Time Remotely Almost Periodic Viscosity Solutions of Hamilton-Jacobi Equations

Abstract

We study some properties of the remotely almost periodic functions. This paper studies viscosity solutions of general Hamilton-Jacobi equations in the time remotely almost periodic case. Existence and uniqueness results are presented under usual hypotheses.

1. Introduction

In this paper we consider the viscosity solutions of first-order Hamilton-Jacobi equations of the form𝜕_𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡),(𝑥,𝑡)∈ℝ^𝑁×ℝ.(1.1) This problem was studied in [1] in the time periodic and almost periodic cases. And papers by Crandall and Lions (see [2–5]) proved the uniqueness and stability of viscosity solutions for a large class of equations, in particular for the initial value problem𝜕_𝑡𝑢+𝐻𝑢(𝑥(𝑥,𝑡,,0𝑢),=𝐷𝑢𝑢)₀=0,(𝑥,𝑡)∈ℝ^𝑁×]0,𝑇[,(𝑥),𝑥∈ℝ^𝑁(1.2) and also for the stationary problem𝐻(𝑥,𝑢,𝐷𝑢)=0,𝑥∈ℝ^𝑁.(1.3) These results were extended by several papers, for example [6, 7].

Now in this paper we study this problem in a more regular condition, that is, in the time remotely almost periodic case. That is, we will look for such viscosity solutions when the Hamiltonian 𝐻 and 𝑓 are continuous functions 𝑓 is remotely almost periodic in 𝑡. The definition of remotely almost periodic was introduced by Sarason in 1984 in [8]. And Zhang and Yang in [9] and Zhang and Jiang in [10] gave such functions' applications.

This paper is structured as follows. In Section 2, we study a new type of almost periodic function—remotely almost periodic function. We present the definitions and prove some properties of such functions. Section 3 proves the uniqueness and existence of time remotely almost periodic viscosity solutions. In Section 3.1, we list some usual hypotheses used for the existence and uniqueness results and present two properties of viscosity solutions. In Section 3.2, we get some theorems for the uniqueness and existence of time remotely almost periodic viscosity solutions. And for the proof of the theorem we give two lemmas which play an important part. In Section 3.3, we concentrate on the asymptotic behaviour of time remotely almost periodic solutions for large frequencies.

In this paper, there are some abbreviations, like BUC, u.s.c, l.s.c, they stand for bounded uniformly continuous, upper semicontinuous, and lower semicontinuous, respectively. For the definition of viscosity subsolution and supersolution the reader can refer to [11].

2. Remotely Almost Periodic Function

It is well known that Bohr almost periodic function space is a Banach space, in which the distance is the supremum of the function. In [8], the author uses the superior limit as the distance in the space and defines a new type of almost periodic function, that is, remotely almost periodic function.

Definition 2.1. Let 𝑓 be a bounded uniformly continuous function on ℝ=(−∞,+∞). We say that 𝑓 is remotely almost periodic if and only if for all 𝜀>0𝑇(𝑓,𝜀)=𝜏∈ℝ∶limsup_|𝑡|→∞||𝑓(𝑡+𝜏)−𝑓(𝑡)||<𝜀(2.1) is relatively dense on ℝ. The number 𝜏∈𝑇(𝑓,𝜀) is called 𝜀 remotely almost period.
And RAP(ℝ) denotes all these functions.

Definition 2.2. Let 𝑓 be a bounded uniformly continuous function on ℝ. We say that 𝑓 oscillates slowly if and only if for every 𝜏∈ℝlim_|𝑡|→∞||𝑓(𝑡+𝜏)−𝑓(𝑡)||=0.(2.2) And SO(ℝ) denotes all these functions.

Next we will prove two propositions.

Proposition 2.3. Assume that 𝑓(𝑡) is remotely almost periodic and denote by 𝐹(𝑡)=∫^𝑡0𝑓(𝑠)𝑑𝑠 a primitive of 𝑓(𝑡). Then 𝐹(𝑡) is remotely almost periodic if and only if 𝐹(𝑡) is bounded.

