Weighted Pseudo Almost-Periodic Functions and Applications to Semilinear Evolution Equations

Abstract

We first give a solution to a key problem concerning the completeness of the space of weighted pseudo almost-periodic functions and then establish a new composition theorem with respect to these functions. Some important remarks with concrete examples are also presented. Moreover, we prove an existence theorem for the weighted pseudo almost-periodic mild solution to the semilinear evolution equation: 𝑥^′(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)), 𝑡∈ℝ, where 𝐴 is the infinitesimal generator of an exponentially stable 𝐶₀-semigroup. An application is also given to illustrate the abstract existence theorem.

1. Introduction

It is well known that periodicity and almost periodicity are natural and important phenomena in the real world. In 2006, Diagana [1] introduced the concept of weighted pseudo almost-periodic functions, which is a generalization of the classical almost-periodic functions of Bohr as well as the vector-valued almost-periodic functions of Bochner (cf., e.g., [1–4]). Recently, weighted pseudo almost-periodic functions are widely investigated and used in the study of differential equations. Many basic properties and applications to several classes of differential equations were established, see, for example, Blot et al. [5] and Diagana [6]. On the other hand, the properties of weighted pseudo almost-periodic functions are more complicated and changeable than the almost-periodic functions and the pseudo almost-periodic functions because the influence of the weight 𝜌 is very strong sometimes. Very recently, we constructed examples to show that the decomposition of weighted pseudo almost-periodic functions is not generally unique [3]. Hence, the space of the weighted pseudo almost-periodic functions may not be a Banach space under the usual supremum norm. Actually, the completeness of the space of these functions is worthy to be studied deeply by new ideas. In this paper, we will present a solution to the fundamental problem. Then, we will study the corresponding composition problem of weighted pseudo almost-periodic functions, as well as the existence of weighted pseudo almost-periodic mild solution to the following semilinear evolution equation: 𝑥^′(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡)),𝑡∈ℝ, (1.1) where 𝐴 is the infinitesimal generator of an exponentially stable 𝐶₀-semigroup.

The paper is organized as follows. In Section 2, we prove a completeness theorem for the space of weighted pseudo almost-periodic functions by introducing a new norm on the space. Moreover, some remarks on weighted pseudo almost-periodic functions are given. In Section 3, we first establish a composition theorem for weighted pseudo almost-periodic functions and then give an existence theorem for the weighted pseudo almost-periodic mild solution to the evolution equation (1.1). An example is presented to illustrate the abstract existence theorem.

2. Completeness Theorem

Throughout this paper, we let 𝕏,𝕐 be Banach spaces and BC(ℝ,𝕏) the Banach space of all 𝕏-valued bounded continuous functions equipped with the supremum norm. 𝐿¹_loc(ℝ) denotes the space of all locally integrable functions on ℝ, and 𝕌 stands for 𝕌∶=𝜌∈𝐿¹_loc(ℝ)∶𝜌(𝑡)>0,a.e.𝑡∈ℝ. (2.1) If 𝜌∈𝕌, then we set 𝜇𝑇,𝜌∶=^𝑇_−𝑇𝜌(𝑡)𝑑𝑡,𝑇>0. (2.2) Define the space 𝕌_∞ as 𝕌_∞∶=𝜌∈𝕌∶lim_𝑇→∞𝜇𝑇,𝜌=∞, (2.3) and define 𝕌_𝑏 as the set of all 𝜌∈𝕌_∞ such that 𝜌 is bounded with inf_𝑡∈ℝ𝜌(𝑡)>0. Obviously, 𝕌_𝑏⊂𝕌_∞⊂𝕌. (2.4)

In what follows we recall some definitions and notations needed in this paper.

Definition 2.1 (S. Bochner). A continuous function 𝑓∶ℝ↦𝕏 is called almost periodic if for each 𝜀>0 there exists an 𝑙(𝜀)>0 such that every interval 𝐼 of length 𝑙(𝜀) contains a number 𝜏 with the property that 𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀,∀𝑡∈ℝ. (2.5) The set of all such functions will be denoted by AP(𝕏) in this paper.
A continuous function 𝑓∶ℝ×𝕐↦𝕏 is said to be almost periodic if 𝑓(𝑡,𝑥) is almost periodic in 𝑡∈ℝ uniformly for all 𝑥∈𝐾, where 𝐾 is any bounded subset of 𝕐. Denote by AP(ℝ×𝕐,𝕏) the set of all such functions.

