Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations

Abstract

This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the form 𝑢′(𝑡)=𝐴(𝑡)𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡)), 𝑡∈ℝ, 𝑢(𝑡+𝑇)=−𝑢(𝑡), 𝑡∈ℝ, where (𝐴(𝑡))_𝑡∈ℝ (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach space 𝑋. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on (𝐴(𝑡))_𝑡∈ℝ, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.

1. Introduction and Motivation

Antiperiodic problems have recently proved to be valuable tools in the modelling of many phenomena in physical processes. For the background on this class of problems we refer the reader to [1–3] and the references therein. For this reason, much attention is attracted by questions of existence of antiperiodic solutions to the various antiperiodic problems represented by linear and nonlinear abstract evolution equations since the work of Okochi [4] in 1988 (see also [5, 6]). The literature related to such problems is quite extensive; see, for instance, Haraux [7] for nonlinear first order evolution equations in Hilbert spaces and Aftabizadeh et al. [8] and Aizicovici and Pavel [9] for second order evolution equations in Hilbert and Banach spaces. In particular, using the maximal monotone property of the derivative operator with antiperiodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings, Liu [10] recently studied the antiperiodic problem for nonlinear evolution equation with nonmonotone perturbation of the form 𝑢^′(𝑡)(𝑡)(𝑡)(𝑡)(),𝑢(𝑇)(0)+𝐴𝑢+𝐺𝑢=𝑓,a.e.𝑡∈0,𝑇=−𝑢 (1) in a real reflexive Banach space 𝑉, where 𝐴 is monotone and 𝐺 is not. For more details about development and applications along this line, see, for example, [1, 11–13] and the references therein. Let us note that equations in the research mentioned above are all autonomous.

Motivated by these works, in this paper we will carry out our investigation to the semilinear nonautonomous parabolic evolution equation having the form 𝑢^′(𝑡)=𝐴(𝑡)𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡)),𝑡∈ℝ, (2) subject to antiperiodic condition 𝑢(𝑡+𝑇)=−𝑢(𝑡),𝑡∈ℝ (3) in the Banach space 𝑋. Here, ℝ stands for the set of real numbers, (𝐴(𝑡))_𝑡∈ℝ (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on 𝑋 and has domains (𝐷(𝐴(𝑡)))_𝑡∈ℝ, and 𝑓 is a given function to be specified later.

Note also that the problem (2)-(3) has been considered by Wang [14] under different situations, in which the author proved the existence of antiperiodic mild (strict) solutions in the case of (2) being a mild (strict) dissipative differential equation by using the modular degree theorem. The line, which we will go along in this study, is that we establish some results concerning the existence of antiperiodic mild solutions to the problem (2)-(3) under new criteria (without the assumption of (2) being dissipative). Then, we also consider the antiperiodic mild solutions to nonautonomous semilinear parabolic evolution equation of neutral type 𝑑[𝑢(𝑡)((𝑡(𝑡)𝑢(𝑡)((𝑡d𝑡−𝐹𝑡,𝑢))]=𝐴+𝑓𝑡,𝑢)),𝑡∈ℝ. (4) Finally, as a sample of possible applications, we give a result on the existence of antiperiodic mild solutions to a partial differential equation with homogeneous Dirichlet boundary condition, whose operator in the linear part generates an evolution family of exponential stability. As can be seen, our results extend and unify many existing results in this area. Banach’s contraction principle, Schauder’s fixed point theorem, and Krasnoselskii’s fixed point theorem, as well as the theory of evolution families such as exponential dichotomy techniques, are employed in our approach. Note also that the conditions we used in this paper are quite different from those in [14].

We organize this paper as follows. We present some definitions and preliminary facts in Section 2. In Section 3, starting with introducing the assumptions that are needed in the proofs of our main results, we then establish some existence theorems of antiperiodic mild solutions to the problem (2)-(3). In Section 4, we extend the result obtained in Section 3 to the neutral problem (4). An example is given to illustrate the theorem in Section 5.

2. Preliminaries

Throughout this paper, 𝑋 is assumed to be a Banach space with norm ‖⋅‖ and ℒ(𝑋) stands for the Banach space of all bounded linear operators from 𝑋 to 𝑋 equipped with its natural topology. Denote by 𝐶_𝑏(ℝ;𝑋) the Banach space of all bounded, continuous functions from ℝ to 𝑋 equipped with the sup norm ‖𝑢‖_{𝐶𝑏(ℝ;𝑋)}=sup{‖𝑢(𝑡)‖;𝑡∈ℝ}, (5) by 𝐿(0,𝑇;𝑋) the Banach space of all Bocher integrable functions from [0,𝑇] to 𝑋 equipped with the norm ‖𝑢‖_{𝐿(0,𝑇;𝑋)}=^𝑇₀‖𝑢(𝑡)‖d𝑡, (6) and by 𝐿_loc(ℝ;𝑋) the set of all locally Bocher integrable functions from ℝ to 𝑋.

A function 𝑢∈𝐶_𝑏(ℝ;𝑋) is said to be 𝑇-antiperiodic if 𝑢(𝑡+𝑇)=−𝑢(𝑡),∀𝑡∈ℝ. (7)

By 𝑃_𝑇𝐴(ℝ;𝑋), we denote the set of all 𝑇-antiperiodic functions from ℝ to 𝑋. It is easy to see that 𝑃_𝑇𝐴(ℝ;𝑋), equipped with the sup norm, is a Banach space. For every 𝑟>0, write Ω_𝑟=𝑢∈𝑃_𝑇𝐴(ℝ,𝑋);‖𝑢‖_{𝑃𝑇𝐴(ℝ;𝑋)}≤𝑟, (8) which is convex closed subset of 𝑃_𝑇𝐴(ℝ,𝑋).

The following lemma provides some useful information on compactness criterion, which can be regarded as an extension of the known Arzéla-Ascoli theorem.

Lemma 1 (see [15, Lemma 3.1]). A set 𝐷⊆𝑃_𝑇𝐴(ℝ;𝑋) is relatively compact in 𝑃_𝑇𝐴(ℝ;𝑋) if 𝐷 is equicontinuous and the set 𝐷(𝑡)∶={𝑢(𝑡);𝑢∈𝐷} is relatively compact in 𝑋 for every 𝑡∈ℝ.

Definition 2. A family of bounded linear operators 𝑈={𝑈(𝑡,𝑠)}_𝑡≥𝑠 on a Banach space 𝑋 is called a strongly continuous evolution family if(1)𝑈(𝑡,𝑟)𝑈(𝑟,𝑠)=𝑈(𝑡,𝑠) and 𝑈(𝑡,𝑡)=𝐼 for all 𝑡≥𝑟≥𝑠 and 𝑡,𝑟,𝑠∈ℝ,(2)the map (𝑡,𝑠)↦𝑈(𝑡,𝑠)𝑥 is continuous for all 𝑥∈𝑋, 𝑡≥𝑠 and 𝑡,𝑠∈ℝ.

