# Measureω,c-Pseudo-Almost Periodic Functions and Lasota-Wazewska Model with Ergodic and Unbounded Oscillating Oxygen Demand

Authors: James Larrouy; Gaston M. N’Guérékata
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

The primary aim of this work is to introduce a new class of functions calledμ-ω,c-pseudo-almost periodic functions. Using the measure theory, we generalize in a natural way some recent works and study some properties of those μ-ω,c-pseudo-almost periodic functions including two new composition results which play a crucial role for the existence of some μ-ω,c-pseudo-almost periodic solutions of certain semilinear differential equations and partial differential equations. We also investigate the existence and uniqueness of the μ-ω,c-pseudo-almost periodic solutions for some models of Lasota-Wazewska equation with measure ω,c-pseudo-almost periodic coefficient and mixed delays.


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## Body

## 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 ω,c-pseudo-almost periodic functions (or μ-ω,c-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 μ-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 ofω,c-periodicity was introduced by Alvarez et al. [2] motivated by the qualitative properties of solutions to the Mathieu linear second-order differential equation
(1)y′′t+a−2qcos2tyt=0,arising in seasonally forced population dynamics. Further on, Alvarez et al. proposed a new concept ofω,c-pseudoperiodicity and proved the existence of positive ω,c-pseudo-periodic solutions to the Lasota-Wazewska equation with ω,c-pseudoperiodic coefficients
(2)y′t=−δyt+hte−atyt−τ,t≥0.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 ω,c-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 μ on ℝ, they defined the concept of μ-pseudo-almost periodic functions as follows: it is said that a function f is μ pseudo-almost periodic if
(3)f=g+φ,whereg is almost periodic and φ is μ-ergodic in the sense that
(4)limr⟶∞1μ−r,r∫−r,rφtdμt=0.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 weightρ is μ-pseudo-almost periodic where the measure μ is absolutely continuous with respect to the Lebesgue measure, and its Radon-Nikodym derivative is ρ:
(5)dμt=ρtdt.In their work, Blot et al. have investigated many important results on the theory ofμ-pseudo-almost periodicity; they studied the completeness and provided a composition theorem on the functional space of μ-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 ofμ-ω,c-ergodic components, and we investigate many important results on the new theory of μ-ω,c-pseudo-almost periodic functions. We study the completeness and the composition theorem on the functional space of μ-ω,c-pseudo-almost periodic functions.The organization of this work is as follows: in the next section, we recall the basic definitions and properties ofμ-pseudo-almost periodic functions. In Section 3, we give the new concept of μ-ω,c-pseudo-almost periodicity and study the convolution product on the spaces of c-bounded functions, μ-ω,c-ergodic functions, and μ-ω,c-pseudo-almost periodic functions. In Section 4, we introduce the concept of ω,c-type compactness, and then we study a composition theorem which plays a crucial role to study the existence of μ-ω,c-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 ω,c-periodicity, and then we study the existence and uniqueness of μ-ω,c-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.LetX,· and Y,· be complex Banach spaces. Throughout this work, Cℝ,X and BCℝ,X (respectively, Cℝ×Y,X and BCℝ×Y,X denote the Banach spaces consisting of all continuous functions and all bounded continuous functions from ℝ to X (respectively, from ℝ×Y to X) equipped with the supremum norm
(6)f∞=supt∈ℝf.Let us first recall the notion ofω,c-periodicity.Definition 1. (see [2]).
Letω>0 and c∈ℂ\0. A function f∈Cℝ,X is said to be ω,c-periodic if
(7)ft+ω=cft,foreacht∈ℝ.In this case,ω is called a c-period of the function f.We denote byPω,cℝ,X the vector space of all ω,c-periodic functions from ℝ to X. One can note that the space Pω,cℝ,X contains the spaces of periodic, antiperiodic, and Bloch periodic functions among others (respectively, taking c=1, c=−1, and c=eikt) (see [16] for more details).Proposition 2. (see [2]).
Letf∈Cℝ,X. Then, f∈Pω,cℝ,X if and only if
(8)ft=ct/ωut,ut∈Pω,1ℝ,X.
Using the principal branch of the complex Logarithm,ct/ω is defined as
(9)ct/ω≔exptωLogc=c∧t,and we will use the notation c∧t≔c∧t=ct/ω.Now, we recall some properties of almost periodic andω,c-pseudo-almost periodic functions.Definition 3.
A functionf∈Cℝ,X is called (Bohr) almost periodic if for each ε>0, there exists l>0, such that for all α∈ℝ, there exists τ∈α,α+l with
(10)supt∈ℝft+τ−ft<ε.
The vector space consisting of all (Bohr) almost periodic functions is denoted byAPX.It is well known that a continuous functionf:ℝ⟶X is almost periodic if and only if the set
(11)fτ:τ∈ℝis relatively compact inBCℝ,X, where the function fτ is defined by
(12)fτt=ft+τ,t∈ℝ.Such numberτ in (10) is called ε-translation number of ft, and we denote by Tεf the set of all ε-translation numbers of f. This set has the following property:Given anyf∈APX,
(1)
Ifτ∈Tεf, then −τ∈Tεf.This concept has been extended by Khalladi et al. [17] as follows:Definition 4. (see [17]).
A functionf∈Cℝ,X is called ω,c-almost periodic if and only if the function
(13)fω,ct≔c−t/ωft,t∈ℝ
belongs toAPX.
The vector space consisting of allω,c-almost periodic functions is denoted by APω,cX.Unless specified otherwise, in the remainder of the paper, we will always assume thatc∈ℂ\0 and ω∈ℝ+⋆∩Tεf. Furthermore, the principal branches are always used for taking powers of complex numbers.In the following, we will keep the notation:fω,ct≔c−t/wft.Remark 5.
Whenc=1, APω,cX=APX.Remark 6.
One can note that in our paper, contrary to the paper [17], ω is not only positive but it belongs also to the set of all ε-translation number of f. This condition yields APω,1X≔APωX=APX.
In order to conserve the periodic structure ofω,c-periodic type functions, we need to use an ω,c-norm which can be defined as
(14)fω,c≔supt∈ℝc∧−tft.
ω,c-norms were introduced in the literature by Alvarez et al. taking the supremum norm not on the whole ℝ but on the principal c-period interval 0,ω of the ω,c-periodic considered function in order to handle the ω,c-periodicity properties of f (see in [2, 3, 16] for more details). We have the following completeness result.Remark 7.
We say thatf is c-bounded when fω,c<∞.Proposition 8. (see [14]).
APω,cX,·ω,c is a Banach space.Proposition 9. (see [18]).
APω,cX is translation invariant and closed under the multiplication with complex scalars.Now, we recall the concept ofμ-pseudo-almost periodic functions introduced by Blot et al. [4].We denote byB the Lebesgue σ-field of ℝ and by M the set of all positive measures μ on B satisfying μℝ=+∞ and μa,b<+∞, for all a,b∈ℝa≤b.Definition 10. (see [4]).
Letμ∈M. A function f∈BCℝ,X is said to be μ-ergodic if
(15)limr⟶∞1μ−r,r∫−r,rftdμt=0.
We denote the space of all such functions byEℝ,X,μ.Definition 11. (see [4]).
Letμ∈M. A function f∈Cℝ,X is said to be μ-pseudo-almost periodic if f is written in the form
(16)f=g+φ,
whereg∈APX and φ∈Eℝ,X,μ.
We denote the space of all such functions byPAPℝ,X,μ.Proposition 12. (see [4]).
Letμ∈M. Then, Eℝ,X,μ,·∞ is a Banach space.In the last section of this work, the following result will be required.Lemma 13.
Iff,g∈PAPℝ,ℂ,μ, then fg∈PAPℝ,ℂ,μ.Proof.
Sincef,g∈PAPℝ,ℂ,μ, then they have following decompositions f=f1+f2 and g=g1+g2 where f1,g1∈APℂ and f2,g2∈Eℝ,ℂ,μ. Then, we have
(17)fg=f1g1+f1g2+g1f2+f2g2.
First, we show that the productf1g1∈APℂ. If we take f1=g1, we have
(18)f12t+τ−f12t=f1t+τ+f1t×f1t+τ−f1t.
It can be easily seen that sincef1 is bounded, then there exists M∈ℝ+ such that
(19)f1≤M.
Thus, it comes the following(20)f12t+τ−f12t≤2Mε≤ε′.
Then,f12∈APℂ. Now, one can note that f1g1=1/4f1+g12−f1−g12. Since f1+g12∈APℂ and f1−g12∈APℂ, then f1g1∈APℂ.
Now, forf1g2+g1f2+f2g2 one has that
(21)1μ−r,r∫−rrf1g2+g1f2+f2g2tdμt≤1μ−r,r∫−rrf1∞g2t+g1∞f2t+f2∞g2tdμt.
And consequently, sincef2,g2∈Eℝ,ℂ,μ, we have
(22)limr⟶+∞1μ−r,r∫−rrf1g2+g1f2+f2g2tdμt=0.
The proof is complete.We end this section recalling the following lemma due to Schwartz [19].Lemma 14.
IfΨ∈CX,Y, then for each compact set K in X and all ε>0, there exists δ>0 such that for any x1,x2∈X, one has
(23)x1∈Kandx1−x2≤δ⇒Ψx1−Ψx2≤ε.


## 3. Measureω,c-Pseudo-Almost Periodic Functions

In this section, we introduce the new concepts ofμ-ω,c-ergodic functions and the μ-ω,c-pseudo-almost periodic functions. The notion of μ-ω,c-pseudo-almost periodic functions is a generalization of μ-pseudo-almost periodic functions introduced by Blot et al. [4] which now becomes the particular case c=1 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 spaceBCℝ,X,cresp.,BCℝ×Y,X,c, where BCℝ,X,c (resp., BCℝ×Y,X,c) denotes the Banach space consisting of all c-bounded continuous functions from ℝ to X (resp., from ℝ×Y to X) equipped with the ω,c-norm ·ω,c defined in Section 2.Remark 15.
One can note that in the casec=1,
(24)BCℝ,X,c,·ω,c=BCℝ,X,·∞,BCℝ×Y,X,c,·ω,c=BCℝ×Y,X,·∞.
Moreover, we have the following result.Theorem 16.
Letc,c′∈ℂ\0 and f∈BCℝ,X,c. Then, f∈BCℝ,X,c′ is and only if c=c′.Remark 17.
It can be easily seen that whenc≠1, the space BCℝ,X,c does not contain the space of constant functions.We begin this part with the following helpful convolution theorem forc-bounded functions.LetLX be the space of bounded linear maps from the complex Banach space X into itself. We denote L1ℝ,LX the Lebesgue space with respect to the Lebesgue measure on ℝ.Remark 18.
One can note that ifh∈L1ℝ,LX, then ϕ·≔c∧·h·∈L1ℝ,LX but ϕ~≔c∧−·ϕ·∈L1ℝ,LX.Theorem 19.
Letf∈BCℝ,X,c and ϕ~≔c∧−·ϕ·∈L1ℝ,LX, and then the convolution product of fåϕ defined by
(25)f⋆ϕt=∫−∞+∞ϕsft−sds,fort∈ℝ
isc-bounded.Proof.
Letf∈BCℝ,X,c. In order to state that f⋆ϕ∈BCℝ,X,c, we consider the function ρn:ℝ⟶Xn∈N,n≥1 defined by
(26)ρnt=∫−nnϕsft−sds.
Observing that(27)c∧−tρnt=∫−nnc∧−sϕsc∧−t+sft−sds.
it is clear thatρn is c-bounded on ℝ. We deduce that ρnt is continuous by using the uniform continuity of f on all compact subsets of ℝ. Consequently, ρnt∈BCℝ,X,c, which means that c∧−tρnt∈BCℝ,X, and from the following inequality:
(28)c∧−tf⋆ϕt−ρnt≤fω,c∫−∞−nc∧−sϕsds+∫n+∞c∧−sϕsds.
We deduce thatlimn⟶+∞c∧−tρnt=c∧−tf⋆ϕt uniformly on ℝ. Therefore, f⋆ϕ∈BCℝ,X,c.


