Abstract
In this work, we present a new composition theorem of \(\mu \)-pseudo almost automorphic functions in the sense of Stepanov satisfying some local Lipschitz conditions. Using this results, we establish an existence result of \(\mu \)-pseudo almost automorphic solutions for some nonautonomous neutral partial evolution equation with Stepanov \(\mu \)-pseudo almost automorphic nonlinearity. An example is shown to illustrate our results.
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Introduction
In this work, we give a new composition theorem of \(\mu \)-pseudo almost automorphic functions in the sense of Stepanov, we suppose that the coefficient function satisfies some local Lipschitz conditions.
Then, we study the existence and uniqueness of \(\mu \)-pseudo almost automorphic mild solutions to the following nonautonomous neutral partial evolution equation:
where \(A(t)\) generates an hyperbolic evolution family \((U(t,s))_{t\ge s}, f:\mathbb {R}\times \mathbb {X}\rightarrow \mathbb {X}\) is a \(\mu \)-pseudo almost automorphic function and \(g:\mathbb {R}\times \mathbb {X}\rightarrow \mathbb {X}\) is Stepanov \(\mu \)-pseudo almost automorphic.
Pseudo almost periodic and automorphic functions have many applications in several problems like functional differential equations, integral equations and partial differential equations. The concept of almost automorphy was first introduced in the literature by Bochner [6], as a natural generalization of the almost periodicity. The notion of weighted pseudo almost automorphy has been introduced for the first time by Blot et al. [3]. More recently, using the measure theory, Blot, Cieutat and Ezzinbi [4] introduced the concept of \(\mu \)-pseudo almost automorphy, as a natural generalization of the classical concept of weighted pseudo almost automorphy. On the other hand, since the work [20] by N’Guéréekata and Pankov, Stepanov-like almost automorphic problems have widely been investigated.
The existence and uniqueness of pseudo almost periodic mild solutions to differential equations in Banach spaces has attracted many researchers [10, 14]. This led many authors to develop composition theorems of such functions [5, 11, 22].
In a recent paper [16], authors gave a result on the existence and uniqueness of pseudo almost periodic solution for the nonautonomous evolution equation (1), where the input function \(g\) is \(S^{p}\)-pseudo almost periodic. For contributions on nonautonomous evolution equations in Banach spaces, see [1, 16, 17].
In this paper, motivated by [4, 5, 15, 16], we use the measure theory to define a Stepanov-ergodic function, we study the composition of \(\mu \)-pseudo almost automorphic functions in the sense of Stepanov and we give a result of existence of \(\mu \)-pseudo almost automorphic mild solution of (1).
The organization of this work is as follows. In “Preliminaries” section, we introduce the basic notations and recall the definition of \(\mu \)-pseudo almost automorphic functions introduced in [5], we also give the new concept of \(S^{p}-\mu \)-pseudo almost automorphic functions and we investigate some properties. We present different composition theorems of Stepanov \(\mu \)-pseudo almost automorphic function in “Composition Theorems” section. In “Evolution Family and Exponential Dichotomy” section, we introduce the basic notations of evolution family and exponential dichotomy. “Pseudo Almost Automorphic Mild Solutions” section is devoted to study the existence and uniqueness of \(\mu \)-pseudo almost automorphic mild solutions of (1). As an illustration, an example of neutral heat equation with \(S^{p}{-}\mu \)-pseudo almost automorphic coefficients is studied under Dirichlet conditions.
Preliminaries
Pseudo Almost Automorphic Functions
Let \((\mathbb {X},\Vert .\Vert )\;\text{ and }\;(\mathbb {Y},\Vert .\Vert )\) be two Banach spaces, and \(BC(\mathbb {R},\mathbb {X})\) (respectively, \(BC(\mathbb {R}\times \mathbb {Y},\mathbb {X})\)) be the space of bounded continuous functions \(f:\mathbb {R}\longrightarrow \mathbb {X}\) (respectively, \(f:\mathbb {R}\times \mathbb {Y}\longrightarrow \mathbb {X}\)). \(BC(\mathbb {R},\mathbb {X})\) equipped with the supremum norm
is a Banach space.
Definition 2.1
[19] A continuous function \(f:\mathbb {R}\mapsto \mathbb {X}\) is said to be almost automorphic if for every sequence of real numbers \((s'_{n})_{n\in \mathbb {N}}\) there exists a subsequence \((s_{n})_{n\in \mathbb {N}}\subset (s'_{n})_{n\in \mathbb {N}}\) and a measurable function \(g\), such that
is well defined for each \(t\in \mathbb {R}\), and
for each \(t\in \mathbb {R}\).
Let \(AA(\mathbb {R},\mathbb {X})\) be the set of all almost automorphic functions from \(\mathbb {R}\) to \(\mathbb {X}\). Then \((AA(\mathbb {R},\mathbb {X}),\Vert .\Vert _{\infty })\) is a Banach space.
Definition 2.2
[13] A continuous function \(f:\mathbb {R}\times \mathbb {R}\mapsto \mathbb {X}\) is said to be bi-almost automorphic if for every sequence of real numbers \((s'_{n})_{n\in \mathbb {N}}\) there exists a subsequence \((s_{n})_{n\in \mathbb {N}}\subset (s'_{n})_{n\in \mathbb {N}}\) and a measurable function \(g\), such that
is well defined for each \(\left( t,s\right) \in \mathbb {R}^{2}\), and
for each \(\left( t,s\right) \in \mathbb {R}^{2}\). Denote by \(bAA(\mathbb {X})\) the set of all such functions.