Proof. When 𝐹(𝑡) is remotely almost periodic, 𝐹(𝑡) is certainly bounded. For the converse, let 𝐹(𝑡) be bounded, without losing general, and assume that 𝐹(𝑡) is a real function. For any 𝜀>0, there exists 𝑡₀>0 large enough; we have 𝐺=sup_|𝑡|>𝑡0𝐹(𝑡)>𝑔=inf_|𝑡|>𝑡0𝐹(𝑡);(2.3) take fixed 𝑡₁ and 𝑡₂, |𝑡₁|>𝑡₀, |𝑡₂|>𝑡₀, and assume that 𝑡₁<𝑡₂, satisfying 𝐹𝑡₁<𝑔+𝜀6,𝐹𝑡²>𝐺−6𝜀.(2.4) Assume that 𝑙=𝑙(𝜀₁) is an interval length of 𝑇(𝑓,𝜀₁), where 𝜀₁=𝜀/6𝑑, 𝑑=|𝑡₁−𝑡₂|. For every 𝛼∈𝑅, take 𝜏∈𝑇(𝑓,𝜀₁)∩[𝛼−𝑡₁,𝛼−𝑡₁+𝑙].
As we already know that 𝑓(𝑡) is remotely almost periodic, then we have limsup_|𝑡|→∞||𝑓(𝑡+𝜏)−𝑓(𝑡)||<𝜀₁;(2.5) that is, for 𝜀₁>0, there exists 𝑡₀>0, and when |𝑡|>𝑡₀, there is ||𝑓(𝑡+𝜏)−𝑓(𝑡)||<𝜀₁,−𝜀₁<𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀₁.(2.6) Now take 𝑠_𝑖=𝑡_𝑖+𝜏(𝑖=1,2),𝐿=𝑙+𝑑. So 𝑠₁,𝑠₂∈[𝛼,𝛼+𝐿], and 𝐹𝑠₂−𝐹𝑠₁==𝐹𝐹𝑡𝑡₂₂−𝐹𝑡₁−^𝑡2_𝑡1𝑓(𝑡)𝑑𝑡+^{𝑡2𝑡+𝜏}_1+𝜏𝑓(𝑡)𝑑𝑡>𝐺−𝑔−−𝜀𝐹𝑡₁+^𝑡2_𝑡1[𝑓(𝑡+𝜏)−𝑓(𝑡)]𝑑𝑡3−𝜀¹𝑑=𝐺−𝑔−𝜀2,(2.7) that is, 𝐹𝑠₁−𝑔+𝐺−𝐹𝑠₂<2𝜀;(2.8) as the formulas in two brackets of previous inequality are both nonnegative, so there are two numbers 𝑠₁ and 𝑠₂ in any interval of length 𝐿 satisfying simultaneously 𝐹𝑠₁<𝑔+𝜀2,𝐹𝑠²>𝐺−2𝜀.(2.9) Now take 𝜀₂=𝜀/2𝐿, and we will prove that when 𝜏∈𝑇(𝑓,𝜀₂), there is 𝜏∈𝑇(𝑓,𝜀). In fact for every 𝑡∈𝑅, we can choose 𝑠₁ and 𝑠₂ in the interval [𝑡,𝑡+𝐿] satisfying 𝐹(𝑠₁)<𝑔+(𝜀/2) and 𝐹(𝑠₂)>𝐺−(𝜀/2). Hence for 𝜏∈𝑇(𝑓,𝜀₂), there are, respectively, limsup_|𝑡|→∞(𝐹(𝑡+𝜏)−𝐹(𝑡))=limsup_|𝑡|→∞𝐹𝑠¹>𝑔−𝑔+𝜀+𝜏−𝐹𝑠₁+^𝑠1_𝑡𝑓(𝑡)𝑑𝑡−^{𝑠1+𝜏𝑡+𝜏}𝑓(𝑡)𝑑𝑡2−𝜀²limsup_|𝑡|→∞𝐿=−𝜀,(𝐹(𝑡+𝜏)−𝐹(𝑡))=limsup_|𝑡|→∞𝐹𝑠²<𝐺−𝐺−𝜀+𝜏−𝐹𝑠₂+^𝑠2_𝑡𝑓(𝑡)𝑑𝑡−^{𝑠2+𝜏𝑡+𝜏}𝑓(𝑡)𝑑𝑡2+𝜀²𝐿=𝜀.(2.10) So for 𝜏∈𝑇(𝑓,𝜀₂), we have 𝜏∈𝑇(𝑓,𝜀); hence 𝐹(𝑡) is remotely almost periodic.

Proposition 2.4. Assume that 𝑓(𝑡) is remotely almost periodic. Then (1/𝑇)∫^{𝑎𝑎+𝑇}𝑓(𝑡)𝑑𝑡 converges as 𝑇→+∞ uniformly with respect to 𝑎∈𝑅. Moreover the limit does not depend on 𝑎, and it is called the average of 𝑓∃⟨𝑓⟩∶=lim_𝑇→+∞1𝑇^{𝑎𝑎+𝑇}𝑓(𝑡)𝑑𝑡,uniformlywithrespectto𝑎∈𝑅.(2.11)