### 3.1. Onμ-ω,c-Ergodicity

First, we introduce the new concept ofμ-ω,c-ergodic functions.Definition 20.
Letμ∈M. A function f∈BCℝ,X,c is said to be μ-ω,c-ergodic if
(29)limr⟶∞1μ−r,r∫−r,rfω,ctdμt=0.
We denote the space of all such functions byEω,cℝ,X,μ.We establish a completeness result forμ-ω,c-ergodic functions.Proposition 21.
Letμ∈M. Then, Eω,cℝ,X,μ,·ω,c is a Banach space.Proof.
It is clear thatEω,cℝ,X,μ is a vector subspace of BCℝ,X,c. We show that Eω,cℝ,X,μ is closed in BCℝ,X,c. Let fn⊂Eω,cℝ,X,μ be a Cauchy sequence converging to f uniformly in ℝ. From μℝ=+∞, it follows that μ−r,r>0 for r sufficiently large. We have that
(30)1μ−r,r∫−r,rfω,ctdμt≤f−fnω,c+1μ−r,r∫−r,rfnω,ctdμt.
Then,(31)limsupr⟶∞1μ−r,r∫−r,rfω,ctdμt≤f−fnω,c,foralln∈ℕ.
Sincelimn⟶+∞f−fnω,c=0, we deduce that
(32)limr⟶∞1μ−r,r∫−r,rfω,ctdμt=0.Now, we characterize the space ofμ-ω,c-ergodic functions with the following theorem.Theorem 22.
Letμ∈M and I be an interval such that μI<∞.
Letf∈BCℝ,X,c, and then following assertions are equivalent:
(1)
f∈Eω,cℝ,X,μ,(2)
limr⟶∞1/μ−r,r\I∫−r,r\Ifω,ctdμt=0,(3)
For anyε>0, μt∈−r,r\I:fω,ct>ε/μ−r,r\I=0.Proof.
First, note that sincef∈BCℝ,X,c, then Λ≔∫Ifω,cdμt<∞. Setting Ir=−r,r for any r>0 and taking r such that I⊂Ir and μIr\I>0, it comes that
(33)1μIr\I∫Ir\Ifω,ctdμt=μIrμIr−μI1μIr∫Irfω,ctdμt−ΛμIr.
Sinceμℝ=∞, we deduce that assertions 3.6 and 3.6 are equivalent.
Now, we set the following:(34)Γrε≔t∈Ir\I:fω,ct>εandΛrε≔t∈Ir\I:fω,ct≤ε.
If 3 holds, from the following equality(35)∫Irfω,ctdμt=∫Γrεfω,ctdμt+∫Λrεfω,ctdμt,
we deduce forr large enough that
(36)1μIr\I∫Ir\Ifω,ctdμt≤fω,cμΓrεμIr\I+ε.
Then, from previous inequality, we have that for allε>0,
(37)limr⟶∞1μIr\I∫Ir\Ifω,ctdμt≤ε,
and consequently, assertion 2 holds.
The last implication is deduced using the following inequality(38)1μIr\I∫Ir\Ifω,ctdμt≥1μIr\I∫Γrεfω,ctdμt≥εμΓrεμIr\I.
Assume that assumption 2 holds, we obtain assumption 3 when makingr⟶+∞.
The proof is complete.Now, we intend to prove thatEω,cℝ,X,μ is translation invariant.Forμ∈M and τ∈ℝ, we denote μτ the positive measure on ℝ,B defined by
(39)μτA=μa+τ:a∈A,forA∈B.We need to formulate the following hypothesis forμ∈M (see [4] for more details), and we also recall two important lemmas.(H1). For all∈ℝ, there exists β>0 and a bounded interval I such that
(40)μa+τ:a∈A≤βμA,whenA∈BsatisfiesA∩I=∅.Lemma 23. (see [4]).
Letμ∈MB. Then, the measures μ and μτ are equivalent for all τ∈ℝ.Lemma 24. (see [4]).
H1 implies that for all α>0,
(41)limsupr⟶+∞μ−r−α,r+αμ−r,r<+∞.In the following, we denote byMB the collection of measures in M satisfying H1.We can prove the following result.Theorem 25.
Letμ∈MB. Then, Eω,cℝ,X,μ is translation invariant.Proof.
Letf∈Eω,cℝ,X,μ and τ∈ℝ. We recall that according to relation (39)
(42)μτA≔μa+τ:a∈A,forA∈B.
We recall that according to Lemma23 it follows that μ and μτ are equivalent.
It comes that(43)1μ−r,r∫−r,rfω,ct+τdμt=μ−r−τ,r+τμ−r,r·1μ−r−τ,r+τ∫−r,rfω,ct+τdμt=μ−r−τ,r+τμ−r,r·1μ−r−τ,r+τ∫−r−τ,r+τfω,ctdμ−τt≤μ−r−τ,r+τμ−r,r·βμ−r−τ,r+τ∫−r−τ,r+τfω,ctdμt,
whereβ>0 is a constant insuring the equivalence between μ and μτ.
Sincef∈Eω,cℝ,X,μ, the proof is complete.We end this section by giving a convolution theorem forμ-ω,c-ergodic functions.Theorem 26.
Letμ∈MB. If f∈Eω,cℝ,X,μ and ϕ~·≔c∧−·ϕ·∈L1ℝ,LX, then the convolution product of fåϕ defined by
(44)f⋆ϕt=∫−∞+∞ϕsft−sds,fort∈ℝ
isμ-ω,c-ergodic.Proof.
Letf∈Eω,cℝ,X,μ. By Theorem 19, f⋆ϕ∈BCℝ,X,c. Now, we set Ir=−r,r for any r≥0.
One can note that there existsγ≥0 such that μIr>0 for all r≥γ. In the other hand, one has
(45)1μIr∫Irf⋆ϕω,ctdμt≤1μIr∫Ir∫−∞+∞ϕ~sc∧−t+sft−sdsdμt,
whereϕ~≔ϕω,c∈L1ℝ,LX.
Applying Fubini’s Theorem, it comes that(46)1μIr∫Irf⋆ϕω,ctdμt≤∫−∞+∞ϕ~sμIr∫Irfω,ct−sdμtds.
Invoking Theorem25, we have that limr⟶+∞1/μIr∫Irfω,ct−sdμt=0,foralls∈ℝ.
Since(47)0≤ϕ~sμIr∫Irfω,ct−sdμt≤ϕ~sfω,c
by the Lebesgue Dominated Convergence Theorem, we conclude that(48)limr⟶+∞1μIr∫Irf⋆ϕω,ctdμt=0.Now, we are ready to define measureω,c-pseudo-almost periodic functions.


### 3.2. Measureω,c-Pseudo-Almost Periodic Function

In this subsection, we introduce the new class of measureω,c-pseudo-almost periodic function, and we study some properties of such functions. Let us define this new notion.Definition 27.
Letμ∈M. A function f∈Cℝ,X is said to be measure ω,c-pseudo-almost periodic (or μ-ω,c-pseudo-almost periodic) if f can be written in the form
(49)f=g+φ,
whereg∈APω,cX and φ∈Eω,cℝ,X,μ.
We denote the space of all such functions byPAPω,cℝ,X,μ.
We will say thatg is the ω,c-almost periodic part of f and φ the μ-ω,c-pseudoergodic perturbation of f.We have the following space inclusions:(50)Pω,cℝ,X⊂APω,cX⊂PAPω,cℝ,X,μ⊂BCℝ,X,c.Remark 28.
Observe thatAPω,cX is a proper subspace of PAPω,cℝ,X,μ since the function
(51)ϕt=0.2tsin2πt+sin25t+e−ttcos2t∈PAP1,0.2ℝ,X,μ,
butϕ∉AP1,0.2X since t↦sin2πt+sin25t+e−ttcos2t∉APX.The following theorem gives a characterization of the measureω,c-pseudo-almost periodic functions.Theorem 29.
Letf∈Cℝ,X. Then, f∈PAPω,cℝ,X,μ if and only if
(52)ft≡c∧tut,withc∧t≔ct/ωandu∈PAPℝ,X,μ.Proof.
Obviously, ifft=c∧tut with u∈PAPℝ,X,μ then f∈PAPω,cℝ,X,μ.
Conversely, letf∈PAPω,cℝ,X,μ. Then, ∃g,φ∈APω,cX×Eω,cℝ,X,μ such that f=g+φ. Therefore, taking ut≔c∧−tft, it comes that u∈PAPℝ,X,μ.In view of Definition27, for any f∈PAPω,cℝ,X,μ, we say that c∧tut is the c-factorization of f.We give the first basic result.Proposition 30.
Letμ∈M. Then, PAPω,cℝ,X,μ is a vector space.Proof.
Obvious.Now, we intend to show thatPAPω,cℝ,X,μ=APω,cX⊕Eω,cℝ,X,μ. In order to prove Proposition 30, we will need following lemma.Lemma 31.
Assumef∈APω,cX, write
(53)Bε≔τ∈ℝ:fω,ct0+τ−fω,ct0<ε,
whereε>0 and t0∈ℝ is fixed. Then, there exists s1,s2,⋯,sm∈ℝ such that
(54)⋃i=1msi+Bε=ℝ.We have following result.Proposition 32.
Letμ∈MB and f∈PAPω,cℝ,X,μ be such that
(55)f=g+φ,
whereg is its ω,c-almost periodic component, and then we have
(56)gℝ⊂fℝ¯.
Therefore,fω,c≥gω,c≥inft∈ℝgω,ct≥inft∈ℝfω,ct.Proof.
Suppose that (56) is not true, then there exists t0∈ℝ, ε>0 such that
(57)gω,ct0−fω,ct≥2ε,t∈ℝ.
Lets1,s2,⋯,sm be as in Lemma 31 and write
(58)τi=si−t0,i=1,2,⋯,m,η=max1≤i≤mτi.
Forr∈ℝ with r>η, we let
(59)Bε,ri≔−r+η−τi,r−η−τi∩t0+Bε,i=1,2,⋯,m,
whereBε is as in Lemma 31. It is clear that
(60)⋃i=1mτi+Bε,ri=−r+η,r−η.
Thus, we obtain(61)2r−η=μ−r+η,r−η≤∑i=1mμτi+Bε,ri=∑i=1mμBε,ri≤m·max1≤i≤mμBε,ri≤m·μ−r,r∩t0+Bε,
since for eachi=1,2,⋯,m,(62)Bε,ri⊂−r,r∩t0+Bε.
Using inequality (57), we have
(63)ϕω,ct=fω,ct−gω,ct≥gω,ct0−fω,ct−gω,ct−gω,ct0>ε,
anyt∈t0+Bε.
This and inequality (61) together give
(64)1μ−r,r∫−r,rϕω,ctdμt≥r−ηmrε⟶εm,asr⟶∞.
This is a contradiction sinceh∈Eω,cℝ,X,μ and establishes our claim (56).We can now establish the uniqueness of the decomposition in Definition27.Theorem 33.
Letμ∈MB. Then, the decomposition (49) is unique.Proof.
Letf∈PAPω,cℝ,X,μ.
Assume thatf admits both decomposition f=g1+φ1 and f=g2+φ2, then 0=g1−g2+φ1−φ2.
Sinceg1−g2∈APω,cX and φ1−φ2∈PAPω,cℝ,X,μ, in view of Proposition 32, we deduce that g1−g2=0 and consequently, φ1=φ2 which proves the uniqueness of the decomposition.From above, it is clear that(65)APω,cX∩Eω,cℝ,X,μ=0.Furthermore, we have following results.Theorem 34.
The spacePAPω,cℝ,X,μ is a translation invariant.Proof.
This is a direct consequence of Proposition9, Theorem 25, and Theorem 33.Theorem 35.
The spacePAPω,cℝ,X,μ is a translation invariant C∗-subalgebra of BCℝ,X,c. Furthermore,
(66)PAPω,cℝ,X,μEω,cℝ,X,μ≅APω,cX.Proof.
We show thatPAPω,cℝ,X,μ is a closed subspace of BCℝ,X,c.
Letfn⊂PAPω,cℝ,X,μ be Cauchy. By proposition 32, the sequence gn⊂APω,cX is Cauchy too and so is φn⊂Eω,cℝ,X,μ. Since APω,cX and Eω,cℝ,X,μ are closed in BCℝ,X,c, there are g∈APω,cX and φ∈Eω,cℝ,X,μ such that gnω,c−gω,c⟶0 and φnω,c−φω,c⟶0 as n⟶∞. Set f=g+φ, then f∈PAPω,cℝ,X,μ and fnω,c−fω,c⟶0 as n⟶0.
The rest of the proof is clear.Now, we show the completeness ofPAPω,cℝ,X,μ with the following result.Theorem 36.
Letμ∈MB. Then, PAPω,cℝ,X,μ,·ω,c is a Banach space.Proof.
Letfn be a Cauchy sequence in PAPω,cℝ,X,μ. Then, given ε>0∃n0∈ℕ such that for all p,q≥n0,
(67)fpt−fqtω,c<ε.
Invoking Theorem29, ∃up,uq∈PAPℝ,X,μ2 such that fpt=c∧tupt and fqt=c∧tuqt for all t∈ℝ and since p,q≥n0, we have
(68)upt−uqt≤fpt−fqtω,c<ε.
Consequently, letun be a Cauchy sequence in PAPℝ,X,μ. Using the completeness of PAPℝ,X,μ, we know that ∃u∈PAPℝ,X,μ such that un−u⟶0 as n⟶∞.
We takeft≔c∧tut. We claim that un−u⟶0 as n⟶∞. And it can be easily seen that
(69)fn−fω,c=supt∈ℝun−u⟶0n⟶∞,
which completes the proof.We end this subsection giving a general convolution theorem for our new class of functions.Theorem 37.
Letμ∈MB. If f∈PAPω,cℝ,X,μ and ϕ~≔c∧−·ϕ·∈L1ℝ,LX, then the convolution product of f⋆ϕ defined by
(70)f⋆ϕt=∫−∞+∞ϕsft−sds,fort∈ℝ
isμ-ω,c-pseudo-almost periodic.Proof.
Letf∈PAPω,cℝ,X,μ and ϕ~≔c∧−·ϕ·∈L1ℝ,LX.
First, note that using Theorem19, f⋆ϕ∈BCℝ,X,c. Furthermore, according to Theorem 29, there exists a u∈PAPℝ,X,μ such that ft≔c∧tut, for any t∈ℝ. It comes that
(71)f⋆ϕt=∫−∞+∞ϕsft−sds=c∧t∫−∞+∞c∧−sϕsc∧−t+sft−sds=c∧t∫−∞+∞ϕ~sut−sds.
Invoking successively [[4], Theorem 22, pp. 511], and our Theorem 29, we have that (71) is μ-ω,c-pseudo-almost periodic. The proof is complete.Example 38.
The unique solution of the heat equation(72)utx,t=uxxx,t,x∈ℝ,t≥0,
with the initial conditionux,0=fx is given by
(73)ux,t=12πt∫−∞+∞e−x−s2/4tfsds,x∈ℝ,t≥0.Ifc∧−·e−·2/4t0∈L1ℝ and f∈PAPω,cℝ,ℝ,μ, then by Theorem 26, the solution
(74)x↦ux,t0,x∈ℝ∈PAPω,cℝ,ℝ,μ.