In what follows, we give the new concept of the ergodic functions developed in [4], which is a generalization of the ergodicity given in [7, 8].
We denote by \(\mathcal {B}\) the Lebesgue \(\sigma \)-field of \(\mathbb {R}\) and by \(\mathcal {M}\) the set of all positive measures \(\mu \) on \(\mathcal {B}\) satisfying \(\mu (\mathbb {R})=+\infty \) and \(\mu ([a,b])<\infty \) for all \(a,b\in \mathbb {R}\) \((a\le b)\).
Definition 2.3
[4] Let \(\mu \in \mathcal {M}\). A bounded continuous function \(f:\mathbb {R}\rightarrow \mathbb {X}\) is said to be \(\mu \)-ergodic if
We denote the space of all such functions by \(\mathcal {E}(\mathbb {R},\mathbb {X},\mu )\).
Definition 2.4
[5] Let \(\mu \in \mathcal {M}\). A continuous function \(f:\mathbb {R}\rightarrow \mathbb {X}\) is said to be \(\mu \)-pseudo almost automorphic if it is written in the form
where \(g\in AA(\mathbb {R},\mathbb {X})\) and \(h\in \mathcal {E}(\mathbb {R},\mathbb {X},\mu )\). The collection of such functions will be denoted by \(\textit{PAA}(\mathbb {R},\mathbb {X},\mu )\).
Theorem 2.5
[4] Let \(\mu \in \mathcal {M}\). Then \((\mathcal {E}(\mathbb {R},\mathbb {X},\mu ),\Vert .\Vert _{\infty })\) is a Banach space.
For \(\mu \in \mathcal {M}\), we formulate the following hypothesis:
- (M1) :
-
\(\limsup \limits _{r\longrightarrow \infty }\frac{2r}{\mu \left[ -r,r\right] }<\infty .\)
- (M2) :
-
For all \(\tau \in \mathbb {R}\), there exist \(\beta >0\) and a bounded interval \(I\) such that
$$\begin{aligned} \mu (\{a+\tau :\; a\in A\})\le \beta \mu (A)\quad \text{ when }\; A\in \mathcal {B}\; \text{ satisfies }\; A\cap I=\emptyset . \end{aligned}$$
The hypothesis (M2) is given in [4].
Definition 2.6
[4] Let \(\mu _{1},\mu _{2}\in \mathcal {M}\). \(\mu _{1}\) is said to be equivalent to \(\mu _{2}\), if there exist constants \(\alpha ,\beta >0\) and a bounded interval \(I\) (eventually \(I=\emptyset \)) such that
Remark
If \(\mu \) is equivalent to the Lebesgue measure, then \(\mu \) satisfies (M1).
Theorem 2.7
[5] Let \(\mu \in \mathcal {M}\) satisfy (M2). Then the space \((\textit{PAA}(\mathbb {R},\mathbb {X},\mu ),\Vert .\Vert _{\infty })\) is a Banach space.
Theorem 2.8
[5] Let \(\mu \in \mathcal {M}\) satisfy (M2). Then \(\textit{PAA}(\mathbb {R},\mathbb {X},\mu )\) is translation invariant that is, if \(f\in \textit{PAA}(\mathbb {R},\mathbb {X},\mu )\) then \(f_{\tau }:=f(.+\tau )\in \textit{PAA}(\mathbb {R},\mathbb {X},\mu )\) for all \(\tau \in \mathbb {R}\).
Definition 2.9
[5] A continuous function \(f:\mathbb {R}\times \mathbb {Y}\rightarrow \mathbb {X}\) is said to be almost automorphic if \(f\left( .,x\right) \in AA(\mathbb {R},\mathbb {X})\), for all \(x\in \mathbb {Y}\). The collection of such functions is denoted by \(AA(\mathbb {R}\times \mathbb {Y},\mathbb {X})\).
Definition 2.10
[5] Let \(\mu \in \mathcal {M}\). A continuous function \(f:\mathbb {R}\times \mathbb {Y}\rightarrow \mathbb {X}\) is said to be \(\mu \)-ergodic if \(f\left( .,x\right) \in \mathcal {E}(\mathbb {R},\mathbb {X},\mu )\), for all \(x\in \mathbb {Y}\). The collection of such functions is denoted by \(\mathcal {E}(\mathbb {R}\times \mathbb {Y},\mathbb {X},\mu )\).
Definition 2.11
[5] Let \(\mu \in \mathcal {M}\). A continuous function \(f:\mathbb {R}\times \mathbb {Y}\rightarrow \mathbb {X}\) is said to be \(\mu \)-pseudo almost automorphic if it is written in the form
where \(g\in AA(\mathbb {R}\times \mathbb {Y},\mathbb {X})\) and \(h\in \mathcal {E}(\mathbb {R}\times \mathbb {Y},\mathbb {X},\mu )\). The collection of such functions is denoted by \(\textit{PAA}(\mathbb {R}\times \mathbb {Y},\mathbb {X},\mu )\).