The set of bounded continuous functions with vanishing mean value is denoted by PAP₀(ℝ,𝕏), that is, PAP₀(ℝ,𝕏)∶=𝜙∈BC(ℝ,𝕏)∶lim_𝑇→∞12𝑇^𝑇_−𝑇𝜙(𝜎)𝑑𝜎=0. (2.6) Let 𝜌∈𝕌_∞, and we define the set of continuous functions with vanishing mean value under weight 𝜌 by PAP₀ℝ,𝕏,𝜌∶=𝜙∈BC(ℝ,𝕏)∶lim_𝑇→∞1𝜇𝑇,𝜌^𝑇_−𝑇𝜙(𝜎)𝜌(𝜎)𝑑𝜎=0, (2.7) and define PAP₀ℝ×𝕐,𝕏,𝜌∶=𝜙∈BC(ℝ×𝕐,𝕏)∶lim_𝑇→∞1𝜇𝑇,𝜌^𝑇_−𝑇𝜙(𝜌(𝜎).𝜎,𝑢)𝑑𝜎=0uniformlyfor𝑢inanyboundedsubsetof𝕐 (2.8) For simplicity of notation, we write PAP₀(𝕏)andPAP₀(𝕏) (2.9) instead of PAP₀(ℝ,𝕏)andPAP₀(ℝ,𝕏), (2.10) respectively.

The following definition is a slight modification of that given by Diagana [1].

Definition 2.2. A function 𝑓∈BC(ℝ,𝕏)(𝑓∈BC(ℝ×𝕏,𝕏)) is called weighted pseudo almost periodic (or 𝜌-pseudo almost periodic) if it can be expressed as 𝑓=𝑔+𝜙, where 𝑔∈AP(𝕏)(𝑔∈AP(ℝ×𝕏,𝕏)) and 𝜙∈PAP₀(𝕏,𝜌)(𝜙∈PAP₀(ℝ×𝕏,𝕏,𝜌)). The set of weighted pseudo almost-periodic functions from ℝ into 𝕏 is denoted by PAP(𝕏,𝜌)(PAP(ℝ×𝕏,𝕏,𝜌)) here.

Remark 2.3. (i) Clearly, AP(𝕏) and PAP₀(𝕏,𝜌) are closed linear subspaces of BC(ℝ,𝕏) under the supremum norm.
(ii) There are a lot of weighted pseudo almost-periodic functions. For example, the following function: 𝑓=sin𝑡+sin√2𝑡+11+𝑡² (2.11) is weighted pseudo almost periodic under weight 𝜌=1+𝑡², see Example 3.6 for another example.
(iii) If one set 𝜌(𝑡)=1, then weighted pseudo almost-periodic functions are exactly the so-called pseudo almost-periodic functions. Furthermore, when the weight 𝜌∈𝕌_𝑏, Diagana’s work showed that the class of functions in PAP₀(𝕏,𝜌) coincides with PAP₀(𝕏). So, the concept of weighted pseudo almost-periodic functions is a generalization of the concept of pseudo almost-periodic functions. From the following example given by us, one will see that this concept is a real generalization.

Example 2.4. Let 𝜌(𝑡)=𝑒^𝑡,𝑡≤0,1,𝑡>0. (2.12) Then the bounded continuous function 𝑓(𝑡)=sin𝑡+sin𝜋𝑡+arctan𝑡 (2.13) is pseudo almost-periodic under this weight 𝜌 but it is not the usual pseudo almost periodic function.
(iv) Suppose 𝑓=𝑔+𝜙∶ℝ↦𝕏 is a weighted pseudo almost-periodic function. From [3] we know that the decomposition of 𝑓 is not generally unique, that is, the function space PAP(𝕏,𝜌) cannot be decomposed as AP(𝕏)PAP₀𝕏,𝜌. (2.14) Therefore, we do not know whether PAP(𝕏,𝜌) is a Banach space under the norm 𝑓(𝑡)=sup_𝑡∈ℝ𝑔(𝑡)+sup_𝑡∈ℝ𝜙(𝑡), (2.15) although AP(𝕏) and PAP₀(𝕏,𝜌) are closed linear subspaces of BC(ℝ,𝕏) under the supremum norm, and hence Banach spaces.

In order to obtain a completeness theorem for the space PAP(𝕏,𝜌) by overcoming the trouble showed in (iii) of the above remark, we use the “modular" idea and endow the weighted pseudo almost-periodic function in PAP(𝕏,𝜌) with a new norm as follows.

Let {𝑔_𝑖+𝜙_𝑖,𝑖∈𝐼} be all the possible decomposition of 𝑓∈PAP(𝕏,𝜌). We define 𝑓_𝜌=inf_𝑖∈𝐼𝑔_𝑖+𝜙_𝑖=inf_𝑖∈𝐼sup_𝑡∈ℝ𝑔_𝑖(𝑡)+sup_𝑡∈ℝ𝜙_𝑖(𝑡). (2.16) Clearly, ‖⋅‖_𝜌 is a norm on PAP(𝕏,𝜌). Moreover, we have the following result.

Theorem 2.5 (completeness theorem). PAP(𝕏,𝜌) is a Banach space under the norm ‖⋅‖_𝜌.

