Abstract
The primary aim of this work is to introduce a new class of functions called --pseudo-almost periodic functions. Using the measure theory, we generalize in a natural way some recent works and study some properties of those --pseudo-almost periodic functions including two new composition results which play a crucial role for the existence of some --pseudo-almost periodic solutions of certain semilinear differential equations and partial differential equations. We also investigate the existence and uniqueness of the --pseudo-almost periodic solutions for some models of Lasota-Wazewska equation with measure -pseudo-almost periodic coefficient and mixed delays.
1. Introduction
Most of the natural phenomena we consider as periodic are in fact almost periodic; in other words, they are periodic up to epsilon. The concept of almost periodic functions was introduced in the literature in the mid-1920s by the Danish mathematician Harald Bohr [1]. It was later generalized in various directions by many researchers [2–12]. As we all know, many phenomena in nature have oscillatory character, and their mathematical models have led to the introduction of certain classes of functions to describe them. Such a class form pseudo-almost periodic functions which is a natural generalization of the concept of almost periodicity (in Bohr’s sense). In this work, we introduce the notion of measure -pseudo-almost periodic functions (or --pseudo-almost periodic functions) with values in a complex Banach space and enlighten their applications throughout the study of a biological model. This work generalizes the concept of -pseudo-almost periodic functions introduced by Blot et al. [4] which already generalizes the class of weighted pseudo-almost periodic functions of Diagana [6, 13]. Here, we investigate many interesting properties of this new class of functions and present new and more general results based on measure theory that extend the existing ones.
The concept of -periodicity was introduced by Alvarez et al. [2] motivated by the qualitative properties of solutions to the Mathieu linear second-order differential equation
arising in seasonally forced population dynamics. Further on, Alvarez et al. proposed a new concept of -pseudoperiodicity and proved the existence of positive -pseudo-periodic solutions to the Lasota-Wazewska equation with -pseudoperiodic coefficients
This equation describes the survival of red blood cells in the blood of an animal. The works of Khalladi et al. [14] have shown that -pseudoperiodic functions can be also solutions time varying impulsive differential equations and linear delayed equations.
The concept of pseudo-almost periodicity was introduced in the literature in the early nineties by Zhang [11, 12, 15], as a natural generalization of the classical almost periodicity in the sense of Bohr. Then, Diagana [6, 13] introduced the concept of weighted pseudo-almost periodicity which generalizes the latter, and the author gave some properties of the space of weighted pseudo-almost periodic functions such as the completeness and a composition theorem. The concept of weighted pseudo-almost periodic functions became an interesting field of dynamical systems that attracted many authors. A few years later, Blot et al. [4] came up with a new concept of weighted pseudo-almost periodic functions under the light of measure theory. Giving a positive measure on , they defined the concept of -pseudo-almost periodic functions as follows: it is said that a function is pseudo-almost periodic if
where is almost periodic and is -ergodic in the sense that
Here, the classical theory of weighted pseudo-almost periodicity became a particular case of Blot et al. approach. Indeed, one can observe that a weighted pseudo-almost periodic function of weight is -pseudo-almost periodic where the measure is absolutely continuous with respect to the Lebesgue measure, and its Radon-Nikodym derivative is :
In their work, Blot et al. have investigated many important results on the theory of -pseudo-almost periodicity; they studied the completeness and provided a composition theorem on the functional space of -pseudo-almost periodic functions. They also gave some applications for evolution equations which include reaction-diffusion systems and partial differential equations.
In this work, we introduce a new class of --ergodic components, and we investigate many important results on the new theory of --pseudo-almost periodic functions. We study the completeness and the composition theorem on the functional space of --pseudo-almost periodic functions.
The organization of this work is as follows: in the next section, we recall the basic definitions and properties of -pseudo-almost periodic functions. In Section 3, we give the new concept of --pseudo-almost periodicity and study the convolution product on the spaces of -bounded functions, --ergodic functions, and --pseudo-almost periodic functions. In Section 4, we introduce the concept of -type compactness, and then we study a composition theorem which plays a crucial role to study the existence of --pseudo-almost periodic solution for a perturbed semilinear system. In Section 5, we propose a more realistic Lasota-Wazewska model than the existing ones due to -periodicity, and then we study the existence and uniqueness of --pseudo-almost periodic solutions for the model, using the completeness and composition results.