Throughout the paper, we assume that (𝐴(𝑡))_𝑡∈ℝ is a family of closed and densely defined operators on 𝑋, which satisfies the conditions of Acquistapace and Terreni (AT₁) and (AT₂).(AT₁)𝐴(𝑡) are linear operators on 𝑋 and there are constants 𝜆₀≥0, 𝜃∈((𝜋/2),𝜋), and 𝐾₁≥0 such that Σ_𝜃∪{0}⊂𝜌(𝐴(𝑡)−𝜆₀) and for all 𝜆∈Σ_𝜃∪{0} and 𝑡∈ℝ, 𝑅𝜆,𝐴(𝑡)−𝜆_0ℒ(𝑋)≤𝐾₁|𝜆|.1+ (9)(AT₂)There are constants 𝐾₂≥0 and 𝛼,𝛽∈(0,1] with 𝛼+𝛽>1 such that for all 𝜆∈Σ_𝜃 and 𝑡,𝑠∈ℝ𝐴(𝑡)−𝜆₀𝑅𝜆,𝐴(𝑡)−𝜆₀×𝑅𝜆₀,𝐴(𝑡)−𝑅𝜆₀,𝐴(𝑠)_ℒ(𝑋)≤𝐾₂|𝑡−𝑠|^𝛼|𝜆|^𝛽. (10)Here Σ_𝜃∶={𝜆∈ℂ⧵{0};|𝜆|≤𝜃}.

Conditions (AT₁) and (AT₂), which are initiated by Acquistapace and Terreni [16, 17] for 𝜆₀=0, are well understood and widely used in the literature.

Remark 3. It should be mentioned that when (𝐴(𝑡))_𝑡∈ℝ has a constant domain 𝐷(𝐴(𝑡)), (AT₂) can be replaced with the following: there exist constants 𝐾₂>0, 0<𝜇≤1 such that (𝐴(𝑡)−𝐴(𝑠))𝑅𝜆₀,𝐴(𝑟)_ℒ(𝑋)≤𝐾₂|𝑡−𝑠|^𝜇 (11) for all 𝑠,𝑡,𝑟∈ℝ (see, e.g., [18, 19]).

By an obvious rescaling from [16, Theorem 2.3] and [20, Theorem 2.1] (see also [17, 21]), it follows that conditions (AT₁) and (AT₂) ensure that there exists a unique evolution family {𝑈(𝑡,𝑠)}_𝑡≥𝑠 on 𝑋 such that(I)𝑈(⋅,𝑠)∈𝐶¹((𝑠,∞),ℒ(𝑋)), 𝜕𝑈(𝑡,𝑠)/𝜕𝑡=𝐴(𝑡)𝑈(𝑡,𝑠) for 𝑡>𝑠, and 𝐴(𝑡)^𝑘𝑈(𝑡,𝑠)_ℒ(𝑋)≤𝐶(𝑡−𝑠)^−𝑘; (12) for 0<𝑡−𝑠≤1, 𝑘=0,1;(II)𝜕⁺𝑈(𝑡,𝑠)𝑥/𝜕𝑠=−𝑈(𝑡,𝑠)𝐴(𝑠)𝑥 for 𝑡>𝑠 and 𝑥∈𝐷(𝐴(𝑠)) with 𝐴(𝑠)𝑥∈𝐷(𝐴(𝑠)).

In this case we say that (𝐴(𝑡))_𝑡∈ℝ generates the evolution family 𝑈.

Definition 4. An evolution family 𝑈={𝑈(𝑡,𝑠)}_𝑡≥𝑠 is called hyperbolic (or has exponential dichotomy) if there are projections 𝒫(𝑡), 𝑡∈ℝ, that are uniformly bounded and strongly continuous in 𝑡, and constants 𝑁, 𝛿>0 such that(a)𝑈(𝑡,𝑠)𝒫(𝑠)=𝒫(𝑡)𝑈(𝑡,𝑠) for all 𝑡≥𝑠;(b)the restriction 𝑈_𝒬(𝑡,𝑠)∶𝒬(𝑠)𝑋→𝒬(𝑡)𝑋 is invertible for all 𝑡≥𝑠 (and we set 𝑈_𝒬(𝑠,𝑡)=𝑈_𝒬(𝑡,𝑠)⁻¹);(c)‖𝑈(𝑡,𝑠)𝒫(𝑠)‖_ℒ(𝑋)≤𝑁𝑒^{−𝛿(𝑡−𝑠)} and ‖𝑈_𝒬(𝑠,𝑡)𝒬(𝑡)‖_ℒ(𝑋)≤𝑁𝑒^{−𝛿(𝑡−𝑠)} for all 𝑡≥𝑠.Here and below 𝒬=𝐼−𝒫. Specially, if 𝒫(𝑡)=𝐼 for 𝑡∈ℝ, then 𝑈 is said to be exponentially stable.

Exponential dichotomy is a classical concept in the study of the long-term behavior of evolution equations; see [22–24] and references therein.

3. The Existence of Antiperiodic Mild Solutions

In this section we establish some existence theorems of antiperiodic mild solutions to the problem (2)-(3).

To prove our main results, we introduce the following assumptions. For sake of brevity, put 𝐵_𝑟∶={𝑥∈𝑋;‖𝑥‖≤𝑟} for some 𝑟>0.(H₁)The evolution family 𝑈={𝑈(𝑡,𝑠)}_𝑡≥𝑠, generated by (𝐴(𝑡))_𝑡∈ℝ, is hyperbolic. Moreover, 𝑈(𝑡+𝑇,𝑠+𝑇)𝒫(𝑠+𝑇)=𝑈(𝑡,𝑠)𝒫(𝑠) for all 𝑡≥𝑠, and 𝑈_𝒬(𝑡+𝑇,𝑠+𝑇)𝒬(𝑠+𝑇)=𝑈_𝒬(𝑡,𝑠)𝒬(𝑠) for all 𝑡≤𝑠.(H₂)The function 𝑓∶ℝ×𝑋→𝑋 satisfies the following conditions.(i)𝑓(⋅,𝑢) is measurable for each 𝑢∈𝑋 and 𝑓(𝑡+𝑇,−𝑢)=−𝑓(𝑡,𝑢) for all 𝑡∈ℝ, 𝑢∈𝑋.(ii)There exists a constant 𝐿_𝑓>0 with 2𝑁𝐿_𝑓<𝛿 such that 𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)≤𝐿_𝑓‖𝑢−𝑣‖ (13) for a.e. 𝑡∈[0,𝑇] and all 𝑢,𝑣∈𝑋.(H₃)(i)The function 𝑓∶ℝ×𝑋→𝑋 is a Carathéodory function; that is, for every 𝑢∈𝑋, 𝑓(⋅,𝑢) is measurable and for a.e. 𝑡∈ℝ, 𝑓(𝑡,⋅) is continuous, and 𝑓(𝑡+𝑇,−𝑢)=−𝑓(𝑡,𝑢) for all 𝑡∈ℝ, 𝑢∈𝑋.(ii)There exists a function Φ_𝑟(⋅)∈𝐿(0,𝑇;ℝ⁺) such that 𝑓(𝑡,𝑢)≤Φ_𝑟(𝑡) (14) for a.e. 𝑡∈[0,𝑇] and all 𝑢∈𝐵_𝑟, and liminf_𝑟→+∞∫^𝑇₀Φ_𝑟(𝑡)d𝑡𝑟=𝜎₁ (15) with 𝑁𝜎₁<1−𝑒^−𝛿𝑇.(H₄)𝑈(𝑡,𝑠) is compact for 𝑡>𝑠.

