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Pseudo almost periodic weak solutions of a semilinear elliptic equation
Advances in Difference Equations volume 2014, Article number: 46 (2014)
Abstract
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.
MSC:35B15, 35J61.
1 Introduction
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 [9]. In early 1970s, Sibuya [10] and Sell [11] investigated almost periodic solutions of some linear partial differential equations. Then Zaidman [12] studied almost periodic functions from into a Banach space. Recent work on almost periodic second order elliptic equations in (see, for instance, Pankov [13], Fournier et al. [14] and N’Guérékata [15]) shows a revival of the interest in this topic.
It is a natural idea to introduce pseudo almost periodic functions on and these functions may have potential use in differential equations with more than one spatial variables. However, to the best of our knowledge, there is no literature systematically treating this yet. So in this paper, we will introduce these functions and study some of their basic properties. To illustrate their potential use, we also study the pseudo almost periodicity of a weak solution of the following semilinear elliptic equation:
where ’s are real constants, f and h are pseudo almost periodic functions.
In order to study (1.1), we need to consider its almost periodic component equation at first, i.e.,
where g and are the almost periodic components of f and h, respectively.
For the existence of bounded weak solutions of equations (1.1) and (1.2), we refer to the work of Fournier et al. [14]. Our main task is the following two steps.
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We firstly show the almost periodicity of a bounded weak solution υ of (1.2). This result is an improvement of [[14], Theorem 3.2] where .
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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 .
Let X be a Banach space with norm . Let denote the space of all continuous functions from to X and denote the space of all bounded functions in . Endowed with the norm
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 ([12])
A function is called almost periodic, if for every positive number ϵ, one may find a relatively dense set in , such that
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 [[13], Proposition 1.1.3]). Let . Since for each , , it is easy to see that the function .
Lemma 2.2 ([[12], Chapter 9])
-
(1)
is a Banach space.
-
(2)
For any , the range is relatively compact.
-
(3)
If , then it is uniformly continuous over .
-
(4)
is almost periodic if and only if it is normal; that is, the set is relatively compact in .
-
(5)
If f and g both are almost periodic on , then for each , the set is relatively dense in .
Lemma 2.3 ([[13], p.10])
-
(1)
For , then
(2.1)where the limit exists uniformly in .
-
(2)
Let . If and , then .
The representation (2.1) has some other useful versions. For example,
uniformly in . For the proof we refer the reader to [16].
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.
Definition 2.4 A function is called almost periodic in and uniformly on compact subsets of Ω, if for every and compact , there exists a relatively dense set in such that
Denote by the set of all such functions.
Lemma 2.5 If , then it is uniformly continuous on for each compact subset .
Proof For , let l be the length associated with the relatively dense set in such that , for all . Since f is uniformly continuous on , there is a such that when , and , one has
Let be such that
Choosing an , then one has and . Therefore, and . It follows from inequalities (2.2) and (2.3) that
The proof is completed. □
3 Pseudo almost periodic functions
Denote by the set of ergodic perturbations for which
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.
Lemma 3.1 Let . Then if and only if
Proof According to the volume formulas of cubes and balls in , we have
where and are positive constants.
For each , the inclusion holds in with the usual Euclidean norm. It follows that
Dividing the inequalities by yields
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 .
Proof If , we prove for each the set is an ergodic zero set in , i.e.,
Suppose to the contrary that there exists such that does not converge to 0 as . Then there exists such that for each n,
Thus, we have
which contradicts the fact that . So (3.3) holds.
On the other hand, suppose that (3.3) is true. That is, for any , there exists such that for ,
Thus,
for . This implies that . □
Lemma 3.4 The following statements hold:
-
(1)
A function is in if and only if is.
-
(2)
For each positive integer n, is in if and only if the norm function is in .
Proof (1) For any , it is obvious that
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 .
Proof If , then there exists an such that . Since g is continuous at , there is a such that implies that . By Definition 2.1, for there exists such that every cube , contains at least one point . Therefore, if , then
This implies that each cube , contains a ball with radius δ on which .
Now, we use (3.2) to show the contradiction. For any positive integer n, we divide into smaller cubes with edge length who do not have common inner point with each other. Then, using the volume formulas and in where is the classical Gamma function, we obtain
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 . □
Definition 3.8 A function is called an ergodic perturbation in and uniformly on compact subsets of Ω, if for each compact , and
uniformly with respect to .
Let denote all the functions f of the form
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 .
Lemma 3.9 Suppose the function is continuous in and uniformly in for a compact . Then
Proof Since K is compact, for arbitrary one can find a finite number, say m, of open balls with center and radius , , such that and
Since each , there is a number such that
It follows from (3.4) and (3.5) that
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.
Now, we consider the problem
where ’s are real constants, , and .
Definition 4.1 The function is a lower solution of (4.1) if, for every ,
The function is an upper solution of (4.1) if, for every ,
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 ([[14], 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 ([[14], Theorem 2.7])
Let . Assume that there exists such that
If, for , is a weak solution of
such that , then
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.
Theorem 4.4 Under the assumptions of Lemma 4.2, we assume further that and there exists such that
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.
From the assumptions, we know that f is almost periodic in and uniformly on the compact set . By [[12], Theorem 9.4], there exist a Banach space Y and an almost periodic function such that
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 .