Proof. As 𝑓(𝑡)∈RAP(𝑅), then 𝑓(𝑡) is bounded, and for all𝜀>0, for all 𝜏∈𝑇(𝑓,𝜀), there exists 𝑠₀>0, when |𝑡|>𝑠₀,|𝑓(𝑡+𝜏)−𝑓(𝑡)|<𝜀. Let 𝐺=sup_𝑡∈𝑅|𝑓(𝑡)|, take 𝜀>0, and assume that 𝑙=𝑙(𝜀/4) is an interval length of 𝑇(𝑓,𝜀/4). Take 𝜏∈𝑇(𝑓,𝜀/4)∩[𝑎,𝑎+𝑙]; then for any 𝑎,𝑠∈𝑅||||^{𝑎𝑎+𝑠}𝑓(𝑡)𝑑𝑡−^𝑠0𝑓(𝑡)𝑑t||||=||||^{𝜏𝜏+𝑠}−^𝑠0+^{𝑎+𝑠𝜏+𝑠}+^𝜏𝑎≤𝑓(𝑡)𝑑𝑡_𝑠0||||=||𝑓(𝑡+𝜏)−𝑓(𝑡)||𝑑𝑡+^{𝑎+𝑠𝜏+𝑠}||𝑓(𝑡)||𝑑𝑡+^𝜏0||𝑓(𝑡)||𝑑𝑡_𝑠00+||𝑓(𝑡+𝜏)−𝑓(𝑡)||𝑑𝑡+^𝑠𝑠₀||𝑓(𝑡+𝜏)−𝑓(𝑡)||𝑑𝑡_{𝑎+𝑠𝜏+𝑠}≤sup_{𝑠0,𝑠}||𝑓(𝑡)||𝑑𝑡+^𝜏0||𝑓(𝑡)||𝑑𝑡<𝜀||𝑓(𝑡+𝜏)−𝑓(𝑡)||⋅𝑠−𝑠₀+2𝐺𝑙+𝑠₀4𝑠−𝑠⁰+2𝐺𝑙+𝑠₀,(2.12) so ||||1𝑇^{𝑎𝑎+𝑇}𝑓(𝑡)𝑑𝑡−1𝑇^𝑇0𝑓(𝑡)𝑑𝑡||||≤𝜀4𝑇𝑇−𝑇⁰+2𝐺𝑙+𝑇⁰||||1𝑇,(2.13)𝑛𝑇^0𝑛𝑇𝑓(𝑡)𝑑𝑡−1𝑇^𝑇0𝑓(𝑡)𝑑𝑡||||=1𝑛|||||^𝑛_𝑘=11𝑇^{𝑘𝑇(𝑘−1)𝑇}≤𝜀𝑓(𝑡)𝑑𝑡−^𝑇0𝑓(𝑡)𝑑𝑡|||||4𝑇𝑇−𝑇⁰+2𝐺𝑙+𝑇⁰𝑇.(2.14) By passing 𝑛→+∞ in (2.14), we get ||||⟨𝑓⟩−1𝑇^𝑇0𝑓(𝑡)𝑑𝑡||||≤𝜀4𝑇𝑇−𝑇⁰+2𝐺𝑙+𝑇⁰𝑇.(2.15) Using triangle inequality from (2.13) and (2.15) we deduce ||||1𝑇^{𝑎𝑎+𝑇}𝑓(𝑡)𝑑𝑡−⟨𝑓⟩||||≤𝜀2𝑇𝑇−𝑇⁰+4𝐺𝑙+𝑇⁰𝑇<𝜀,(2.16) if only 𝑇>(8𝐺(𝑙+𝑇₀)/𝜀)−𝑇₀. That is, when 𝑇→∞,(1/𝑇)∫^{𝑎𝑎+𝑇}𝑓(𝑡)𝑑𝑡 converges at ⟨𝑓⟩ uniformly with respect to 𝑎∈𝑅. Moreover notice the identical equation 1𝑇^{𝑎𝑎+𝑇}𝑓(𝑡)𝑑𝑡=1𝑇^𝑇0𝑓(𝑡+𝑎)𝑑𝑡.(2.17) This means that the limit does not depend on 𝑎.

3. Remotely Almost Periodic Viscosity Solutions

In this section we get some results for remotely almost periodic viscosity solutions.

Definition 3.1. One says that 𝑢∶ℝ^𝑁×ℝ→ℝ is remotely almost periodic in 𝑡 uniformly with respect to 𝑥 if 𝑢 is bounded and uniformly continuous in 𝑡 uniformly with respect to 𝑥 and for all 𝜀>0, and there exists𝑙(𝜀)>0 such that all intervals of length 𝑙(𝜀) contain a number 𝜏 which is 𝜀 remotely almost periodic for 𝑢(𝑥,⋅), for all𝑥∈ℝ^𝑁limsup_|𝑡|→∞||𝑢(𝑥,𝑡+𝜏)−𝑢(𝑥,𝑡)||<𝜀,∀(𝑥,𝑡)∈ℝ^𝑁×ℝ.(3.1)

3.1. Some Hypotheses and Theorems

In this section we list some usual hypotheses used for the uniqueness and existence results and present two properties of viscosity solutions.

