# Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

Authors: Zhong-Hua Wu
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

Through the use of the measure theory, evolution family, “Acquistapace–Terreni” condition, and Hölder inequality, the core objective of this work is to seek to analyze whether there is uniqueμ-pseudo almost periodic solution to a functional evolution equation with Stepanov forcing terms in a Banach space. Certain adequate conditions are derived guaranteeing there is unique μ-pseudo almost periodic solution to the equation by Lipschitz condition and contraction mapping principle. Finally, an example is used to demonstrate our theoretical findings.


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## Body

## 1. Introduction

Periodicity is a common phenomenon in the natural and social sciences that varies over time. Periodicity can be caused by business and economic activities, and it differs from trend changes in that it is not a gradual development in one direction, but rather a series of rising and falling oscillations. Sunrise and sunset, the cycle of the seasons, and the moon changing from deficit to surplus are all cyclical phenomena. In epidemiology, the cycle refers to the frequency of disease according to a certain interval of time, the regular ups and downs of fluctuations; every several years, there is an epidemic peak phenomenon. In physics, it refers to the reciprocal motion of a mass in simple harmonic motion, which is always cyclic, with each cycle taking the same amount of time and being strictly periodic. In mathematics, a function whose output value repeats periodically is called a periodic function. The motion of the planets is not regular and elliptical, because it can be influenced by other stars; the transmission of radio waves can also be disturbed. Thus, the periodicity will no longer be maintained. In such a case, the almost period would better reflect the variation of the above phenomenon with time. In mathematics, the almost periodic function is a class of functions with approximately periodic properties, which is the extension of the continuous periodic function. Different periodic functions have different periods, and their sums, differences, or products are not necessarily periodic functions anymore, although the almost periodic function may not have strict periodicity, but can have some better properties than the periodic function. In theory, all periodic functions in any paradigm do not constitute a Banach space, while the probable periodic functions in the upper bound paradigm constitute a Banach space. This also shows that the probable periodic function will be more widely used than the periodic function, and it is more practical to discuss the probable periodic solution of the equation.In fact, strict periodic changes are difficult to appear in nature. In real life, there are not many strict periodic phenomena such as seven days a week. On the contrary, almost periodic changes are more likely to accurately describe the change law of nature. Almost periodic phenomena are more common than periodic phenomena, and almost periodic phenomena are closer to reality than periodic phenomena. In reality, almost periodic phenomena are more likely to occur.Almost periodic function can better reflect the change law of things in real life and explain the phenomena in real life. It is a phenomenon easier to see in natural science and social science. Almost periodic results are more common than periodic results. Almost periodic function has many other good properties, and its practical application is more and more widely in real life. The almost periodic property of the solution of the equation has attracted the attention of many scholars. Discussing the almost periodic property of the solution of the equation has more extensive practical significance and practical application value. Therefore, it has become an important topic in mathematical research.In 1992, Zhang proposed the idea of pseudo almost periodic functions in the academic paper [1]. After that, in 2006, the idea of weighted pseudo almost periodic functions was presented by Diagana in the literature [2]. As an extension of the concept of pseudo almost periodic functions and weighted pseudo almost periodic functions, Blot et al. presented the idea of μ-pseudo almost functions in 2013 [3]. Since then, many scholars