# Existence and Uniqueness of Pseudo Almost Automorphic Mild Solutions to Some Classes of Partial Hyperbolic Evolution Equations

Authors: Zhanrong Hu; Zhen Jin
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

We will establish an existence and uniqueness theorem of pseudo almost automorphic mild solutions to the following partial hyperbolic evolution equation(d/dt)[u(t)+f(t,Bu(t))]=Au(t)+g(t,Cu(t)),  t∈ℝ, under some assumptions. To illustrate our abstract result, a
concrete example is given.


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## Body

## 1. Introduction

Let(𝕏,∥⋅∥) be a Banach space, and let (𝕏α,∥⋅∥α) for 0<α<1 be an arbitrary abstract intermediate Banach
space between D(A) (domain of the operator A) and 𝕏.
In this paper, we deal with sufficient conditions for the existence and
uniqueness of pseudo almost automorphic mild solutions to
equation(1.1)ddt[u(t)+f(t,Bu(t))]=Au(t)+g(t,Cu(t)),t∈ℝ,where A is a sectorial linear operator on a Banach
space 𝕏 and σ(A)∩iℝ=∅, B,C are bounded linear operators on 𝕏α, f:ℝ×𝕏↦𝕏β, g:ℝ×𝕏↦𝕏 are jointly continuous functions. This turns
out to be a nontrivial problem due to the complexity and importance of pseudo
almost automorphic functions (e.g., see [1–8] and the references therein).Upon making some additional assumptions, it will be
shown that (1.1) admits a unique 𝕏α-valued pseudo almost automorphic mild
solution. Applications include the study of pseudo almost automorphic mild
solutions to some nonlinear heat equation:(1.2)ut(t,x)=uxx(t,x)+au(t,x)+g(t,q(x)u(t,x)),t∈ℝ,x∈[0,π],u(t,0)=u(t,π)=0,t∈ℝ, where a∈ℝ is a constant, and g,q are continuous functions.As a natural and important generalization of almost
automorphy as well as pseudo almost periodicity, pseudo almost automorphy has
recently been investigated (for more details, see [1–3]). Moreover, Liang et al.
in [2] established a
composition theorem about pseudo almost automorphic functions; Xiao et al.
in [3] obtained
sufficient conditions for the existence and uniqueness of pseudo almost
automorphic mild solutions to semilinear differential equations in Banach
spaces. In particular, they proved in [3] that the space (PAA(ℝ,𝕏),∥⋅∥∞) of pseudo almost automorphic functions,
endowed with sup norm, is a Banach space, thus pushed the door open to study pseudo
almost automorphic mild solutions to various differential equations.The method used here was firstly developed in
[9], in which Diagana established an existence and uniqueness theorem of pseudo almost
periodic mild solutions to (1.1) under some similar assumptions. Combining
bounded invariance theorem of pseudo almost automorphic functions established
in [6] with
composition theorem of pseudo almost automorphic functions and completeness of
the space (PAA(ℝ,𝕏),∥⋅∥∞),
this paper extends the results in [9], since such function as g(t,u)=usin(1/(2+cost+cos2t))+maxk∈ℤ{e−(t±k2)2}cosu,t∈ℝ,u∈𝕏 is not pseudo almost periodic, while they are
pseudo almost automorphic.It should be pointed out that one can also make use of
(1.1) to study several types of evolution equations including partial
functional differential equations, integro-differential equations, and reaction
diffusion equations.


## 2. Preliminaries

In this
section, we collect some preliminary facts from [9–11] that will be used later. Throughout this paper, ℕ, ℤ, ℝ,
and ℂ stand for the sets of positive integer,
integer, real and complex numbers; (𝕏,∥⋅∥), (𝕐,∥⋅∥𝕐) stand for Banach spaces. If A is a linear operator on 𝕏,
then ρ(A), σ(A), D(A), ker(A), R(A) stand for the resolvent set, spectrum, domain,
kernel, and range of A.
The space B(𝕏,𝕐) denotes the Banach space of all bounded linear
operators from 𝕏 into 𝕐 equipped with natural norm ∥A∥B(𝕏,𝕐)=supx∈𝕏,x≠0(∥Ax∥𝕐/∥x∥).
If 𝕐=𝕏,
it is simply denoted by B(𝕏) with ∥A∥B(𝕏)=supx∈𝕏,x≠0(∥Ax∥/∥x∥).