First let us list some hypotheses in the stationary case:∀0<𝑅<+∞∀𝑥,∈∃ℝ𝛾_𝑅𝑁>0∶𝐻(𝑥,𝑢,𝑝)−𝐻(𝑥,𝑣,𝑝)≥𝛾_𝑅(𝑢−𝑣),∀𝑅>0,∃𝑚_𝑅,−𝑅≤𝑣≤𝑢≤𝑅,𝑝∈ℝ^𝑁,(3.2),lim_𝑧→0𝑚_𝑅(𝑧)=0∶||𝐻(𝑥,𝑢,𝑝)−∀𝐻𝑥,(𝑦𝑦,∈𝑢,ℝ𝑝)_𝑁||≤𝑚_𝑅||𝑥−𝑦||⋅1+||𝑝||,∀0<𝑅<+∞,lim_{||𝑝||→+∞},−𝑅≤𝑢≤𝑅,𝑝∈ℝ^𝑁,(3.3)∀0<𝑅<+∞,𝐻𝐻(i𝑥s,u𝑢n,i𝑝f)or=m+ly∞co,ntuinnuifooursmolnyfℝor_𝑁(𝑥,𝑢)∈ℝ^𝑁×[−𝑅,𝑅],(3.4)∃𝑀>0∶𝐻(𝑥,−𝑀,0)≤0≤𝐻(𝑥,𝑀,0),×∀𝑥[−∈𝑅ℝ,𝑅_𝑁]×𝐵_𝑅,(3.5).(3.6) From [1] we know that hypotheses (3.2), (3.3) or (3.4), (3.5), (3.6) ensure the existence of a unique solution for the stationary equation (1.3). And more regularly (3.2) can be replaced by𝐻(𝑥,𝑢,𝑝)−𝐻(𝑥,𝑣,𝑝)≥0,∀𝑥∈ℝ^𝑁,𝑣≤𝑢,𝑝∈ℝ^𝑁(3.7) (which comes to taking 𝛾_𝑅=0 in (3.2)).

When the Hamiltonian is time dependent the corresponding assumptions are∀0<𝑅<∀+𝑥∞∈,ℝ_𝑁∃𝛾_𝑅>0∶𝐻(𝑥,𝑡,𝑢,𝑝)−𝐻(𝑥,𝑡,𝑣,𝑝)≥𝛾_𝑅(𝑢−𝑣),∀𝑅>0,∃𝑚_𝑅,0≤𝑡≤𝑇,−𝑅≤𝑣≤𝑢≤𝑅,𝑝∈ℝ^𝑁,(3.8)∀𝑥,𝑦∈ℝ_𝑁∶||𝐻(𝑥,𝑡,𝑢,𝑝)−𝐻(𝑦,𝑡,𝑢,𝑝)||≤𝑚_𝑅||𝑥−𝑦||⋅1+||𝑝||,∀0<𝑅<+∞,,𝑡𝐻∈i[s0,u𝑇ni]fo,r−m𝑅ly≤co𝑢nt≤in𝑅uo,u𝑝so∈nℝℝ^𝑁_𝑁,wherelim_𝑧→0𝑚_𝑅(𝑧)=0,(3.9)∃𝑀>0∶𝐻(𝑥,𝑡,−𝑀,0)≤0≤𝐻(𝑥,𝑡,𝑀,0),×∀[𝑥0,∈𝑇ℝ]×_𝑁[−𝑅,𝑅]×𝐵_𝑅,(3.10),𝑡∈[0,𝑇].(3.11) Now we present two results of viscosity solutions (see [1, 6, 7]).

Theorem 3.2. Assume that (3.8), (3.9), (3.10), and (3.11) hold (with 𝛾_𝑅∈ℝ, for all 𝑅>0). Then for every 𝑢₀∈BUC(ℝ^𝑁) there is a unique viscosity solution 𝑢∈BUC(ℝ^𝑁×[0,𝑇]) of (1.2), for all 𝑇>0.

Theorem 3.3. Let 𝑢 be a bounded time periodic viscosity u.s.c. subsolution of 𝜕_𝑡𝑢+𝐻(𝑥,𝑡,𝑢,𝐷𝑢)=𝑓(𝑥,𝑡) in ℝ^𝑁×ℝ and 𝑣 a bounded time periodic viscosity l.s.c. supersolution of 𝜕_𝑡𝑣+𝐻(𝑥,𝑡,𝑣,𝐷𝑣)=𝑔(𝑥,𝑡) in ℝ^𝑁×ℝ, where 𝑓,𝑔∈BUC(ℝ^𝑁×ℝ) and 𝐻 are 𝑇 periodic such that (3.8), (3.9), and (3.10) hold. Then one has sup_{𝑥∈ℝ𝑁}(𝑢(𝑥,𝑡)−𝑣(𝑥,𝑡))≤sup_𝑠≤𝑡^tssup_{𝑥∈ℝ𝑁}(𝑓(𝑥,𝜎)−𝑔(𝑥,𝜎))𝑑𝜎.(3.12) Moreover, the hypothesis (3.9) can be replaced by 𝑢∈𝑊^1,∞(ℝ^𝑁×ℝ) or 𝑣∈𝑊^1,∞(ℝ^𝑁×ℝ).

3.2. Uniqueness and Existence of Time Remotely Almost Periodic Viscosity Solutions

In this section we establish uniqueness and existence results for time remotely almost periodic viscosity solutions. For the uniqueness we have the more general result.