have conducted extensive research on the properties of such functions and made advantage of these properties to set up the existence of μ-pseudo almost periodic solutions for partial functional nonautonomous equation [4, 5], partial functional differential equations [6, 7], partial differential equation [8], nonautonomous integrodifferential equations [9], and semilinear fractional integrodifferential equations [10]. In 2007, Diagana proposed the conception of the Stepanov pseudo almost periodic function in the literature [11]. Then, in 2010, Diagana et al. presented the conception of weighted Stepanov-like pseudo almost periodic function in the literature [12], as the generalization of Stepanov pseudo almost periodic functions. In the literature [13], the authors Es-sebbar and Ezzinbi proposed the notion of Stepanov μ-pseudo almost periodic function. In [14], the authors studied whether there is unique μ-pseudo almost periodic solution for a parabolic evolution equation and the equation’s forced terms are assumed to be μ-Stepanov pseudo almost periodic. In [15], the authors established the convolution and composition theorems of the μ-Stepanov pseudo almost periodic functions and verified whether there is unique μ-pseudo almost periodic mild solution of fractional integrodifferential equation.Inspired by the above, in this work, letK,⋅ be a Banach space and 1<p<∞; by the measure theory, we mostly prove whether there is unique μ-pseudo almost periodic solution to a functional evolution equation
(1)ddtzt−ψt,C1~tzt=Atzt−ψt,C1~tzt+gt,zt,C2~tzt,for t∈ℝ, where Att∈ℝ fulfills the Acquistapace–Terreni condition, which is exponentially stable, and Qt,s created by At and ψ:ℝ×K⟶K, is μ-pseudo almost periodic and g:ℝ×K×K⟶K is Stepanov-like μ-pseudo almost periodic functions; C1~t,C2~tt∈ℝ are families of (perhaps unbounded) linear operators. The topic about whether there is unique μ-pseudo almost periodic solution to functional evolution equation with Stepanov forcing terms of (1) is untreated in the past work, which is one of the crucial incentives of this study.The following is an outline for this work. Section2 introduces some essential concepts and basic properties. Section 3 is dedicated to demonstrating whether there are unique μ-pseudo almost periodic solutions of (1). To give you an illustration, in the last part, we provide a case supporting our conclusion.


## 2. Preliminaries

Throughout the course of this project,K,⋅ and ℍ,⋅ℍ are two Banach spaces. BCℝ,K (respectively, BCℝ×ℍ,K) is the Banach space of bounded continuous functions from ℝ into K (respectively, jointly bounded continuous functions ψ:ℝ×ℍ⟶K) as a result of its norm by
(2)ψ∞=supt∈ℝψt.We signify byD the Lebesgue D^-field of ℝ and by N the set of all positive measures μ on D such that μℝ=+∞ and μc,d<+∞ for all c,d∈ℝc≤d.Fromμ∈N, the following assumptions are assumed.(H1) For allc,d,ande∈ℝ, such that 0≤c<d≤e, there exist ν^0≥0 and α0≥0 satisfying ∣ν^∣≥ν^0⇒μc+ν^,d+ν^≥α0μν^,e+ν^.(H2) For allν^∈ℝ, there exist α~>0 and a bounded interval I satisfying
(3)μc+ν^:c∈B≤α~μB,ifB∈DsuchthatB∩I=∅.Definition 1 (see [16]).
Letψt∈Cℝ,K,ψt be referred to as almost periodic if, for any ε>0, the set Eψ,ε=κ^:ψt+κ^−ψtforallt∈ℝ is relatively dense; that is, for any ε>0, it is possible to locate a real number l=lε>0; for any interval with length lε, there exists a number κ^=κ^ε in this interval satisfying ψt+κ^−ψt for all t∈ℝ. We denote the space of all such functions by APℝ,X.
Consider the functionF, which is equal to
(4)Fx=sinx+sin2x.F is an almost periodic function, while F is not periodic. See almost periodic function F below in Figure 1.Definition 2 (see [17]).
A continuous functionψ:ℝ×ℍ⟶K is regarded as almost periodic in t uniformly in relation to u~inℍ, if for any compact subset K⊂ℍ and every ϵ>0, there exists l>0 such that every interval of length l, it is possible to find a number ν^ in this interval with the feature that
(5)ψt+ν^,u~−ψt,u~<ϵ,fort,u~∈ℝ×K.Figure 1
Almost periodic functionF.The union of such functions will be referred to asAPUℝ×ℍ;K.Proposition 3 (see [16]).
Ifψ,ψ1 and ψ2∈APK, then
(6)ψ1+ψ2∈APK.(i)
λψ∈APK for any scalar λ(ii)
ψα∈APK where ψα is the right translation function of ψψα:ℝ⟶K is characterised by ψα⋅=ψ⋅+α(iii)
The rangeRψ≔ψt:t∈ℝ is a little package in K; thus, ψ is bounded in norm(iv)
Ifψn⟶ψ uniformly on ℝ where each ψn∈APK, then ψ∈APK tooDefinition 4 (see [3]).