Proof. Suppose {𝑓_𝑛} is a Cauchy sequence in PAP(𝕏,𝜌), relative to the norm ‖⋅‖_𝜌. Then we can choose a subsequence {𝑓_𝑛𝑖} with 𝑓_𝑛𝑖+1−𝑓_𝑛𝑖𝜌≤12^𝑖+1,𝑖=1,2,…. (2.17) Set ℎ₀=𝑓_𝑛1,ℎ_𝑖=𝑓_𝑛𝑖+1−𝑓_𝑛𝑖for𝑖=1,2,…. (2.18) Then by the definition of the norm ‖⋅‖_𝜌, we can decompose each ℎ_𝑖 as ℎ_𝑖=𝑔_𝑖+𝜙_𝑖, where 𝑔_𝑖∈AP(ℝ,𝕏) and 𝜙_𝑖∈PAP₀(𝕏,𝜌) with ℎ_𝑖𝜌≥𝑔_𝑖+𝜙_𝑖−12^𝑖+1,𝑖=0,1,2,…. (2.19) Observing that 𝑓_𝑛𝑖=𝑓_𝑛1+𝑓_𝑛2−𝑓_𝑛1+⋯+𝑓_𝑛𝑖−𝑓_𝑛𝑖−1=ℎ₀+ℎ₁+⋯+ℎ_𝑖−1=𝑔₀+𝑔₁+⋯+𝑔_𝑖−1+𝜙₀+𝜙₁+⋯+𝜙_𝑖−1=^𝑖−1_𝑘=0𝑔_𝑘+^𝑖−1_𝑘=0𝜙_𝑘,𝑖=0,1,2,…, (2.20) we claim that 𝑓_𝑛𝑖 converges to a weighted pseudo almost-periodic function. Actually, it follows from (2.17) and (2.19) that 𝑔_𝑖+𝜙_𝑖≤ℎ_𝑖𝜌+12^𝑖+1≤𝑓_𝑛𝑖+1−𝑓_𝑛𝑖𝜌+12^𝑖+1≤12^𝑖,𝑖=1,2,…. (2.21) So 𝑔_𝑖≤12^𝑖,𝑖=1,2,…. (2.22) Therefore ^∞_𝑘=0𝑔_𝑘≤^∞_𝑘=0𝑔_𝑘≤𝑔₀+^∞_𝑘=112^𝑘=𝑔₀+1. (2.23) This, together with the fact that AP(ℝ,𝕏) is a Banach space and lim_𝑖→∞‖𝑔_𝑖‖=0, implies that 𝑔∶=∑^∞_𝑘=0𝑔_𝑘 exists and 𝑔∈AP(ℝ,𝕏). The same arguments indicate that 𝜙∶=∑^∞_𝑘=0𝜙_𝑘 exists and 𝜙∈PAP(𝕏,𝜌). Thus, by (2.20) we have lim_𝑖→∞𝑓_𝑛𝑖=^∞_𝑘=0𝑔_𝑘+^∞_𝑘=0𝜙_𝑘=𝑔+𝜙. (2.24) Therefore, 𝑓_𝑛𝑖 converges to 𝑓=𝑔+𝜙, which belongs to PAP(𝕏,𝜌). In other words, {𝑓_𝑛} is a Cauchy sequence having a convergent subsequence. Consequently, {𝑓_𝑛} is a convergent sequence under the norm ‖⋅‖_𝜌. This means that PAP(𝕏,𝜌) is a Banach space under the norm ‖⋅‖_𝜌.

Remark 2.6. Clearly, the norm ‖⋅‖₁ coincides with the sup norm when 𝑓 is a pseudo almost-periodic function, as a result of the uniqueness of decomposition of pseudo almost-periodic functions.

3. Composition Theorem and Existence Theorem

To obtain the existence of weighted pseudo almost periodic mild solutions to the semilinear evolution equation (1.1), we first establish a new composition theorem as follows.

Theorem 3.1 (composition theorem). Let 𝑓∈PAP(ℝ×𝕏,𝕏,𝜌) and𝑓=𝑔+𝜙 with 𝑔∈AP(ℝ×𝕏,𝕏) and 𝜙∈PAP₀(ℝ×𝕏,𝜌). Assume that 𝑓(𝑡,𝑥) is uniformly continuous in every bounded subset 𝐾⊂𝕏 uniformly for 𝑡∈ℝ. If 𝑥∈PAP(𝕏,𝜌) and 𝑥(𝑡)=𝛼(𝑡)+𝛽(𝑡), with 𝛼(𝑡)∈AP(𝕏), and 𝛽(𝑡)∈PAP₀(𝕏,𝜌), then 𝑔(⋅,𝛼(⋅))∈AP(𝕏),𝑓(⋅,𝑥(⋅))−𝑔(⋅,𝛼(⋅))∈PAP₀𝕏,𝜌, (3.1) and furthermore 𝑓(⋅,𝑥(⋅))∈PAP(𝕏,𝜌).