Proposition 3.4. Let 𝑢 a bounded u.s.c. viscosity subsolution of 𝜕_𝑡𝑢+𝐻(𝑥,𝑡,𝑢,𝐷𝑢)=𝑓(𝑥,𝑡), in ℝ^𝑁×ℝ and 𝑣 a bounded l.s.c. viscosity supersolution of 𝜕_𝑡𝑣+𝐻(𝑥,𝑡,𝑣,𝐷𝑣)=𝑔(𝑥,𝑡), in ℝ^𝑁×ℝ where 𝑓,𝑔∈BUC(ℝ^𝑁×ℝ) and (3.8), (3.9), (3.10) hold uniformly for 𝑡∈ℝ. Then one has for all 𝑡∈ℝsup_{𝑥∈ℝ𝑁}(𝑢(𝑥,𝑡)−𝑣(𝑥,𝑡))₊≤𝑒^−𝛾𝑡^𝑡−∞sup_{𝑥∈ℝ𝑁}(𝑓(𝑥,𝜎)−𝑔(𝑥,𝜎))₊𝑑𝜎.(3.13) Moreover hypotheses (3.9) can be replaced by 𝑢∈𝑊^1,∞(ℝ^𝑁×ℝ) or 𝑣∈𝑊^1,∞(ℝ^𝑁×ℝ).

The proof of this proposition is similar to Proposition 6.5 in [1]. Hence we do not prove it here.

Before we concentrate on the existence part, let us see two important lemmas first. Now take ℎ(𝑡)=∫^𝑡−∞𝑒^{𝛾(𝜎−𝑡)}𝑓(𝜎)𝑑𝜎, where 𝛾>0 is a constant, 𝑡∈ℝ.

Lemma 3.5. If 𝑓(𝑡)∈SO(ℝ), then ℎ(𝑡)∈SO(ℝ).

Proof. As 𝑓(𝑡)∈SO(ℝ), so for every 𝜏∈ℝlim_|𝑡|→∞||𝑓(𝑡+𝜏)−𝑓(𝑡)||=0.(3.14) Now for every 𝜏∈ℝ||ℎ(𝑡+𝜏)−ℎ(𝑡)||==||||||||^{𝑡+𝜏−∞}𝑒^{𝛾(𝜎−𝑡−𝜏)}𝑓(𝜎)𝑑𝜎−^𝑡−∞𝑒^{𝛾(𝜎−𝑡)}𝑓(𝜎)𝑑𝜎_0−∞||||=||||𝑒^𝛾𝜎𝑓(𝑡+𝜎+𝜏)𝑑𝜎−^0−∞𝑒^𝛾𝜎𝑓(𝑡+𝜎)𝑑𝜎_0−∞||||≤𝑒^𝛾𝜎[𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)]𝑑𝜎_0−∞||||≤sup_𝜎𝑒^𝛾𝜎||𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)||𝑑𝜎||𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)||⋅1𝛾,(3.15) hence lim_|𝑡|→∞||ℎ(𝑡+𝜏)−ℎ(𝑡)||≤limsup_|𝑡|→∞||𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)||⋅1𝛾=0.(3.16) Since we already know that 𝑓(𝑡)∈BUC(ℝ), we deduce also that ℎ(𝑡)∈BUC(ℝ). That is, ℎ(𝑡)∈SO(ℝ).

Lemma 3.6. If 𝑓(𝑡)∈RAP(ℝ), then ℎ(𝑡)∈RAP(ℝ).