Letμ∈N. A continuous bounded function ψ:ℝ⟶K is regarded as μ-ergodic if
(7)limϖ⟶+∞1μϖ^∫ϖ^ψtdμt=0,where ϖ^≔−ϖ,ϖ. All of these functions’ space is denoted by Eℝ;K,μ.Definition 5 (see [3]).
Letμ∈N. A continuous function ψ:ℝ×K⟶ℍ is regarded as μ-ergodic in t∈ℝuniformly in relation to x∈K, if the two following statements are satisfied:
(i)
For allx∈K, ψ.,x∈Eℝ;K,μ(ii)
ψ is uniformly continuous on every compact set K in K in terms of the second variable xThe union of such functions will be referred to asEUℝ×K;ℍ,μ.Proposition 6 (see [13]).
If a sequenceψnn of μ-ergodic functions uniformly converges to a function ψ, then ψ is also μ-ergodic.Theorem 7 (see [3]).
Letμ∈N. Then, Eℝ;K,μ,⋅∞ is a Banach space.Definition 8 (see [3]).
Letμ∈N. A continuous function ψ:ℝ⟶K is called μ-pseudo almost periodic if it is able to create it as
(8)ψ=ψ1+ψ2,
whereψ1∈APℝ;K and ψ2∈Eℝ;K,μ. Indicate with PAPℝ;K,μ the area occupied by such functions.Definition 9 (see [3]).
Letμ∈N. A continuous function ψ:ℝ×K⟶ℍ is called μ-pseudo almost periodic in t∈ℝ uniformly in relation to x∈K, if it is possible to write it in the form
(9)ψ=ψ1+ψ2,where ψ1∈APUℝ×K;ℍ and ψ2∈EUℝ×K;ℍ,μ. The set of such functions is represented by the symbol PAPUℝ×K;ℍ,μ.Theorem 10 (see [3]).
Letμ∈N, ψ∈PAPUℝ×ℍ;K,μ, and u∈PAPℝ;ℍ,μ. Assume for all bounded subset B of ℍ, f is bounded on ℝ×B. Then, the function
(10)t↦ψt,ut∈PAPℝ;K,μ.Definition 11 (see [12]).
Let1≤p<∞. The space BSpℝ;K of all Stepanov bounded functions, in conjunction with the exponent p, all measurable functions are included in this category f on ℝ in regard to value K such that ψb∈L∞ℝ,Lp0,1;K.BSpℝ;K. This is a Banach space when the norm is followed
(11)ψSp=ψbL∞ℝ,Lp=supt∈ℝ∫tt+1ψϱpdϱ1/p<∞.
A functionψ∈Llocpℝ;K is Stepanov bounded (Sp-bounded) if ψSp<∞. It goes without saying that Lpℝ;K⊂BSpℝ;K⊂Llocpℝ;K.Definition 12 (see [12]).
A functionψ∈BSpℝ;K is called almost periodic in the sense of Stepanov (Sp-almost periodic) if for each ϵ>0. There is such a thing as a positive number l as a result each length l interval includes a number c; there exists ν^∈c,c+lε satisfying
(12)∫t,t+1ψϱ+ν∧−ψϱpdϱ1/p<ε,t∈ℝ.
LetAPpℝ;KSp−APℝ;K be the collection of all Sp-almost periodic functions.Definition 13 (see [13]).
Letμ∈N. A function ψ∈BSpℝ;K is referred to as Sp-μ-ergodic if
(13)limϖ⟶+∞1μϖ^∫ϖ^∫tt+1ψϱpdϱ1/pdμt=0.
We mean byEpℝ;K,μ the functions.Definition 14 (see [13]).
Letμ∈N. A function ψ∈BSpℝ;K is called a Sp-μ-pseudo almost periodic (or Stepanov μ-pseudo almost periodic) if it can be stated in this way ψ=ψ1+ψ2, where ψ1∈APpℝ;K and ψ2∈Epℝ;K,μ. PAPpℝ;K,μ or Sp−PAPℝ;K,μ denotes a group of functions that are analogous.Definition 15 (see [14]).