## 3.1. Onμ-ω,c-Ergodicity

First, we introduce the new concept ofμ-ω,c-ergodic functions.Definition 20.
Letμ∈M. A function f∈BCℝ,X,c is said to be μ-ω,c-ergodic if
(29)limr⟶∞1μ−r,r∫−r,rfω,ctdμt=0.
We denote the space of all such functions byEω,cℝ,X,μ.We establish a completeness result forμ-ω,c-ergodic functions.Proposition 21.
Letμ∈M. Then, Eω,cℝ,X,μ,·ω,c is a Banach space.Proof.
It is clear thatEω,cℝ,X,μ is a vector subspace of BCℝ,X,c. We show that Eω,cℝ,X,μ is closed in BCℝ,X,c. Let fn⊂Eω,cℝ,X,μ be a Cauchy sequence converging to f uniformly in ℝ. From μℝ=+∞, it follows that μ−r,r>0 for r sufficiently large. We have that
(30)1μ−r,r∫−r,rfω,ctdμt≤f−fnω,c+1μ−r,r∫−r,rfnω,ctdμt.
Then,(31)limsupr⟶∞1μ−r,r∫−r,rfω,ctdμt≤f−fnω,c,foralln∈ℕ.
Sincelimn⟶+∞f−fnω,c=0, we deduce that
(32)limr⟶∞1μ−r,r∫−r,rfω,ctdμt=0.Now, we characterize the space ofμ-ω,c-ergodic functions with the following theorem.Theorem 22.
Letμ∈M and I be an interval such that μI<∞.
Letf∈BCℝ,X,c, and then following assertions are equivalent:
(1)
f∈Eω,cℝ,X,μ,(2)
limr⟶∞1/μ−r,r\I∫−r,r\Ifω,ctdμt=0,(3)
For anyε>0, μt∈−r,r\I:fω,ct>ε/μ−r,r\I=0.Proof.
First, note that sincef∈BCℝ,X,c, then Λ≔∫Ifω,cdμt<∞. Setting Ir=−r,r for any r>0 and taking r such that I⊂Ir and μIr\I>0, it comes that
(33)1μIr\I∫Ir\Ifω,ctdμt=μIrμIr−μI1μIr∫Irfω,ctdμt−ΛμIr.
Sinceμℝ=∞, we deduce that assertions 3.6 and 3.6 are equivalent.
Now, we set the following:(34)Γrε≔t∈Ir\I:fω,ct>εandΛrε≔t∈Ir\I:fω,ct≤ε.
If 3 holds, from the following equality(35)∫Irfω,ctdμt=∫Γrεfω,ctdμt+∫Λrεfω,ctdμt,
we deduce forr large enough that
(36)1μIr\I∫Ir\Ifω,ctdμt≤fω,cμΓrεμIr\I+ε.
Then, from previous inequality, we have that for allε>0,
(37)limr⟶∞1μIr\I∫Ir\Ifω,ctdμt≤ε,
and consequently, assertion 2 holds.
The last implication is deduced using the following inequality(38)1μIr\I∫Ir\Ifω,ctdμt≥1μIr\I∫Γrεfω,ctdμt≥εμΓrεμIr\I.
Assume that assumption 2 holds, we obtain assumption 3 when makingr⟶+∞.
The proof is complete.Now, we intend to prove thatEω,cℝ,X,μ is translation invariant.Forμ∈M and τ∈ℝ, we denote μτ the positive measure on ℝ,B defined by
(39)μτA=μa+τ:a∈A,forA∈B.We need to formulate the following hypothesis forμ∈M (see [4] for more details), and we also recall two important lemmas.(H1). For all∈ℝ, there exists β>0 and a bounded interval I such that
(40)μa+τ:a∈A≤βμA,whenA∈BsatisfiesA∩I=∅.Lemma 23. (see [4]).
Letμ∈MB. Then, the measures μ and μτ are equivalent for all τ∈ℝ.Lemma 24. (see [4]).
H1 implies that for all α>0,
(41)limsupr⟶+∞μ−r−α,r+αμ−r,r<+∞.In the following, we denote byMB the collection of measures in M satisfying H1.We can prove the following result.Theorem 25.
Letμ∈MB. Then, Eω,cℝ,X,μ is translation invariant.Proof.
Letf∈Eω,cℝ,X,μ and τ∈ℝ. We recall that according to relation (39)
(42)μτA≔μa+τ:a∈A,forA∈B.
We recall that according to Lemma23 it follows that μ and μτ are equivalent.
It comes that(43)1μ−r,r∫−r,rfω,ct+τdμt=μ−r−τ,r+τμ−r,r·1μ−r−τ,r+τ∫−r,rfω,ct+τdμt=μ−r−τ,r+τμ−r,r·1μ−r−τ,r+τ∫−r−τ,r+τfω,ctdμ−τt≤μ−r−τ,r+τμ−r,r·βμ−r−τ,r+τ∫−r−τ,r+τfω,ctdμt,
whereβ>0 is a constant insuring the equivalence between μ and μτ.
Sincef∈Eω,cℝ,X,μ, the proof is complete.We end this section by giving a convolution theorem forμ-ω,c-ergodic functions.Theorem 26.
Letμ∈MB. If f∈Eω,cℝ,X,μ and ϕ~·≔c∧−·ϕ·∈L1ℝ,LX, then the convolution product of fåϕ defined by
(44)f⋆ϕt=∫−∞+∞ϕsft−sds,fort∈ℝ
isμ-ω,c-ergodic.Proof.
Letf∈Eω,cℝ,X,μ. By Theorem 19, f⋆ϕ∈BCℝ,X,c. Now, we set Ir=−r,r for any r≥0.
One can note that there existsγ≥0 such that μIr>0 for all r≥γ. In the other hand, one has
(45)1μIr∫Irf⋆ϕω,ctdμt≤1μIr∫Ir∫−∞+∞ϕ~sc∧−t+sft−sdsdμt,
whereϕ~≔ϕω,c∈L1ℝ,LX.
Applying Fubini’s Theorem, it comes that(46)1μIr∫Irf⋆ϕω,ctdμt≤∫−∞+∞ϕ~sμIr∫Irfω,ct−sdμtds.
Invoking Theorem25, we have that limr⟶+∞1/μIr∫Irfω,ct−sdμt=0,foralls∈ℝ.
Since(47)0≤ϕ~sμIr∫Irfω,ct−sdμt≤ϕ~sfω,c
by the Lebesgue Dominated Convergence Theorem, we conclude that(48)limr⟶+∞1μIr∫Irf⋆ϕω,ctdμt=0.Now, we are ready to define measureω,c-pseudo-almost periodic functions.