Pseudo Almost Automorphy in the Sense of Stepanov
Definition 2.12
[18] The Bochner transform \(f^{b}(t,s)\;\text{ for }\;t\in \mathbb {R}\;\text{ and }\;s\in [0,1]\) of a function \(f:\mathbb {R}\longrightarrow \mathbb {X}\) is defined by
Definition 2.13
[18] Let \(1\le p<\infty \). The space \(BS^{p}(\mathbb {R},\mathbb {X})\) of all Stepanov bounded (or \(S^{p}\)-bounded) functions with the exponent \(p\) consists of all measurable functions \(f\) on \(\mathbb {R}\) with value in \(\mathbb {X}\) such that \(f^{b}\in L^{\infty }\left( \mathbb {R},L^{p}\left( \left( 0,1\right) ,\mathbb {X}\right) \right) \). This is a Banach space with the norm
Remark
A function \(f\in L_{\mathrm{loc}}^{p}(\mathbb {R},\mathbb {X})\) is Stepanov bounded if
It is obvious that
Definition 2.14
[18] A function \(f\in BS^{p}(\mathbb {R},\mathbb {X})\), is said to be almost automorphic in the sense of Stepanov (or \(S^{p}\)-almost automorphic) if for every sequence of real numbers \((s'_{n})_{n\in \mathbb {N}}\) there exist a subsequence \((s_{n})_{n\in \mathbb {N}}\subset (s'_{n})_{n\in \mathbb {N}}\) and a function \(g\in L_{loc}^{p}(\mathbb {R},\mathbb {X})\) such that
and
as \(n\rightarrow +\infty \) pointwise on \(\mathbb {R}\). The collection of such functions will be denoted by \(AA^{p}(\mathbb {R},\mathbb {X})\).
In other words, a function \(f\in L_{loc}^{p}(\mathbb {R},\mathbb {X})\) is said to be \(S^{p}\)-almost automorphic if its Bochner transform \(f^{b}:\mathbb {R}\longrightarrow L^{p}([0,1],\mathbb {X})\) is almost automorphic.
We introduce the following notion of ergodicity:
Definition 2.15
Let \(\mu \in \mathcal {M}\). A function \(f\in BS^{p}(\mathbb {R},\mathbb {X})\), is said to be \(S^{p}\)-\(\mu \)-ergodic if
The collection of such functions is denoted by \(\mathcal {E}^{p}(\mathbb {R},\mathbb {X},\mu )\).
Definition 2.16
Let \(\mu \in \mathcal {M}\). A function \(f\in BS^{p}(\mathbb {R},\mathbb {X})\), is said to be \(S^{p}\)-\(\mu \)-pseudo almost automorphic if it can be decomposed as \(f=g+\phi \), where \(g\in AA^{p}(\mathbb {R},\mathbb {X})\) and \(\phi \in \mathcal {E}^{p}(\mathbb {R},\mathbb {X},\mu )\). The collection of such functions is denoted by \(\textit{PAA}^{p}(\mathbb {R},\mathbb {X},\mu )\).
Theorem 2.17
Let \(\mu \in \mathcal {M}\) satisfy (M2). If \(f\in \mathcal {E}(\mathbb {R},\mathbb {X},\mu )\), then \(f\in \mathcal {E}^{p}(\mathbb {R},\mathbb {X},\mu )\) for all \(p>1\).
Proof
Let \(f\in \mathcal {E}(\mathbb {R},\mathbb {X},\mu )\), since \(\mu \) is a \(\sigma \)- finite measure, then by Hölder inequality and Fubini’s theorem
Hence
Since \(\mathcal {E}(\mathbb {R},\mathbb {X},\mu )\) is invariant by translation, then
for all \(s\in [0,1]\). The Lebesgue Dominated Convergence Theorem implies that
\(\square \)
Corollary 2.18
Let \(\mu \in \mathcal {M}\) satisfy (M2). If \(f\in \textit{PAA}(\mathbb {R},\mathbb {X},\mu )\), then \(f\in \textit{PAA}^{p}(\mathbb {R},\mathbb {X},\mu )\) for all \(p>1\).
Theorem 2.19
Let \(\mu \in \mathcal {M}\) satisfy (M2). Then \(\textit{PAA}^{p}(\mathbb {R},\mathbb {X},\mu )\) is invariant by translation, that is, \(f\in \textit{PAA}^{p}(\mathbb {R},\mathbb {X},\mu )\) implies \(f_{\tau }\in PAA^{p}(\mathbb {R},\mathbb {X},\mu )\), for all \(\tau \in \mathbb {R}\).