Remark 5. From (H₁) it is clear that 𝑈(𝑡+𝑇,𝑠+𝑇)=𝑈(𝑡,𝑠) for all 𝑡≥𝑠.

Definition 6. A mild solution to (2) is a function 𝑢∈𝐶_𝑏(ℝ;𝑋) satisfying the integral equation 𝑢(𝑡)=𝑈(𝑡,𝑠)𝑢(𝑠)+^𝑡_𝑠𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏 (16) for all 𝑡>𝑠 and 𝑠∈ℝ.

Before stating the existence theorem, we first prove the following lemma.

Lemma 7. Let the assumption (𝐻₁) be satisfied. Given 𝑔∈𝐿_𝑙𝑜𝑐(ℝ;𝑋), suppose that 𝑔(𝑠+𝑇)=−𝑔(𝑠) for a.e. 𝑠∈ℝ. Define Φ𝑔(𝑡)∶=^𝑡_−∞𝑈()𝒫(𝜏)𝑔(𝜏)−𝑡,𝜏d𝜏_𝑡+∞𝑈_𝒬(𝑡,𝜏)𝒬(𝜏)𝑔(𝜏)d𝜏,𝑡∈ℝ. (17) Then (Φ𝑔)(𝑡) is well defined for each 𝑡∈ℝ and Φ𝑔 belongs to 𝑃_𝑇𝐴(ℝ;𝑋).

Proof. From (c) and our assumptions on 𝑔 it follows that for each 𝑡∈ℝ,Φ𝑔(𝑡)≤^𝑡_−∞𝑈()𝒫(𝜏)𝑔(𝜏+𝑡,𝜏)d𝜏_𝑡+∞𝑈_𝒬(𝑡,𝜏)𝒬(𝜏)𝑔(𝜏)d𝜏≤𝑁^𝑡_−∞𝑒^{−𝛿(𝑡−𝜏)}𝑔(𝜏)d𝜏+𝑁_𝑡^+∞𝑒^{𝛿(𝑡−𝜏)}𝑔(𝜏≤𝑁)d𝜏1−𝑒^{−𝛿𝑇𝑡}_𝑡−𝑇𝑔(𝜏)d𝜏+_𝑡^𝑡+𝑇𝑔(𝜏=)d𝜏2𝑁1−𝑒^−𝛿𝑇𝑔_{𝐿((0,𝑇);𝑋)}, (18) which implies that (Φ𝑔)(𝑡) is well defined for 𝑡∈ℝ and Φ𝑔 is bounded.
To prove Φ𝑔 belongs to 𝑃_𝑇A(ℝ;𝑋), we first verify the continuity of Φ𝑔. For fixed 𝑡₀∈ℝ and ℎ∈ℝ, we obtain upon changing of variables that Φ𝑔𝑡₀+ℎ−Φ𝑔𝑡₀=^𝑡0_−∞𝑈𝑡₀𝒫()𝑔()−+ℎ,𝜏+ℎ𝜏+ℎ𝜏+ℎd𝜏_𝑡+∞0𝑈_𝒬𝑡₀𝒬()𝑔()−+ℎ,𝜏+ℎ𝜏+ℎ𝜏+ℎd𝜏_𝑡0−∞𝑈𝑡₀𝒫(𝜏)𝑔(𝜏)+,𝜏d𝜏_𝑡+∞0𝑈_𝒬𝑡₀,𝜏𝒬(𝜏)𝑔(𝜏)d𝜏. (19) Thus, we have Φ𝑔𝑡₀+ℎ−Φ𝑔𝑡₀≤^𝑡0_−∞𝑈𝑡₀𝒫()𝑔()(𝜏)++ℎ,𝜏+ℎ𝜏+ℎ𝜏+ℎ−𝑔d𝜏_𝑡+∞0𝑈_𝒬𝑡₀𝒬()×𝑔()(𝜏)++ℎ,𝜏+ℎ𝜏+ℎ𝜏+ℎ−𝑔d𝜏_𝑡0−∞𝑈𝑡₀𝒫()()𝒫(𝜏)(𝜏+ℎ,𝜏+ℎ𝜏+ℎ−𝑈𝑡,𝜏×𝑔)d𝜏+_𝑡^+∞₀𝑈_𝒬𝑡₀+ℎ,𝜏+ℎ𝒬(𝜏+ℎ)−𝑈𝑡₀𝒬(𝜏)(𝜏,𝜏×𝑔)d𝜏∶=𝐼₁+𝐼₂+𝐼₃+𝐼₄. (20) Noticing that ‖𝑈(𝑡₀+ℎ,𝜏+ℎ)𝒫(𝜏+ℎ)‖≤𝑁𝑒^{−𝛿(𝑡0−𝜏)}, we have 𝐼₁≤𝑁1−𝑒^{−𝛿𝑇𝑇}₀𝑔(𝑠+ℎ)−𝑔(𝑠)d𝑠, (21) which yields lim_ℎ→0𝐼₁=0. An analogue argument shows lim_ℎ→0𝐼₂=0.
Since {(𝑡,𝜏)∶−∞<𝜏≤𝑡<+∞}∋(𝑡,𝜏)⟶𝑈(𝑡,𝜏)𝒫(𝜏) (22) is strongly continuous, we obtain lim_ℎ→0𝑈𝑡₀+ℎ,𝜏+ℎ𝒫(𝜏+ℎ)𝑔(𝜏)=𝑈𝑡₀,𝜏𝒫(𝜏)𝑔(𝜏) (23) for all 𝜏≤𝑡₀, 𝜏∈ℝ. This, together with (c), yields that 𝑈𝑡₀+ℎ,𝜏+ℎ𝒫(𝜏+ℎ)−𝑈𝑡₀,𝜏𝒫(𝜏)𝑔(𝜏)≤2𝑁𝑒^{−𝛿(𝑡0−𝜏)}𝑔_{𝐶𝑏(ℝ;𝑋)} (24) for all 𝜏≤𝑡₀, 𝜏∈ℝ. Now, using the Lebesgue dominated convergence theorem we obtain lim_ℎ→0𝐼₃=0. Similarly, we can show that lim_ℎ→0𝐼₄=0.
Next, it remains to prove that Φ𝑔 is 𝑇-antiperiodic. Noting that 𝑔(𝜏+𝑇)=−𝑔(𝜏) for a.e. 𝜏∈ℝ, we have Φ𝑔(𝑡+𝑇)=^𝑡+𝑇_−∞𝑈()𝒫(𝜏)𝑔(𝜏)−𝑡+𝑇,𝜏d𝜏_{+∞𝑡+𝑇}𝑈_𝒬(𝑡+𝑇,𝜏)𝒬(𝜏)𝑔(𝜏)d𝜏=−^𝑡_−∞𝑈()𝒫(𝜏)𝑔(𝜏)+𝑡,𝜏d𝜏_𝑡+∞𝑈_𝒬()𝒬(𝜏)𝑔(𝜏)(𝑡)𝑡,𝜏d𝜏=−Φ𝑔 (25) for any 𝑡∈ℝ. Therefore, we can conclude that Φ𝑔 belongs to 𝑃_𝑇𝐴(ℝ;𝑋). This completes the proof.