Letμ∈N A function ψ:ℝ×ℍ⟶K with ψ.,y∈Llocpℝ;K for each y∈ℍ is called Sp-μ-pseudo almost periodic in y∈ℍ uniformly in t∈ℝ if it may be presented as ψ=ψ1+ψ2, where ψ1∈APpℝ×ℍ;K and ψ2∈Epℝ×ℍ;K,μ. The space occupied by such functions will be indicated by PAPUpℝ×ℍ;K,μ or Sp−PAPℝ×ℍ;K,μ.Theorem 16 (see [14]).
Letμ∈N satisfy (H2), if ψ∈PAPℝ;K,μ then f∈PAPpℝ;K,μ.Theorem 17 (see [18]).
Assume thatψ∈BSpℝ,K. Then, ψ∈Epℝ,K,μ if and only if, for whatever reason ϵ>0(14)limϖ⟶+∞μt∈ϖ^;∫tt+1ψϱpdϱ1/p>ϵμϖ^=0.
Then, by Propositions3 and 6, as a consequence, we arrive to the following conclusion.Proposition 18.
Letψnn∈ℕ⊂PAPℝ;K,μ be a sequence of functions. If ψn uniformly converges to some ψ, then ψ∈PAPℝ;K,μ.


## 3. Main Results

In the next, we are devoted whether there is uniqueμ-pseudo almost periodic mild solution to (1). Throughout the remainder of this paper, the following hypotheses will be required.(H3) There exist constantsμ0≥0, ϑ∈π/2,π, L1,L2≥0, and α1,β1∈0,1 with α1+β1>1 such that
(15)Σϑ∪0⊂ρAx−μ0,Rλ,Ax−μ0≤L21+∣λ∣,Ax−μ0Rλ,Ax−μ0Rμ0,Ax−Rμ0,Ay≤L1x−yα1λ−β1,forx,y∈R, λ∈Σϑ≔λ∈C−0:∣argλ∣≤ϑ.(H4) The evolution familyQx,κ^ produced by Ax is exponentially stable, namely, there are constants K0,β^>0 in order for
(16)Qx,κ^≤K0e−β∧x−κ∧,for all x≥κ^. And the function ℝ×ℝ↦K,x,κ^↦Qx,κ^s∈APℝ×ℝ,K uniformly for all s in any bounded subset of K.(H5) The linear operatorsC1~t,C2~t:K⟶K satisfy
(17)t↦C1~tandt↦C2~t∈APℝ,BK,K.Let(18)ω≔maxsupt∈ℝC1~tBK,K,supt∈ℝC2~tBK,K.(H6) The functionψ∈PAPUℝ×ℍ;ℍ,μ and there exists Lψ>0 which satisfies
(19)ψt,s1−ψt,s2≤Lψs1−s2,for all t∈ℝ and for each s1,s2∈ℍ.(H7) The functiong∈PAPUpℝ×K×K;K,μ and there exists L3>0 that satisfies gt,m1,n1−gt,m2,n2≤L3m1−m2+n1−n2, for all t∈ℝ and for m1,n1,m2,n2∈K×K.Definition 19.
A continuous functionz:ℝ⟶K is referred to as a μ-pseudo almost periodic mild solution of (1), if for each t≥t0 and t0∈ℝ,
(20)zt−ψt,C1~tzt=Qt,t0zt0−ψt0,C1~t0zt0+∫t0tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ,t∈ℝ.
It can be shown that a functionz is a mild solution of (1) for each t∈ℝ and by the expression
(21)zt=ψt,C1~tzt+∫−∞tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ,since
(22)zt0=ψt0,C1~t0zt0+∫−∞t0Qt0,ϱgϱ,zϱ,C2~ϱzϱdϱ,Qt,t0zt0=Qt,t0ψt0,C1~t0zt0+∫−∞t0Qt,ϱgϱ,zϱ,C2~ϱzϱdϱ=Qt,t0ψt0,C1~t0zt0+∫−∞tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ−∫t0tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ=Qt,t0ψt0,C1~t0zt0+zt−ψt,C1~tzt−∫t0tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ.It follows that(23)zt=ψt,C1~tzt+∫−∞tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ.Theorem 20.