Proof. The main result in [8] proved that 𝑓(𝑡)∈RAP(ℝ) is the closed subalgebra in 𝐶(ℝ) created by AP(ℝ) and SO(ℝ). Hence, if 𝑓(𝑡)∈RAP(ℝ), for every 𝜀>0, take 𝜀₁=𝛾⋅𝜀, there exists 𝑔₁,𝑔₂∈𝐴𝑃(ℝ) and 𝜑₁,𝜑₂∈SO(ℝ); hence ‖‖𝑓−𝑔₁+𝜑₁+𝑔₂𝜑₂‖‖<𝜀¹4.(3.17) If 𝜑₂=0, consider a number 𝜏 which is an 𝜀₁/2 remotely almost period of 𝑔₁: ||ℎ(𝑡+𝜏)−ℎ(𝑡)||=≤||||^0−∞𝑒^𝛾𝜎[𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)]𝑑𝜎_0−∞||||≤𝑒^𝛾𝜎||𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)||𝑑𝜎_0−∞+𝑒^𝛾𝜎||𝑓(𝑡+𝜎+𝜏)−𝑔₁(𝑡+𝜎+𝜏)+𝜑₁(𝑡+𝜎+𝜏)||𝑑𝜎_0−∞+𝑒^𝛾𝜎||𝑔₁(𝑡+𝜎+𝜏)−𝑔₁(𝑡+𝜎)||𝑑𝜎_0−∞+𝑒^𝛾𝜎||𝜑₁(𝑡+𝜎+𝜏)−𝜑₁(𝑡+𝜎)||𝑑𝜎_0−∞<𝜀𝑒^𝛾𝜎||𝑓(𝑡+𝜎)−𝑔₁(𝑡+𝜎)+𝜑₁(𝑡+𝜎)||𝑑𝜎₁4𝛾+2𝜀𝛾¹+^0−∞𝑒^𝛾𝜎||𝜑₁(𝑡+𝜎+𝜏)−𝜑₁(𝑡+𝜎)||𝑑𝜎+𝜀¹4𝛾.(3.18) By using Lemma 3.5 we deduce limsup_|𝑡|→∞||ℎ(𝑡+𝜏)−ℎ(𝑡)||<𝜀¹4𝛾+2𝜀𝛾¹+4𝜀𝛾¹=𝜀.(3.19) Thus this proves that any 𝜀₁/2 remotely almost period of 𝑔₁ is an 𝜀 remotely almost period of ℎ.
If 𝜑₂≠0, assume that 𝛿=min{𝜀₁/4,𝜀₁/(4⋅‖𝜑₂‖)}, and take number 𝜏 which is a common 𝛿 remotely almost period of 𝑔₁ and 𝑔₂. We will prove that 𝜏 is an 𝜀₁/2 remotely almost period of (𝑔₁+𝜑₁+𝑔₂𝜑₂), and an 𝜀 remotely almost period of ℎ: ||𝑔₂(𝑡≤+||𝜎𝑔+𝜏)𝜑₂(𝑡+𝜎+𝜏)−𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)₂||+||𝑔(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎+𝜏)−𝑔₂(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎)₂||≤‖‖𝑔(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎)−𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)₂||‖‖⋅||𝜑₂(𝑡+𝜎+𝜏)−𝜑₂(𝑡+𝜎)||+‖‖𝜑₂‖‖⋅||𝑔₂(𝑡+𝜎+𝜏)−𝑔₂(𝑡+𝜎)||.(3.20) We have limsup_|𝑡|→∞||𝑔₂(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎+𝜏)−𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)||<𝜀¹4.(3.21) Hence limsup_|𝑡|→∞||−𝑔₁𝑔(𝑡+𝜎+𝜏)+𝜑₁(𝑡+𝜎+𝜏)+𝑔₂(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎+𝜏)₁≤limsup_|𝑡|→∞(𝑡+𝜎)+𝜑₁(𝑡+𝜎)+𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)||+limsup_|𝑡|→∞||𝑔₁(𝑡+𝜎+𝜏)−𝑔₁(𝑡+𝜎)||+limsup_|𝑡|→∞||𝜑₁(𝑡+𝜎+𝜏)−𝜑₁(𝑡+𝜎)||<𝜀||𝑔₂(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎+𝜏)−𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)₁||||𝑓(𝑡≤+𝜎||4𝑓+(+𝑡𝜏+𝜀)4¹−𝜎=+𝑓𝜀(𝜏2𝑡¹)+,−𝜎)𝑔||₁+||𝑔(𝑡+𝜎+𝜏)+𝜑₁(𝑡+𝜎+𝜏)+𝑔₂(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎+𝜏)₁||−(𝑡+𝑔𝜎+𝜏)+𝜑₁(𝑡+𝜎+𝜏)+𝑔₂(𝑡+𝜎+𝜏)𝜑₂(𝑡+𝜎+𝜏)₁+||𝑔(𝑡+𝜎)+𝜑₁(𝑡+𝜎)+𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)₁||(𝑡+𝜎)+𝜑₁(𝑡+𝜎)+𝑔₂(𝑡+𝜎)𝜑₂(𝑡+𝜎)−𝑓(𝑡+𝜎)||.(3.22) So we have limsup_|𝑡|→∞||ℎ(𝑡+𝜏)−ℎ(𝑡)||=limsup_|𝑡|→∞||||^0−∞≤limsup_|𝑡|→∞𝑒^𝛾𝜎[𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)]𝑑𝜎||||||𝑓(𝑡+𝜎+𝜏)−𝑓(𝑡+𝜎)||⋅1<𝜀₁𝛾4+𝜀2¹+𝜀4¹⋅𝛾1=𝜀.(3.23) Since we already know that 𝑓(𝑡)∈BUC(ℝ), we deduce also that ℎ(𝑡)∈BUC(ℝ). So this proves that ℎ(𝑡)∈RAP(ℝ).

Now we concentrate on the existence part.

Proposition 3.7. Assume that 𝑓∶ℝ→ℝ is remotely almost periodic and that the Hamiltonian 𝐻=𝐻(𝑥,𝑧,𝑝) satisfying the hypotheses (3.2), (3.3), (3.5), and there exists𝑀>0 such that 𝐻(𝑥,−𝑀,0)≤𝑓(𝑡)≤𝐻(𝑥,𝑀,0), for all (𝑥,𝑡)∈ℝ^𝑁×ℝ. Then there is a time remotely almost periodic viscosity solution in BUC(ℝ^𝑁×ℝ) of 𝜕_𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡), in ℝ^𝑁×ℝ.