## 3.2. Measureω,c-Pseudo-Almost Periodic Function

In this subsection, we introduce the new class of measureω,c-pseudo-almost periodic function, and we study some properties of such functions. Let us define this new notion.Definition 27.
Letμ∈M. A function f∈Cℝ,X is said to be measure ω,c-pseudo-almost periodic (or μ-ω,c-pseudo-almost periodic) if f can be written in the form
(49)f=g+φ,
whereg∈APω,cX and φ∈Eω,cℝ,X,μ.
We denote the space of all such functions byPAPω,cℝ,X,μ.
We will say thatg is the ω,c-almost periodic part of f and φ the μ-ω,c-pseudoergodic perturbation of f.We have the following space inclusions:(50)Pω,cℝ,X⊂APω,cX⊂PAPω,cℝ,X,μ⊂BCℝ,X,c.Remark 28.
Observe thatAPω,cX is a proper subspace of PAPω,cℝ,X,μ since the function
(51)ϕt=0.2tsin2πt+sin25t+e−ttcos2t∈PAP1,0.2ℝ,X,μ,
butϕ∉AP1,0.2X since t↦sin2πt+sin25t+e−ttcos2t∉APX.The following theorem gives a characterization of the measureω,c-pseudo-almost periodic functions.Theorem 29.
Letf∈Cℝ,X. Then, f∈PAPω,cℝ,X,μ if and only if
(52)ft≡c∧tut,withc∧t≔ct/ωandu∈PAPℝ,X,μ.Proof.
Obviously, ifft=c∧tut with u∈PAPℝ,X,μ then f∈PAPω,cℝ,X,μ.
Conversely, letf∈PAPω,cℝ,X,μ. Then, ∃g,φ∈APω,cX×Eω,cℝ,X,μ such that f=g+φ. Therefore, taking ut≔c∧−tft, it comes that u∈PAPℝ,X,μ.In view of Definition27, for any f∈PAPω,cℝ,X,μ, we say that c∧tut is the c-factorization of f.We give the first basic result.Proposition 30.
Letμ∈M. Then, PAPω,cℝ,X,μ is a vector space.Proof.
Obvious.Now, we intend to show thatPAPω,cℝ,X,μ=APω,cX⊕Eω,cℝ,X,μ. In order to prove Proposition 30, we will need following lemma.Lemma 31.
Assumef∈APω,cX, write
(53)Bε≔τ∈ℝ:fω,ct0+τ−fω,ct0<ε,
whereε>0 and t0∈ℝ is fixed. Then, there exists s1,s2,⋯,sm∈ℝ such that
(54)⋃i=1msi+Bε=ℝ.We have following result.Proposition 32.
Letμ∈MB and f∈PAPω,cℝ,X,μ be such that
(55)f=g+φ,
whereg is its ω,c-almost periodic component, and then we have
(56)gℝ⊂fℝ¯.
Therefore,fω,c≥gω,c≥inft∈ℝgω,ct≥inft∈ℝfω,ct.Proof.
Suppose that (56) is not true, then there exists t0∈ℝ, ε>0 such that
(57)gω,ct0−fω,ct≥2ε,t∈ℝ.
Lets1,s2,⋯,sm be as in Lemma 31 and write
(58)τi=si−t0,i=1,2,⋯,m,η=max1≤i≤mτi.
Forr∈ℝ with r>η, we let
(59)Bε,ri≔−r+η−τi,r−η−τi∩t0+Bε,i=1,2,⋯,m,
whereBε is as in Lemma 31. It is clear that
(60)⋃i=1mτi+Bε,ri=−r+η,r−η.
Thus, we obtain(61)2r−η=μ−r+η,r−η≤∑i=1mμτi+Bε,ri=∑i=1mμBε,ri≤m·max1≤i≤mμBε,ri≤m·μ−r,r∩t0+Bε,
since for eachi=1,2,⋯,m,(62)Bε,ri⊂−r,r∩t0+Bε.
Using inequality (57), we have
(63)ϕω,ct=fω,ct−gω,ct≥gω,ct0−fω,ct−gω,ct−gω,ct0>ε,
anyt∈t0+Bε.
This and inequality (61) together give
(64)1μ−r,r∫−r,rϕω,ctdμt≥r−ηmrε⟶εm,asr⟶∞.
This is a contradiction sinceh∈Eω,cℝ,X,μ and establishes our claim (56).We can now establish the uniqueness of the decomposition in Definition27.Theorem 33.
Letμ∈MB. Then, the decomposition (49) is unique.Proof.
Letf∈PAPω,cℝ,X,μ.
Assume thatf admits both decomposition f=g1+φ1 and f=g2+φ2, then 0=g1−g2+φ1−φ2.
Sinceg1−g2∈APω,cX and φ1−φ2∈PAPω,cℝ,X,μ, in view of Proposition 32, we deduce that g1−g2=0 and consequently, φ1=φ2 which proves the uniqueness of the decomposition.From above, it is clear that(65)APω,cX∩Eω,cℝ,X,μ=0.Furthermore, we have following results.Theorem 34.
The spacePAPω,cℝ,X,μ is a translation invariant.Proof.
This is a direct consequence of Proposition9, Theorem 25, and Theorem 33.Theorem 35.
The spacePAPω,cℝ,X,μ is a translation invariant C∗-subalgebra of BCℝ,X,c. Furthermore,
(66)PAPω,cℝ,X,μEω,cℝ,X,μ≅APω,cX.Proof.
We show thatPAPω,cℝ,X,μ is a closed subspace of BCℝ,X,c.
Letfn⊂PAPω,cℝ,X,μ be Cauchy. By proposition 32, the sequence gn⊂APω,cX is Cauchy too and so is φn⊂Eω,cℝ,X,μ. Since APω,cX and Eω,cℝ,X,μ are closed in BCℝ,X,c, there are g∈APω,cX and φ∈Eω,cℝ,X,μ such that gnω,c−gω,c⟶0 and φnω,c−φω,c⟶0 as n⟶∞. Set f=g+φ, then f∈PAPω,cℝ,X,μ and fnω,c−fω,c⟶0 as n⟶0.
The rest of the proof is clear.Now, we show the completeness ofPAPω,cℝ,X,μ with the following result.Theorem 36.
Letμ∈MB. Then, PAPω,cℝ,X,μ,·ω,c is a Banach space.Proof.
Letfn be a Cauchy sequence in PAPω,cℝ,X,μ. Then, given ε>0∃n0∈ℕ such that for all p,q≥n0,
(67)fpt−fqtω,c<ε.
Invoking Theorem29, ∃up,uq∈PAPℝ,X,μ2 such that fpt=c∧tupt and fqt=c∧tuqt for all t∈ℝ and since p,q≥n0, we have
(68)upt−uqt≤fpt−fqtω,c<ε.
Consequently, letun be a Cauchy sequence in PAPℝ,X,μ. Using the completeness of PAPℝ,X,μ, we know that ∃u∈PAPℝ,X,μ such that un−u⟶0 as n⟶∞.
We takeft≔c∧tut. We claim that un−u⟶0 as n⟶∞. And it can be easily seen that
(69)fn−fω,c=supt∈ℝun−u⟶0n⟶∞,
which completes the proof.We end this subsection giving a general convolution theorem for our new class of functions.Theorem 37.
Letμ∈MB. If f∈PAPω,cℝ,X,μ and ϕ~≔c∧−·ϕ·∈L1ℝ,LX, then the convolution product of f⋆ϕ defined by
(70)f⋆ϕt=∫−∞+∞ϕsft−sds,fort∈ℝ
isμ-ω,c-pseudo-almost periodic.Proof.
Letf∈PAPω,cℝ,X,μ and ϕ~≔c∧−·ϕ·∈L1ℝ,LX.
First, note that using Theorem19, f⋆ϕ∈BCℝ,X,c. Furthermore, according to Theorem 29, there exists a u∈PAPℝ,X,μ such that ft≔c∧tut, for any t∈ℝ. It comes that
(71)f⋆ϕt=∫−∞+∞ϕsft−sds=c∧t∫−∞+∞c∧−sϕsc∧−t+sft−sds=c∧t∫−∞+∞ϕ~sut−sds.
Invoking successively [[4], Theorem 22, pp. 511], and our Theorem 29, we have that (71) is μ-ω,c-pseudo-almost periodic. The proof is complete.Example 38.
The unique solution of the heat equation(72)utx,t=uxxx,t,x∈ℝ,t≥0,
with the initial conditionux,0=fx is given by
(73)ux,t=12πt∫−∞+∞e−x−s2/4tfsds,x∈ℝ,t≥0.Ifc∧−·e−·2/4t0∈L1ℝ and f∈PAPω,cℝ,ℝ,μ, then by Theorem 26, the solution
(74)x↦ux,t0,x∈ℝ∈PAPω,cℝ,ℝ,μ.


## 4. Jointly Continuous Case

This section is devoted to the study of a composition result well suited for the introducedω,c-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 withω,c-periodicity where c≠1 since ω,c periodic type functions are not bounded on ℝ (i.e., 1-bounded on ℝ) but c-bounded on ℝ.With the following definition, we propose a new concept of compactness for subset well suited forω,c-periodic calculus.Definition 39.
LetK be a nonempty set. We say that K is an ω,c-type compact subset of Y if and only if following assumptions are satisfied:
(1)
K is compact(2)
Everyk∈K admits following decomposition k≔c∧−·y∈K where y∈Y
One can note that a compact subset ofAPX is in fact an ω,c-type subset of APω,cX since if K is a compact subset of APX, we have the following equality:
(75)Σ≔ut:t∈ℝ,u∈K=c∧tutω,c:t∈ℝ,c∧tut∈APω,cX,u∈K,
for anyc∈ℂ−0.


### 4.1. Onω,c-Almost Periodic Functions Depending on a Parameter

Throughout this section, we introduce a new concept ofω,c-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 functionF∈Cℝ×Y,X is called (Yoshizawa) almost periodic in t∈ℝ uniformly in y∈Y if for each ε>0 and any compact K⊂Y, and there exists L>0, such that for all β∈ℝ, there exists τ∈β,β+L with
(76)supt∈ℝsupy∈KFt+τ,y−Ft,y<ε,
for allt∈ℝ and all y∈K.
The collection of such functions will be denoted byAPY,X.Such numberτ in (76) is called ε-translation number of Ft,y, and we denote by TεF,K the set of all ε-translation numbers of F for y∈K. This set has the following properties:For a fixed compact setK,
(1)
Anε-translation number is also an ε′-translation number if ε′>ε, and hence we have the inclusion TεF,K⊂Tε′F,K(2)
Ifτ∈TεF,K, then −τ∈TεF,K(3)
Ifτ1,τ2∈Tε1F,K×Tε2F,K, then τ1+τ2∈Tε1+ε2F,KIn what follows, we assume thatω∈ℝ+⋆∩TεF,K.In [17], authors have introduced two concepts of ω,c-almost periodic functions in the case of jointly continuous functions, but in this paper, it uses a novel approach.Definition 41.
A functionF∈Cℝ×Y,X is called ω,c-almost periodic in t∈ℝ uniformly in y∈Y if for each ε>0 and any ω,c-type compact subset K of Y, there exists L>0, such that for all β∈ℝ, there exists τ∈β,β+L with
(77)supt∈ℝsupy∈Kc∧−t−τFt+τ,y−c∧−tFt,y<ε,⇔supt∈ℝsupy∈KFω,ct+τ,y−Fω,ct,y<ε,
for allt∈ℝ and all y∈K, where Fω,ct,·≔c−t/ωFt,·.
The space of all such functions will be denoted byAPω,cY,X.In the following, we use the notation:Fω,ct,·≔c−t/wFt,·.Remark 42.
Whenc=1, APω,cY,X=APY,X.Proposition 43.
APω,cY,X,·ω,c is a Banach space.We need to develop some tools in order to propose a composition theorem for measureω,c-pseudo-almost periodic functions.We give the following results.Lemma 44.
IfK is an ω,c-type compact subset of APω,cY, then
(78)Σ≔uω,ct:t∈ℝ,u∈APω,cY,uω,c∈K
is a relatively compact subset ofY.Proof.
Letε>0.
SinceK is compact, it is also precompact; thus, there exists u1ω,c,⋯,umω,c a finite ω,c-type subset of APω,cY (i.e., a finite subset of APY) such that
(79)K⊂⋃1≤i≤mz∈APω,cY:z−uiω,c≤ε2.
Since,uiω,cℝ is relatively compact in X for all i=1,⋯,m, then ⋃1≤i≤muiω,cℝ is also relatively compact and consequently, there exists a finite subset of ℝt1,⋯,tk such that
(80)⋃1≤i≤muiω,cℝ⊂⋃1≤i≤m⋃1≤j≤ky∈Y:y−uiω,ctj≤ε2.
Ify∈Y, there exists z∈K and t∈ℝ such that y=zt, and there exists i∈1,⋯,m such that z−uiω,c≤ε/2 and consequently y−uiω,ct≤ε/2.
Now, using the previous inclusion, there existsp∈1,⋯,m and j∈1,⋯,k such that
(81)uiω,ct−upω,ctj≤ε2.
It comes that(82)y−uiω,ctj≤y−uiω,ct+uiω,ct−upω,ctj≤2ε2=ε.This proves thatΣ⊂⋃1≤i≤m⋃1≤j≤ky∈Y:y−uiω,ctj≤ε/2 or in other words, Σ is precompact, and since Y is complete, we obtain that Σ is relatively compact.Now, for a given functionF∈APω,cY,X, we define Nemytskii’s superposition operator NF:APω,cY⟶APω,cX such that t↦ut↦NFu≔t↦Ft,ut.The first main result of this section is the following theorem.Theorem 45.
LetF∈APω,cY,X. Then, the Nemytskii superposition operator NF is continuous from APω,cY into APω,cX.Proof.
LetK be an ω,c-type compact subset of APω,cY let u~∈K and ε>0.
We setΣ≔uω,ct:t∈ℝ,u∈APω,cY,uω,c∈K. According to Lemma 44, the closure Σ¯ is compact.
SinceF∈APω,cY,X, there exists l>0 such that for α∈ℝ, there exists −τ∈α,α+l satisfying
(83)Fω,ct+τ,y−Fω,ct,y≤ε3,forally,t∈Σ×ℝ.
SinceΣ¯×0,l is compact, then Fω,c is uniformly continuous on it and consequently, there exists δ>0 such that, for all t1,t2∈0,l and for all y1,y2∈Σ¯(84)y1−y2≤δ,t1−t2≤δ⇒Fω,ct1,y1−Fω,ct2,y2≤ε3.
And this implies that(85)y1−y2≤δ⇒Fω,ct,y1−Fω,ct,y2≤ε3,forallt∈0,l,
ifv~∈K satisfies v~−u~∞≤δ⇔c∧tv~t−c∧tu~tω,c≤δ.
We setut≔c∧tu~t and vt≔c∧tv~t for all t∈ℝ.
Then, we haveu,v∈APω,cY, and using (83) and (85), we obtain, for all t∈ℝ(86)Fω,ct,ut−Fω,ct,vt≤Fω,ct,ut−Fω,ct−τ,ut+Fω,ct−τ,ut−Fω,ct−τ,vt+Fω,ct−τ,vt−Fω,ct,vt≤3ε3=ε.
And so, by taking the supremum on thet∈ℝ, we obtain NFu−NFvω,c≤ε.
This proves that the restriction ofJf to K is continuous for all ω,c-type compact subset K of APω,cY. And since APω,cY,·ω,c and APω,cX,·ω,c are Banach Spaces, this proves the continuity of JF on APω,cY.The following proposition is a generalization of Cieutat, Fatajou, and N’Gu´er´ekata´s Theorem in [21] which becomes the particular case c=1 of our result.Proposition 46.
LetF:ℝ×Y⟶X be a continuous function. Then, F∈APω,cY,X if and only if the following conditions hold:
(1)
For ally∈Y, Fω,c·,y∈APX(2)
Fω,c is uniformly continuous on each ω,c-type compact set K in Y with respect to the second variable, namely, for each ω,c-type compact set K in Y, for all ε>0, there exists δ>0 such that for all y1,y2∈K, one has(87)y1−y2≤δ⇒supt∈ℝFω,ct,y1−Fω,ct,y2≤ε.