Letμ∈N. If g∈PAPUpℝ×W×K;K,μ satisfies the following condition
(24)gt,m1,r1−gt,m2,r2X≤L3m1−m2W+r1−r2X,for all t∈ℝ and m1,m2∈W and r1,r2∈K. If φ∈PAPℝ;K,μ moreover ϕ∈PAPℝ;W,μ, then g.,ϕ.,φ.∈PAPpℝ;K,μ.Proof.
Sinceg∈PAPUpℝ×K,K,μ, then g=ζ+h and ϕ=u+v,φ=x+y, where ζ∈APUpℝ×K×K,K,h∈Epℝ×K×K,K,μ,u,x∈APℝ,K and v,y∈Eℝ,K,μ. Decompose g as follows:
(25)Ψt=ζt,ut,xt+gt,ϕt,φt−ζt,ut,xt=ζt,ut,xt+gt,ϕt,φt−gt,ut,xt+ht,ut,xt.
Let us rewriteΨt=Ft+Φt+Δt, where
(26)Ft=ζt,ut,xt,Φt=gt,ϕt,φt−gt,ut,xt,Δt=ht,ut,xt.
We haveFt∈APpℝ,X through the composition of Sp-almost periodic functions in [19]. So it can be proved Φt and Δt belong to Epℝ×K,K,μ. Obviously, Φt∈BSpℝ,X. Actually, we get
(27)∫t,t+1Φϱpdϱ1/p≤∫t,t+1L3ϕϱ−uϱ+φϱ−xϱpdϱ1/p≤L3∫t,t+1vϱpdϱ1/p+∫t,t+1yϱpdϱ1p,foranyt∈ℝ.
Sincev,y is bounded, then Φt∈BSpℝ×X,X. Let Aϖ^ϵ and Mϖ^,ϵ be denoted by
(28)Aϖ^ϵ=t∈ϖ^;∫t,t+1Φϱpdϱ1/p>ϵ,Mϖ^,ϵ=t∈ϖ^;∫t,t+1vϱpdϱ1/p+∫t,t+1yϱpdϱ1/p>2ϵ.
From (27), we obtain Mϖ^,β^⊂Aϖ^ϵ and μMϖ^,β^≤μAϖ^ϵ infers that
(29)μt∈ϖ^;∫t,t+1Φϱpdϱ1/p>ϵμϖ^≤μt∈ϖ^;∫t,t+1vϱpdϱ1/p>ϵμϖ^+μt∈ϖ^;∫t,t+1yϱpdϱ1/p>ϵμϖ^.
From Theorem16, we get v,y∈Epℝ,K,μ. Since v,y∈Epℝ,K,μ and Theorem 17, we derive that
(30)limϖ⟶+∞μt∈ϖ^;∫t,t+1vϱpdϱ1/p>ϵμϖ^=limϖ⟶+∞μt∈ϖ^;∫t,t+1yϱpdϱ1/p>ϵμϖ^=0,
which deduces that(31)limϖ⟶+∞μt∈ϖ^;∫t,t+1Φϱpdϱ1/p>ϵμϖ^=0.
This deduces from Theorem17 that Φt∈Epℝ×K,K,μ. Now, we show that Δt∈Epℝ,K,μ. From u,x∈APℝ,K then uℝ¯,xℝ¯ is compact; moreover, ζb is uniformly continuous with respect to ℝ×0,1×uℝ¯orxℝ¯. We infer that hb is uniformly continuous with respect to ℝ×0,1×uℝ¯orxℝ¯, namely, for ϵ>0, let β^>0 such that for a1,b1,a2,b2∈uℝ¯×xℝ¯ with a1−a2+b1−b2<β^ we get
(32)∫t,t+1hϱ,a1,b1−hϱ,a2,b2pdϱ1/p<ε2,foranyt∈ℝ.
Becauseuℝ¯×xℝ¯ is compact, there exist finite balls Oθ with center βθ,γθ∈uℝ×xℝ,θ=1,2⋯n such that for any a2,b2∈uℝ¯×xℝ¯ we get βθ−a2+γθ−b2<β^ for 1≤θ≤n. Let
(33)Bθ=t∈ℝ,ut−βθ+xt−γθ<β^,θ=1,2,3,⋯,n,D1=B1,Dθ=Bθ∪i=1θ−1Bi,θ=1,2,3,⋯,n.