Proof
It suffices to show that \(\mathcal {E}^{p}(\mathbb {R},\mathbb {X},\mu )\) is invariant by translation. Let \(f\in \mathcal {E}^{p}(\mathbb {R},\mathbb {X},\mu )\) and \(F\left( t\right) =\big (\int _{t}^{t+1}\left\| f\left( s\right) \right\| ^{p}ds\big )^{\frac{1}{p}}\), then \(F\in \mathcal {E}\left( \mathbb {R},\mathbb {R},\mu \right) \), but since \(\mathcal {E}\left( \mathbb {R},\mathbb {R},\mu \right) \) is invariant by translation [4], then
We deduce that \(f\left( .+a\right) \in \mathcal {E}^{p}(\mathbb {R},\mathbb {X},\mu )\). \(\square \)
Definition 2.20
Let \(AA{}^{p}(\mathbb {R}\times \mathbb {Y},\mathbb {X})\) denote the space of functions \(f:\mathbb {R}\times \mathbb {Y}\rightarrow \mathbb {X}\) such that \(f(.,y)\in AA{}^{p}(\mathbb {R},\mathbb {X})\), for each \(y\in \mathbb {Y}\), \(\mathcal {E}^{p}(\mathbb {R\times \mathbb {Y}},\mathbb {X},\mu )\) denote the space of functions \(f:\mathbb {R}\times \mathbb {Y}\rightarrow \mathbb {X}\) such that \(f(.,y)\in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \), for each \(y\in \mathbb {Y}\). Let us set
Now we introduce the space of \(\mu \)-Stepanov bounded functions:
Definition 2.21
Let \(1\le p<\infty \). The space \(BS^{p}(\mathbb {R},\mathbb {X},\mu )\) of all \(\mu \)-Stepanov bounded (or \(\mu -S^{p}\)-bounded) functions with the exponent \(p\) consists of all measurable functions \(f\) on \(\mathbb {R}\) with value in \(\mathbb {X}\) such that
Remark
A function \(f\in L_{\mathrm{loc}}^{p}(\mathbb {R},\mathbb {X},\mu )\) is \(\mu -S^{p}\) bounded if
It is obvious that
Let \(\mu \in \mathcal {M}\), we introduce the following hypothesis:
- (M3) :
-
\(\sup \limits _{t\in \mathbb {R}}\mu \left[ t,t+1\right] <\infty .\)
Example
If \(\mu \) is absolutely continuous with respect to Lebesgue measure with a bounded Radon–Nikodym derivative, then (M3) naturally holds.
Proposition 2.22
Let \(\mu \in \mathcal {M}\) satisfy (M3), then constant functions belong to \(BS^{p}(\mathbb {R},\mathbb {X},\mu )\).
Proof
Let \(f\left( s\right) =M\) be a constant function. Then
\(\square \)
Composition Theorems
Definition 3.1
Let \(UC\left( \mathbb {R}\times \mathbb {\mathbb {X}},\mathbb {X}\right) \) denote the set of all uniformly continuous functions \(f:\mathbb {R}\times \mathbb {X}\longrightarrow \mathbb {X}\), i.e for each compact set \(K\) in \(\mathbb {X}\) and for each \(\varepsilon >0,\) there exists \(\delta >0\) such that
for all \(t\in \mathbb {R}\) and \(u,v\in K\) with \(\left\| u-v\right\| \le \delta \).
Definition 3.2
Let \(UC^{p}\left( \mathbb {R}\times \mathbb {\mathbb {X}},\mathbb {X}\right) \) denote the set of all \(BS^{p}\)-uniformly continuous functions \(f:\mathbb {R}\times \mathbb {X}\longrightarrow \mathbb {X}\), i.e there is a non-negative function \(L\in BS^{1}\left( \mathbb {R},\mathbb {R},\mu \right) \) such that for each compact set \(K\) in \(\mathbb {X}\) and for each \(\varepsilon >0,\) there exists \(\delta >0\) such that
for all \(t\in \mathbb {R}\) and \(u,v\in K\) with \(\left\| u-v\right\| \le \delta \).
Lemma 3.3
[13] Assume that
If \(u\in AA^{p}\left( \mathbb {R},\mathbb {X}\right) \) and \(K=\overline{\{u\left( t\right) :\;t\in \mathbb {R}\}}\) is compact, Then
Lemma 3.4
Assume that \(\alpha (.)\in AA^{p}\left( \mathbb {R},\mathbb {X}\right) ,\) \(K=\overline{\{\alpha \left( t\right) :\,\, t\in \mathbb {R}\}}\) is a compact subset of \(\mathbb {X}\), \(h\in \mathcal {E}^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X},\mu \right) \cap UC^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and let \(\mu \in \mathcal {M}\) satisfy (M1). Then \(h\left( .,\alpha \left( .\right) \right) \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \).
