(𝑡,𝜉)=𝑎(𝑡)𝜕𝜕𝜉²²Δ𝑧𝑡_𝑖𝑧𝑧((𝑡𝑡,,𝜉0))+=𝑏𝑧(𝑡()0𝑧,𝜋(𝑡),𝜉=)0+,𝑤𝑡(𝑡∈,𝜉ℝ),,𝑡∈ℝ,𝜉∈[0,𝜋],,𝜉=^𝜋0𝑝𝑡_𝑖,𝜂,𝜉𝑧𝑡_𝑖,𝜂𝑑𝜂,𝜉∈[0,𝜋],𝑖∈ℤ,(4.22) when 𝑎(⋅)∈AP(ℝ) is a positive function. In fact, in this case the evolution family 𝑈(⋅,⋅) is given by𝑈(𝑡,𝑠)=𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑆^𝑡𝑠𝑎(𝜏)𝑑𝜏,𝑠≤𝑡,(4.23) and we can argue as above to study this system.

Acknowledgments

The authors wish to thank the referees for their comments. H. R. Henríquez is supported in part by Conicyt under Grant FONDECYT no. 1090009. B. De Andrade is partially supported by CNPQ/Brazil under Grant 100994/2011-3.