Thus,(34)ℝ=∪θ=1nBθ,ℝ=∪θ=1nDθ,Di∩Dj=∅i≠j.
Define functione1,e2:ℝ⟶K by e1t=βθ,e2t=γθ for t∈Bθ and θ=1,2,3,⋯,n. So ut−e1t+xt−e2t<β^ for t∈ℝ. Therefore, we have
(35)∫t,t+1hϱ,uϱ,xϱ−hϱ,e1ϱ,e2ϱpdϱ1/p≤4×21/pϵ.Since h∈Epℝ×K,K,μ, then there exists ϖ0>0 such that for ϖ>ϖ0: for every θ=1,2,3,⋯,n(36)1μϖ^∫ϖ^∫t,t+1∩Dθhϱ,βθ,γθpdϱ1/pdμt≤4×21/pn1/pϵ.
Then,(37)1μϖ^∫ϖ^∫t,t+1hϱ,uϱ,xϱpdϱ1/pdμt=1μϖ^∫ϖ^∑θ=1n∫t,t+1∩Dθhϱ,uϱ,xϱ−hϱ,βθ,γθ+hϱ,βθ,γθpdϱ1/pdμt≤21+1/pμϖ^∫ϖ^∑θ=1n∫t,t+1∩Dθhϱ,uϱ,xϱ−hϱ,βθ,γθpdϱ1pdμt+21+1/pμϖ^∫ϖ^∑θ=1n∫t,t+1∩Dθhϱ,βθ,γθpdϱ1/pdμt≤21+1/pμϖ^∫ϖ^∫t,t+1hϱ,uϱ,xϱ−hϱ,e1ϱ,e2ϱpdϱ1/pdμt+21+1/pμϖ^∫ϖ^∑θ=1n∫t,t+1∩Dθhϱ,βθ,γθpdϱ1/pdμt≤ϵ.
It follows thatΔt∈Epℝ,K,μ. The proof is completed.Lemma 21.
Letμ∈N. Assume that H3−H4 hold. If Φ is an Sp-μ-pseudo almost periodic function, then
(38)ΓΦt=∫−∞tQt,ϱΦϱdϱ,t∈ℝis a μ-pseudo almost periodic function.Proof.
SinceΦ∈SpPAPℝ;K,μ, then we have the following decomposition Φ=Φ1+Φ2 where Φ1∈SpAPℝ;K and Φ2∈Epℝ;K,μ. We define
(39)ΓΦ1t=∫−∞tQt,ϱΦ1ϱdϱ,ΓΦ2t=∫−∞tQt,ϱΦ2ϱdϱ,t∈ℝ.
Since there exist1/p+1/q=1, where p>1, the proof is as follows. Let us show that ΓΦ1t∈APℝ;K.
We define(40)ΓΦ1θt=∫t−θt−θ+1Qt,ϱΦ1ϱdϱ,θ∈ℕ,t∈ℝ.
By virtue of the Hölder inequality and formula (16), we get
(41)ΓΦ1θt=∫t−θt−θ+1Qt,ϱΦ1ϱdϱ≤∫t−θt−θ+1Qt,ϱΦ1ϱdϱ≤K0∫t−θt−θ+1e−qβ∧t−ϱdϱ1/q∫t−θt−θ+1Φ1ϱpdϱ1/p≤K0eqβ∧−1qβ^qe−θβ∧Φ1Sp.
Since∑θ=1∞e−β∧θ is convergent, so take advantage of the Weierstrass theorem, we obtain that the sequence of functions ∑θ=1∞ΓΦ1θt is convergent uniformly on ℝ; moreover, ∑θ=1∞ΓΦ1θt=ΓΦ1t then ΓΦ1. is continuous and
(42)ΓΦ1t≤∑θ=1∞ΓΦ1θt≤K0eqβ∧−1qβ^qΦ1Sp∑θ=1∞e−β∧θ,forallt∈ℝ.