2. Terminology and Definitions
In this section, we review a few notations, definitions, and lemmas which will be utilized throughout this paper.
Let and be complex Banach spaces. Throughout this work, and (respectively, and denote the Banach spaces consisting of all continuous functions and all bounded continuous functions from to (respectively, from to ) equipped with the supremum norm
Let us first recall the notion of -periodicity.
Definition 1. (see [2]). Let and . A function is said to be -periodic if
In this case, is called a -period of the function .
We denote by the vector space of all -periodic functions from to . One can note that the space contains the spaces of periodic, antiperiodic, and Bloch periodic functions among others (respectively, taking , , and ) (see [16] for more details).
Proposition 2. (see [2]). Let . Then, if and only if Using the principal branch of the complex Logarithm, is defined as and we will use the notation .
Now, we recall some properties of almost periodic and -pseudo-almost periodic functions.
Definition 3. A function is called (Bohr) almost periodic if for each , there exists , such that for all , there exists with The vector space consisting of all (Bohr) almost periodic functions is denoted by .
It is well known that a continuous function is almost periodic if and only if the set
is relatively compact in , where the function is defined by
Such number in (10) is called -translation number of , and we denote by the set of all -translation numbers of . This set has the following property:
Given any , (1)If , then .
This concept has been extended by Khalladi et al. [17] as follows:
Definition 4. (see [17]). A function is called -almost periodic if and only if the function
belongs to .
The vector space consisting of all -almost periodic functions is denoted by .
Unless specified otherwise, in the remainder of the paper, we will always assume that and . Furthermore, the principal branches are always used for taking powers of complex numbers.
In the following, we will keep the notation: .
Remark 5. When , .
Remark 6. One can note that in our paper, contrary to the paper [17], is not only positive but it belongs also to the set of all -translation number of . This condition yields .
In order to conserve the periodic structure of -periodic type functions, we need to use an -norm which can be defined as
-norms were introduced in the literature by Alvarez et al. taking the supremum norm not on the whole but on the principal -period interval of the -periodic considered function in order to handle the -periodicity properties of (see in [2, 3, 16] for more details). We have the following completeness result.
Remark 7. We say that is -bounded when .
Proposition 8. (see [14]). is a Banach space.
Proposition 9. (see [18]). is translation invariant and closed under the multiplication with complex scalars.
Now, we recall the concept of -pseudo-almost periodic functions introduced by Blot et al. [4].
We denote by the Lebesgue -field of and by the set of all positive measures on satisfying and , for all
Definition 10. (see [4]). Let . A function is said to be -ergodic if We denote the space of all such functions by .
Definition 11. (see [4]). Let . A function is said to be -pseudo-almost periodic if is written in the form
where and .
We denote the space of all such functions by .
Proposition 12. (see [4]). Let . Then, is a Banach space.
In the last section of this work, the following result will be required.
Lemma 13. If , then .
Proof. Since , then they have following decompositions and where and . Then, we have
First, we show that the product . If we take , we have
It can be easily seen that since is bounded, then there exists such that
Thus, it comes the following
Then, . Now, one can note that . Since and , then .
Now, for one has that
And consequently, since , we have
The proof is complete.
We end this section recalling the following lemma due to Schwartz [19].
Lemma 14. If , then for each compact set in and all , there exists such that for any , one has
3. Measure -Pseudo-Almost Periodic Functions
In this section, we introduce the new concepts of --ergodic functions and the --pseudo-almost periodic functions. The notion of --pseudo-almost periodic functions is a generalization of -pseudo-almost periodic functions introduced by Blot et al. [4] which now becomes the particular case of our work. It is also a generalization of the concept of weighted pseudo-almost periodicity given by Diagana [6, 13] and consequently, this work generalizes that of Zhang [11, 12, 15] on the classical pseudo-almost periodicity.
Here, we introduce the space , where (resp., ) denotes the Banach space consisting of all -bounded continuous functions from to (resp., from to X) equipped with the -norm defined in Section 2.
