Pseudo almost periodic weak solutions of a semilinear elliptic equation
© Ji and Zhang; licensee Springer. 2014
Received: 11 December 2013
Accepted: 8 January 2014
Published: 29 January 2014
In this paper, pseudo almost periodic functions on , with N an integer larger than 1, are introduced and some basic properties of them are studied. As an application, we investigate the pseudo almost periodicity of a weak solution of the semilinear elliptic equation . In addition, a pseudo almost periodic forced pendulum equation is considered as an example.
The notion of pseudo almost periodic functions on ℝ was introduced by Zhang [1–3] in the early 1990s and it is a natural generalization of Bohr almost periodicity. In the past 20 years, these functions have attracted much attention and have been applied to qualitative analysis for various kinds of equations (see [4–8] and the references therein).
The notion of Bohr almost periodicity is also suitable for functions on , with . This notion was deeply studied for numerical-valued functions in Bochner’s fundamental paper . In early 1970s, Sibuya  and Sell  investigated almost periodic solutions of some linear partial differential equations. Then Zaidman  studied almost periodic functions from into a Banach space. Recent work on almost periodic second order elliptic equations in (see, for instance, Pankov , Fournier et al.  and N’Guérékata ) shows a revival of the interest in this topic.
where ’s are real constants, f and h are pseudo almost periodic functions.
where g and are the almost periodic components of f and h, respectively.
We firstly show the almost periodicity of a bounded weak solution υ of (1.2). This result is an improvement of [, Theorem 3.2] where .
Then, we show that the bounded weak solution u of (1.1) is pseudo almost periodic by proving that is an ergodic perturbation.
The organization of this paper is as follows. In Section 2 we recall some basic facts on Bohr almost periodic functions on . Section 3 is devoted to the pseudo almost periodic functions on . In the last section, we investigate pseudo almost periodicity of a weak solution of (1.1) and consider a pseudo almost periodic forced pendulum equation as an example.
2 Bohr almost periodic functions
Let or ℂ, be the n-dimensional real or complex Euclidean space with norm and production for any . Let N be an integer larger than 1. A closed ball in (of center and radius ) is defined by . The cubes in are defined by and where . We denote the Lebesgue measure of a set by .
is a Banach space.
We say a set is relatively dense in if there exists a number such that the intersection is non-empty, for all .
Definition 2.1 ()
We denote by the space of all such almost periodic functions.
contains the space, , which consists of functions where , and , and n is any natural number (see [, Proposition 1.1.3]). Let . Since for each , , it is easy to see that the function .
Lemma 2.2 ([, Chapter 9])
is a Banach space.
For any , the range is relatively compact.
If , then it is uniformly continuous over .
is almost periodic if and only if it is normal; that is, the set is relatively compact in .
If f and g both are almost periodic on , then for each , the set is relatively dense in .
Lemma 2.3 ([, p.10])
- (1)For , then(2.1)
where the limit exists uniformly in .
Let . If and , then .
uniformly in . For the proof we refer the reader to .
Let n be a positive integer and Ω be a subset of . We denote by the set of all jointly continuous functions from to X. We give the following definition.
Denote by the set of all such functions.
Lemma 2.5 If , then it is uniformly continuous on for each compact subset .
The proof is completed. □
3 Pseudo almost periodic functions
It is easy to see that is a closed subspace of and it contains the space of functions which vanish at infinite as a proper subspace.
As (2.1), (3.1) also has another version.
where and are positive constants.
Letting in the formula above, we will get the equivalence of equations (3.1) and (3.2). □
We use (3.1) or (3.2) in different occasions for convenience. For example, (3.1) will be used in Theorem 4.6 and (3.2) in Lemma 3.6.
Definition 3.2 A closed subset C of is said to be an ergodic zero set in if as .
Theorem 3.3 A function is in if and only if for each , the set is an ergodic zero set in .
which contradicts the fact that . So (3.3) holds.
for . This implies that . □
A function is in if and only if is.
For each positive integer n, is in if and only if the norm function is in .
Since the is arbitrary, Theorem 3.3 implies the conclusion.
(2) By (1), if and only if , , the latter is equivalent to , which, again by (1), is equivalent to . □
Definition 3.5 A function is called pseudo almost periodic if , where and . The functions g and φ are called the almost periodic component and ergodic perturbation of f, respectively. Denote by the space of all such pseudo almost periodic functions.