Proof
For any fixed \(\varepsilon >0\), let \(\delta >0\) such that (3) holds. Then there exist \(\alpha _{1}\dots \alpha _{k}\in K\) such that
For each \(t\in \mathbb {R}\), there exists \(\alpha _{i\left( t\right) }\), \(1\le i\left( t\right) \le k\) such that \(\left\| \alpha \left( t\right) -\alpha _{i\left( t\right) }\right\| \le \delta \). Then we get
which gives
Noting that \(h(.,\alpha _{i})\in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) ,\) \(i=1\dots k,\) and using hypothesis (M1) one has
Therefore,
i.e., \(h\left( .,\alpha \left( .\right) \right) \in \mathcal {E}^{p} \left( \mathbb {R},\mathbb {X},\mu \right) \). \(\square \)
Remark
If \(\mu \) is absolutely continuous with respect to the Lebesgue measure with a Radon Nikodym derivative \(\rho \), then (M1) which was used in Lemma 3.4 is equivalent to the condition:
A similar result was given in [22], Lemma3.1] if \(\rho \) satisfies
However, if for example \(\rho (t)=e^{t}\), one cannot apply [22], Lemma3.1] since the condition (5) is not verified, in fact:
While the condition (4) holds since
Another example where one cannot apply [22], Lemma 3.1] is when \(\mu \) has a Radon–Nikodym derivative \(\rho \) defined as follows:
One has
In fact
and
Therefore
and then \(\mu \in \mathcal {M}\). In addition \(\mu \) satisfies (M3), in fact for \(t\ge 0\) we have
it follows that
On one hand by (7), the condition (4) holds since
On other hand the condition (5) does not hold, in fact
Then
Therefore
Lemma 3.5
Let \(\mu \in \mathcal {M}\) and \(f\in BS^{p}\left( \mathbb {R},\mathbb {X}\right) ,\) then \(f\in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \) if and only if for any \(\varepsilon >0\)
Proof
Since \(t\longrightarrow \left( \int _{t}^{t+1}\left\| f\left( s\right) \right\| ^{p}ds\right) ^{\frac{1}{p}}\in \mathcal {E}\left( \mathbb {R},\mathbb {X},\mu \right) \), then Lemma 3.5 is a direct result of [4], Theorem2.13]. \(\square \)
Theorem 3.6
Let \(\mu \in \mathcal {M}\) and \(f=g+h\in PAA^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X},\mu \right) \) with \(g\in AA^{p}(\mathbb {R}\times \mathbb {X},\mathbb {X})\cap UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and \(h\in \mathcal {E}^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X},\mu \right) .\) We suppose that there exists a non-negative function \(L\left( .\right) \in BS^{1}\left( \mathbb {R},\mathbb {R},\mu \right) \) with \(p>1\) such that for all \(u,v\in L_{loc}^{p}\left( \mathbb {R},\mathbb {X}\right) \) and \(t\in \mathbb {R},\)
Assume that \(\mu \) satisfies (M1)–(M3).
If \(x=\alpha +\beta \in PAA^{p}\left( \mathbb {R},\mathbb {X},\mu \right) ,\) with \(\alpha \in AA^{p}\left( \mathbb {R},\mathbb {X}\right) \), \(\beta \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \) and \(K=\overline{\left\{ \alpha \left( t\right) :\, t\in \mathbb {R}\right\} }\) is compact, then \(f\left( .,x\left( .\right) \right) \in PAA^{p}\left( \mathbb {R},\mathbb {X},\mu \right) .\)
Proof
We have the following decomposition
where \(G\left( t\right) =g\left( t,\alpha \left( t\right) \right) \), \(F\left( t\right) =f\left( t,x\left( t\right) \right) -f\left( t,\alpha \left( t\right) \right) \) and \(H\left( t\right) =h\left( t,\alpha \left( t\right) \right) \). Since \(g\in UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and \(K=\overline{\left\{ \alpha \left( t\right) :\, t\in \mathbb {R}\right\} }\) is compact, it follows from Lemma 3.3 that \(g\left( t,\alpha \left( t\right) \right) \in AA^{p}\left( \mathbb {R},\mathbb {X}\right) \). First we prove that \(F\left( .\right) \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \). By Lemma 3.5 we have for all \(\varepsilon >0\)
where
Let \(\varepsilon >0\), we have
Therefore
Thus \(F\left( .\right) \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \).
Next we prove that \(H\left( .\right) \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \). From (8), we can see that \(f\in UC^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \). Using Proposition 2.22, it is easy to see that \(g\in UC^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and then \(h=f-g\in UC^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \). It follows from Lemma 3.4 that \(h\left( .,\alpha \left( .\right) \right) \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \). \(\square \)
Theorem 3.7
Let \(\mu \in \mathcal {M}\) and \(f=g+h\in PAA^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X},\mu \right) \), \(p>1\) with \(g\in AA^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \cap UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \), \(h\in \mathcal {E}^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X},\mu \right) .\) We suppose that there exists a non negative function \(L\left( .\right) \in BS^{r}\left( \mathbb {R},\mathbb {R}\right) \cap BS^{1}\left( \mathbb {R},\mathbb {R},\mu \right) \) with \(r\ge \max \big \{ p,\frac{p}{p-1}\big \} \) such that for all \(u,v\in \mathbb {X}\) and \(t\in \mathbb {R},\)
Assume that \(\mu \) satisfies (M1)–(M3).
If \(x=\alpha +\beta \in PAA^{p}\left( \mathbb {R},\mathbb {X},\mu \right) ,\) with \(\alpha \in AA^{p}\left( \mathbb {R},\mathbb {X}\right) \), \(\beta \in \mathcal {E}^{p}\left( \mathbb {R},\mathbb {X},\mu \right) \) and \(K=\overline{\left\{ \alpha \left( t\right) :\, t\in \mathbb {R}\right\} }\) is compact, then there exists \(q\in [1,p)\) such that \(f\left( .,x\left( .\right) \right) \in PAA^{q}\left( \mathbb {R},\mathbb {X},\mu \right) .\)
Proof
Since \(r\ge \frac{p}{p-1},\) there exists \(q\in [1,p)\) such that \(r=\frac{pq}{p-q}\). Let
Then \(p',q'>1\) and \(\frac{1}{p'}+\frac{1}{q'}=1\).