Now we are ready to state the first main result.

Theorem 8. Let (𝐻₁) and (𝐻₂) hold. Then the problem (2)-(3) has a unique 𝑇-antiperiodic mild solution.

Proof. Set, for 𝑢∈𝑃_𝑇𝐴(ℝ;𝑋), 𝑔(⋅)=𝑓(⋅,𝑢(⋅)). It easily follows from (H₂) that the function 𝑔 satisfies the conditions of Lemma 7. From this, we obtain that the mapping Γ, defined by(Γ𝑢)(𝑡)=^𝑡_−∞𝑈()𝒫(𝜏)𝑓((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑡+∞𝑈_𝒬(𝑡,𝜏)𝒬(𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏,𝑡∈ℝ,𝑢∈𝑃_𝑇𝐴(ℝ;𝑋), (26) is well defined and maps 𝑃_𝑇𝐴(ℝ;𝑋) into itself.
To prove the theorem, we first show that Γ has a unique fixed point in 𝑃_𝑇𝐴(ℝ,𝑋). Let 𝑢,𝑣∈𝑃_𝑇𝐴(ℝ;𝑋). Then, by (H₂) we have 𝑡)−(𝑡≤‖(Γ𝑢)(Γ𝑣)()‖_𝑡−∞𝑈()𝒫(𝜏𝑓(𝜏((𝜏+𝑡,𝜏)𝜏,𝑦)−𝑓𝜏,𝑣))d𝜏_𝑡+∞𝑈_𝒬()𝒬(𝜏)𝑓((𝜏((𝜏≤𝑡,𝜏𝜏,𝑢))−𝑓𝜏,𝑣))d𝜏2𝑁𝐿_𝑓𝛿‖𝑢−𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)}. (27) Consequently, ‖Γ𝑢−Γ𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)}≤2𝑁𝐿_𝑓𝛿‖𝑢−𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)}, (28) which, together with our assumption 2𝑁𝐿_𝑓<𝛿, implies that Γ is a strict contraction on 𝑃_𝑇𝐴(ℝ;𝑋). Thus, using the Banach contraction principle we conclude that Γ has a unique fixed point in 𝑃_𝑇𝐴(ℝ,𝑋).
To the end of the proof, we will prove that 𝑢∈𝑃_𝑇𝐴(ℝ,𝑋) is a mild solution of (2) if and only if it is a fixed point of Γ.
We first suppose that 𝑢∈𝑃_𝑇𝐴(ℝ;𝑋) is a mild solution of (2); that is, 𝑢 satisfies the integral equation 𝑢(𝑡)=𝑈(𝑡,𝑠)𝑢(𝑠)+^𝑡_𝑠𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏 (29) for all 𝑡>𝑠 and 𝑠∈ℝ. From this and (c), it immediately follows that 𝒫(𝑡)𝑢(𝑡)=^𝑡_−∞𝑈()𝒫(𝜏)𝑓((𝜏𝒬(𝑡)𝑢(𝑡)=𝑡,𝜏𝜏,𝑢))d𝜏,𝑡∈ℝ,_𝑡+∞𝑈_𝒬(𝑡,𝜏)𝒬(𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏,𝑡∈ℝ. (30) So, one has 𝑢(𝑡)=(𝒫(𝑡)(𝑡𝑢(𝑡)=+𝒬))_𝑡−∞𝑈()𝒫(𝜏)𝑓((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑡+∞𝑈_𝒬(𝑡,𝜏)𝒬(𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏,𝑡∈ℝ. (31) This proves that 𝑢 is a fixed point of Γ.
Conversely, if 𝑢∈𝑃_𝑇𝐴(ℝ;𝑋) is a fixed point of Γ, then 𝑢 satisfies the integral equations 𝑢(𝑠)=^𝑠_−∞𝑈()𝒫(𝜏)𝑓((𝜏−𝑠,𝜏𝜏,𝑢))d𝜏_𝑠+∞𝑈_𝒬(𝑠,𝜏)𝒬(𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏,𝑠∈ℝ. (32) For any 𝑡>𝑠, 𝑠∈ℝ, we obtain upon multiplying both sides of (32) by 𝑈(𝑡,𝑠) that 𝑈(𝑡,𝑠)𝑢(𝑠)=^𝑠_−∞𝑈()𝒫(𝜏)𝑓((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑠+∞𝑈_𝒬()𝒬(𝜏)𝑓((𝜏=𝑡,𝜏𝜏,𝑢))d𝜏_𝑡−∞𝑈()𝒫(𝜏)𝑓((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑡𝑠𝑈()𝒫(𝜏)𝑓((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑡+∞𝑈_𝒬()𝒬(𝜏)𝑓((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑡𝑠𝑈_𝒬()𝒬(𝜏)𝑓((𝜏(𝑡)−𝑡,𝜏𝜏,𝑢))d𝜏=𝑢^𝑡_𝑠𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏, (33) which implies that 𝑢 is a mild solution to problem (2).
Now, according to the discussed above we deduce that the problem (2)-(3) has a unique 𝑇-antiperiodic mild solution. The proof is completed.

Now we are in a position to prove our second existence result of antiperiodic mild solutions for the problem (2)-(3).

Theorem 9. Let (𝐻₁), (𝐻₃), and (𝐻₄) hold with 𝒫(𝑡)=𝐼 for 𝑡∈ℝ. Then the problem (2)-(3) has at least one 𝑇-antiperiodic mild solution.