Remark 15. One can note that in the case , Moreover, we have the following result.
Theorem 16. Let and . Then, is and only if .
Remark 17. It can be easily seen that when , the space does not contain the space of constant functions.
We begin this part with the following helpful convolution theorem for -bounded functions.
Let be the space of bounded linear maps from the complex Banach space into itself. We denote the Lebesgue space with respect to the Lebesgue measure on .
Remark 18. One can note that if , then but .
Theorem 19. Let and , and then the convolution product of defined by is -bounded.
Proof. Let . In order to state that , we consider the function defined by Observing that it is clear that is -bounded on . We deduce that is continuous by using the uniform continuity of on all compact subsets of . Consequently, , which means that , and from the following inequality: We deduce that uniformly on . Therefore, .
3.1. On --Ergodicity
First, we introduce the new concept of --ergodic functions.
Definition 20. Let . A function is said to be --ergodic if We denote the space of all such functions by .
We establish a completeness result for --ergodic functions.
Proposition 21. Let . Then, is a Banach space.
Proof. It is clear that is a vector subspace of . We show that is closed in . Let be a Cauchy sequence converging to uniformly in . From , it follows that for sufficiently large. We have that Then, Since , we deduce that
Now, we characterize the space of --ergodic functions with the following theorem.
Theorem 22. Let and be an interval such that .
Let , and then following assertions are equivalent:
(1),(2),(3)For any , .
Proof. First, note that since , then . Setting for any and taking such that and , it comes that
Since , we deduce that assertions and are equivalent.
Now, we set the following:
If 3 holds, from the following equality
we deduce for large enough that
Then, from previous inequality, we have that for all ,
and consequently, assertion 2 holds.
The last implication is deduced using the following inequality
Assume that assumption 2 holds, we obtain assumption 3 when making .
The proof is complete.
Now, we intend to prove that is translation invariant.
For and , we denote the positive measure on defined by
We need to formulate the following hypothesis for (see [4] for more details), and we also recall two important lemmas.
(H1). For all , there exists and a bounded interval such that
Lemma 23. (see [4]). Let . Then, the measures and are equivalent for all
Lemma 24. (see [4]). implies that for all ,
In the following, we denote by the collection of measures in satisfying .
We can prove the following result.
Theorem 25. Let . Then, is translation invariant.
Proof. Let and . We recall that according to relation (39)
We recall that according to Lemma 23 it follows that and are equivalent.
It comes that
where is a constant insuring the equivalence between and .
Since , the proof is complete.
We end this section by giving a convolution theorem for --ergodic functions.
Theorem 26. Let . If and , then the convolution product of defined by is --ergodic.
Proof. Let . By Theorem 19, . Now, we set for any .
One can note that there exists such that for all . In the other hand, one has
where .
Applying Fubini’s Theorem, it comes that
Invoking Theorem 25, we have that
Since
by the Lebesgue Dominated Convergence Theorem, we conclude that
Now, we are ready to define measure -pseudo-almost periodic functions.
3.2. Measure -Pseudo-Almost Periodic Function
In this subsection, we introduce the new class of measure -pseudo-almost periodic function, and we study some properties of such functions. Let us define this new notion.
Definition 27. Let . A function is said to be measure -pseudo-almost periodic (or --pseudo-almost periodic) if can be written in the form
where and .
We denote the space of all such functions by .
We will say that is the -almost periodic part of and the --pseudoergodic perturbation of .
We have the following space inclusions:
Remark 28. Observe that is a proper subspace of since the function but since
The following theorem gives a characterization of the measure -pseudo-almost periodic functions.
Theorem 29. Let . Then, if and only if
Proof. Obviously, if with then .
Conversely, let . Then, such that . Therefore, taking , it comes that .
In view of Definition 27, for any we say that is the -factorization of .
We give the first basic result.
Proposition 30. Let . Then, is a vector space.
Proof. Obvious.
Now, we intend to show that . In order to prove Proposition 30, we will need following lemma.
Lemma 31. Assume , write where and is fixed. Then, there exists such that
We have following result.