We have the following decomposition
where \(G\left( t\right) =g\left( t,\alpha \left( t\right) \right) \), \(F\left( t\right) =\left( t,x\left( t\right) \right) -f\left( t,\alpha \left( t\right) \right) \) and \(H\left( t\right) =h\left( t,\alpha \left( t\right) \right) \). Since \(g\in UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and \(K=\overline{\left\{ \alpha \left( t\right) :\, t\in \mathbb {R}\right\} }\) is compact, it follows from Lemma 3.3 that \(g\left( t,\alpha \left( t\right) \right) \in AA^{p}\left( \mathbb {R},\mathbb {X}\right) \). First we prove that \(F\left( .\right) \in \mathcal {E}^{q}\left( \mathbb {R},\mathbb {X},\mu \right) \).
Therefore
Thus \(F\left( .\right) \in \mathcal {E}^{q}\left( \mathbb {R},\mathbb {X},\mu \right) \).
Next we prove that \(H\left( .\right) \in \mathcal {E}^{q}\left( \mathbb {R},\mathbb {X},\mu \right) \). We have
and \(g\in UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \). Then using Proposition 2.22, it is easy to see that \(f,g\in UC^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and then \(h=f-g\in UC^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \). It follows from Lemma 3.4 that \(h\left( .,\alpha \left( .\right) \right) \in \mathcal {E}^{p} \left( \mathbb {R},\mathbb {X},\mu \right) \subset \mathcal {E}^{q}\left( \mathbb {R},\mathbb {X},\mu \right) \). \(\square \)
Evolution Family and Exponential Dichotomy
Definition 4.1
[12, 21] A family of bounded linear operators \((U(t,s))_{t\ge s}\), on a Banach space \(\mathbb {\mathbb {X}}\) is called a strongly continuous evolution family if
-
1.
\(U(t,r)U(r,s)=U(t,s)\) and \(U(s,s)=I\), for all \(t\ge r\ge s\) and \(t,r,s\in \mathbb {R}\),
-
2.
The map \((t,s)\rightarrow U(t,s)x\) is continuous for all \(x\in \mathbb {X}\), \(t\ge s\) and \(t,s\in \mathbb {R}\).
Definition 4.2
[12, 21] An evolution family \((U(t,s))_{t\ge s}\) on a Banach space \(\mathbb {X}\) is called hyperbolic (or has exponential dichotomy) if there exist projections \(P(t),t\in \mathbb {R}\), uniformly bounded and strongly continuous in \(t\), and constants \(M>0\), \(\delta >0\) such that
-
1.
\(U(t,s)P(s)=P(t)U(t,s)\), for \(t\ge s\) and \(t,s\in \mathbb {R}\),
-
2.
The restriction \(U_{Q}(t,s):Q(s)\mathbb {X}\rightarrow Q(t)\mathbb {X}\) of \(U(t,s)\) is invertible for \(t\ge s\) and \(t,s\in \mathbb {R}\) ( and we set \(U_{Q}(t,s)=U(s,t)^{-1})\).
-
3.
3.
$$\begin{aligned} \Vert U(t,s)P(s)\Vert \le Me^{-\delta (t-s)} \end{aligned}$$(9)and
$$\begin{aligned} \Vert U_{Q}(s,t)Q(t)\Vert \le Me^{-\delta (t-s)}, \end{aligned}$$(10)for \(t\ge s\) and \(t,s\in \mathbb {R}\).
Here and below we set \(Q:=I-P\).
Definition 4.3
Given a hyperbolic evolution family, we define its so-called Green’s function by
Pseudo Almost Automorphic Mild Solutions
In this section, we investigate the existence and uniqueness of \(\mu \)-pseudo almost automorphic mild solutions of Eq. (1).
Before starting our main result in this section, we recall the definition of the mild solution to Eq. (1) and we make the following assumptions:
- (H0) :
-
There exist constants \(\lambda _{0}\ge 0\), \(\theta \in (\frac{\pi }{2},\pi )\), \(L,K\ge 0\), and \(\alpha ,\beta \in (0,1]\) with \(\alpha +\beta >1\) such that
$$\begin{aligned} \Sigma _{\theta }\cup \{0\}\subset \rho (A(t)-\lambda _{0}),\ \ \ \ \Vert R(\lambda ,A(t)-\lambda _{0})\Vert \le \frac{K}{1+|\lambda |} \end{aligned}$$and
$$\begin{aligned} \left\| \left( A(t)-\lambda _{0}\right) R\left( \lambda ,A(t)-\lambda _{0}\right) \left[ R(\lambda _{0},A(t))-R(\lambda _{0},A(s)\right] \right\| \le L|t-s|^{\alpha }|\lambda |^{-\beta }, \end{aligned}$$for \(t,s\in \mathbb {R}\) and \(\lambda \in \Sigma _{\theta }:=\{\lambda \in \mathbb {C}\setminus \{0\},|\arg \lambda |\le \theta \}\).
- (H1) :
-
The evolution family \((U(t,s))_{t\ge s}\) generated by \(A(t)\) has an exponential dichotomy with constants \(M>0\), \(\delta >0\), dichotomy projections \(P(t),\ t\in \mathbb {R}\) and Green’s function \(\Gamma (t,s)\).
- (H2) :
-
\(t\rightarrow R(\lambda _{0},A(t))\in AA(\mathbb {R},B(\mathbb {X}))\).
We point out that assumption (H0) is usually called “Acquistapace-Terreni” condition, which was firstly introduced in [1] and widely used to investigate nonautonomous evolution equations.