Proof. Let us define the mapping Γ by (Γ𝑢)(𝑡)=^𝑡_−∞𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏,𝑡∈ℝ,𝑢∈𝑃_𝑇𝐴(ℝ;𝑋). (34) We first notice, thanks to assumptions (H₁) and (H₃) (i) and Lemma 7, that Γ is well defined and maps 𝑃_𝑇𝐴(ℝ;𝑋) into itself.
Next, by applying Schauder’s fixed point theorem we show that Γ has at least one fixed point in 𝑃_𝑇𝐴(ℝ,𝑋). From (H₃) (ii) it is easy to see that there exists some 𝑘₀>0 such that 𝑁1−𝑒^{−𝛿𝑇𝑇}₀Φ_𝑘0(𝜏)d𝜏≤𝑘₀. (35) Using this, a direct calculation yields that, for every 𝑢∈Ω_𝑘0 and all 𝑡∈ℝ, ‖(Γ𝑢)(𝑡)‖≤^𝑡_−∞𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏≤𝑁^𝑡_−∞𝑒^{−𝛿(𝑡−𝜏)}𝑓((𝜏≤𝑁𝜏,𝑢))d𝜏1−𝑒^{−𝛿𝑇𝑇}₀𝑓((𝜏≤𝑁𝜏,𝑢))d𝜏1−𝑒^{−𝛿𝑇𝑇}₀Φ_𝑘0(𝜏)d𝜏≤𝑘₀, (36) which implies that Γ𝑢∈Ω_𝑘0 for every 𝑢∈Ω_𝑘0.
In the sequel, we show that Γ is completely continuous on Ω_𝑘0. The proof will be divided into two steps.
Step 1. Γ is continuous on Ω_𝑘0.
Take 𝑢₁,𝑢₂∈Ω_𝑘0. Then it follows from (H₃) (i) that Γ𝑢₁(𝑡)−Γ𝑢₂(𝑡≤_𝑡−∞)𝑈(𝑡,𝜏)𝑓𝜏,𝑢₁(𝜏)−𝑓𝜏,𝑢₂(𝜏)d𝜏≤𝑁^𝑡_−∞𝑒^{−𝛿(t−𝜏)}𝑓𝜏,𝑢₁(𝜏)−𝑓𝜏,𝑢₂(𝜏)≤𝑁d𝜏1−𝑒^{−𝛿𝑇𝑇}₀𝑓𝜏,𝑢₁(𝜏)−𝑓𝜏,𝑢₂(𝜏)d𝜏. (37) This, together with the Lebesgue dominated convergence theorem and the continuity of 𝑓 with respect to second variable, shows that Γ𝑢₁(𝑡)−Γ𝑢₂(𝑡≤𝑁)1−𝑒^{−𝛿𝑇𝑇}₀𝑓𝜏,𝑢₁(𝜏)−𝑓𝜏,𝑢₂(𝜏)d𝜏⟶0as𝑢₁⟶𝑢₂, (38) which implies the continuity of Γ.
Step 2. Γ is a compact operator on Ω_𝑘0.
For each 𝜀>0, set Γ_𝜀𝑢(𝑡)=^𝑡−𝜀_−∞𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏,𝑢∈Ω_𝑘0. (39) From (H₄) it follows that, for each 𝑡∈ℝ and 𝜀>0, the set ^𝑡−𝜀_−∞𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏;𝑢∈Ω_𝑘0=𝑈(𝑡,𝑡−𝜀)^𝑡−𝜀_−∞𝑈(𝑡−𝜀,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏;𝑢∈Ω_𝑘0 (40) is relatively compact in 𝑋. Thus, for each 𝑡∈ℝ and 𝜀>0, the set {(Γ_𝜀𝑢)(𝑡);𝑢∈Ω_𝑘0} is also relatively compact in 𝑋. Then, for every 𝑢∈Ω_𝑘0 and 𝑡∈ℝ, as (Γ𝑢)(𝑡)−Γ_𝜀𝑢(𝑡≤_{𝑡𝑡−𝜀})𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏⟶0as𝜀⟶0 (41) in 𝑋, we conclude, in view of the total boundedness, that for each 𝑡∈ℝ, the set {(Γ𝑢)(𝑡);𝑢∈Ω_𝑘0} is relatively compact in 𝑋.
Next, we will show that {Γ𝑢;𝑢∈Ω_𝑘0}⊂𝑃_𝑇𝐴(ℝ;𝑋) is equicontinuous. Taking 𝑡,𝑠∈ℝ with 𝑡>𝑠, we have (𝑡)−(𝑠)=Γ𝑢)(Γ𝑢)(_𝑡𝑠𝑈()𝑓((𝜏+𝑡,𝜏𝜏,𝑢))d𝑠_{𝑠𝑠−𝜅}(𝑈()(𝑓((𝜏+𝑡,𝜏−𝑈𝑠,𝜏))𝜏,𝑢))d𝜏_{𝑠−𝜅𝑡−𝑀}(𝑈()(𝑓((𝜏+𝑡,𝜏−𝑈𝑠,𝜏))𝜏,𝑢))d𝜏_{𝑡−𝑀−∞}(𝑈(𝑡,𝜏)−𝑈(𝑠,𝜏))𝑓(𝜏,𝑢(𝜏))d𝜏=𝐽₁+𝐽₂+𝐽₃+𝐽₄, (42) where 𝜅, 𝑀 are positive constants yet to be determined.
Given 𝜖>0. We first note that there exist 𝜂₁>𝜅, 𝜅>0 small enough such that 𝐽₁≤^𝑡_𝑠Φ_𝑘0(𝜏)𝜖d𝜏≤4whenever𝑡−𝑠≤𝜂₁,𝐽₂≤2𝑁^𝑠_𝑠−𝜅Φ_𝑘0(𝜏)𝜖d𝜏≤4. (43) For 𝐽₄, one can take a 𝑀>(𝜂₁+𝜅) big enough which is independent of 𝑡 and 𝑠 such that 𝐽₄≤^𝑡−𝑀_−∞‖(𝑈(𝑡,𝜏)−𝑈(𝑠,𝜏))‖_ℒ(𝑋)𝑓(𝜏,𝑢(𝜏))d𝜏≤𝑁^𝑡−𝑀_−∞𝑒^{−𝛿(𝑡−𝜏)}+𝑒^{−𝛿(𝑠−𝜏)}Φ_𝑘0(𝜏)≤𝑁d𝜏1+𝑒^𝛿𝜂1𝑒^−𝛿𝑀1−𝑒^{−𝛿𝑇𝑇}₀Φ_𝑘0(𝜏)𝜖d𝜏≤4. (44) For such fixed 𝜅, 𝑀, it is easy to find that there exists a 𝑑 big enough such that |𝑀−𝜅|≤𝑑𝑇, which, together with (H₃) (ii) and (c), yields that ^𝑠−𝜅_𝑡−𝑀‖𝑈(𝑠−𝜅,𝜏)‖_ℒ(𝑋)𝑓(𝜏,𝑢(𝜏))d𝜏≤d𝑁^𝑇₀Φ_𝑘0(𝜏)d𝜏. (45) Therefore, from the continuity of 𝑈(𝑡,𝑠) for 𝑡>𝑠 in the uniform operator topology it follows that 𝐽₃≤^𝑠−𝜅_𝑡−𝑀‖(𝑈(𝑡,𝜏)−𝑈(𝑠,𝜏))‖_ℒ(𝑋)𝑓((𝜏≤𝜏,𝑢))d𝜏_{𝑠−𝜅𝑡−𝑀}‖(𝑈(𝑡,𝑠−𝜅)−𝑈(𝑠,𝑠−𝜅))‖_ℒ(𝑋)×‖𝑈(𝑠−𝜅,𝜏)‖_ℒ(𝑋)𝑓((𝜏𝑈()(𝜏,𝑢))d𝜏≤𝑑𝑁‖(𝑡,𝑠−𝜅−𝑈𝑠,𝑠−𝜅))‖_ℒ(𝑋)^𝑇₀Φ_𝑘0(𝜏)≤𝜖d𝜏4 (46) whenever 𝑡−𝑠≤𝜂₂, where 𝜂₂ is small enough.
Thus, from the arguments above one can deduce that there exist 𝜂=min{𝜂₁,𝜂₂} such that ‖(Γ𝑢)(𝑡)−(Γ𝑢)(𝑠)‖≤𝜖 (47) whenever 𝑡−𝑠≤𝜂 and 𝑢∈Ω_𝑘0, which implies that the set {Γ𝑢;𝑢∈Ω_𝑘0} is equicontinuous. Consequently, Γ is compact operator on Ω_𝑘0 due to Lemma 1.
Now, applying Schauder’s fixed point theorem, we deduce that Γ has at least one fixed point 𝑢∈𝑃_𝑇𝐴(ℝ,𝑋). Moreover, following from the same idea as the last part of the proof in Theorem 8, we obtain that 𝑢 is a 𝑇-antiperiodic mild solution of the problem (2)-(3). This completes the proof.

Remark 10. Theorems 8 and 9 generalize corresponding results for antiperiodic problems due to [12]. Note in particular that Theorems 8 and 9 cover results in [12].