### 2.1. Sectorial Linear Operators and Analytic Semigroups

Definition 2.1.
A linear operatorA:D(A)⊂𝕏↦𝕏 (not necessarily densely defined) is said to
be sectorial if the following hold. There exist constants ω∈ℝ, θ∈(π/2,π),
and M>0 such that(2.1)ρ(A)⊃Sθ,ω:={λ∈ℂ:λ≠ω,|arg(λ−ω)|<θ},∥R(λ,A)∥≤M|λ−ω|,λ∈Sθ,ω,where R(λ,A)=(λI−A)−1 for each λ∈ρ(A).Remark 2.2.
IfA is sectorial, then it generates an analytic
semigroup (T(t))t≥0,
which maps (0,∞) into B(𝕏) such that there exist M0,M1>0 with(2.2)∥T(t)∥≤M0eωt,t>0,∥t(A−ωI)T(t)∥≤M1eωt,t>0.Definition 2.3.
A semigroup(T(t))t≥0 is said to be hyperbolic, if there exist a
projection P and constants M,δ>0 such that each T(t) commutes with P, ker(P) is invariant with respect to T(t), T(t):R(Q)→R(Q) is invertible and(2.3)∥T(t)Px∥≤Me−δt∥x∥fort≥0,∥T(t)Qx∥≤Meδt∥x∥fort≤0,where Q:=I−P and T(t):=(T(−t))−1 for t≤0.Recall that if
a semigroup (T(t))t≥0 is analytic, then (T(t))t≥0 is hyperbolic if and only if σ(A)∩iℝ=∅ (see, e.g., [11, Definition 1.14 and Proposition 1.15,
page 305]).


### 2.2. Intermediate Banach Spaces

Definition 2.4.
Letα∈(0,1).
A Banach space (𝕏α,∥⋅∥α) is said to be an intermediate space between D(A) and 𝕏,
if D(A)⊂𝕏α⊂𝕏 and there is a constant C>0 such that(2.4)∥x∥α≤C∥x∥1−α∥x∥Aα,x∈D(A),where ∥⋅∥A is the graph norm of A.Concrete examples of𝕏α include D(Aα) for α∈(0,1),
the domains of the fractional powers of A,
the real interpolation spaces DA(α,∞), α∈(0,1),
defined as follows:(2.5)DA(α,∞):={x∈𝕏:[x]α=sup0<t≤1∥t1−α(A−ωI)e−ωtT(t)x∥<∞},∥x∥α=∥x∥+[x]α,and the abstract Hölder spaces DA(α):=D(A)¯∥⋅∥α.Lemma 2.5.
For the hyperbolic analytic semigroup(T(t))t≥0,
there exist constants C(α)>0,  δ>0,M(α)>0,
and γ>0 such that(2.6)∥T(t)Qx∥α≤C(α)eδt∥x∥fort≤0,∥T(t)Px∥α≤M(α)t−αe−γt∥x∥fort>0.Lemma 2.6 (see [9]).
Let0<α<β<1.
For the hyperbolic analytic semigroup (T(t))t≥0,
there exist constants c>0,δ>0,
and γ>0 such that(2.7)∥AT(t)Qx∥α≤ceδt∥x∥βfort≤0,∥AT(t)Px∥α≤ctβ−α−1e−γt∥x∥βfort>0.


