ISRN Mathematical Analysis

Volume 2011 (2011), Article ID 415358, 13 pages

http://dx.doi.org/10.5402/2011/415358

## Time Remotely Almost Periodic Viscosity Solutions of Hamilton-Jacobi Equations

^{1}School of Mathematics, Shandong University, Jinan 250100, China^{2}School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 6 December 2010; Accepted 6 January 2011

Academic Editor: A. Carpio

Copyright © 2011 Shilin Zhang and Daxiong Piao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 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 and also for the stationary problem 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 and are continuous functions is remotely almost periodic in . 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 be a bounded uniformly continuous function on . We say that is remotely almost periodic if and only if for all
is relatively dense on . The number is called remotely almost period.

And denotes all these functions.

*Definition 2.2. * Let be a bounded uniformly continuous function on . We say that oscillates slowly if and only if for every
And denotes all these functions.

Next we will prove two propositions.

Proposition 2.3. *Assume that is remotely almost periodic and denote by a primitive of . Then is remotely almost periodic if and only if is bounded.*

*Proof. *When is remotely almost periodic, is certainly bounded. For the converse, let be bounded, without losing general, and assume that is a real function. For any , there exists large enough; we have
take fixed and , , , and assume that , satisfying
Assume that is an interval length of , where , . For every , take .

As we already know that is remotely almost periodic, then we have
that is, for , there exists , and when , there is
Now take . So , and
that is,
as the formulas in two brackets of previous inequality are both nonnegative, so there are two numbers and in any interval of length satisfying simultaneously
Now take , and we will prove that when , there is . In fact for every , we can choose and in the interval satisfying and . Hence for , there are, respectively,
So for , we have ; hence is remotely almost periodic.

Proposition 2.4. *Assume that is remotely almost periodic. Then converges as uniformly with respect to . Moreover the limit does not depend on , and it is called the average of *

*Proof. *As , then is bounded, and for all, for all , there exists , when . Let , take , and assume that is an interval length of . Take ; then for any
so
By passing in (2.14), we get
Using triangle inequality from (2.13) and (2.15) we deduce
if only . That is, when converges at uniformly with respect to . Moreover notice the identical equation
This means that the limit does not depend on .

#### 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 is remotely almost periodic in uniformly with respect to if is bounded and uniformly continuous in uniformly with respect to and for all , and there exists such that all intervals of length contain a number which is remotely almost periodic for , for all

##### 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: 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 by (which comes to taking in (3.2)).

When the Hamiltonian is time dependent the corresponding assumptions are 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 , for all ). Then for every there is a unique viscosity solution of (1.2), for all .*

Theorem 3.3. *Let be a bounded time periodic viscosity u.s.c. subsolution of in and a bounded time periodic viscosity l.s.c. supersolution of in , where and are periodic such that (3.8), (3.9), and (3.10) hold. Then one has
**
Moreover, the hypothesis (3.9) can be replaced by or .*

##### 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 a bounded u.s.c. viscosity subsolution of , in and a bounded l.s.c. viscosity supersolution of , in where and (3.8), (3.9), (3.10) hold uniformly for . Then one has for all **
Moreover hypotheses (3.9) can be replaced by or .*

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 , where is a constant, .

Lemma 3.5. *If , then .*

*Proof. *As , so for every
Now for every
hence
Since we already know that , we deduce also that . That is, .

Lemma 3.6. *If , then .*

*Proof. *The main result in [8] proved that is the closed subalgebra in created by and . Hence, if , for every , take , there exists and ; hence
If , consider a number which is an remotely almost period of :
By using Lemma 3.5 we deduce
Thus this proves that any remotely almost period of is an remotely almost period of .

If , assume that , and take number which is a common remotely almost period of and . We will prove that is an remotely almost period of , and an remotely almost period of :
We have
Hence
So we have
Since we already know that , we deduce also that . So this proves that .

Now we concentrate on the existence part.

Proposition 3.7. *Assume that is remotely almost periodic and that the Hamiltonian satisfying the hypotheses (3.2), (3.3), (3.5), and there exists such that , for all . Then there is a time remotely almost periodic viscosity solution in of , in .*

*Proof. *We consider the unique viscosity solution of the problem
for all . Such a solution exists by Theorem 3.2. Next we will prove that for all , converges to a remotely almost periodic viscosity solution of , in . Similar to the proof of Proposition 6.6 in [1], we obtain by fixing and large enough
By passing we have , and therefore
As is remotely almost periodic, using Lemma 3.6 we deduce
Since we already know that , for all , by time remotely almost periodicity we deduce also that .

Now we will study the time remotely almost periodic viscosity solutions of for Hamiltonians satisfying (3.7). We introduce also the stationary equation We have the following theorem for the existence of time remotely almost periodic viscosity solution.

Theorem 3.8. *Assume that Hamiltonian satisfies hypotheses (3.7), (3.4), (3.5), and is a time remotely almost periodic function such that is bounded on . 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 is a bounded viscosity of (3.29). We deduce that is a Lipschitz function as the Hamiltonian satisfies (3.4). For any , take . By Propositions 3.4 and 3.7 we can construct the family of time remotely almost periodic solutions for
Similar to Theorems 4.1 and 6.1 in [1], we can extract a sequence which converges uniformly on compact sets of towards a bounded Lipschitz solution of (3.28). Next we will prove that is remotely almost periodic. By the hypotheses and Proposition 2.3 we deduce that is remotely almost periodic, and thus, for all , there is such that any interval of length contains an remotely almost period of . Take an interval of length and an remotely almost period of in this interval. We have for all
After passing to the limit for one gets , and hence
By using the uniform continuity of , we can prove exactly in the same manner that is continuous in uniformly with respect to . 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 where is a remotely almost periodic function. For all notice that , for all is remotely almost periodic and has the same average as . Now suppose that such a hypothesis exists

Theorem 3.9. *Let be a Hamiltonian satisfying (3.7), (3.3), (3.5), (3.34) where is remotely almost periodic function. Suppose also that there is a bounded l.s.c viscosity supersolution of (3.29), that is bounded, and denote by the minimal stationary l.s.c. viscosity supersolution of (3.29), the time remotely almost periodic l.s.c. viscosity supersolution of (3.33). Then the sequence converges uniformly on towards and , for all .*

*Proof. *As is remotely almost periodic, we introduce , , which is also remotely almost periodic. Similar to Theorem 5.1 in [1] and by using Theorem 3.3 we deduce that
and similarly , for all . We have for all
and after passing to the limit for one gets for all
Finally we deduce that for all .

#### Acknowledgment

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

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