Let . Then is a weak solution of the equation
Note that both u and satisfy the condition in (4.3). We can take the constants and as two functions in , then Lemma 4.3 implies that
i.e., . The proof is completed. □
Remark 4.5 Theorem 4.4 improves [[14], Theorem 3.2] where the nonlinear term , which benefits from the applications of the notion of almost periodic dominance (see [[12], Chapter 9]) and Lemma 2.2(5).
4.2 Pseudo almost periodic weak solution
Now we assume that f is pseudo almost periodic in and uniformly on compact subsets of ℝ and h is also pseudo almost periodic. Let
where g and are the almost periodic components, φ and are ergodic perturbations.
In the following, we shall use the auxiliary nonnegative function , , such that for some ,
where is the multiple index with the ’s being nonnegative integers, , , does not depend on R. The function originates from [[17], Lemma 4.1] and has been used in [[13], page 159] for another purpose.
To consider (4.1), we also need to consider its almost periodic component, i.e. the equation
Theorem 4.6 Assume that are lower and upper solutions of (4.1) and (4.4), respectively. Let f be continuous in and uniformly in for each compact . If there exists such that
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).
Proof It follows from Lemma 3.6 that (4.5) is also valid when f is replaced with g. In fact, for each pair of s and t satisfying the condition in (4.5), is the almost periodic component of the pseudo almost periodic function . By Lemma 3.6, for each there exists a sequence such that
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.
Let , then w is a weak solution of the equation
Since and , . Thus, by the definition of a weak solution, we have
Now we will give some estimations for terms on the left side of (4.6). For the first term, we have
Since is nonnegative, . Using integration by parts, it is easy to see that
where whose j th element is 1 and the others are 0, is the symbol of infinite of the same order.
For the second term in (4.6), similar to the last formulas, we have
where is the zero index.
For the third term, it is implied by (4.5) that
Multiplying (4.6) by and using the estimations for -, we obtain for sufficiently large R,
where C is a positive constant.
Since , we have
By Lemma 2.2(2), the range of υ is relatively compact. We know from Lemma 2.5 that is uniformly continuous in . Since is continuous in and uniformly in , then so is . Now by Lemma 3.9, we have
It follows from (4.8)-(4.10) that
Note that , as . Then (4.11) implies that
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
Consider the 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 .
A special case of (4.12) was treated in [[14], Example 3.3] and [[15], Example 7.1(b)], where , h is almost periodic and .
Proof Let g and are the almost periodic components of f and h, respectively. We need to consider the following almost periodic equation together:
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). □
References
Zhang CY: Integration of vector-valued pseudo almost periodic functions. Proc. Am. Math. Soc. 1994, 121: 167-174. 10.1090/S0002-9939-1994-1186140-8
Zhang CY: Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl. 1994, 151: 62-76.
Zhang CY: Pseudo almost periodic solutions of some differential equations, II. J. Math. Anal. Appl. 1995, 192: 543-561. 10.1006/jmaa.1995.1189
Li HX, Huang FL, Li JY: Composition of pseudo almost-periodic functions and semilinear differential equations. J. Math. Anal. Appl. 2001, 255: 436-446. 10.1006/jmaa.2000.7225
Cieutat P, Fatajou S, N’Guérékata GM: Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations. Appl. Anal. 2010, 89: 11-27. 10.1080/00036810903397503
Dads EA, Ezzinbi K: Existence of positive pseudo almost periodic solution for a class of functional equations arising in epidemic problems. Cybern. Syst. Anal. 1994, 30: 900-910. 10.1007/BF02366449
Xu B, Yuan R: The existence of positive almost periodic type solutions for some neutral nonlinear integral equation. Nonlinear Anal., Theory Methods Appl. 2005, 60: 669-684. 10.1016/j.na.2004.09.043
Diagana T: Pseudo Almost Periodic Functions in Banach Spaces. Nova Science Publishers, New York; 2007.
Bochner S: Abstrakte fastperiodische funktionen. Acta Math. 1933, 61: 149-184. 10.1007/BF02547790
Sibuya Y: Almost periodic solutions of Poisson’s equation. Proc. Am. Math. Soc. 1971, 28: 195-198.
Sell GR: Almost periodic solutions of linear partial differential equations. J. Math. Anal. Appl. 1973, 42: 302-312. 10.1016/0022-247X(73)90139-X
Zaidman S: Almost Periodic Functions in Abstract Spaces. Pitman, Boston; 1985.
Pankov A: Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations. Kluwer Academic, Dordrecht; 1990. English Edition
Fournier G, Szulkin A, Willem M:Semilinear elliptic equations in with almost periodic or unbounded forcing term. SIAM J. Math. Anal. 1996, 27: 1653-1660. 10.1137/S0036141094278699
N’Guérékata GM, Pankov A: Almost periodic elliptic equations: sub- and super-solutions. Operator Theory: Advances and Applications 228. In Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Birkhäuser, Basel; 2013:275-292.
Shubin MA: Differential and pseudodifferential operators in spaces of almost periodic functions. Math. USSR Sb. 1974, 24(4):547-573. 10.1070/SM1974v024n04ABEH001923
Shubin MA: Almost periodic functions and partial differential operators. Russ. Math. Surv. 1978, 33(2):1-52. 10.1070/RM1978v033n02ABEH002303
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The authors acknowledge support from the NSF of China (no. 11071048).
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Ji, D., Zhang, C. Pseudo almost periodic weak solutions of a semilinear elliptic equation. Adv Differ Equ 2014, 46 (2014). https://doi.org/10.1186/1687-1847-2014-46
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DOI: https://doi.org/10.1186/1687-1847-2014-46