### 4.2. Measureω,c-Pseudo-Almost Periodic Functions Depending on a Parameter and Composition Principle

In this section, we extend our new concept of measureω,c-pseudo-almost periodic functions to that of measure ω,c-pseudo-almost periodic functions depending on a parameter.Here, we propose a concept ofμ-ω,c-ergodicity for the jointly continuous functions case.Definition 47.
Letμ∈M. A function F∈BCℝ×Y,X,c is said to be μ-ω,c-ergodic in t uniformly with respect to y∈Y if the two following conditions are true:
(1)
For ally∈Y, Fω,c·,y∈Eℝ,X,μ(2)
Fω,c is uniformly continuous on each ω,c-type compact set K in Y with respect to the second variable, namely, for each ω,c-type compact set K in Y, for all ε>0, there exists δ>0 such that for all y1,y2∈K, one has(88)y1−y2≤δ⇒supt∈ℝFω,ct,y1−Fω,ct,y2≤ε.
We denote the space of all such functions byEω,c,3ℝ×Y,X,μ.Remark 48.
Whenc=1, we write Eℝ×Y,X,μ instead of Eω,1,3ℝ×Y,X,μ.Now, we are able to introduce the new concept of measureω,c-pseudo-almost periodic functions depending on a parameter.Definition 49.
Letμ∈M. A function F∈Cℝ×Y,X is said to be μ-ω,c-pseudo-almost periodic in t uniformly with respect to y∈Y if F is written in the form
(89)F=G+Φ,
whereG∈APω,cY,X and Φ∈Eω,c,3ℝ×Y,X,μ.PAPω,cℝ×Y,X,μ denotes the set of such that functions.The following inclusion hold(90)APω,cY,X⊂PAPω,cℝ×Y,X,μ⊂BCℝ×Y,X,c.Remark 50.
Whenc=1, we write PAPℝ×Y,X,μ instead of PAPω,1ℝ×Y,X,μ.
As in the previous section, we propose a characterization result which holds forω,c-almost periodic, μ-ω,c-ergodic and μ-ω,c-pseudo-almost periodic functions in t uniformly with respect to y∈Y.Theorem 51.
LetF∈Cℝ×Y,X.
Then,F∈PAPω,cℝ×Y,X,μ (resp., APω,cY,X or Eω,c,3ℝ×Y,X,μ if and only if
(91)Ft,y≡c∧tut,y,
withc∧t≔ct/ω and u∈PAPℝ×Y,X,μ (resp., APY,X or Eℝ×Y,X,μ).Proof.
The proof is similar to the one of Theorem29.Using Proposition46 and Definition 47, one can obtain following.Theorem 52.
Letμ∈M and F:ℝ×Y⟶ be μ-ω,c-ergodic in t uniformly with respect to y∈Y. Then,
(1)
For ally∈Y, Fω,c·,y∈PAPℝ,X,μ(2)
Fω,c is uniformly continuous on each ω,c-type compact set K in Y with respect to the second variableWe are now in a position to give a composition theorem for measureω,c-pseudo-almost periodic functions. We have the following theorem.Theorem 53.
Letμ∈M, F∈PAPω,cℝ×Y,X,μ and y∈PAPω,cℝ,Y,μ. Assume that the following hypothesis holds.
For all bounded subsetΩ of Y, F is c-bounded on ℝ×Ω (i.e., Fω,c is bounded on ℝ×Ω).
Then,t↦Ft,yω,ct∈PAPω,cℝ×Y,X,μ.Proof.
First note that the functiont↦Ft,yω,ct is continuous and by Hypothesis (10), it is c-bounded. Since F∈PAPω,cℝ×Y,X,μ by Theorem 5, there exists
(92)F~∈PAPℝ×Y,X,μsuchthatFt,yt≔c∧tF~t,yt,∀t∈ℝ.
Now sinceyω,c∈PAPℝ,Y,μ, and using Theorem 4.10 in [4], we deduce that the function
(93)t↦F~t,yω,ct∈PAPℝ×Y,X,μ.
In conclusion, invoking again Theorem51, we showed that
(94)t↦Ft,yω,ct∈PAPω,cℝ×Y,X,μ.
The proof is complete.The following theorem will be very useful in the sequel.Corollary 54.
Letμ∈M, Ψ∈CX,Y, ω>0, and c∈ℂ−0.
Assume that for all bounded subsetB of X, Ψ is c-bounded on B, then if x∈PAPω,cℝ,X,μ,
(95)t↦Ψxω,ct≔c∧tΨ~xω,ct∈PAPω,cℝ,X,μ.Proof.
This is direct consequence of Theorem5 with Ft,xω,ct=Ψxω,ct.


## 4.1. Onω,c-Almost Periodic Functions Depending on a Parameter

Throughout this section, we introduce a new concept ofω,c-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 functionF∈Cℝ×Y,X is called (Yoshizawa) almost periodic in t∈ℝ uniformly in y∈Y if for each ε>0 and any compact K⊂Y, and there exists L>0, such that for all β∈ℝ, there exists τ∈β,β+L with
(76)supt∈ℝsupy∈KFt+τ,y−Ft,y<ε,
for allt∈ℝ and all y∈K.
The collection of such functions will be denoted byAPY,X.Such numberτ in (76) is called ε-translation number of Ft,y, and we denote by TεF,K the set of all ε-translation numbers of F for y∈K. This set has the following properties:For a fixed compact setK,
(1)
Anε-translation number is also an ε′-translation number if ε′>ε, and hence we have the inclusion TεF,K⊂Tε′F,K(2)
Ifτ∈TεF,K, then −τ∈TεF,K(3)
Ifτ1,τ2∈Tε1F,K×Tε2F,K, then τ1+τ2∈Tε1+ε2F,KIn what follows, we assume thatω∈ℝ+⋆∩TεF,K.In [17], authors have introduced two concepts of ω,c-almost periodic functions in the case of jointly continuous functions, but in this paper, it uses a novel approach.Definition 41.
A functionF∈Cℝ×Y,X is called ω,c-almost periodic in t∈ℝ uniformly in y∈Y if for each ε>0 and any ω,c-type compact subset K of Y, there exists L>0, such that for all β∈ℝ, there exists τ∈β,β+L with
(77)supt∈ℝsupy∈Kc∧−t−τFt+τ,y−c∧−tFt,y<ε,⇔supt∈ℝsupy∈KFω,ct+τ,y−Fω,ct,y<ε,
for allt∈ℝ and all y∈K, where Fω,ct,·≔c−t/ωFt,·.
The space of all such functions will be denoted byAPω,cY,X.In the following, we use the notation:Fω,ct,·≔c−t/wFt,·.Remark 42.
Whenc=1, APω,cY,X=APY,X.Proposition 43.
APω,cY,X,·ω,c is a Banach space.We need to develop some tools in order to propose a composition theorem for measureω,c-pseudo-almost periodic functions.We give the following results.Lemma 44.
IfK is an ω,c-type compact subset of APω,cY, then
(78)Σ≔uω,ct:t∈ℝ,u∈APω,cY,uω,c∈K
is a relatively compact subset ofY.Proof.
Letε>0.
SinceK is compact, it is also precompact; thus, there exists u1ω,c,⋯,umω,c a finite ω,c-type subset of APω,cY (i.e., a finite subset of APY) such that
(79)K⊂⋃1≤i≤mz∈APω,cY:z−uiω,c≤ε2.
Since,uiω,cℝ is relatively compact in X for all i=1,⋯,m, then ⋃1≤i≤muiω,cℝ is also relatively compact and consequently, there exists a finite subset of ℝt1,⋯,tk such that
(80)⋃1≤i≤muiω,cℝ⊂⋃1≤i≤m⋃1≤j≤ky∈Y:y−uiω,ctj≤ε2.
Ify∈Y, there exists z∈K and t∈ℝ such that y=zt, and there exists i∈1,⋯,m such that z−uiω,c≤ε/2 and consequently y−uiω,ct≤ε/2.
Now, using the previous inclusion, there existsp∈1,⋯,m and j∈1,⋯,k such that
(81)uiω,ct−upω,ctj≤ε2.
It comes that(82)y−uiω,ctj≤y−uiω,ct+uiω,ct−upω,ctj≤2ε2=ε.This proves thatΣ⊂⋃1≤i≤m⋃1≤j≤ky∈Y:y−uiω,ctj≤ε/2 or in other words, Σ is precompact, and since Y is complete, we obtain that Σ is relatively compact.Now, for a given functionF∈APω,cY,X, we define Nemytskii’s superposition operator NF:APω,cY⟶APω,cX such that t↦ut↦NFu≔t↦Ft,ut.The first main result of this section is the following theorem.Theorem 45.
LetF∈APω,cY,X. Then, the Nemytskii superposition operator NF is continuous from APω,cY into APω,cX.Proof.
LetK be an ω,c-type compact subset of APω,cY let u~∈K and ε>0.
We setΣ≔uω,ct:t∈ℝ,u∈APω,cY,uω,c∈K. According to Lemma 44, the closure Σ¯ is compact.
SinceF∈APω,cY,X, there exists l>0 such that for α∈ℝ, there exists −τ∈α,α+l satisfying
(83)Fω,ct+τ,y−Fω,ct,y≤ε3,forally,t∈Σ×ℝ.
SinceΣ¯×0,l is compact, then Fω,c is uniformly continuous on it and consequently, there exists δ>0 such that, for all t1,t2∈0,l and for all y1,y2∈Σ¯(84)y1−y2≤δ,t1−t2≤δ⇒Fω,ct1,y1−Fω,ct2,y2≤ε3.
And this implies that(85)y1−y2≤δ⇒Fω,ct,y1−Fω,ct,y2≤ε3,forallt∈0,l,
ifv~∈K satisfies v~−u~∞≤δ⇔c∧tv~t−c∧tu~tω,c≤δ.
We setut≔c∧tu~t and vt≔c∧tv~t for all t∈ℝ.
Then, we haveu,v∈APω,cY, and using (83) and (85), we obtain, for all t∈ℝ(86)Fω,ct,ut−Fω,ct,vt≤Fω,ct,ut−Fω,ct−τ,ut+Fω,ct−τ,ut−Fω,ct−τ,vt+Fω,ct−τ,vt−Fω,ct,vt≤3ε3=ε.
And so, by taking the supremum on thet∈ℝ, we obtain NFu−NFvω,c≤ε.
This proves that the restriction ofJf to K is continuous for all ω,c-type compact subset K of APω,cY. And since APω,cY,·ω,c and APω,cX,·ω,c are Banach Spaces, this proves the continuity of JF on APω,cY.The following proposition is a generalization of Cieutat, Fatajou, and N’Gu´er´ekata´s Theorem in [21] which becomes the particular case c=1 of our result.Proposition 46.
LetF:ℝ×Y⟶X be a continuous function. Then, F∈APω,cY,X if and only if the following conditions hold:
(1)
For ally∈Y, Fω,c·,y∈APX(2)
Fω,c is uniformly continuous on each ω,c-type compact set K in Y with respect to the second variable, namely, for each ω,c-type compact set K in Y, for all ε>0, there exists δ>0 such that for all y1,y2∈K, one has(87)y1−y2≤δ⇒supt∈ℝFω,ct,y1−Fω,ct,y2≤ε.