## 2.1. Sectorial Linear Operators and Analytic Semigroups

Definition 2.1.
A linear operatorA:D(A)⊂𝕏↦𝕏 (not necessarily densely defined) is said to
be sectorial if the following hold. There exist constants ω∈ℝ, θ∈(π/2,π),
and M>0 such that(2.1)ρ(A)⊃Sθ,ω:={λ∈ℂ:λ≠ω,|arg(λ−ω)|<θ},∥R(λ,A)∥≤M|λ−ω|,λ∈Sθ,ω,where R(λ,A)=(λI−A)−1 for each λ∈ρ(A).Remark 2.2.
IfA is sectorial, then it generates an analytic
semigroup (T(t))t≥0,
which maps (0,∞) into B(𝕏) such that there exist M0,M1>0 with(2.2)∥T(t)∥≤M0eωt,t>0,∥t(A−ωI)T(t)∥≤M1eωt,t>0.Definition 2.3.
A semigroup(T(t))t≥0 is said to be hyperbolic, if there exist a
projection P and constants M,δ>0 such that each T(t) commutes with P, ker(P) is invariant with respect to T(t), T(t):R(Q)→R(Q) is invertible and(2.3)∥T(t)Px∥≤Me−δt∥x∥fort≥0,∥T(t)Qx∥≤Meδt∥x∥fort≤0,where Q:=I−P and T(t):=(T(−t))−1 for t≤0.Recall that if
a semigroup (T(t))t≥0 is analytic, then (T(t))t≥0 is hyperbolic if and only if σ(A)∩iℝ=∅ (see, e.g., [11, Definition 1.14 and Proposition 1.15,
page 305]).


## 2.2. Intermediate Banach Spaces

Definition 2.4.
Letα∈(0,1).
A Banach space (𝕏α,∥⋅∥α) is said to be an intermediate space between D(A) and 𝕏,
if D(A)⊂𝕏α⊂𝕏 and there is a constant C>0 such that(2.4)∥x∥α≤C∥x∥1−α∥x∥Aα,x∈D(A),where ∥⋅∥A is the graph norm of A.Concrete examples of𝕏α include D(Aα) for α∈(0,1),
the domains of the fractional powers of A,
the real interpolation spaces DA(α,∞), α∈(0,1),
defined as follows:(2.5)DA(α,∞):={x∈𝕏:[x]α=sup0<t≤1∥t1−α(A−ωI)e−ωtT(t)x∥<∞},∥x∥α=∥x∥+[x]α,and the abstract Hölder spaces DA(α):=D(A)¯∥⋅∥α.Lemma 2.5.
For the hyperbolic analytic semigroup(T(t))t≥0,
there exist constants C(α)>0,  δ>0,M(α)>0,
and γ>0 such that(2.6)∥T(t)Qx∥α≤C(α)eδt∥x∥fort≤0,∥T(t)Px∥α≤M(α)t−αe−γt∥x∥fort>0.Lemma 2.6 (see [9]).
Let0<α<β<1.
For the hyperbolic analytic semigroup (T(t))t≥0,
there exist constants c>0,δ>0,
and γ>0 such that(2.7)∥AT(t)Qx∥α≤ceδt∥x∥βfort≤0,∥AT(t)Px∥α≤ctβ−α−1e−γt∥x∥βfort>0.