Proposition 32. Let and be such that where is its -almost periodic component, and then we have Therefore,
Proof. Suppose that (56) is not true, then there exists , such that
Let be as in Lemma 31 and write
For with , we let
where is as in Lemma 31. It is clear that
Thus, we obtain
since for each Using inequality (57), we have
any .
This and inequality (61) together give
This is a contradiction since and establishes our claim (56).
We can now establish the uniqueness of the decomposition in Definition 27.
Theorem 33. Let . Then, the decomposition (49) is unique.
Proof. Let .
Assume that admits both decomposition and , then
Since and , in view of Proposition 32, we deduce that and consequently, which proves the uniqueness of the decomposition.
From above, it is clear that
Furthermore, we have following results.
Theorem 34. The space is a translation invariant.
Proof. This is a direct consequence of Proposition 9, Theorem 25, and Theorem 33.
Theorem 35. The space is a translation invariant -subalgebra of . Furthermore,
Proof. We show that is a closed subspace of .
Let be Cauchy. By proposition 32, the sequence is Cauchy too and so is . Since and are closed in , there are and such that and as Set , then and as .
The rest of the proof is clear.
Now, we show the completeness of with the following result.
Theorem 36. Let . Then, is a Banach space.
Proof. Let be a Cauchy sequence in . Then, given such that for all ,
Invoking Theorem 29, such that and for all and since , we have
Consequently, let be a Cauchy sequence in . Using the completeness of , we know that such that as .
We take . We claim that as . And it can be easily seen that
which completes the proof.
We end this subsection giving a general convolution theorem for our new class of functions.
Theorem 37. Let . If and , then the convolution product of defined by is --pseudo-almost periodic.
Proof. Let and .
First, note that using Theorem 19, . Furthermore, according to Theorem 29, there exists a such that , for any . It comes that
Invoking successively [[4], Theorem 22, pp. 511], and our Theorem 29, we have that (71) is --pseudo-almost periodic. The proof is complete.
Example 38. The unique solution of the heat equation with the initial condition is given by
If and , then by Theorem 26, the solution
4. Jointly Continuous Case
This section is devoted to the study of a composition result well suited for the introduced -periodicity concept. The main results of this section are Theorems 45 and 51. But first, let us define some new notions.
First of all, reader should be aware that the already known concept of compactness for subsets seems to be irrelevant when it comes to deal with -periodicity where since periodic type functions are not bounded on (i.e., -bounded on ) but -bounded on .
With the following definition, we propose a new concept of compactness for subset well suited for -periodic calculus.
Definition 39. Let be a nonempty set. We say that is an -type compact subset of if and only if following assumptions are satisfied: (1) is compact(2)Every admits following decomposition where One can note that a compact subset of is in fact an -type subset of since if is a compact subset of , we have the following equality: for any .
4.1. On -Almost Periodic Functions Depending on a Parameter
Throughout this section, we introduce a new concept of -almost periodic function in the jointly continuous case. Then, we study some properties and establish some results as the continuity of Nemytskii’s superposition operator.
Definition 40 (see [20]). A function is called (Yoshizawa) almost periodic in uniformly in if for each and any compact , and there exists , such that for all , there exists with
for all and all .
The collection of such functions will be denoted by .
Such number in (76) is called -translation number of , and we denote by the set of all -translation numbers of for . This set has the following properties:
For a fixed compact set , (1)An -translation number is also an -translation number if , and hence we have the inclusion (2)If , then (3)If , then
In what follows, we assume that .
In [17], authors have introduced two concepts of -almost periodic functions in the case of jointly continuous functions, but in this paper, it uses a novel approach.
Definition 41. A function is called -almost periodic in uniformly in if for each and any -type compact subset of , there exists , such that for all , there exists with
for all and all , where .
The space of all such functions will be denoted by .
In the following, we use the notation: .
Remark 42. When , .
Proposition 43. is a Banach space.
We need to develop some tools in order to propose a composition theorem for measure -pseudo-almost periodic functions.
We give the following results.
Lemma 44. If is an -type compact subset of , then is a relatively compact subset of .