Proof. Write 𝑓(𝑡,𝑥(𝑡))=𝑔(𝑡,𝛼(𝑡))+𝑓(𝑡,𝑥(𝑡))−𝑓(𝑡,𝛼(𝑡))+𝜙(𝑡,𝛼(𝑡)). (3.2) Then, we claim that(a)𝑔(𝑡,𝛼(𝑡))∈AP(𝕏), (b)𝐹(𝑡)∶=𝑓(𝑡,𝑥(𝑡))−𝑓(𝑡,𝛼(𝑡))∈PAP₀(𝕏,𝜌), (c)𝜙(𝑡,𝛼(𝑡))∈PAP₀(𝕏,𝜌). Actually, by the theorem of composition of almost-periodic functions, it is easy to see that (a) holds.
Next, let us prove (b).
Clearly, 𝐹(𝑡) is bounded and continuous. By the uniformly continuity of 𝑓(𝑡,𝑥), we know that for any 𝜀>0, there exists 𝛿>0 such that 𝛽(𝑡)∶=‖𝑥(𝑡)−𝛼(𝑡)‖<𝛿 (3.3) implies that 𝑓(𝑡,𝑥(𝑡))−𝑓(𝑡,𝛼(𝑡))<𝜀. (3.4) Set 𝐸₁∶=𝑡∈ℝ∶𝛽(𝑡)<𝛿,𝐸₂∶=𝑡∈ℝ∶𝛽(𝑡)≥𝛿. (3.5) Then 𝐸₁ and 𝐸₂ are measurable sets and 𝐸₁∪𝐸₂=ℝ. In addition, ‖𝐹(𝑡)‖<𝜀 on the set 𝐸₁. Observe 1𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}𝜌(𝑠)1𝑑𝑠=𝛿𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}(𝑠)≤1𝛿𝜌𝑑𝑠𝛿𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}𝛽(𝑠𝜌(𝑠)≤1)𝑑𝑠𝛿𝜇𝑇,𝜌^𝑇_−𝑇𝛽(𝑠)𝜌(𝑠)𝑑𝑠. (3.6) Using 𝛽(𝑡)∈PAP₀(𝕏,𝜌), we get lim_𝑇→∞1𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}𝜌(𝑠)𝑑𝑠=0. (3.7) Moreover, for any 𝑇>0, 1𝜇𝑇,𝜌^𝑇_−𝑇‖𝐹(𝑠𝜌(𝑠)=1)‖𝑑𝑠𝜇𝑇,𝜌_{𝐸1∩[−𝑇,𝑇]}‖𝐹(𝑠𝜌(𝑠)1)‖𝑑𝑠+𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}‖𝐹(𝑠𝜌(𝑠)≤𝜀)‖𝑑𝑠𝜇𝑇,𝜌_{𝐸1∩[−𝑇,𝑇]}𝜌(𝑠)𝑑𝑠+sup_𝑡∈ℝ‖𝐹(𝑡)‖𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}𝜌(𝑠)𝑑𝑠≤𝜀+sup_𝑡∈ℝ‖𝐹(𝑡1)‖𝜇𝑇,𝜌_{𝐸2∩[−𝑇,𝑇]}𝜌(𝑠)𝑑𝑠. (3.8) Therefore, we have lim_𝑇→∞1𝜇𝑇,𝜌^𝑇_−𝑇‖𝐹(𝑠)‖𝜌(𝑠)𝑑𝑠=0. (3.9) That means that (b) is true.
Finally, we prove (c).
Since 𝜙(𝑡,𝛼(𝑡)) is continuous in [−𝑇,𝑇], it is uniformly continuous in [−𝑇,𝑇]. Set 𝐼=𝛼([−𝑇,𝑇]). Then 𝐼 is compact in 𝕏. So one can find finite open balls 𝐵_𝑘(𝑘=1,2,…,𝑚) with center 𝑥_𝑘∈𝐼 and radius 𝛿 small enough such that 𝐼⊂⋃^𝑚_𝑘=1𝐵_𝑘 and 𝜙(𝑡,𝛼(𝑡))−𝜙𝑡,𝑥_𝑘<𝜀2,𝛼(𝑡)∈𝐵_𝑘,𝑡∈[−𝑇,𝑇]. (3.10) The set 𝑂_𝑘={𝑡∈[−𝑇,𝑇]∶𝛼(𝑡)∈𝐵_𝑘} is open in [−𝑇,𝑇] and [−𝑇,𝑇]=⋃^𝑚_𝑘=1𝑂_𝑘. Let 𝑉₁=𝑂₁,𝑉_𝑘=𝑂_𝑘⧵^𝑘−1_𝑗=1𝑂_𝑗(2≤𝑘≤𝑚). (3.11) Then 𝑉_𝑖∩𝑉_𝑗=∅ when 𝑖≠𝑗,1≤𝑖,𝑗≤𝑚, and 𝑉_𝑘⊂𝑂_𝑘,^𝑚_𝑘=1𝑉_𝑘=[−𝑇,𝑇]. (3.12) On the other hand, by 𝜙∈PAP₀(ℝ×𝕏,𝕏,𝜌), we see that there exists 𝑇₀>0 such that 1𝜇𝑇,𝜌^𝑇_−𝑇𝜙𝑠,𝑥_𝑘𝜌(𝑠)𝜀𝑑𝑠<2𝑚,for𝑇≥𝑇₀,𝑘∈{1,2,…,𝑚}. (3.13) Hence for 𝑇≥𝑇₀, 1𝜇𝑇,𝜌^𝑇_−𝑇𝜙((𝑠𝜌(𝑠)=1𝑠,𝛼))𝑑𝑠𝜇𝑇,𝜌^𝑚_{𝑘=1𝑉𝑘∩[−𝑇,𝑇]}𝜙((𝑠𝜌(𝑠)≤1𝑠,𝛼))𝑑𝑠𝜇𝑇,𝜌^𝑚_{𝑘=1𝑉𝑘∩[−𝑇,𝑇]}𝜙(𝑠,𝛼(𝑠))−𝜙𝑠,𝑥_𝑘𝜌(𝑠)+1𝑑𝑠𝜇𝑇,𝜌^𝑚_{𝑘=1𝑉𝑘∩[−𝑇,𝑇]}𝜙𝑠,𝑥_𝑘𝜌(𝑠)≤1𝑑𝑠𝜇𝑇,𝜌^𝑚_{𝑘=1𝑉𝑘∩[−𝑇,𝑇]}𝜀2𝜌(𝑠)𝑑𝑠+^𝑚_𝑘=11𝜇𝑇,𝜌^𝑇_−𝑇𝜙𝑠,𝑥_𝑘𝜌(𝑠)≤𝜀𝑑𝑠2𝜀+𝑚2𝑚=𝜀. (3.14) Thus, (c) holds.
Since PAP₀(𝕏,𝜌) is a linear space, we have 𝑓(⋅,𝑥(⋅))∈PAP(𝕏,𝜌), by (a), (b), and (c).