Proof. We consider the unique viscosity solution of the problem 𝑢𝜕_𝑡𝑢_𝑛+𝐻𝑥,𝑢_𝑛,𝐷𝑢_𝑛=𝑓(𝑡),(𝑥,𝑡)∈ℝ^𝑁×]−𝑛,+∞_𝑛[,(𝑥,−𝑛)=0,𝑥∈ℝ^𝑁(3.24) for all 𝑛≥1. Such a solution exists by Theorem 3.2. Next we will prove that for all 𝑡∈ℝ, (𝑢_𝑛(𝑡))_{𝑛≥−𝑡} converges to a remotely almost periodic viscosity solution of 𝜕_𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡), in ℝ^𝑁×ℝ. Similar to the proof of Proposition 6.6 in [1], we obtain by fixing 𝑡∈ℝ and 𝑛 large enough ||𝑢_𝑛(𝑥,𝑡)−𝑢_𝑛(𝑥,𝑡+𝜏)||≤2𝑀⋅𝑒^{−𝛾(𝑡−𝑡𝑛)}+𝑒^−𝛾𝑡^𝑡𝑡_𝑛𝑒^𝛾𝜎||𝑓(𝜎+𝜏)−𝑓(𝜎)||𝑑𝜎.(3.25) By passing 𝑛→+∞ we have 𝑡_𝑛→−∞, and therefore ||𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡+𝜏)||≤^𝑡−∞𝑒^{−𝛾(𝑡−𝜎)}||𝑓(𝜎+𝜏)−𝑓(𝜎)||𝑑𝜎.(3.26) As 𝑓 is remotely almost periodic, using Lemma 3.6 we deduce limsup_|𝑡|→∞||𝑢(𝑥,𝑡)−𝑢(𝑥,𝑡+𝜏)||≤limsup_|𝑡|→∞^𝑡−∞𝑒^{𝛾(𝜎−𝑡)}||𝑓(𝜎+𝜏)−𝑓(𝜎)||𝑑𝜎<𝜀.(3.27) Since we already know that 𝑢∈BUC(ℝ^𝑁×[𝑎,𝑏]), for all 𝑎,𝑏∈ℝ,𝑎≤𝑏, by time remotely almost periodicity we deduce also that 𝑢∈BUC(ℝ^𝑁×ℝ).

Now we will study the time remotely almost periodic viscosity solutions of𝜕_𝑡𝑢+𝐻(𝑥,𝑢,𝐷𝑢)=𝑓(𝑡),(𝑥,𝑡)∈ℝ^𝑁×ℝ,(3.28) for Hamiltonians satisfying (3.7). We introduce also the stationary equation𝐻(𝑥,𝑢,𝐷𝑢)=⟨𝑓⟩∶=1𝑇^𝑇0𝑓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑁.(3.29) We have the following theorem for the existence of time remotely almost periodic viscosity solution.

Theorem 3.8. Assume that Hamiltonian H=𝐻(𝑥,𝑧,𝑝) satisfies hypotheses (3.7), (3.4), (3.5), sup{|𝐻(𝑥,0,0)|∶𝑥∈ℝ}=𝐶<+∞ and 𝑓 is a time remotely almost periodic function such that 𝐹(𝑡)=∫^𝑡0{𝑓(𝜎)−⟨𝑓⟩}𝑑𝜎 is bounded on ℝ. Then there is a bounded Lipschitz time remotely almost periodic viscosity solution of (3.28) if and only if there is a bounded viscosity solution of (3.29).

Proof. Assume that 𝑉 is a bounded viscosity of (3.29). We deduce that 𝑉 is a Lipschitz function as the Hamiltonian satisfies (3.4). For any 𝛼>0, take 𝑀_𝛼=‖𝑉‖_𝐿^∞_(ℝ^𝑁₎+(1/𝛼)(𝐶+‖𝑓‖_𝐿^∞_(ℝ)). By Propositions 3.4 and 3.7 we can construct the family of time remotely almost periodic solutions 𝑣_𝛼 for 𝛼𝑣_𝛼−𝑉(𝑥)+𝜕_𝑡𝑣_𝛼+𝐻𝑥,𝑣_𝛼,𝐷𝑣_𝛼=𝑓(𝑡),(𝑥,𝑡)∈ℝ^𝑁×ℝ.(3.30) Similar to Theorems 4.1 and 6.1 in [1], we can extract a sequence which converges uniformly on compact sets of ℝ^𝑁×ℝ towards a bounded Lipschitz solution 𝑣 of (3.28). Next we will prove that 𝑣 is remotely almost periodic. By the hypotheses and Proposition 2.3 we deduce that 𝐹 is remotely almost periodic, and thus, for all 𝜀>0, there is 𝑙(𝜀/2) such that any interval of length 𝑙(𝜀/2) contains an 𝜀/2 remotely almost period of 𝐹. Take an interval of length 𝑙(𝜀/2) and 𝜏 an 𝜀/2 remotely almost period of 𝐹 in this interval. We have for all 𝛼>0,(𝑥,𝑡)∈ℝ^𝑁×ℝ||𝑣_𝛼(𝑥,𝑡+𝜏)−𝑣_𝛼(𝑥,𝑡)||≤||||sup_𝑠≤𝑡^𝑡𝑠=||||sup_𝑠≤𝑡{𝑓(𝜎+𝜏)−𝑓(𝜎)}𝑑𝜎||||^{𝑡+𝜏𝑠+𝜏}=||||sup_𝑠≤𝑡(𝑓(𝜎)−⟨𝑓⟩)𝑑𝜎−^𝑡𝑠(𝑓(𝜎)−⟨𝑓⟩)𝑑𝜎||||≤2sup_𝑡{(𝐹(𝑡+𝜏)−𝐹(𝑡))−(𝐹(𝑠+𝜏)−𝐹(𝑠))}||||||𝐹(𝑡+𝜏)−𝐹(𝑡)||.(3.31) After passing to the limit for 𝛼↘0 one gets |𝑣(𝑥,𝑡+𝜏)−𝑣(𝑥,𝑡)|≤2sup_𝑡|𝐹(𝑡+𝜏)−𝐹(𝑡)|, and hence limsup_|𝑡|→∞||𝑣(𝑥,𝑡+𝜏)−𝑣(𝑥,𝑡)||≤2limsup_|𝑡|→∞||𝐹(𝑡+𝜏)−𝐹(𝑡)||≤𝜀.(3.32) By using the uniform continuity of 𝐹, we can prove exactly in the same manner that 𝑣 is continuous in 𝑡 uniformly with respect to 𝑥. The converse implication follows similarlyTheorem 4.1 in [1]; here we do not prove it.