The decomposition for f above is unique. Otherwise, let where is almost periodic and is an ergodic perturbation. Then . If does not hold, then by Lemma 2.3(2), . But since both φ and are ergodic perturbations, . This is a contradiction.
Lemma 3.6 If and g is its almost periodic component, then . Therefore .
This implies that each cube , contains a ball with radius δ on which .
Note that the last term in the formula above is independent of n. Letting above, because , we get a contradiction to (3.2). □
Theorem 3.7 is a Banach space.
Proof Obviously is a normed linear subspace of , and we only need to show it is complete. Let be a Cauchy sequence, and be the almost periodic component and ergodic perturbation of , respectively. By Lemma 3.6, is Cauchy too. So is . Since and are closed in , there exist and such that and as . Set . Then and , as . □
uniformly with respect to .
where and .
Let K be a subset of Ω. A function is said to be continuous in and uniformly in , if for any given and , there exists a , such that and imply that for all .
for . By the arbitrariness of ϵ, we get the conclusion. □
4 Almost periodic and pseudo almost periodic weak solutions
In this section, we simplify the notations for ℝ-valued function spaces in the previous sections by omitting the value space. For example, is replaced with . Also some new notations will be used in this section, i.e., is the set of all continuous differentiable functions; denotes the set all smooth functions with compact support; denotes the local Sobolev space of functions which and their first generalized derivatives are locally square integrable; and () is the Lebesgue measurable (locally) bounded function space.
where ’s are real constants, , and .
For the existence of a weak solution to (4.1) and its relationship with the nonlinear term f and forcing term h, we refer to the following two lemmas.
Lemma 4.2 ([, Theorem 2.5])
Assume that α is a lower and β an upper solution of (4.1) and . Then problem (4.1) has a weak solution such that .
Lemma 4.3 ([, Theorem 2.7])
In the remaining of this section, we will consider the almost periodicity and pseudo almost periodicity of the bounded weak solution of problem (4.1) under the assumption that f and h have the corresponding properties.
4.1 Almost periodic weak solution
To show the pseudo almost periodicity of a weak solution u of problem (4.1), the process consists of two parts: obtaining its almost periodic component and then the ergodic perturbation. This subsection is devoted to obtaining the almost periodic component.
where essinf (esssup) denotes the essential infimum (essential supremum) of a function. We also assume that f is almost periodic in and uniformly on compact subsets of ℝ, and h is almost periodic. Then the weak solution u of (4.1) in Lemma 4.2 is unique and almost periodic.
Proof The uniqueness of u follows from Lemma 4.3.
For an arbitrary , it follows from Lemma 2.2(5) that is relatively dense in . We will show that , and thus u is almost periodic on .
i.e., . The proof is completed. □
Remark 4.5 Theorem 4.4 improves [, Theorem 3.2] where the nonlinear term , which benefits from the applications of the notion of almost periodic dominance (see [, Chapter 9]) and Lemma 2.2(5).
4.2 Pseudo almost periodic weak solution
where g and are the almost periodic components, φ and are ergodic perturbations.
then problem (4.1) has a unique pseudo almost periodic weak solution such that whose almost periodic component is a weak solution of (4.4).
Lemmas 4.2 and 4.3 imply the existence and uniqueness of weak solutions and of (4.1) and (4.4), respectively, such that and . Moreover, Theorem 4.4 implies that υ is almost periodic.
Now if we can show that , then the proof is completed.
where is the zero index.
where C is a positive constant.
i.e., . From Lemma 3.4, it follows that . □
Remark 4.7 Theorem 4.6 can also be comprehended in the following way. If the nonlinear term g and forcing term of (4.4) are perturbed by ergodic perturbations which maintain the monotonicity of nonlinear term and the existence of upper and lower solutions, then its almost periodic weak solution is also perturbed by an ergodic perturbation.
4.3 Pseudo almost periodic forced pendulum equation
where ’s are real constants, f and h both are pseudo almost periodic. If there exists an such that , , then (4.12) has a unique pseudo almost periodic weak solution such that .
It follows from Lemma 3.6 that g and also satisfy the relationship that , .
If is sufficient small (<ϵ), then and are lower and upper solutions of both (4.12) and (4.13).
Since is strictly positive when .
Now we can apply Theorem 4.6 and get the conclusion that (4.12) has a unique pseudo almost periodic weak solution u such that and its almost periodic component is a weak solution of (4.13). □
The authors acknowledge support from the NSF of China (no. 11071048).
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