The following assumption on the weight 𝜌∈𝕌_∞ will be used in our next investigation.

(H₀) For a 𝜔>0, 𝑃(𝜔)∶=lim_𝑇→+∞1𝜇𝑇,𝜌^𝑇_−𝑇𝑒^{−𝜔(𝑡+𝑇)}𝜌(𝑡)𝑑𝑡=0,_𝑇𝜎𝑒^{−𝜔(𝑡−𝜎)}𝜌(𝑡)𝑑𝑡≤𝑀𝜌(𝜎) (3.15)(3.16) for some constant 𝑀>0 and 𝑇 is sufficiently large.

Remark 3.2. (i) Since 𝜌(𝑡)∈𝕌_∞, condition (3.15) is satisfied when sup_𝑇>0^𝑇_−𝑇𝑒^{−𝜔(𝑡+𝑇)}𝜌(𝑡)𝑑𝑡<+∞. (3.17) Condition (3.16) is satisfied when 𝜌(𝑡) is nonincreasing. Therefore, if the weight 𝜌 is bounded and nonincreasing, then both (3.15) and (3.16) are satisfied.
(ii) Besides the weight mentioned in (i) above, there are lots of weights satisfying the assumption (𝐻₀). By giving the following example, we present two unbounded weights satisfying the assumption (𝐻₀).

Example 3.3. (1) Take 𝜌(𝑡)=𝑒𝜔_𝑡,𝜔>𝜔>0. Then we have 0≤𝑃(𝜔)=lim_𝑇→+∞1𝜇𝑇,𝜌^𝑇_−𝑇𝑒^{−𝜔(𝑡+𝑇)}𝜌(𝑡)𝑑𝑡=lim_𝑇→+∞𝑒^−𝜔𝑇𝜇𝑇,𝜌^𝑇_−𝑇𝑒^−(𝜔−𝜔_)𝑡𝑑𝑡=lim_𝑇→+∞−𝜔𝜔𝑒𝜔−⁽𝜔_{−2𝜔)𝑇}−𝑒⁻𝜔_𝑇𝑒𝜔_𝑇−𝑒⁻𝜔_𝑇≤lim_𝑇→+∞𝜔𝑒⁻²𝜔_𝑇−𝑒^−2𝜔𝑇𝜔−𝜔=0. (3.18) Therefore 𝑃(𝜔) satisfies condition (3.15). Furthermore, ^𝑇_𝜎𝑒^{−𝜔(𝑡−𝜎)}𝜌(𝑡)𝑒𝑑𝑡=^𝜔𝜎𝑒^−(𝜔−𝜔_)𝜎−𝑒^−(𝜔−𝜔_)𝑇𝜔≤𝑒𝜔𝜔−^𝜎𝜔(𝜎)𝜔−∶=𝑀𝜌, (3.19) where 𝑀=1/(𝜔−𝜔). So (3.16) is satisfied.
(2) Take 𝜌(𝑡)=1+𝑡². Then for every 𝜔>0, we obtain 0≤𝑃(𝜔)=lim_𝑇→+∞𝑒^−𝜔𝑇𝜇𝑇,𝜌^𝑇_−𝑇𝑒^−𝜔𝑡1+𝑡²𝑑𝑡=lim_𝑇→+∞𝑒^−𝜔𝑇𝜇−𝜔𝑇,𝜌²𝑡²+2𝜔𝑡+2+𝜔²𝜔³𝑒^{−𝜔𝑡𝑇}_−𝑇≤lim_𝑇→+∞𝜔²𝑇²−2𝜔𝑇+2+𝜔²2𝑇+𝑇³/3𝜔³=0. (3.20) Moreover, ^𝑇_𝜎𝑒^{−𝜔(𝑡−𝜎)}𝜌(𝑡)𝑑𝑡=^𝑇_𝜎𝑒^{−𝜔(𝑡−𝜎)}1+𝑡²𝑑𝑡=𝑒^𝜔𝜎−𝜔²𝑡²+2𝜔𝑡+2+𝜔²𝜔³𝑒^{−𝜔𝑡𝑇}_𝜎≤𝜔²𝜎²+2𝜔𝜎+2+𝜔²𝜔³≤𝜔²+1𝜎²+2+2𝜔²𝜔³≤2𝜔²+1𝜔³𝜎²(𝜎),+1∶=𝑀𝜌 (3.21) where 𝑀=2(𝜔²+1)/𝜔³ is a constant.