3.3. Asymptotic Behaviour for Large Frequencies

In this section we study the asymptotic behaviour of time remotely almost periodic viscosity solutions of𝜕_𝑡𝑢_𝑛+𝐻𝑥,𝑢_𝑛,𝐷𝑢_𝑛=𝑓_𝑛(𝑡),(𝑥,𝑡)∈ℝ^𝑁×ℝ,(3.33) where 𝑓∶ℝ→ℝ is a remotely almost periodic function. For all 𝑛≥1 notice that 𝑓_𝑛(𝑡)=𝑓(𝑛𝑡), for all 𝑡∈ℝ is remotely almost periodic and has the same average as 𝑓. Now suppose that such a hypothesis exists∃𝑀>0suchthat𝐻(𝑥,−𝑀,0)≤𝑓(𝑡),∀(𝑥,𝑡)∈ℝ^𝑁×ℝ.(3.34)

Theorem 3.9. Let 𝐻=𝐻(𝑥,𝑧,𝑝) be a Hamiltonian satisfying (3.7), (3.3), (3.5), (3.34) where 𝑓 is remotely almost periodic function. Suppose also that there is a bounded l.s.c viscosity supersolution 𝑉≥−𝑀 of (3.29), that 𝑡→𝐹(𝑡)=∫^𝑡0{𝑓(𝑠)−⟨𝑓⟩}𝑑𝑠 is bounded, and denote by 𝑉 the minimal stationary l.s.c. viscosity supersolution of (3.29), 𝑣_𝑛 the time remotely almost periodic l.s.c. viscosity supersolution of (3.33). Then the sequence (𝑣_𝑛)_𝑛 converges uniformly on ℝ^𝑁×ℝ towards 𝑉 and ‖𝑣_𝑛−𝑉‖_{𝐿∞(ℝ𝑁×ℝ)}≤(2/𝑛)‖𝐹‖_𝐿^∞_(ℝ), for all 𝑛≥1.

Proof. As 𝑣_𝑛=sup_𝛼>0𝑣_𝑛,𝛼 is remotely almost periodic, we introduce 𝑤_𝑛,𝛼(𝑥,𝑡)=𝑣_𝑛,𝛼(𝑥,𝑡/𝑛), (𝑥,𝑡)∈ℝ^𝑁×ℝ, which is also remotely almost periodic. Similar to Theorem 5.1 in [1] and by using Theorem 3.3 we deduce that 𝑤_𝑛,𝛼(𝑥,𝑡)−𝑉_𝛼(𝑥)≤sup_𝑠≤𝑡1𝑛^𝑡𝑠(𝑓(𝜎)−⟨𝑓⟩)𝑑𝜎≤2𝑛‖𝐹‖^𝐿∞(ℝ),(3.35) and similarly 𝑉_𝛼(𝑥)−𝑤_𝑛,𝛼(𝑥,𝑡)≤(2/𝑛)‖𝐹‖_𝐿^∞_(ℝ), for all 𝑛≥1. We have for all 𝑛≤1||𝑤_𝑛,𝛼(𝑥,𝑡)−𝑉_𝛼(𝑥)||≤2𝑛‖𝐹‖^𝐿∞(ℝ),(3.36) and after passing to the limit for 𝛼↘0 one gets for all (𝑥,𝑡)∈ℝ^𝑁×ℝ||𝑤_𝑛(𝑥,𝑡)−𝑉(𝑥)||≤2𝑛‖𝐹‖^𝐿∞(ℝ).(3.37) Finally we deduce that ‖𝑣_𝑛−𝑉‖_𝐿^∞_(ℝ^𝑁_×ℝ)≤(2/𝑛)‖𝐹‖_𝐿^∞_(ℝ^𝑁_×ℝ) for all 𝑛≥1.

Acknowledgment

This work was supported by National Science Foundation of China (Grant no. 11001152).