Next, we can demonstrate thatΓΦ1.∈APℝ;K. Since Φ1∈SpAPℝ;K, let ϵ>0; afterwards, there is lϵ>0 such that each length interval lϵ contains ν^ based on the estimate
(43)supt∈ℝ∫tt+1Φ1ϱ+ν∧−Φ1ϱpdϱ1/p<ϵ1ϵ,where ϵ1=qβ^qeβ∧−1/K0eqβ∧−1,
(44)ΓΦ1θt+ν^−ΓΦ1θt=∫t+ν^−θt+ν^−θ+1Qt+ν^,ϱΦ1ϱdϱ−∫t−θt−θ+1Qt,ϱΦ1ϱdϱ≤∫t−θt−θ+1Qt,ϱΦ1ϱ+ν^−Φ1ϱdϱ≤K0∫t−θt−θ+1e−qβ∧t−ϱdϱ1/q∫t−θt−θ+1Φ1ϱ+ν∧−Φ1ϱpdϱ1/p≤K0eqβ∧−1qβ^qe−θβ∧ϵ1ϵ.
Hence,(45)∑θ=1∞ΓΦ1θt+ν^−ΓΦ1θt<ϵϵ1K0eqβ∧−1qβ^q∑θ=1∞e−θβ∧=ϵ.Therefore, ∑θ=1∞ΓΦ1θ.∈APℝ,K for all θ∈ℕ and ΓΦ1.∈APℝ;K.
Next, we show thatΓΦ2.∈Eℝ;K,μ.
Letϖ>0, Φ2∈Epℝ;K,μ; we get that
(46)limϖ⟶+∞1μϖ^∫ϖ^∫ϱϱ+1Φ2σpdσ1/pdμϱ=0.
First, we prove thatΓΦ2∈BCℝ;K. As a matter of fact, it is similar to above work of ΓΦ1. Then, we demonstrate that ΓΦ2∈Eℝ;K,μ. We observe
(47)ΓΦ2θt=∫t−θt−θ+1Qt,ϱΦ2ϱdϱ,θ∈ℕ,t∈ℝ.
Taking advantage of the Hölder inequality and applying formula (16), considering that
(48)ΓΦ2θt=∫t−θt−θ+1Qt,ϱΦ2ϱdϱ≤∫t−θt−θ+1Qt,ϱΦ2ϱdϱ≤K0∫t−θt−θ+1e−qβ∧t−ϱdϱ1/q∫t−θt−θ+1Φ2ϱpdϱ1/p≤K0eqβ∧−1qβ^qe−θβ∧Φ2Sp,
it follows that(49)1μϖ^∫ϖ^ΓΦ2θtdμt≤K0eqβ∧−1qβ^qe−θβ∧1μϖ^∫ϖ^Φ2tSpdμt,limϖ⟶+∞1μϖ^∫ϖ^ΓΦ2θtdμt≤K0eqβ∧−1qβ^qe−θβ∧.limϖ^⟶+∞1μϖ^∫ϖ^Φ2tSpdμt=0.
Hence,ΓΦ2θ∈Eℝ,K,μ. For another, since ∑θ=1∞e−β∧θ is convergent, so applying the Weierstrass theorem, then the series, as a result, ∑θ=1∞ΓΦ2θt is convergent uniformly on ℝ.
What is more,∑θ=1∞ΓΦ2θt=ΓΦ2t; therefore, ∑θ=1∞ΓΦ2θt∈Eℝ,K,μ, so ΓΦ2t∈Eℝ,K,μ from Proposition 18.Theorem 22.
Letμ∈N and suppose that (H3)–(H7) hold. If Lψω+2K0L3/β^1+ω<1, then equation (1) has a unique μ-pseudo almost periodic mild solution.Proof.
Define the operatorΠ as follows:
(50)Πzt=ψt,C1~tzt+∫−∞tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ.
Forzt∈PAPℝ;K,μ, by Lemma 21 in [5], we get that C1~tzt is μ-pseudo almost periodic; by (H6) and Theorem 10, it is simple to demonstrate that ψt,C1~tzt is part of PAPℝ;K,μ. We will demonstrate this now that ∫−∞tQt,ϱgϱ,zϱ,C2~ϱzϱdϱ is part of PAPℝ;K,μ; by using Theorem 20 and hypothesis (H7), we can easily deduce that gϱ,zϱ,C2~ϱzϱ is part of PAPpℝ;K,μ. Thus, Π is mapping from PAPℝ;K,μ into itself.