Definition 3.4. A mild solution to (1.1) is a continuous function 𝑥(𝑡)∶ℝ↦𝕏 satisfying 𝑥(𝑡)=𝑇(𝑡−𝑎)𝑥(𝑎)+^𝑡_𝑎𝑇(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠 (3.22) for all 𝑡≥𝑎 and all 𝑎∈ℝ.

Theorem 3.5. Let 𝐴 generate an exponentially stable 𝐶₀- semigroup {𝑇(𝑡)}_𝑡≥0, that is, ‖𝑇(𝑡)‖≤𝐾𝑒^−𝜔𝑡,∀𝑡≥0, (3.23) for some positive constants 𝐾 and 𝜔. Let the weight 𝜌 satisfy (𝐻₀) for the 𝜔 in (3.23). Assume that(𝐻₁)𝑓=𝑔+𝜙∶ℝ×𝕏↦𝕏 is weighted pseudo almost periodic, with 𝑔∈𝐴𝑃(ℝ×𝕏,𝕏) and 𝜙∈PAP₀(ℝ×𝕏,𝕏,𝜌).(𝐻₂) for every 𝑡∈ℝ,𝑥,𝑦∈𝕏, 𝑓(𝑡,𝑥)−𝑓𝑡,𝑦≤𝐿_𝑓𝑥−𝑦,𝑔(𝑡,𝑥)−𝑔𝑡,𝑦≤𝐿_𝑔𝑥−𝑦, (3.24) for some constants 𝐿_𝑓 and 𝐿_𝑔 with (2𝐿_𝑔+𝐿_𝑓)𝐾/𝜔<1.Then (1.1) admits a unique weighted pseudo almost-periodic mild solution.