Proof. Let
Since is compact, it is also precompact; thus, there exists a finite -type subset of (i.e., a finite subset of ) such that
Since, is relatively compact in for all then is also relatively compact and consequently, there exists a finite subset of such that
If , there exists and such that , and there exists such that and consequently
Now, using the previous inclusion, there exists and such that
It comes that
This proves that or in other words, is precompact, and since is complete, we obtain that is relatively compact.
Now, for a given function , we define Nemytskii’s superposition operator such that
The first main result of this section is the following theorem.
Theorem 45. Let . Then, the Nemytskii superposition operator is continuous from into .
Proof. Let be an -type compact subset of let and
We set . According to Lemma 44, the closure is compact.
Since , there exists such that for , there exists satisfying
Since is compact, then is uniformly continuous on it and consequently, there exists such that, for all and for all And this implies that
if satisfies .
We set and for all .
Then, we have , and using (83) and (85), we obtain, for all And so, by taking the supremum on the , we obtain
This proves that the restriction of to is continuous for all -type compact subset of . And since and are Banach Spaces, this proves the continuity of on .
The following proposition is a generalization of Cieutat, Fatajou, and N’Gu´er´ekata´s Theorem in [21] which becomes the particular case of our result.
Proposition 46. Let be a continuous function. Then, if and only if the following conditions hold: (1)For all , (2) is uniformly continuous on each -type compact set in with respect to the second variable, namely, for each -type compact set in , for all , there exists such that for all , one has
4.2. Measure -Pseudo-Almost Periodic Functions Depending on a Parameter and Composition Principle
In this section, we extend our new concept of measure -pseudo-almost periodic functions to that of measure -pseudo-almost periodic functions depending on a parameter.
Here, we propose a concept of --ergodicity for the jointly continuous functions case.
Definition 47. Let . A function is said to be --ergodic in uniformly with respect to if the two following conditions are true: (1)For all , (2) is uniformly continuous on each -type compact set in with respect to the second variable, namely, for each -type compact set in , for all , there exists such that for all , one hasWe denote the space of all such functions by .
Remark 48. When , we write instead of .
Now, we are able to introduce the new concept of measure -pseudo-almost periodic functions depending on a parameter.
Definition 49. Let . A function is said to be --pseudo-almost periodic in uniformly with respect to if is written in the form where and .
denotes the set of such that functions.
The following inclusion hold
Remark 50. When , we write instead of .
As in the previous section, we propose a characterization result which holds for -almost periodic, --ergodic and --pseudo-almost periodic functions in uniformly with respect to .
Theorem 51. Let .
Then, (resp., or if and only if
with and (resp., or ).
Proof. The proof is similar to the one of Theorem 29.
Using Proposition 46 and Definition 47, one can obtain following.
Theorem 52. Let and be --ergodic in uniformly with respect to . Then, (1)For all , (2) is uniformly continuous on each -type compact set in with respect to the second variable
We are now in a position to give a composition theorem for measure -pseudo-almost periodic functions. We have the following theorem.
Theorem 53. Let , and . Assume that the following hypothesis holds.
For all bounded subset of , is -bounded on (i.e., is bounded on ).
Then, .
Proof. First note that the function is continuous and by Hypothesis (10), it is -bounded. Since by Theorem 5, there exists Now since , and using Theorem 4.10 in [4], we deduce that the function In conclusion, invoking again Theorem 51, we showed that The proof is complete.
The following theorem will be very useful in the sequel.
Corollary 54. Let , , , and .
Assume that for all bounded subset of , is -bounded on , then if ,
Proof. This is direct consequence of Theorem 5 with .
5. Application: Measure -Pseudo-Almost Periodic Solutions to a Lasota-Wazewska Model
First, Wazewska-Czyzewska and Lasota [22] proposed in 1976 the delay logistic equations with one constant concentrated delay
in order to describe the survival of red blood cells in an animal. Here, denotes the number of red blood cells at time , is the probability of death of a red blood cell, and are positive constants related to the production of red blood cells per unit time, and is the time required to produce a red blood cell. Few years later, Gopalsamy and Trofimchuk [23] obtained that the Lasota-Wazewska model with one discrete delay
has a globally attractive almost periodic solution under some additional assumptions.