Remark 11. (i) It can be easily shown that if 𝑢 is antiperiodic with period 𝑇, then it is periodic with period 2𝑇. Hence, from the arguments of Theorems 8 and 9 we can also obtain the existence results of 2𝑇-periodic solutions of the problem (2)-(3).
(ii) It is clear that if 𝑢 is periodic with period 2𝑇, 𝑢 may or may not be antiperiodic with period 𝑇.

Additional information is contained in the following. We consider the following nonautonomous semilinear parabolic evolution equation with periodic condition 𝑢^′(𝑡)(𝑡)𝑢(𝑡)((𝑡𝑢()(𝑡)=𝐴+𝑓𝑡,𝑢)),𝑡∈ℝ,𝑡+𝑇=𝑢,𝑡∈ℝ. (48)

From the arguments of Theorems 8 and 9 it is easy to see that if(1)the hypotheses in Theorem 8 are satisfied except that the antiperiodic on 𝑓 is replaced by the following 𝑓(𝑡+𝑇,𝑢)=𝑓(𝑡,𝑢)∀𝑡∈ℝ,𝑢∈𝑋, (49) then there exists a unique 𝑇-periodic mild solution for the problem (48);(2)the hypotheses in Theorem 9 are satisfied except that the antiperiodic conditions on 𝑓 are replaced by (49), then there exists at least a 𝑇-periodic mild solution for the problem (48).

The following remark indicates one motivation of the present paper.

Remark 12. As in [25], under certain conditions, the existence result is valid for the case of antiperiodic solutions, while there is no such a result in the periodic case. It is also noted that in dealing with the existence of certain problems, there is an essential difference between the periodic solutions and antiperiodic solutions (see also [26] for more details).

4. Neutral Problems

In this section, it is assumed that (𝐴(𝑡))_𝑡∈ℝ has a constant domain 𝐷 and verifies the conditions of Acquistapace and Terreni (AT₁) and (AT₂) with 𝜆₀=0.

If the hypothesis (H₁) is satisfied, then it follows readily that 𝑡↦𝐴(𝑡) is periodic. Also, from Remark 3 it is easy to see that there exist constants 𝐾₂>0, 0<𝜇≤1 such that 𝐴(𝑡)𝐴⁻¹(0)_ℒ(𝑋)≤1+𝐾₂|𝑡|^𝜇 (50) for all 𝑡∈ℝ. Therefore, we deduce that 𝐴(𝑡)𝐴⁻¹(0)_ℒ(𝑋)≤1+𝐾₂𝑇^𝜇 (51) for all 𝑡∈ℝ.

Let 𝑋¹ denote the Banach space 𝐷 endowed with the graph norm ‖𝑢‖₁=‖𝐴(0)𝑢‖ for 𝑢∈𝑋¹. By 𝑃_𝑇𝐴(ℝ;𝑋¹), we denote the set of all 𝑇-antiperiodic functions from ℝ to 𝑋¹. It is clear that 𝑃_𝑇𝐴(ℝ;𝑋¹), equipped with the sup norm, is a Banach space.

In this section, we extend the result obtained in Section 3 to the antiperiodic problem of neutral type (4).

Definition 13. A mild solution to (4) is a function 𝑢∈𝐶_𝑏(ℝ;𝑋) satisfying the integral equation 𝑢(𝑡)(𝑢(𝑠)((𝑠((𝑡+=𝑈𝑡,𝑠)[−𝐹𝑠,𝑢))]+𝐹𝑡,𝑢))_𝑡𝑠𝑈()𝐴(𝜏)𝐹((𝜏+𝑡,𝜏𝜏,𝑢))d𝜏_𝑡𝑠𝑈(𝑡,𝜏)𝑓(𝜏,𝑢(𝜏))d𝜏 (52) for all 𝑡>𝑠 and 𝑠∈ℝ.

Remark 14. It will be seen later that the last two terms on right side in (52), being integrals in sense of Bocher (see [27]), are reasonable.

To prove the existence of antiperiodic mild solutions to the problem (4), let us introduce the following assumptions:(H₅)the function 𝐹∶ℝ×𝑋→𝑋¹ is continuous and 𝐹(𝑡+𝑇,−𝑢)=−𝐹(𝑡,𝑢) for all 𝑡∈ℝ, 𝑢∈𝑋. Moreover,(i)there exists a constant 𝐿_𝐹 such that ‖𝐹(𝑡,𝑢)−𝐹(𝑡,𝑣)‖₁≤𝐿_𝐹‖𝑢−𝑣‖; (53) for all 𝑡∈ℝ, 𝑢,𝑣∈𝑋;(ii)there exists a nondecreasing function Ψ∶ℝ⁺→ℝ⁺ such that ‖𝐹(𝑡,𝑢)‖₁≤Ψ(‖𝑢‖); (54) for all 𝑡∈ℝ, 𝑢∈𝑋, and liminf_𝑟→+∞Ψ(𝑟)𝑟=𝜎₂. (55)(H₆)Consider 𝐿_𝐹‖𝐴⁻¹(0)‖_ℒ(𝑋)+(2𝑁(1+𝐾₂𝑇^𝜇)𝐿_𝐹+2𝑁𝐿_𝑓)/𝛿<1.(H₇)(‖𝐴⁻¹(0)‖_ℒ(𝑋)+(𝑁(1+𝐾₂𝑇^𝜇))/𝛿)𝜇^′+𝑁𝜎₁/(1−𝑒^−𝛿𝑇)<1, where 𝜇^′∶=max{𝜎₂,𝐿_𝐹}.

Theorem 15. Under the hypotheses (𝐻₁), (𝐻₂), (𝐻₅) (i), and (𝐻₆), the problem (4) has a unique 𝑇-antiperiodic mild solution.