Proof. Define a nonlinear operator ℱ on PAP(𝕏,𝜌) by (ℱ𝑥)(𝑡)=^𝑡_−∞𝑇(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈ℝ. (3.25) Fix 𝑥(𝑡)∈PAP(𝕏,𝜌). Then 𝑥(𝑡)=𝛼(𝑡)+𝛽(𝑡) (3.26) with 𝛼(𝑡)∈AP(𝕏),𝛽(𝑡)∈PAP₀𝕏,𝜌. (3.27) By Theorem 3.1, we know that there exist two functions (𝑡)((𝑡(𝕏),𝜑(𝑡)((𝑡((𝑡((𝑡𝑔=𝑔𝑡,𝛼))∈AP=𝑓𝑡,𝑥))−𝑓𝑡,𝛼))+𝜑𝑡,𝛼))∈PAP₀𝕏,𝜌 (3.28)(3.29) such that 𝑓(⋅,𝑥(⋅))=𝑔(⋅)+𝜑(⋅). (3.30) Therefore (ℱ𝑥)(𝑡) can be expressed as (ℱ𝑥)(𝑡)=𝐺(𝑡)+Φ(𝑡), where 𝐺(𝑡)∶=^𝑡_−∞𝑇(𝑡−𝑠)𝑔(𝑠)𝑑𝑠,Φ(𝑡)∶=^𝑡_−∞𝑇(𝑡−𝑠)𝜑(𝑠)𝑑𝑠. (3.31) From (3.28) and the fact that {𝑇(𝑡)} is exponentially stable, it follows that 𝐺(𝑡)∈AP(𝕏).
Next, we show that Φ(𝑡)∈PAP₀(𝕏,𝜌).
By (3.29), we see that 𝜑(𝑡) is bounded on ℝ, that is, sup_𝑡∈ℝ‖𝜑(𝑡)‖<+∞, and 𝜑 is continuous. Hence, for any 𝑇>0, we have 1𝜇𝑇,𝜌^𝑇_−𝑇‖Φ(𝑠𝜌(𝑠)1)‖𝑑𝑠=𝜇𝑇,𝜌^𝑇_−𝑇^𝑠_−∞𝑇()𝜑(𝜎)𝜌(𝑠)≤1𝑡−𝜎𝑑𝜎𝑑𝑠𝜇𝑇,𝜌^𝑇_−𝑇^𝑠_−∞‖𝑇(𝜑(𝜎(𝑠)≤1𝑠−𝜎)‖)𝑑𝜎𝜌𝑑𝑠𝜇𝑇,𝜌^𝑇_−𝑇^𝑠_−∞𝐾𝑒^{−𝜔(𝑠−𝜎)}𝜑(𝜎(𝑠)≤1)𝑑𝜎𝜌𝑑𝑠𝜇𝑇,𝜌^𝑇_−𝑇^−𝑇_−∞+^𝑠_−𝑇𝐾𝑒^{−𝜔(𝑠−𝜎)}𝜑(𝜎)𝑑𝜎𝜌(𝑠)𝑑𝑠∶=𝐼+𝐽, (3.32) where 1𝐼=𝜇𝑇,𝜌^𝑇_−𝑇^−𝑇_−∞𝐾𝑒^{−𝜔(𝑠−𝜎)}𝜑(𝜎(𝑠)1)𝑑𝜎𝜌𝑑𝑠,𝐽=𝜇𝑇,𝜌^𝑇_−𝑇^𝑠_−𝑇𝐾𝑒^{−𝜔(𝑠−𝜎)}𝜑(𝜎)𝑑𝜎𝜌(𝑠)𝑑𝑠. (3.33) Clearly, 1𝐼≤𝜇𝑇,𝜌sup_𝑡∈ℝ𝜑(𝜎)^𝑇_−𝑇^−𝑇_−∞𝐾𝑒^{−𝜔(𝑠−𝜎)}𝜌(𝑠)≤𝐾𝑑𝜎𝑑𝑠,𝜔𝜇𝑇,𝜌sup_𝑡∈ℝ𝜑(𝜎)^𝑇_−𝑇𝑒^{−𝜔(𝑠+𝑇)}𝜌(𝑠)𝑑𝑠. (3.34) This, together with the assumption (𝐻₀), implies that 𝐼⟶0as𝑇⟶+∞. (3.35) Moreover, by Fubini’s theorem and assumption 𝐻₀, we get 𝐾𝐽=𝜇𝑇,𝜌^𝑇_−𝑇𝜑(𝜎)^𝑇_𝜎𝑒^{−𝜔(𝑠−𝜎)}𝜌(𝑠)≤𝑑𝑠𝑑𝜎𝑀𝐾𝜇𝑇,𝜌^𝑇_−𝑇𝜑(𝜎)𝜌(𝜎)𝑑𝜎. (3.36) Thus, (3.29) implies that 𝐽⟶0as𝑇⟶+∞. (3.37) So, Φ(𝑡)∈PAP₀(𝕏,𝜌).
Consequently, (ℱ𝑥)(𝑡)=𝐺(𝑡)+Φ(𝑡) is a weighted pseudo almost-periodic function on ℝ. This means that ℱ maps PAP(𝕏,𝜌) into PAP(𝕏,𝜌).
Now, let 𝑥,𝑦∈PAP(𝕏,𝜌) with 𝑥=𝛼_𝑥+𝛽_𝑥,𝑦=𝛼_𝑦+𝛽_𝑦, (3.38) where 𝛼_𝑥,𝛼_𝑦∈AP(𝕏),𝛽_𝑥,𝛽_𝑦∈PAP₀𝕏,𝜌 (3.39) are periodic and vanishing mean value components of weighted pseudo almost-periodic functions 𝑥,𝑦. Then by Theorem 3.1, we have 𝑔⋅,𝛼_𝑥(⋅),𝑔⋅,𝛼_𝑦(⋅(𝕏),𝑓((⋅)∈AP⋅,𝑥))−𝑔⋅,𝛼_𝑥(⋅),𝑓⋅,𝑦(⋅)−𝑔⋅,𝛼_𝑦(⋅)∈PAP₀𝕏,𝜌. (3.40) Observe ℱ𝑥(𝑡)−ℱ𝑦(𝑡)=^𝑡_−∞𝑇(𝑡−𝑠)𝑔𝑠,𝛼_𝑥(𝑠)−𝑔𝑠,𝛼_𝑦(𝑠)+𝑑𝑠_𝑡−∞𝑇(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))−𝑔𝑠,𝛼_𝑥(𝑠)−𝑓𝑠,𝑦(𝑠)−𝑔𝑠,𝛼_𝑦(𝑠)𝑑𝑠. (3.41) We know from the arguments in the above paragraph that the two integrals on the right side are in, respectively, AP(𝕏) and PAP₀(𝕏,𝜌). Therefore, ℱ𝑥−ℱ𝑦_𝜌≤sup_𝑡∈ℝ^𝑡_−∞𝑇(𝑡−𝑠)𝑔𝑠,𝛼_𝑥(𝑠)−𝑔𝑠,𝛼_𝑦(𝑠)𝑑𝑠+sup_𝑡∈ℝ^𝑡_−∞𝑇(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))−𝑔𝑠,𝛼_𝑥(𝑠)−𝑓𝑠,𝑦(𝑠)−𝑔𝑠,𝛼_𝑦(𝑠)≤𝑑𝑠2𝐿_𝑔𝛼_𝑥−𝛼_𝑦+𝐿_𝑓𝑥−𝑦sup_𝑡∈ℝ^𝑡_−∞‖𝑇(≤𝑡−𝑠)‖𝑑𝑠2𝐿_𝑔+𝐿_𝑓𝛼_𝑥−𝛼_𝑦+𝛽_𝑥−𝛽_𝑦sup_𝑡∈ℝ^𝑡_−∞𝐾𝑒^{−𝜔(𝑡−𝑠)}=𝑑𝑠2𝐿_𝑔+𝐿_𝑓𝐾𝜔𝛼_𝑥−𝛼_𝑦+𝛽_𝑥−𝛽_𝑦. (3.42) Since the inequality is true for any decomposition of 𝑥(𝑡),𝑦(𝑡), we can take the infimum and then obtain ℱ𝑥−ℱ𝑦_𝜌≤2𝐿_𝑔+𝐿_𝑓𝐾𝜔𝑥−𝑦_𝜌. (3.43) Hence ℱ is a contraction under the assumption that (2𝐿_𝑔+𝐿_𝑓)𝐾/𝜔<1.
By the contraction mapping theorem, the mapping ℱ has a unique fixed point 𝑥(𝑡)∈PAP(𝕏,𝜌), and this fixed point satisfies the integral equation 𝑥(𝑡)=^𝑡_−∞𝑇(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠, (3.44) for all 𝑡∈ℝ. Fixing 𝑎∈ℝ, we have 𝑥(𝑎)=^𝑎_−∞𝑇(𝑎−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠. (3.45) Since 𝑇(𝑡−𝑠)=𝑇(𝑡−𝑎)(𝑎−𝑠),𝑡≥𝑎≥𝑠, (3.46) it follows that 𝑥(𝑡) satisfies (3.22). Hence 𝑥(𝑡) is a mild solution to (1.1).
On the other hand, let 𝑦(𝑡) be a pseudo almost-periodic mild solution to (1.1). Then 𝑦(𝑡) satisfies (3.22), with 𝑥 replaced by 𝑦. Letting 𝑎→−∞ yields 𝑦(𝑡)=^𝑡_−∞𝑇(𝑡−𝑠)𝑓𝑠,𝑦(𝑠)𝑑𝑠,𝑡∈ℝ. (3.47) Since 𝑦(𝑡) is bounded on ℝ and {𝑇(𝑡)}_𝑡≥0 is exponentially stable, hence 𝑥(𝑡)≡𝑦(𝑡) on ℝ.
In conclusion, 𝑥(𝑡) is the unique mild solution to (1.1), which ends the proof.