Letz,v∈PAPℝ;K,μ. We deduce that
(51)Πzt−Πvt≤ψt,C1~tzt−ψt,C1~tvt+∫−∞tQt,ϱgϱ,zϱ,C2~ϱzϱ−gϱ,vϱ,C2~ϱvϱdϱ≤LψC1~tzt−C1~tvt+K0L3∫−∞te−β∧t−ϱ1+C2~ϱzϱ−vϱdϱ≤LψC1~tz−v∞+2K0L31+C2~ϱ∫−∞te−β∧t−ϱdϱz−v∞≤Lψω+2K0β^1+ωL3z−v∞.
Thus,(52)Πz−Πv∞≤Lψω+2K0β^1+ωL3z−v∞.
As a result, this establishes thatΠ is a contraction map. We may infer that Π has a single fixed point in PAPℝ;K,μ such that Πz=z. Consequently, (1) has a unique μ-pseudo almost periodic mild solution.


## 4. Example

As an example, take a look at the differential equations listed below.(53)ddzuz,x−ψz,c1z,xuz,x=d2dx2uz,x−ψz,c1z,xuz,x+−3+sinmz+sinnzuz,x−ψz,c1z,xuz,x+gz,uz,x,c2z,xuz,x,z∈ℝ,x∈0,1,(54)uz,0−ψz,c1z,0uz,0=uz,1−ψz,c1z,1uz,1=0,z∈ℝ,where m,n∈ℝ and m/n∉ℚ,ψ,g:ℝ×K↦K, and c1,c2:ℝ×0,1↦ℝ are continuous functions. So as to rephrase this equation in form 1.1, we set the spaces K=L20,1 provided with the norm ·2. The operator A:DA⊂K⟶K is defined by
(55)Au=u′′u∈DA.Furthermore,A is the infinitesimally small generator C0-semigroup Tzz≥0 with Tz≤e−z (K=β^=1) for z≥0.We consider the operatorAz given by
(56)DAz=DA,Azv=A−3+sinmz+sinnzv,forallv∈DA.Thus,DAz=DA. Furthermore,
(57)Az−As=sinmz−sinms+sinnz−sinns≤m+nz−s,for every s,z∈ℝ, and hence, (H3) holds. Then, an evolution family Qz,sz≥s generated by Az with
(58)Qz,sv=Tz−sexp∫sz−3+sinmν^+sinnν^dν^v.SinceQz,s≤e−z−s, then it is very clear to know that Az fulfill the hypothesis (H4) with K0=1,β^=1. At the same time, we define
(59)C1~zu=c1z,xu,forallz∈ℝ,C2~zu=c2z,xu,forallz∈ℝ.c1z,.,c2z,.:ℝ×0,1↦ℝ are continuous and almost periodic in z∈ℝ uniformly in x∈0,1. Then, Equation (53) can be represented as the abstract equation (1).Study the measureμ with respect to its Radon-Nikodym derivative of
(60)ρz=ez,ifz≤0,1,ifz>0. Then, μ∈N satisfy H1 and H2. If we take the assumption that ψ,g satisfy (H6) and (H7), therefore, all assumptions of Theorem 22 are fulfilled; thus, the evolution equation (53) has a unique μ-pseudo almost periodic solution if Lψ,L3 is sufficiently small.


## 5. Conclusions and the Future Work

The primary objective of this research is to find out whether there is uniqueμ-pseudo almost periodic solution to a functional evolution equation with Stepanov forcing components in a Banach space using measure theory, evolution family, and Hölder inequality. The Lipschitz condition and the contraction mapping theory are used to develop certain suitable conditions that guarantee whether there is unique μ-pseudo almost periodic solutions to the problem. As a future work, we intend to study whether there is unique Stepanov-like pseudo almost periodic solution of neural network and discrete equation. We will also use the Hölder inequality to consider whether there is a unique Stepanov-like pseudo almost periodic solution of discrete equation. Almost periodic problems play a very important basic role in engineering applications and mathematical research.



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