## 4.2. Measureω,c-Pseudo-Almost Periodic Functions Depending on a Parameter and Composition Principle

In this section, we extend our new concept of measureω,c-pseudo-almost periodic functions to that of measure ω,c-pseudo-almost periodic functions depending on a parameter.Here, we propose a concept ofμ-ω,c-ergodicity for the jointly continuous functions case.Definition 47.
Letμ∈M. A function F∈BCℝ×Y,X,c is said to be μ-ω,c-ergodic in t uniformly with respect to y∈Y if the two following conditions are true:
(1)
For ally∈Y, Fω,c·,y∈Eℝ,X,μ(2)
Fω,c is uniformly continuous on each ω,c-type compact set K in Y with respect to the second variable, namely, for each ω,c-type compact set K in Y, for all ε>0, there exists δ>0 such that for all y1,y2∈K, one has(88)y1−y2≤δ⇒supt∈ℝFω,ct,y1−Fω,ct,y2≤ε.
We denote the space of all such functions byEω,c,3ℝ×Y,X,μ.Remark 48.
Whenc=1, we write Eℝ×Y,X,μ instead of Eω,1,3ℝ×Y,X,μ.Now, we are able to introduce the new concept of measureω,c-pseudo-almost periodic functions depending on a parameter.Definition 49.
Letμ∈M. A function F∈Cℝ×Y,X is said to be μ-ω,c-pseudo-almost periodic in t uniformly with respect to y∈Y if F is written in the form
(89)F=G+Φ,
whereG∈APω,cY,X and Φ∈Eω,c,3ℝ×Y,X,μ.PAPω,cℝ×Y,X,μ denotes the set of such that functions.The following inclusion hold(90)APω,cY,X⊂PAPω,cℝ×Y,X,μ⊂BCℝ×Y,X,c.Remark 50.
Whenc=1, we write PAPℝ×Y,X,μ instead of PAPω,1ℝ×Y,X,μ.
As in the previous section, we propose a characterization result which holds forω,c-almost periodic, μ-ω,c-ergodic and μ-ω,c-pseudo-almost periodic functions in t uniformly with respect to y∈Y.Theorem 51.
LetF∈Cℝ×Y,X.
Then,F∈PAPω,cℝ×Y,X,μ (resp., APω,cY,X or Eω,c,3ℝ×Y,X,μ if and only if
(91)Ft,y≡c∧tut,y,
withc∧t≔ct/ω and u∈PAPℝ×Y,X,μ (resp., APY,X or Eℝ×Y,X,μ).Proof.
The proof is similar to the one of Theorem29.Using Proposition46 and Definition 47, one can obtain following.Theorem 52.
Letμ∈M and F:ℝ×Y⟶ be μ-ω,c-ergodic in t uniformly with respect to y∈Y. Then,
(1)
For ally∈Y, Fω,c·,y∈PAPℝ,X,μ(2)
Fω,c is uniformly continuous on each ω,c-type compact set K in Y with respect to the second variableWe are now in a position to give a composition theorem for measureω,c-pseudo-almost periodic functions. We have the following theorem.Theorem 53.
Letμ∈M, F∈PAPω,cℝ×Y,X,μ and y∈PAPω,cℝ,Y,μ. Assume that the following hypothesis holds.
For all bounded subsetΩ of Y, F is c-bounded on ℝ×Ω (i.e., Fω,c is bounded on ℝ×Ω).
Then,t↦Ft,yω,ct∈PAPω,cℝ×Y,X,μ.Proof.
First note that the functiont↦Ft,yω,ct is continuous and by Hypothesis (10), it is c-bounded. Since F∈PAPω,cℝ×Y,X,μ by Theorem 5, there exists
(92)F~∈PAPℝ×Y,X,μsuchthatFt,yt≔c∧tF~t,yt,∀t∈ℝ.
Now sinceyω,c∈PAPℝ,Y,μ, and using Theorem 4.10 in [4], we deduce that the function
(93)t↦F~t,yω,ct∈PAPℝ×Y,X,μ.
In conclusion, invoking again Theorem51, we showed that
(94)t↦Ft,yω,ct∈PAPω,cℝ×Y,X,μ.
The proof is complete.The following theorem will be very useful in the sequel.Corollary 54.
Letμ∈M, Ψ∈CX,Y, ω>0, and c∈ℂ−0.
Assume that for all bounded subsetB of X, Ψ is c-bounded on B, then if x∈PAPω,cℝ,X,μ,
(95)t↦Ψxω,ct≔c∧tΨ~xω,ct∈PAPω,cℝ,X,μ.Proof.
This is direct consequence of Theorem5 with Ft,xω,ct=Ψxω,ct.


## 5. Application: Measureω,c-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
(96)N′t=−μNt+pe−rNt−τ,in order to describe the survival of red blood cells in an animal. Here,Nt denotes the number of red blood cells at time t, μ is the probability of death of a red blood cell, p and r are positive constants related to the production of red blood cells per unit time, and τ 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
(97)x′t=−αtxt+βte−νxt−τhas 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
(98)y′t=−αtyt+∑j=1majte−ωjt∫−∞tKjt−sysds+∑i=1nbite−βit∑j=1pyt−τij,t∈ℝ.The aim here is to study the existence and uniqueness of a generalized Lasota-Wazewska model withμ-ω,c-pseudo-almost periodic coefficients and with mixed delay which is in the form:
(99)y′t=−αtyt+∑j=1majte−c∧−tωjt∫−∞tKjt−sysds+∑i=1nbite−c∧−tβit∑j=1pyt−τij,t∈ℝ,whereyt stands for the number of red blood cells at time t, and αt is the average part of red blood cells population being destroyed in time t. For all 1≤j≤m and 1≤i≤n, ajt and bit are the connected with demand for oxygen at time t, ωjt and βit characterize excitability of haematopoietic system at time t, Kj is the probability kernel of the distributed delays, and τij is the time required to produce a red blood cell. One can note that we consider in our new approach the μ-ω,c-pseudo-almost periodic for the connected with demand for oxygen at time t and the μ-pseudo-almost periodic for the excitability of haematopoietic system at time t 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 PAPω,cℝ,ℝ+,μ.Notice that we restrict ourselves toℝ+-valued functions since only nonnegative solutions are biologically meaningful.