Proof. From (H₅) (i) note that 𝐹(⋅,𝑢(⋅))∈𝑃_𝑇𝐴(ℝ;𝑋¹) for each 𝑢∈𝑃_𝑇𝐴(ℝ;𝑋). Define, for 𝑢∈𝑃_𝑇𝐴(ℝ;𝑋), (𝑡)((𝑡+Γ𝑢∶=𝐹𝑡,𝑢))_𝑡−∞𝑈()𝒫(𝜏)𝐴(𝜏)𝐹((𝜏−𝑡,𝜏𝜏,𝑢))d𝜏_𝑡+∞𝑈_𝒬(𝑡,𝜏)𝒬(𝜏)𝐴(𝜏)𝐹(𝜏,𝑢(𝜏))d𝜏,𝑡∈ℝ. (56) Then by (c) and (51) a direct calculation gives (𝑡≤‖𝐹((𝑡+Γ𝑢)𝑡,𝑢))‖_𝑡−∞‖𝑈()𝒫(𝜏)𝐴(𝜏)𝐹((𝜏+𝑡,𝜏𝜏,𝑢))‖d𝜏_𝑡+∞𝑈_𝒬()𝒬(𝜏)𝐴(𝜏)𝐹((𝜏≤𝐴𝑡,𝜏𝜏,𝑢))d𝜏₋₁(0)_ℒ(𝑋)‖𝐹(𝑡,𝑢(𝑡))‖₁+𝑁^𝑡_−∞𝑒^{−𝛿(𝑡−𝜏)}𝐴(𝜏)𝐴⁻¹(0)_ℒ(𝑋)×‖𝐹(𝜏,𝑢(𝜏))‖₁d𝜏+𝑁_𝑡^+∞𝑒^{𝛿(𝑡−𝜏)}𝐴(𝜏)𝐴⁻¹(0)_ℒ(𝑋)×‖𝐹(𝜏,𝑢(𝜏))‖₁≤𝐴d𝜏₋₁(0)_ℒ(𝑋)+2𝑁1+𝐾₂𝑇^𝜇𝛿×‖𝐹(⋅,𝑢(⋅))‖_{𝑃𝑇𝐴(ℝ;𝑋}¹₎. (57) This proves that (Γ𝑢)(𝑡) is well defined for 𝑡∈ℝ and Γ𝑢 is bounded. Moreover, noticing (H₅) (i) and using a similar argument with that in Lemma 7 one can show easily that Γ maps 𝑃_𝑇𝐴(ℝ;𝑋) into itself.
Let us assume that the mapping Γ is defined the same as in Theorem 8. Then from the proof of Theorem 8 with (H₂) it follows that Γ is well defined and maps 𝑃_𝑇𝐴(ℝ;𝑋) into itself.
Now, consider the mapping Γ+Γ. We see, from the arguments above, that Γ+Γ maps 𝑃_𝑇𝐴(ℝ;𝑋) into itself. Also, for 𝑢,𝑣∈𝑃_𝑇𝐴(ℝ;𝑋), 𝑡∈ℝ, as Γ𝑢+Γ𝑢(𝑡)−Γ𝑣+Γ𝑣(𝑡)≤𝐿_𝐹𝐴⁻¹(0)_ℒ(𝑋)‖𝑢(𝑡)(𝑡+−𝑣)‖2𝑁1+𝐾₂𝑇^𝜇𝐿_𝐹+2𝑁𝐿_𝑓𝛿‖𝑢−𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)}≤𝐿_𝐹𝐴⁻¹(0)_ℒ(𝑋)+2𝑁1+𝐾₂𝑇^𝜇𝐿_𝐹+2𝑁𝐿_𝑓𝛿×‖𝑢−𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)} (58) in view of (H₂) and (H₅) (i), we conclude that Γ+Γ is a strict contraction on 𝑃_𝑇𝐴(ℝ;𝑋) due to (H₆). This allows us to obtain, in view of the contraction mapping principle, that Γ+Γ has a unique fixed point 𝑃_𝑇𝐴(ℝ,𝑋). Moreover, an application of the same idea as the last part of the proof in Theorem 8 justifies that 𝑢∈𝑃_𝑇𝐴(ℝ,𝑋) is a mild solution of (4) if and only if it is a fixed point of Γ+Γ. The proof is then completed.

The following fixed point theorem plays a key role in the proof of our subsequent result; see, for example, [28].

Lemma 16. Krasnoselskii’s fixed point theorem: let 𝐸 be a Banach space and 𝐵 be a nonempty closed convex subset of 𝐸, and let 𝐹₁, 𝐹₂ be maps of 𝐵 into 𝐸 such that 𝐹₁𝑥+𝐹₂𝑦∈𝐵 for every pair 𝑥,𝑦∈𝐵. If 𝐹₁ is a strict contraction and 𝐹₂ is completely continuous, then the equation 𝐹₁𝑥+𝐹₂𝑥=𝑥 has a solution on 𝐵.

Theorem 17. Under the hypotheses (𝐻₁), (𝐻₇), and (𝐻₃)–(𝐻₅) with 𝒫(𝑡)=𝐼 for 𝑡∈ℝ, the problem (4) has at least one 𝑇-antiperiodic mild solution.

Proof. From our hypotheses on 𝐹, 𝑓 and (H₇), it is easy to see that there exists a 𝑘₀>0 such that 𝐴⁻¹(0)_ℒ(𝑋)+𝑁1+𝐾₂𝑇^𝜇𝛿Ψ𝑘₀+𝑁1−𝑒^{−𝛿𝑇𝑇}₀Φ_𝑘0(𝜏)d𝜏≤𝑘₀. (59)
Assume that the mapping Γ is defined as in Theorem 9. Let us define the mapping Γ by Γ𝑢(𝑡)∶=𝐹(𝑡,𝑢(𝑡))+^𝑡_−∞𝑈(𝑡,𝜏)𝐴(𝜏)𝐹(𝜏,𝑢(𝜏))d𝜏,𝑢∈𝑃_𝑇𝐴(ℝ;𝑋). (60) Note, thanks to the proofs of Theorem 9 with assumptions (H₁) and (H₃) (i) and Theorem 15 with assumption (H₅) (i), that Γ and Γ are well defined and map 𝑃_𝑇𝐴(ℝ;𝑋) into itself. Moreover, for every pair 𝑢,𝑣∈Ω_𝑘0 and 𝑡∈ℝ, a direct calculation yields 𝑡)+(𝑡≤𝐴(Γ𝑢)(Γ𝑣)₋₁(0)_ℒ(𝑋)‖𝐹(𝑡,𝑢(𝑡))‖₁+𝑁^𝑡_−∞𝑒^{−𝛿(𝑡−𝜏)}𝐴(𝜏)𝐴⁻¹(0)_ℒ(𝑋)×‖𝐹(𝜏,𝑢(𝜏))‖₁d𝜏+𝑁^𝑡_−∞𝑒^{−𝛿(𝑡−𝜏)}𝑓((𝜏≤𝐴𝜏,𝑢))d𝜏₋₁(0)_ℒ(𝑋)+𝑁1+𝐾₂𝑇^𝜇𝛿Ψ𝑘₀+𝑁1−𝑒^{−𝛿𝑇𝑇}₀Φ_𝑘0(𝜏)d𝜏≤𝑘₀, (61) in view of (59), which implies that Γ𝑢+Γ𝑣∈Ω_𝑘0 for every pair 𝑢,𝑣∈Ω_𝑘0.
To obtain the fixed points of Γ+Γ, we will use Krasnoselskii’s fixed point theorem. In what follows, we show that Γ and Γ satisfy the conditions of Lemma 16. For 𝑢,𝑣∈Ω_𝑘0, from (H₅) (i) we infer that Γ𝑢(𝑡)−Γ𝑣(𝑡)≤𝐿_𝐹𝐴⁻¹(0)_ℒ(𝑋)‖𝑢(𝑡)(𝑡+𝑁−𝑣)‖1+𝐾₂𝑇^𝜇𝐿_𝐹𝛿‖𝑢−𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)}≤𝐿_𝐹𝐴⁻¹(0)_ℒ(𝑋)+𝑁1+𝐾₂𝑇^𝜇𝐿_𝐹𝛿×‖𝑢−𝑣‖_{𝑃𝑇𝐴(ℝ;𝑋)} (62) for all 𝑡∈ℝ, which together with (H₇) yields that Γ is a contraction on Ω_𝑘0. On the other hand, by a similar proof with that in Theorem 9 the mapping Γ is completely continuous on Ω_𝑘0. Hence, applying Lemma 16 we deduce that Γ+Γ has at least one fixed point 𝑢∈Ω_𝑘0, which is a 𝑇-antiperiodic mild solution to (4) due to the same idea as the last part of the proof in Theorem 8. This completes the proof of theorem.