Example 3.6. Let 𝕏=𝐿²(0,1), and 𝐷(𝐴)∶=𝑥∈𝐶¹[0,1];𝑥^′isabsolutelycontinuouson[0,1],𝑥^′′(0)(1),(𝑡)∈𝕏,𝑥=𝑥=0𝐴𝑥=𝑥^′′(𝑡),𝑡∈(0,1),𝑥∈𝐷(𝐴). (3.48) Then, 𝐴 generates a 𝐶₀-semigroup 𝑇(𝑡) on 𝕏 with ‖𝑇(𝑡)‖≤𝑒^−𝜋2𝑡,𝑡≥0. (3.49) Let 𝑓(𝑡,𝑥)=𝑥cos𝑡+cos√+12𝑡1+𝑡²()()𝜌(𝑡)=cos𝑥=𝑔𝑡,𝑥+𝜙𝑡,𝑥,𝑡∈ℝ,𝑥∈𝕏,1+𝑡². (3.50) Then, 𝑔∈AP(ℝ×𝕏,𝕏),𝜙∈PAP₀(ℝ×𝕏,𝕏,𝜌), and 𝑓,𝑔 satisfy (𝐻₂) with 𝐿_𝑓=2,𝐿_𝑔=2,𝐾=1,𝜔=𝜋². (3.51) By Theorem 3.5, the corresponding equation (1.1) has a unique weighted pseudo almost-periodic mild solution under the weight 𝜌(𝑡)=(1+𝑡²).

Acknowledgments

This work was supported by the NSF of China (11071042, 11026100, 11171210), the Chinese Academy of Sciences, the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Laboratory of Mathematics for Nonlinear Science at the Fudan University.