### 5.1. Existence and Uniqueness ofμ-ω,c-Pseudo-Almost Periodic Solution to the Model

In what follows, given ac-bounded continuous function f defined on ℝ, fω,c¯ and fω,c¯ are defined by
(100)fω,c¯=supt∈ℝfω,ct=supt∈ℝc∧−tft,andfω,c¯=inft∈ℝfω,ct=inft∈ℝc∧−tft.Remark 55.
Ifc=1, we use the notations
(101)f¯≔fω,1¯=supt∈ℝftandf¯≔fω,1¯=inft∈ℝft.First, we give sufficient conditions which ensures existence and uniqueness ofμ-ω,c-pseudo-almost periodic solution of (99).C10<c≤1C2α∈APℝ+C3βi,ωj∈PAPℝ,ℝ+,μ, for all i,j∈1,n×1,mC4aj,bi∈PAPω,cℝ,ℝ+,μ, for all i,j∈1,n×1,mC5∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯/α¯<1C6 For all 1≤j≤m, Kjω,c:0,+∞⟶ℝ+ are continuous, integrable, and
(102)∫0∞Kjω,cudu=1,and∫0∞Kjω,cueλudu<+∞,whereλ is a sufficiently non negative small constant.Lemma 56.
Letf∈PAPω,cℝ,ℝ+,μ and g∈PAPℝ,ℝ+,μ. If c>0, then fg∈PAPω,cℝ,ℝ+,μ.Proof.
According to Theorem29, there exists a unique u∈PAPℝ,ℝ+,μ such that
(103)ft≔c∧tut,
for allt∈ℝ. Using Lemma 13, it is clear that u×g∈PAPℝ,ℝ+,μ. Then,
(104)fgt≔c∧tutgt,whereug∈PAPℝ,ℝ+,μ.
Invoking Theorem29, we complete the proof.Now, we can establish following lemma.Lemma 57.
Letμ∈MB. For all x·∈PAPω,cℝ,ℝ+,μ, the function
(105)ψi:t↦ψixω,ct=ate−c∧−tωit∑j=1pxt−τij=c∧taω,cte−ωit∑j=1pc∧−τijxω,ct−τij
belongs toPAPω,cℝ,ℝ+,μ for all 1≤i≤n.Proof.
First, we can say that the function(106)t↦xt−τij∈PAPω,cℝ,ℝ+,μ,
for all1≤i≤n and 1≤j≤p. Then, according to Proposition 30(107)t↦∑j=1pxt−τij∈PAPω,cℝ,ℝ+,μ.
Furthermore, by Lemma5,
(108)t↦ωit∑j=1pxt−τij∈PAPω,cℝ,ℝ+,μ.
for all1≤i≤n. Now, using the fact that the function x↦e−x is Lipschitzian and bounded, and aω,c∈PAPℝ,ℝ+,μ is also bounded then invoking the Corollary 54, it is clear that
(109)ψi:t↦c∧taω,cte−c∧−tωit∑j=1pxt−τij∈PAPω,cℝ,ℝ+,μ,
for all1≤i≤n.By using conditionC6 and Theorem 37, we can deduce the following Lemma.Lemma 58.
Suppose thatH1 and C6 hold, if x∈PAPω,cℝ,ℝ+,μ, then the function defined by
(110)t↦∫−∞tKjt−sxsds∈PAPω,cℝ,ℝ+,μ,
for all1≤j≤m.Theorem 59.
Suppose thatH1 and C1–C4 are satisfied. Then, the nonlinear operator Γ defined for each x∈PAPω,cℝ,ℝ+,μ by
(111)Γxt=∫−∞te−∫stαξdξ∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τijds
mapsPAPω,cℝ,ℝ+,μ into itself.Proof.
Using Lemmas13, 5, 5, and 5 and Corollary 54, then the function χ defined by
(112)χs=∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τij=c∧s∑j=1majω,cse−c∧−sωjs∫−∞sKjs−σxσdσ+∑i=1nbiω,cse−c∧−sβis∑j=1pxs−τij
is measureω,c-pseudo-almost periodic.
Consequently, we can writeχ=χ1+χ2 where χ1∈APω,cℝ+ and χ2∈Eω,cℝ,ℝ+,μ. It follows that
(113)Γχt≔∫−∞te−∫stαξdξχsds=Γχ1t+Γχ2t.
Let us show thatΓχ1∈APω,cℝ+.
We recall that by applying conditionC1 to the model (99), α is almost periodic (i.e., ω,c-almost periodic with constant c=1. Now, in the view of the almost periodicity of the function α and the ω,c-almost periodicity of the function χ1, there exists a number lε such that in any interval δ,δ+lε one finds a number η, such that
(114)supξ∈ℝαξ+η−αξ<εandsupξ∈ℝχ1ω,cξ+η−χ1ω,cξ<ε.
It comes that(115)Γχ1t+η−Γχ1t=∫−∞t+ηe−∫st+ηαξdξχ1sds−∫−∞te−∫stαξdξχ1sds=∫−∞t+ηe−∫s−ηtαξ+ηdξχ1sds−∫−∞te−∫stαξdξχ1sds=∫−∞te−∫stαξ+ηdξχ1s+ηds−∫−∞te−∫stαξdξχ1sds=∫−∞te−∫stαξ+ηdξχ1s+ηds−∫−∞te−∫stαξdξχ1s+ηds+∫−∞te−∫stαξdξχ1s+ηds−∫−∞te−∫stαξdξχ1sds.
So, there existsδ∈0,1 such that
(116)Γχ1ω,ct+η−Γχ1ω,ct≤χ1ω,c∫−∞te−∫stαξ+ηdξ−e−∫stαξdξds+∫−∞te−∫stαξdξχ1ω,cs+η−χ1ω,csds≤χ1ω,c∫−∞te−∫stαξ+ηdξ−e−∫stαξdξds+ε∫−∞te−∫stαξdξds≤χ1ω,c∫−∞te−∫stαξ+ηdξ−e−∫stαξdξds+ε∫−∞te−t−sα¯ds≤χ1ω,c∫−∞te−∫stαξ+ηdξ+δ∫stαξdξ−∫stαξ+ηdξ×∫stαξ+η−αξdξds+ε∫−∞te−t−sα¯ds≤χ1ω,c∫−∞te−∫stαξ+ηdξe−δ∫stαξdξ−∫stαξ+ηdξ∫stαξ+η−αξdξds+ε∫−∞te−t−sα¯ds,≤εχ1ω,c∫−∞te−t−sα¯e−δ∫stαξdξ−∫stαξ+ηdξt−sds+ε∫−∞te−t−sα¯ds≤εχ1ω,c∫−∞te−t−sα¯t−sds+ε∫−∞te−t−sα¯ds≤εχ1ω,cα¯2+ε∫−∞te−t−sα¯ds≤εχ1ω,cα¯2+εα¯=χ1ω,cα¯2+1α¯ε.
This proves thatΓχ1∈APω,cℝ+. Now, let us show that Γχ2∈Eω,cℝ,ℝ+,μ. We have that
(117)limr⟶+∞1μ−r,r∫−rr∫−∞te−∫stαξdξχ2sdsω,cdμt≤limr⟶+∞1μ−r,r∫−rr∫−∞te−t−sα¯χ2sω,cdsdμt≤limr⟶+∞1μ−r,r∫−rr∫0∞e−α¯ξχ2t−ξω,cdξdμt=limr⟶+∞∫0∞e−α¯ξ1μ−r,r∫−rrχ2t−ξω,cdμtdξ.
By the Lebesgue Dominated Convergence Theorem andH1, we obtain that
(118)limr⟶+∞1μ−r,r∫−rr∫−∞te−∫stαξdξχ2sdsω,cdμt=0.
Then,Γχ2∈Eω,cℝ,ℝ+,μ and consequently, Γ∈PAPω,cℝ,ℝ+,μ.Theorem 60.
Assume thatH1 and C1–C6 hold, then the Lasota-Wazewska model with mixed delays (99) possesses a unique measure ω,c-pseudo-almost periodic solution y, and we have yω,c in the region
(119)R=ψ∈PAPω,cℝ,ℝ+,μ,LB≤ψω,c≤UB,
where(120)UB=∑j=1majω,c¯+∑i=1nbiω,c¯α¯andLB=∑j=1majω,c¯e−ωj¯UB+∑i=1nbiω,c¯e−pξβi¯UBα¯.Proof.
First, we proves that the operatorΓ is a mapping from R to R. We set
(121)γjs=ajse−c∧−sωjs∫−∞sKjs−σxσdσ,θis=bise−c∧−sβis∑j=1pxs−τij.
In fact, we have(122)Γxtω,c≤∫−∞te−∫stαξdξ∑j=1mγjs+∑i=1nθisω,cds=∫−∞te−∫stαξdξ∑j=1mγjs+∑i=1nθisω,cds≤∫−∞te−∫stαξdξ∑j=1majω,c¯+∑i=1nbiω,c¯ds≤∫−∞te−α¯t−s∑j=1majω,c¯+∑i=1nbiω,c¯ds=∑j=1majω,c¯+∑i=1nbiω,c¯α¯.
In the other hand, if we set(123)ξ=maxc∧τi,jfori,j∈1,n×1,p,δjs=c∧−sωjs∫−∞sKjs−σxσdσ,forj∈1,m,ϕis=c∧−sβis∑j=1pxs−τijfori∈1,n,
then, we have forx∈R(124)Γxtω,c=∫−∞te−∫stαξdξ∑j=1majse−δjs+∑i=1nbise−ϕisdsω,c≥c∧−t∫−∞te−∫stαξdξ∑j=1majse−ωj¯∫−∞sKjω,cs−σxω,cσdσ+∑i=1nbise−βi¯∑j=1pc∧−τijxω,cs−τijds≥c∧−t∫−∞te−∫stαξdξ∑j=1majse−ωj¯UB∫−∞sKjω,cs−σdσ+∑i=1nbise−βi¯UB∑j=1pc∧−τijds≥c∧−t∫−∞te−∫stαξdξ∑j=1majse−ωj¯UB+∑i=1nbise−pξβi¯UBds≥∫−∞tc∧−se−∫stαξdξ∑j=1majse−ωj¯UB+∑i=1nbise−pξβi¯UBds=∫−∞te−∫stαξdξ∑j=1majω,cse−ωj¯UB+∑i=1nbiω,cse−pξβi¯UBds≥∫−∞te−α¯t−s∑j=1majω,c¯e−ωj¯UB+∑i=1nbiω,c¯e−pξβi¯UBds=∑j=1majω,c¯e−ωj¯UB+∑i=1nbiω,c¯e−pξβi¯UBα¯,
which implies that the operatorΓ is a mapping from R to itself. To end the proof, it suffices to prove that Γ is a contraction mapping. Let x,y∈R. Then,
(125)Γxt−Γytω,c=∫−∞te−∫stαξdξ∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijdsω,c≤∫−∞te−∫stαξdξ∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijω,cds=∫−∞te−∫stαξdξ∑j=1majω,cse−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbiω,cse−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijds≤∫−∞te−∫stαξdξ∑j=1majω,c¯e−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbiω,c¯e−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijds≤∫−∞te−∫stαξdξ∑j=1majω,c¯e−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbiω,c¯e−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijds.
Obviously, foru,v∈0,+∞,
(126)e−u−e−v≤u−v.
Then it comes that(127)Γxt−Γytω,c≤∫−∞te−∫stαξdξ∑j=1majω,c¯c∧−sωjs∫−∞sKjs−σxσ−yσdσ+∑i=1nbiω,c¯c∧−sβis∑j=1pxs−τij−ys−τijds≤∫−∞te−∫stαξdξ∑j=1majω,cωj¯∫−∞sKjω,cs−σdσx−yω,c+pξx−yω,c∑i=1nbiω,cβi¯ds=∫−∞te−∫stαξdξ∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯dsx−yω,c≤∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯α¯x−yω,c,
which implies (invokingC5) that the mapping Γ is a contraction mapping of R. Consequently, Γ possesses a unique fixed point x⋆∈R. Hence, x⋆ is the unique measure ω,c-pseudo-almost periodic solution of Equation (99) in R.


### 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(128)y′t=−αtyt+∑j=13ajte−c∧−tωjt∫−∞tKjt−sysds+∑i=13bite−c∧−tβityt−τi,whereαt=8+cos25t+cos2t, c=0.9, and ω=1,
(129)a1ta2ta3t=0.9∧t1+0.25cos22t+0.25cos2πt+0.51+t20.9∧t0.5+0.25cos23t+0.25cos2πt+11+t20.9∧t0.5+0.25cos25t+0.25cos22t+e−t2cos2t,ω1tω2tω3t=0.125cos22t+0.125cos2πt+0.251+t20.125cos22t+0.125cos2πt+0.251+t20.250cos22t+0.25e−t2cos2t,b1tb2tb3t=0.9∧t1+0.25cos25t+0.25cos2πt+0.5e−t2cos2t0.9∧t1+0.25cos25t+0.25cos2πt+0.5e−t2cos2t0.9∧t1+0.25cos25t+0.25cos2πt+0.5e−t2cos2t,β1tβ2tβ3t=0.125cos22t+0.125cos2πt+0.251+t20.125cos22t+0.125cos2πt+0.251+t20.125cos22t+0.125cos2πt+0.25e−t2cos2t.τ1=τ2=τ3=1, Kj=0.9∧te−t. Then,
(130)∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯α¯≤34<1.If the Radon-Nikodym derivativeρ of the measure μ is ρt=esint with respect to the Lebesgue measure on ℝ (i.e. dμ=ρtdt), then μ∈MB, since
(131)μ−r,r=∫−rresint⟶+∞,ifr⟶+∞,(132)μτ+a≤e2μA,∀τ∈ℝanda∈A.Hence, conditionsC1 – C6 and H1 are satisfied then according to the Theorem 60, the Lasota-Wazewska model with a mixed delays (14) has a unique μ-ω,c-pseudo-almost periodic solution in the region R=y∈PAPω,cℝ,ℝ+,μ,LB≤yω,c≤UB where
(133)UB=∑j=1majω,c¯+∑i=1nbiω,c¯α¯=32,(134)LB=∑j=1maj¯ω,ce−ωj¯UB+∑i=1nbi¯ω,ce−pξβi¯UBα¯≤e−1/23/2+0.5e−1/23/2+0.5e−1/23/2+e−0.91/23/2+e−0.91/23/2+e−0.91/23/210=e−3/45+3e−0.9/410≈0.246.