Remark 18. Let us note that in Theorems 8 and 15, exponential dichotomy on evolution equations 𝑈 is involved. However, as can be seen from the proofs of Theorems 9 and 17, such condition is not enough to obtain our desired results and therefore is replaced by the special one: 𝑈 is exponentially stable.

5. Application

In this section, we give an example to illustrate our abstract results, which do not aim at generality but indicate how our theorems can be applied to concrete problem.

Consider the antiperiodic problem for partial differential equation in the form 𝜕𝑢()(𝑡)𝜕𝜕𝑡𝑡,𝑥=𝑎²𝑢(𝑡,𝑥)𝜕𝑥²((,[],𝑢()()[],+𝑔𝑡,𝑥,𝑢𝑡,𝑥))𝑡∈ℝ,𝑥∈0,𝜋𝑡+𝑇,𝑥=−𝑢𝑡,𝑥,𝑡∈ℝ,𝑥∈0,𝜋 (63) supplemented with homogeneous Dirichlet boundary condition 𝑢(𝑡,0)=𝑢(𝑡,𝜋)=0(𝑡∈ℝ), where 𝑔∶ℝ×[0,𝜋]×ℝ→ℝ, 𝑎∶ℝ→ℝ are given functions which will be specified later.

Here, our objective is to show the existence of 𝑇-antiperiodic solutions for the antiperiodic problem (63).

Take 𝑋=(𝐿²(0,𝜋);(⋅,⋅)₂) and define the operator 𝐵∶𝐷(𝐵)⊂𝑋→𝑋 by 𝐵𝑢=𝜕²𝑢/𝜕𝑥², 𝑢∈𝐷(𝐵), where 𝐷(𝐵)=𝐻²(0,𝜋)∩𝐻¹₀(0,𝜋). (64) It is well-known that 𝐵 has a discrete spectrum and its eigenvalues are −𝑛², 𝑛∈ℕ⁺ with the corresponding normalized eigenvectors 𝑦_𝑛(𝑥)=(2/𝜋)sin(𝑛𝑥).

Assume that 𝑎 is a Hölder continuous function such that 𝑎(𝑡+𝑇)=𝑎(𝑡)∀𝑡∈ℝ,inf_𝑡∈ℝ𝑎(𝑡)≥𝑐>0. (65) Take 𝑔(𝑡,𝑥,𝑢(𝑡,𝑥))=𝑏(𝑡)sin𝑢(𝑡,𝑥), where 𝑏(𝑡+𝑇)=𝑏(𝑡)fora.e.𝑡∈ℝ,𝑏|_[0,𝑇]∈𝐿0,𝑇;ℝ⁺. (66)

Define 𝐷(𝐴(𝑡(𝐵)𝐴(𝑡)(𝑡)(𝐴(𝑡,𝑢(𝑡𝑥)(),𝑓((𝑡𝑥)((.))=𝐷,𝑡∈ℝ,𝑢=𝑎𝐵𝑢,𝑢∈𝐷)))(=𝑢𝑡,𝑥𝑡,𝑢))(=𝑔𝑡,𝑥,𝑢𝑡,𝑥)) (67)

It follows from [19, Lemma 6.1 in Chapter 7] that there are constants 𝜃∈(𝜋/2,𝜋) and 𝐾₁≥0 such that 𝐴(𝑡) satisfy Σ_𝜃{{0};|𝜆|}∪{0}(𝐴(𝑡,‖𝑅((𝑡≤𝐾∶=𝜆∈ℂ⧵≤𝜃⊂𝜌))𝜆,𝐴))‖¹1+|𝜆| (68) for all 𝜆∈Σ_𝜃∪{0} and 𝑡∈ℝ. Moreover, we note that for 𝜆∈Σ_𝜃∪{0}, 𝑡,𝑠∈ℝ, 𝑢∈𝑋, 𝑅(𝜆,𝐴(𝑡))𝑢=^∞_𝑛=11𝜆+𝑛²𝑎(𝑡)𝑢,𝑦_𝑛𝑦_𝑛,𝐴(𝑡)𝑅((𝑡𝑅((𝑡((𝑠𝑢=𝜆,𝐴))[0,𝐴))−𝑅0,𝐴))]_∞𝑛=11𝜆+𝑛²𝑎(𝑡)𝑎(𝑡)−𝑎(𝑠)𝑎(𝑠)𝑢,𝑦_𝑛𝑦_𝑛 (69) from which we see that for, 𝜆∈Σ_𝜃∪{0}, 𝑡,𝑠∈ℝ‖𝐴(𝑡)𝑅((𝑡𝑅((𝑡((𝑠≤‖𝑅((𝑡⋅|𝑎(𝑡)(𝑠𝜆,𝐴))[0,𝐴))−𝑅0,𝐴))]‖𝜆,𝐴))‖−𝑎)|𝑎(𝑠). (70) Accordingly, 𝐴(𝑡) satisfy the conditions (AT₁) and (AT₂). Thus, the family (𝐴(𝑡))_𝑡∈ℝ generates an evolution family 𝑈: 𝑈(𝑡,𝑠)𝑢=^∞_𝑛=1𝑒^{−𝑛2∫𝑡𝑠𝑎(𝜏)d𝜏}𝑢,𝑦_𝑛𝑦_𝑛,for−∞≤𝑠≤𝑡<∞,𝑢∈𝑋. (71) A direct calculation gives ‖𝑈(𝑡,𝑠)‖≤𝑒^{−𝑐(𝑡−𝑠)}for−∞≤𝑠≤𝑡<+∞. (72)

Also, 𝑈 is compact due to the boundedness of 𝐴(𝑡)𝑈(𝑡,𝑠) for −∞≤𝑠<𝑡<+∞ (cf. [19, Theorem 6.1 in Chapter 5]). Moreover, it is easy to verify that 𝑈(𝑡+𝑇,𝑠+𝑇)=𝑈(𝑡,𝑠) for all 𝑡≥𝑠.

On the other hand, observe that 𝑓∶ℝ×𝑋→𝑋 is a Carathéodory function, 𝑓(𝑡+𝑇,−𝑢)=−𝑓(𝑡,𝑢) for all 𝑡∈ℝ, 𝑢∈𝑋, and 𝑓(𝑡,𝑢)≤√𝜋𝑏(𝑡)fora.e.𝑡∈[0,𝑇]andall𝑢∈𝑋, (73) which implies that hypothesis (H₃) is satisfied with Φ_𝑟(𝑡)=√𝜋𝑏(𝑡),𝜎₁=0. (74)

Therefore, the antiperiodic problem (63) can be transformed into the abstract problem (2)-(3) and assumptions (H₁), (H₃), and (H₄) hold with 𝒫(𝑡)=𝐼 for 𝑡∈ℝ. Hence, (63) has at least one 𝑇-antiperiodic mild solution due to Theorem 9.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundations of China (no. 11271309, 11101202), the Specialized Research Fund for the Doctoral Program of Higher Education (20114301110001), and Hunan Provincial Natural Science Foundation of China (12JJ2001). The authors also want to express their thanks to the anonymous referees for their suggestions and comments that improved the quality of the paper.