- (H3) :
-
\(f=f_{1}+f_{2}\in PAA(\mathbb {R}\times \mathbb {X},\mathbb {X},\mu )\), with \(f_{1}\in AA\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \cap UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and \(f_{2}\in \mathcal {E}(\mathbb {R}\times \mathbb {X},\mathbb {X},\mu )\). Assume that \(f\) is bounded on \(\mathbb {R}\times B\) for each bounded subset \(B\) of \(\mathbb {X}\) and there exists a constant \(L_{f}\) such that for all \(u,v\in \mathbb {\mathbb {X}}\) and for all \(t\in \mathbb {R}\):
$$\begin{aligned} \Vert f(t,u)-f(t,v)\Vert \le L_{f}\Vert u-v\Vert . \end{aligned}$$ - (H4) :
-
\(g=g_{1}+g_{2}\in PAA{}^{p}(\mathbb {R}\times \mathbb {\mathbb {X}},\mathbb {\mathbb {X}},\mu )\), with \(g_{1}\in AA^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \cap UC\left( \mathbb {R}\times \mathbb {X},\mathbb {X}\right) \) and \(g_{2}\in \mathcal {E}^{p}\left( \mathbb {R}\times \mathbb {X},\mathbb {X},\mu \right) \). Assume that there exists a non-negative function \(L\in BS^{r}\left( \mathbb {R},\mathbb {R}\right) \cap BS^{1}\left( \mathbb {R},\mathbb {R},\mu \right) \) with \(r\ge \max \left\{ p,\frac{p}{p-1}\right\} \) such that for all \(u,v\in \mathbb {\mathbb {X}}\) and for all \(t\in \mathbb {R}\):
$$\begin{aligned} \Vert g(t,u)-g(t,v)\Vert \le L\left( t\right) \Vert u-v\Vert . \end{aligned}$$
Definition 5.1
A mild solution to Eq. (1) is a continuous function \(u:\mathbb {R}\rightarrow \mathbb {\mathbb {X}}\) satisfying
Theorem 5.2
[13] Let assumptions (H0)–(H1) hold and \(u\) be a bounded mild solution of (1) on \(\mathbb {R}\), then for all \(t\in \mathbb {R}\)
Lemma 5.3
[2] Assume that (H0)–(H2) hold. Then \(\Gamma \in bAA\left( \mathbb {X}\right) \).
Theorem 5.4
Let \(\mu \in \mathcal {M}\) satisfy (M2). Assume that (H0)–(H2) hold, if \(h\in PAA^{p}(\mathbb {R},\mathbb {\mathbb {X}},\mu ),\) for \(p>1\), then
belongs to \(PAA(\mathbb {R},\mathbb {\mathbb {X}},\mu )\).
Proof
Since \(h\in PAA^{p}(\mathbb {R},\mathbb {\mathbb {X}},\mu )\), we can write \(h=h_{1}+h_{2}\), where \(h_{1}\in AA^{p}(\mathbb {R},\mathbb {\mathbb {X}})\) and \(h_{2}\in \mathcal {E}^{p}(\mathbb {R},\mathbb {\mathbb {X}},\mu )\). By [13] and using Lemma 5.3, we have \(\int _{\mathbb {R}}\Gamma \left( t,s\right) h_{1}\left( s\right) ds\in AA(\mathbb {R},\mathbb {X})\). To complete the proof, we will prove that \(\int _{\mathbb {R}}\Gamma \left( t,s\right) h_{2}\left( s\right) ds\in \mathcal {E}(\mathbb {R},\mathbb {\mathbb {X}},\mu )\). Let us consider for each \(t\in \mathbb {R}\) and \(n\in \mathbb {N}\):
We have
Multiply both sides of the inequality by \(\frac{1}{\mu ([-r,r])}\) and integrating, we obtain
Since \(h_{2}\in \mathcal {E}^{p}(\mathbb {R},\mathbb {\mathbb {X}},\mu )\) and \(\mu \) satisfies (M2), then by Theorem 2.19, \(\mathcal {E}^{p}(\mathbb {R},\mathbb {\mathbb {X}},\mu )\) is invariant by translation and the left side of the inequality goes to \(0\) when \(r\) goes to infinity. Therefore
From
we deduce that \(\sum _{n\ge 0}\Phi _{n}\) converges uniformly to
it follows that
Using the same argument, we show that
We conclude that \(\int _{\mathbb {R}}\Gamma \left( t,s\right) h_{2}\left( s\right) ds\in \mathcal {E}(\mathbb {R},\mathbb {\mathbb {X}},\mu )\). \(\square \)
Lemma 5.5
Let \(\mu \in \mathcal {M}\) satisfy (M1)–(M3). Assume that (H0)–(H4) hold. The operator \(\Lambda \) defined by
maps \(PAA(\mathbb {R},\mathbb {\mathbb {X}},\mu )\) to \(PAA(\mathbb {R},\mathbb {\mathbb {X}},\mu )\).
Proof
We can easily obtain this result from Theorems 3.7 and 5.4.\(\square \)
Theorem 5.6
Let \(\mu \in \mathcal {M}\) satisfy (M1)–(M3). Assume that (H0)–(H4) hold. Then Eq. (1) admits a unique \(\mu \)-pseudo almost automorphic mild solution if
where \(\dfrac{1}{r}+\dfrac{1}{r'}=1\).