## 5.1. Existence and Uniqueness ofμ-ω,c-Pseudo-Almost Periodic Solution to the Model

In what follows, given ac-bounded continuous function f defined on ℝ, fω,c¯ and fω,c¯ are defined by
(100)fω,c¯=supt∈ℝfω,ct=supt∈ℝc∧−tft,andfω,c¯=inft∈ℝfω,ct=inft∈ℝc∧−tft.Remark 55.
Ifc=1, we use the notations
(101)f¯≔fω,1¯=supt∈ℝftandf¯≔fω,1¯=inft∈ℝft.First, we give sufficient conditions which ensures existence and uniqueness ofμ-ω,c-pseudo-almost periodic solution of (99).C10<c≤1C2α∈APℝ+C3βi,ωj∈PAPℝ,ℝ+,μ, for all i,j∈1,n×1,mC4aj,bi∈PAPω,cℝ,ℝ+,μ, for all i,j∈1,n×1,mC5∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯/α¯<1C6 For all 1≤j≤m, Kjω,c:0,+∞⟶ℝ+ are continuous, integrable, and
(102)∫0∞Kjω,cudu=1,and∫0∞Kjω,cueλudu<+∞,whereλ is a sufficiently non negative small constant.Lemma 56.
Letf∈PAPω,cℝ,ℝ+,μ and g∈PAPℝ,ℝ+,μ. If c>0, then fg∈PAPω,cℝ,ℝ+,μ.Proof.
According to Theorem29, there exists a unique u∈PAPℝ,ℝ+,μ such that
(103)ft≔c∧tut,
for allt∈ℝ. Using Lemma 13, it is clear that u×g∈PAPℝ,ℝ+,μ. Then,
(104)fgt≔c∧tutgt,whereug∈PAPℝ,ℝ+,μ.
Invoking Theorem29, we complete the proof.Now, we can establish following lemma.Lemma 57.
Letμ∈MB. For all x·∈PAPω,cℝ,ℝ+,μ, the function
(105)ψi:t↦ψixω,ct=ate−c∧−tωit∑j=1pxt−τij=c∧taω,cte−ωit∑j=1pc∧−τijxω,ct−τij
belongs toPAPω,cℝ,ℝ+,μ for all 1≤i≤n.Proof.
First, we can say that the function(106)t↦xt−τij∈PAPω,cℝ,ℝ+,μ,
for all1≤i≤n and 1≤j≤p. Then, according to Proposition 30(107)t↦∑j=1pxt−τij∈PAPω,cℝ,ℝ+,μ.
Furthermore, by Lemma5,
(108)t↦ωit∑j=1pxt−τij∈PAPω,cℝ,ℝ+,μ.
for all1≤i≤n. Now, using the fact that the function x↦e−x is Lipschitzian and bounded, and aω,c∈PAPℝ,ℝ+,μ is also bounded then invoking the Corollary 54, it is clear that
(109)ψi:t↦c∧taω,cte−c∧−tωit∑j=1pxt−τij∈PAPω,cℝ,ℝ+,μ,
for all1≤i≤n.By using conditionC6 and Theorem 37, we can deduce the following Lemma.Lemma 58.
Suppose thatH1 and C6 hold, if x∈PAPω,cℝ,ℝ+,μ, then the function defined by
(110)t↦∫−∞tKjt−sxsds∈PAPω,cℝ,ℝ+,μ,
for all1≤j≤m.Theorem 59.
Suppose thatH1 and C1–C4 are satisfied. Then, the nonlinear operator Γ defined for each x∈PAPω,cℝ,ℝ+,μ by
(111)Γxt=∫−∞te−∫stαξdξ∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τijds
mapsPAPω,cℝ,ℝ+,μ into itself.Proof.
Using Lemmas13, 5, 5, and 5 and Corollary 54, then the function χ defined by
(112)χs=∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τij=c∧s∑j=1majω,cse−c∧−sωjs∫−∞sKjs−σxσdσ+∑i=1nbiω,cse−c∧−sβis∑j=1pxs−τij
is measureω,c-pseudo-almost periodic.
Consequently, we can writeχ=χ1+χ2 where χ1∈APω,cℝ+ and χ2∈Eω,cℝ,ℝ+,μ. It follows that
(113)Γχt≔∫−∞te−∫stαξdξχsds=Γχ1t+Γχ2t.
Let us show thatΓχ1∈APω,cℝ+.
We recall that by applying conditionC1 to the model (99), α is almost periodic (i.e., ω,c-almost periodic with constant c=1. Now, in the view of the almost periodicity of the function α and the ω,c-almost periodicity of the function χ1, there exists a number lε such that in any interval δ,δ+lε one finds a number η, such that
(114)supξ∈ℝαξ+η−αξ<εandsupξ∈ℝχ1ω,cξ+η−χ1ω,cξ<ε.
It comes that(115)Γχ1t+η−Γχ1t=∫−∞t+ηe−∫st+ηαξdξχ1sds−∫−∞te−∫stαξdξχ1sds=∫−∞t+ηe−∫s−ηtαξ+ηdξχ1sds−∫−∞te−∫stαξdξχ1sds=∫−∞te−∫stαξ+ηdξχ1s+ηds−∫−∞te−∫stαξdξχ1sds=∫−∞te−∫stαξ+ηdξχ1s+ηds−∫−∞te−∫stαξdξχ1s+ηds+∫−∞te−∫stαξdξχ1s+ηds−∫−∞te−∫stαξdξχ1sds.
So, there existsδ∈0,1 such that
(116)Γχ1ω,ct+η−Γχ1ω,ct≤χ1ω,c∫−∞te−∫stαξ+ηdξ−e−∫stαξdξds+∫−∞te−∫stαξdξχ1ω,cs+η−χ1ω,csds≤χ1ω,c∫−∞te−∫stαξ+ηdξ−e−∫stαξdξds+ε∫−∞te−∫stαξdξds≤χ1ω,c∫−∞te−∫stαξ+ηdξ−e−∫stαξdξds+ε∫−∞te−t−sα¯ds≤χ1ω,c∫−∞te−∫stαξ+ηdξ+δ∫stαξdξ−∫stαξ+ηdξ×∫stαξ+η−αξdξds+ε∫−∞te−t−sα¯ds≤χ1ω,c∫−∞te−∫stαξ+ηdξe−δ∫stαξdξ−∫stαξ+ηdξ∫stαξ+η−αξdξds+ε∫−∞te−t−sα¯ds,≤εχ1ω,c∫−∞te−t−sα¯e−δ∫stαξdξ−∫stαξ+ηdξt−sds+ε∫−∞te−t−sα¯ds≤εχ1ω,c∫−∞te−t−sα¯t−sds+ε∫−∞te−t−sα¯ds≤εχ1ω,cα¯2+ε∫−∞te−t−sα¯ds≤εχ1ω,cα¯2+εα¯=χ1ω,cα¯2+1α¯ε.
This proves thatΓχ1∈APω,cℝ+. Now, let us show that Γχ2∈Eω,cℝ,ℝ+,μ. We have that
(117)limr⟶+∞1μ−r,r∫−rr∫−∞te−∫stαξdξχ2sdsω,cdμt≤limr⟶+∞1μ−r,r∫−rr∫−∞te−t−sα¯χ2sω,cdsdμt≤limr⟶+∞1μ−r,r∫−rr∫0∞e−α¯ξχ2t−ξω,cdξdμt=limr⟶+∞∫0∞e−α¯ξ1μ−r,r∫−rrχ2t−ξω,cdμtdξ.
By the Lebesgue Dominated Convergence Theorem andH1, we obtain that
(118)limr⟶+∞1μ−r,r∫−rr∫−∞te−∫stαξdξχ2sdsω,cdμt=0.
Then,Γχ2∈Eω,cℝ,ℝ+,μ and consequently, Γ∈PAPω,cℝ,ℝ+,μ.Theorem 60.
Assume thatH1 and C1–C6 hold, then the Lasota-Wazewska model with mixed delays (99) possesses a unique measure ω,c-pseudo-almost periodic solution y, and we have yω,c in the region
(119)R=ψ∈PAPω,cℝ,ℝ+,μ,LB≤ψω,c≤UB,
where(120)UB=∑j=1majω,c¯+∑i=1nbiω,c¯α¯andLB=∑j=1majω,c¯e−ωj¯UB+∑i=1nbiω,c¯e−pξβi¯UBα¯.Proof.
First, we proves that the operatorΓ is a mapping from R to R. We set
(121)γjs=ajse−c∧−sωjs∫−∞sKjs−σxσdσ,θis=bise−c∧−sβis∑j=1pxs−τij.
In fact, we have(122)Γxtω,c≤∫−∞te−∫stαξdξ∑j=1mγjs+∑i=1nθisω,cds=∫−∞te−∫stαξdξ∑j=1mγjs+∑i=1nθisω,cds≤∫−∞te−∫stαξdξ∑j=1majω,c¯+∑i=1nbiω,c¯ds≤∫−∞te−α¯t−s∑j=1majω,c¯+∑i=1nbiω,c¯ds=∑j=1majω,c¯+∑i=1nbiω,c¯α¯.
In the other hand, if we set(123)ξ=maxc∧τi,jfori,j∈1,n×1,p,δjs=c∧−sωjs∫−∞sKjs−σxσdσ,forj∈1,m,ϕis=c∧−sβis∑j=1pxs−τijfori∈1,n,
then, we have forx∈R(124)Γxtω,c=∫−∞te−∫stαξdξ∑j=1majse−δjs+∑i=1nbise−ϕisdsω,c≥c∧−t∫−∞te−∫stαξdξ∑j=1majse−ωj¯∫−∞sKjω,cs−σxω,cσdσ+∑i=1nbise−βi¯∑j=1pc∧−τijxω,cs−τijds≥c∧−t∫−∞te−∫stαξdξ∑j=1majse−ωj¯UB∫−∞sKjω,cs−σdσ+∑i=1nbise−βi¯UB∑j=1pc∧−τijds≥c∧−t∫−∞te−∫stαξdξ∑j=1majse−ωj¯UB+∑i=1nbise−pξβi¯UBds≥∫−∞tc∧−se−∫stαξdξ∑j=1majse−ωj¯UB+∑i=1nbise−pξβi¯UBds=∫−∞te−∫stαξdξ∑j=1majω,cse−ωj¯UB+∑i=1nbiω,cse−pξβi¯UBds≥∫−∞te−α¯t−s∑j=1majω,c¯e−ωj¯UB+∑i=1nbiω,c¯e−pξβi¯UBds=∑j=1majω,c¯e−ωj¯UB+∑i=1nbiω,c¯e−pξβi¯UBα¯,
which implies that the operatorΓ is a mapping from R to itself. To end the proof, it suffices to prove that Γ is a contraction mapping. Let x,y∈R. Then,
(125)Γxt−Γytω,c=∫−∞te−∫stαξdξ∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijdsω,c≤∫−∞te−∫stαξdξ∑j=1majse−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbise−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijω,cds=∫−∞te−∫stαξdξ∑j=1majω,cse−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbiω,cse−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijds≤∫−∞te−∫stαξdξ∑j=1majω,c¯e−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbiω,c¯e−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijds≤∫−∞te−∫stαξdξ∑j=1majω,c¯e−c∧−sωjs∫−∞sKjs−σxσdσ−e−c∧−sωjs∫−∞sKjs−σyσdσ+∑i=1nbiω,c¯e−c∧−sβis∑j=1pxs−τij−e−c∧−sβis∑j=1pys−τijds.
Obviously, foru,v∈0,+∞,
(126)e−u−e−v≤u−v.
Then it comes that(127)Γxt−Γytω,c≤∫−∞te−∫stαξdξ∑j=1majω,c¯c∧−sωjs∫−∞sKjs−σxσ−yσdσ+∑i=1nbiω,c¯c∧−sβis∑j=1pxs−τij−ys−τijds≤∫−∞te−∫stαξdξ∑j=1majω,cωj¯∫−∞sKjω,cs−σdσx−yω,c+pξx−yω,c∑i=1nbiω,cβi¯ds=∫−∞te−∫stαξdξ∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯dsx−yω,c≤∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯α¯x−yω,c,
which implies (invokingC5) that the mapping Γ is a contraction mapping of R. Consequently, Γ possesses a unique fixed point x⋆∈R. Hence, x⋆ is the unique measure ω,c-pseudo-almost periodic solution of Equation (99) in R.


## 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(128)y′t=−αtyt+∑j=13ajte−c∧−tωjt∫−∞tKjt−sysds+∑i=13bite−c∧−tβityt−τi,whereαt=8+cos25t+cos2t, c=0.9, and ω=1,
(129)a1ta2ta3t=0.9∧t1+0.25cos22t+0.25cos2πt+0.51+t20.9∧t0.5+0.25cos23t+0.25cos2πt+11+t20.9∧t0.5+0.25cos25t+0.25cos22t+e−t2cos2t,ω1tω2tω3t=0.125cos22t+0.125cos2πt+0.251+t20.125cos22t+0.125cos2πt+0.251+t20.250cos22t+0.25e−t2cos2t,b1tb2tb3t=0.9∧t1+0.25cos25t+0.25cos2πt+0.5e−t2cos2t0.9∧t1+0.25cos25t+0.25cos2πt+0.5e−t2cos2t0.9∧t1+0.25cos25t+0.25cos2πt+0.5e−t2cos2t,β1tβ2tβ3t=0.125cos22t+0.125cos2πt+0.251+t20.125cos22t+0.125cos2πt+0.251+t20.125cos22t+0.125cos2πt+0.25e−t2cos2t.τ1=τ2=τ3=1, Kj=0.9∧te−t. Then,
(130)∑j=1majω,cωj¯+pξ∑i=1nbiω,cβi¯α¯≤34<1.If the Radon-Nikodym derivativeρ of the measure μ is ρt=esint with respect to the Lebesgue measure on ℝ (i.e. dμ=ρtdt), then μ∈MB, since
(131)μ−r,r=∫−rresint⟶+∞,ifr⟶+∞,(132)μτ+a≤e2μA,∀τ∈ℝanda∈A.Hence, conditionsC1 – C6 and H1 are satisfied then according to the Theorem 60, the Lasota-Wazewska model with a mixed delays (14) has a unique μ-ω,c-pseudo-almost periodic solution in the region R=y∈PAPω,cℝ,ℝ+,μ,LB≤yω,c≤UB where
(133)UB=∑j=1majω,c¯+∑i=1nbiω,c¯α¯=32,(134)LB=∑j=1maj¯ω,ce−ωj¯UB+∑i=1nbi¯ω,ce−pξβi¯UBα¯≤e−1/23/2+0.5e−1/23/2+0.5e−1/23/2+e−0.91/23/2+e−0.91/23/2+e−0.91/23/210=e−3/45+3e−0.9/410≈0.246.



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*Source: 9558928-2022-07-26.xml*