## 3. Pseudo Almost Automorphic Functions

In this
section, we recall some recent results on almost automorphic functions and
pseudo almost automorphic functions. LetC(ℝ,𝕏) denote the collection of continuous functions
from ℝ into 𝕏.
Let BC(ℝ,𝕏) denote the Banach space of all 𝕏-valued bounded continuous functions equipped
with the sup norm ∥u∥∞:=supt∈ℝ∥u(t)∥ for each u∈BC(ℝ,𝕏).
Similarly, C(ℝ×𝕏,𝕏) denotes the collection of continuous functions
from ℝ×𝕏 into 𝕏, BC(ℝ×𝕏,𝕏) denotes the collection of all bounded
continuous functions f:ℝ×𝕏↦𝕏.Definition 3.1 ( see[8]).
A functionf∈C(ℝ,𝕏) is said to be almost automorphic if for every
sequence of real numbers (σn)n∈ℕ, there exists a subsequence (sn)n∈ℕ such that(3.1)limm→+∞limn→+∞f(t+sn−sm)=f(t)foreacht∈ℝ.This limit means that(3.2)g(t)=limn→+∞f(t+sn)is well defined for each t∈ℝ, and(3.3)f(t)=limn→+∞g(t−sn)for each t∈ℝ. The collection of all such functions will be denoted by AA(ℝ,𝕏).Theorem 3.2 (see [7]).
Assumef,g:ℝ↦𝕏 are almost automorphic, and λ is any scalar. Then the following holds true:(1)
f+g,λf,fτ(t):=f(t+τ) and f^(t):=f(−t) are almost automorphic;(2)
the rangeRf of f is precompact, so f is bounded;(3)
if{fn} is a sequence of almost automorphic functions
and fn→f uniformly on ℝ,
then f is almost automorphic.Theorem 3.3 (see [7]).
If one
equipsAA(ℝ,𝕏) with the sup norm ∥u∥∞=supt∈ℝ∥u(t)∥,
then AA(ℝ,𝕏) turns out to be a Banach space.One sets(3.4)AA0(ℝ,𝕏):={f∈BC(ℝ,𝕏):limr→∞12r∫−rr∥f(s)∥ds=0}.Definition 3.4 (see [2]).
A functionf∈BC(ℝ,𝕏) is said to be pseudo almost automorphic if it
can be decomposed as f=g+ϕ,
where g∈AA(ℝ,𝕏) and ϕ∈AA0(ℝ,𝕏).
The collection of such functions will be denoted by PAA(ℝ,𝕏).Theorem 3.5 (see [3]).
If one
equipsPAA(ℝ,𝕏) with the sup norm ∥u∥∞=supt∈ℝ∥u(t)∥,
then PAA(ℝ,𝕏) turns out to be a Banach space.Definition 3.6 (see [2]).
A functionf∈C(ℝ×𝕏,𝕏) is said to be almost automorphic if f∈C(ℝ×𝕏,𝕏) is almost automorphic in t∈ℝ uniformly for all x∈K,
where K is any bounded subset of 𝕏. That is to say, for every sequence of real
numbers (σn)n∈ℕ there exists a subsequence (sn)n∈ℕ such that(3.5)g(t,x)=limn→+∞f(t+sn,x)is well defined in t∈ℝ, for all x∈K,
and(3.6)f(t,x)=limn→+∞g(t−sn,x)for each t∈ℝ and x∈K. Denote by AA(ℝ×𝕏,𝕏) the collection of all such functions.One also defines AA0(ℝ×𝕏,𝕏) as the collection of functions f∈BC(ℝ×𝕏,𝕏) such that(3.7)limr→∞12r∫−rr∥f(t,x)∥dt=0uniformly for any bounded subset
of 𝕏.Definition 3.7 (see [2]).
A functionf∈BC(ℝ×𝕏,𝕏) is said to be pseudo almost automorphic if it
can be decomposed as f=g+ϕ,
where g∈AA(ℝ×𝕏,𝕏) and ϕ∈AA0(ℝ×𝕏,𝕏).
The collection of such functions will be denoted by PAA(ℝ×𝕏,𝕏).Theorem 3.8 (see [2]).
Assumef=g+ϕ∈PAA(ℝ×𝕏,𝕏) with g(t,x)∈AA(ℝ×𝕏,𝕏), ϕ(t,x)∈AA0(ℝ×𝕏,𝕏),
satisfying the following conditions:(1)
g(t,x) is uniformly continuous in any bounded subset K⊂𝕏 uniformly for t∈ℝ;(2)
f(t,x) is uniformly continuous in any bounded subset K⊂𝕏 uniformly for t∈ℝ.  If x(⋅)∈PAA(ℝ,𝕏),
then f(⋅,x(⋅))∈PAA(ℝ,𝕏).Remark 3.9.
If𝕏α for α∈(0,1) is an intermediate space between D(A) and 𝕏, BC(ℝ,𝕏α), AA(ℝ,𝕏α), PAA(ℝ,𝕏α) are equipped with the α-sup norm: ∥u∥∞,α=supt∈ℝ∥u(t)∥α,
then they all constitute Banach spaces in view of Definition 2.4, Theorems 3.3
and 3.5.Theorem 3.10 (see [6]).
Letu∈PAA(ℝ,𝕐), B∈B(𝕐,𝕏).
If v(t):=Bu(t) for each t∈ℝ,
then v∈PAA(ℝ,𝕏).