Proof
Define the nonlinear operator \(\Lambda \) on \(BC(\mathbb {R},\mathbb {\mathbb {X}})\) by :
Let \(u\in PAA(\mathbb {R},\mathbb {\mathbb {X}},\mu )\), using Lemma 5.5 and [5], Theorem5.7], we deduce that \(\Lambda \) is well defined and maps \(PAA(\mathbb {R},\mathbb {\mathbb {X}},\mu )\) into itself. Let \(u,v\in PAA(\mathbb {R},\mathbb {\mathbb {X}},\mu )\). It follows that for each \(t\in \mathbb {R}\):
where
and
Since \(L\in BS^{r}\left( \mathbb {R},\mathbb {R}\right) \), we get
Similarly we have
Thus
By the well known contraction principle, we can show that \(\Lambda \) has a unique fixed point
which satisfies
\(\square \)
Application
Let \(\mu \) be a measure with a Radon–Nikodym derivative \(\rho \) defined by
Since
then by [4], \(\mu \) satisfies (M2). In addition, \(\mu \) satisfies (M1) since
The fact that the derivative \(\rho \) is bounded implies that \(\mu \) satisfies (M3).
To illustrate the above results we examine the existence of \(\mu \)-pseudo almost automorphic solutions to the following model:
with
and
where \(\alpha ,\, a,\, c:\mathbb {R}\rightarrow \mathbb {R}\) are almost automorphic functions such that \(\alpha (t)\le -M<0\), for all \(t\in \mathbb {R}\), \(b\in \mathcal {E}\left( \mathbb {R},\mathbb {R},\mu \right) \) and \(d\in \mathcal {E}^{2}\left( \mathbb {R},\mathbb {R},\mu \right) \). The functions \(\psi ,\varphi :\mathbb {R}\rightarrow \mathbb {R}\) are bounded Lipschitz continuous. It is clear that \(f\) belongs to \(PAA(\mathbb {R}\times \mathbb {R},\mathbb {R},\mu )\) and satisfies:
where \(L_{f}=L_{\psi }\left| a\right| _{\infty }+L_{\varphi }\left| b\right| _{\infty }\). We can see also that \(g\) belongs to \(PAA{}^{2}(\mathbb {R}\times \mathbb {R},\mathbb {R},\mu )\) and satisfies:
where \(L\left( t\right) =L_{\psi }\left| c\left( t\right) \right| +L_{\varphi }\left| d\left( t\right) \right| \). The boundedness of \(\rho \) implies that \(L\in BS^{2}\left( \mathbb {R},\mathbb {R}\right) \cap BS^{1}\left( \mathbb {R},\mathbb {R},\mu \right) \).
To represent the system (13) in the abstract form (1), we choose the space \(\mathbb {\mathbb {X}}=L^{2}\left( [0,\pi ],\mathbb {R}\right) \), endowed with its natural topology. We also consider the operator \(A:D(A)\subset \mathbb {\mathbb {X}}\longrightarrow \mathbb {\mathbb {X}}\), given by
where
Let us set for \(t\in \mathbb {R}\) and \(\xi \in \left[ 0,\pi \right] \):
Using (14) and (15), it is clear that \(F\) and \(G\) satisfies (H3) and (H4) with \(p=r=2\). Moreover, it is well known ([9]) that \(A\) is the generator of an analytic \(C_{0}\)-semigroup \(\{T(t)\}_{t\ge 0}\) on \(\mathbb {\mathbb {X}}\) with \(\Vert T(t)\Vert \le e^{-t}\), for \(t\ge 0\).
Define a family of linear operators \(A(t)\) by:
Equation (13) takes the following abstract form
The operators \(A(t)\) generate an evolution family \((U(t,s))_{t\ge s}\) given by
with
It follows that \((U(t,s))_{t\ge s}\) has an exponential dichotomy. Let \((s_{n}^{\prime })_{n\ge 0}\) be a real sequence, then there is a subsequence \((s_{n})_{n\ge 0}\subseteq (s_{n}^{\prime })_{n\ge 0}\) and a real measurable function \(t\rightarrow \tilde{\alpha }(t)\) such that for all \(t\in \mathbb {R}\)
Consider \(\tilde{A}(t):=A+\tilde{\alpha }(t)\), then we have
It follows that
Similarly, we show that \(\big \Vert R\big (\lambda ,\tilde{A}\left( t-s_{n}\right) \big )-R\big (\lambda ,A\left( t\right) \big )\big \Vert \rightarrow 0.\) Therefore, the family \(A\left( t\right) \) satisfies (H2). Consequently all assumptions \(\mathbf (H0) {-}\mathbf (H4) \) are satisfied, by Theorem 5.6 we deduce that (13) has a unique \(\mu \)-pseudo almost automorphic mild solution on \(\mathbb {R}\), under the condition
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Akdad, AN., Essebbar, B. & Ezzinbi, K. Composition Theorems of Stepanov \(\mu \)-Pseudo Almost Automorphic Functions and Applications to Nonautonomous Neutral Evolution Equations. Differ Equ Dyn Syst 25, 397–416 (2017). https://doi.org/10.1007/s12591-015-0246-x
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DOI: https://doi.org/10.1007/s12591-015-0246-x
Keywords
- Pseudo almost automorphic solutions
- Stepanov almost automorphy
- Ergodic perturbations
- Neutral evolution equations