Recently, Cherif and Miraoui [24] investigate the existence, the uniqueness, the global attractivity, and the exponential stability of the measure pseudo-almost periodic solutions for the following Lasota-Wazewska model with measure pseudo-almost periodic coefficients and mixed delays
The aim here is to study the existence and uniqueness of a generalized Lasota-Wazewska model with --pseudo-almost periodic coefficients and with mixed delay which is in the form:
where stands for the number of red blood cells at time , and is the average part of red blood cells population being destroyed in time . For all and , and are the connected with demand for oxygen at time , and characterize excitability of haematopoietic system at time , is the probability kernel of the distributed delays, and is the time required to produce a red blood cell. One can note that we consider in our new approach the --pseudo-almost periodic for the connected with demand for oxygen at time and the -pseudo-almost periodic for the excitability of haematopoietic system at time since it is more realistic for the description of the physical and biological phenomena.
The method consists to reduce the existence of the unique solution for the Lasota-Wazewska model (99) to the search for the existence of the unique fixed point of an appropriate operator on the Banach space .
Notice that we restrict ourselves to -valued functions since only nonnegative solutions are biologically meaningful.
5.1. Existence and Uniqueness of --Pseudo-Almost Periodic Solution to the Model
In what follows, given a -bounded continuous function defined on , and are defined by
Remark 55. If , we use the notations
First, we give sufficient conditions which ensures existence and uniqueness of --pseudo-almost periodic solution of (99).
, for all
, for all
For all , are continuous, integrable, and
where is a sufficiently non negative small constant.
Lemma 56. Let and . If , then .
Proof. According to Theorem 29, there exists a unique such that for all . Using Lemma 13, it is clear that . Then, Invoking Theorem 29, we complete the proof.
Now, we can establish following lemma.
Lemma 57. Let . For all , the function belongs to for all .
Proof. First, we can say that the function for all and . Then, according to Proposition 30Furthermore, by Lemma 5, for all . Now, using the fact that the function is Lipschitzian and bounded, and is also bounded then invoking the Corollary 54, it is clear that for all
By using condition and Theorem 37, we can deduce the following Lemma.
Lemma 58. Suppose that and hold, if , then the function defined by for all
Theorem 59. Suppose that and are satisfied. Then, the nonlinear operator defined for each by maps into itself.
Proof. Using Lemmas 13, 5, 5, and 5 and Corollary 54, then the function defined by
is measure -pseudo-almost periodic.
Consequently, we can write where and . It follows that
Let us show that
We recall that by applying condition to the model (99), is almost periodic (i.e., -almost periodic with constant . Now, in the view of the almost periodicity of the function and the -almost periodicity of the function , there exists a number such that in any interval one finds a number , such that
It comes that
So, there exists such that
This proves that Now, let us show that We have that
By the Lebesgue Dominated Convergence Theorem and , we obtain that
Then, and consequently,
Theorem 60. Assume that and – hold, then the Lasota-Wazewska model with mixed delays (99) possesses a unique measure -pseudo-almost periodic solution , and we have in the region where
Proof. First, we proves that the operator is a mapping from to . We set In fact, we have In the other hand, if we set then, we have for which implies that the operator is a mapping from to itself. To end the proof, it suffices to prove that is a contraction mapping. Let . Then, Obviously, for , Then it comes that which implies (invoking ) that the mapping is a contraction mapping of . Consequently, possesses a unique fixed point . Hence, is the unique measure -pseudo-almost periodic solution of Equation (99) in .
5.2. Example
In order to illustrate some features of our theoretical study, we will apply our main results to a special system and demonstrate the efficiencies of our criteria.
We consider the following Lasota-Wazewska model with mixed delays
where , , and ,
, . Then,
If the Radon-Nikodym derivative of the measure is with respect to the Lebesgue measure on (i.e. ), then , since
Hence, conditions – and are satisfied then according to the Theorem 60, the Lasota-Wazewska model with a mixed delays (14) has a unique --pseudo-almost periodic solution in the region where
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest and no data used.