## 4. Pseudo Almost Automorphic Mild Solutions

In this
section, combining Theorem3.10 with composition theorem of pseudo almost
automorphic functions (Theorem 3.8) and completeness of the space (PAA(ℝ,𝕏),∥⋅∥∞) (Theorem 3.5), we will establish the existence
and uniqueness theorem of pseudo almost automorphic mild solutions to (1.1)
under the following assumptions.  (H1)
For0<α<β<1, 𝕏β↪𝕏α↪𝕏 are continuously embedded and there exist k0>0,k(α)>0 such that ∥u∥≤k0∥u∥β, ∥u∥α≤k(α)∥u∥β for each u∈𝕏β.(H2)
A is a sectorial linear operator on a Banach
space 𝕏 and σ(A)∩iℝ=∅.(H3)
f=ℳ+φ∈PAA(ℝ×𝕏,𝕏β) with ℳ∈AA(ℝ×𝕏,𝕏β), φ∈AA0(ℝ×𝕏,𝕏β) and g=𝒩+ψ∈PAA(ℝ×𝕏,𝕏) with 𝒩∈AA(ℝ×𝕏,𝕏), ψ∈AA0(ℝ×𝕏,𝕏).(H4)
The functionsf,g are uniformly Lipschitz with respect to the
second argument in the sense that: there exists L>0 such that(4.1)∥f(t,x)−f(t,y)∥β≤L∥x−y∥,∥g(t,x)−g(t,y)∥≤L∥x−y∥, for all t∈ℝ and x,y∈𝕏.(H5)
ℳ,𝒩 are uniformly continuous in any bounded subset K⊂𝕏 uniformly for t∈ℝ.(H6)
The operatorsB,C∈B(𝕏α,𝕏) and ω¯=max(∥B∥B(𝕏α,𝕏),∥C∥B(𝕏α,𝕏)).Definition 4.1 (see [9]).
A functionu∈BC(ℝ,𝕏α) is said to be a mild solution to (1.1) if s→AT(t−s)Pf(s,Bu(s)) is integrable on (−∞,t), s→AT(t−s)Qf(s,Bu(s)) is integrable on (t,+∞) and(4.2)u(t)=−f(t,Bu(t))−∫−∞tAT(t−s)Pf(s,Bu(s))ds+∫t+∞AT(t−s)Qf(s,Bu(s))ds+∫−∞tT(t−s)Pg(s,Cu(s))ds−∫t+∞T(t−s)Qg(s,Cu(s))ds for each t∈ℝ.Lemma 4.2.
Let
assumptions (H1)–(H6) hold. Consider the nonlinear operatorΛ1 defined by(4.3)(Λ1u)(t)=∫−∞tAT(t−s)Pf(s,Bu(s))ds−∫t+∞AT(t−s)Qf(s,Bu(s))ds.Then, Λ1 maps PAA(ℝ,𝕏α) into itself.Proof.
From (H1), (H3), (H4), (H5), we deduce thatf=ℳ+φ∈PAA(ℝ×𝕏,𝕏) with ℳ∈AA(ℝ×𝕏,𝕏), φ∈AA0(ℝ×𝕏,𝕏),
satisfying the following conditions.
(1)
ℳ(t,x) is uniformly continuous in any bounded subset K⊂𝕏 uniformly for t∈ℝ.(2)
f(t,x) is uniformly Lipschitz with respect to the
second argument in the sense that there exists k0L>0 such that(4.4)∥f(t,x)−f(t,y)∥≤k0L∥x−y∥,for all t∈ℝ and x,y∈𝕏.
Letu∈PAA(ℝ,𝕏α).
Since B∈B(𝕏α,𝕏),
it follows from Theorem 3.10 that Bu(⋅)∈PAA(ℝ,𝕏).
Setting h(⋅)=f(⋅,Bu(⋅)) and applying Theorem 3.8, we get that h∈PAA(ℝ,𝕏). Using (H1) and Theorem 3.10, we obtain that h∈PAA(ℝ,𝕏β). Now, write h=ξ+η where ξ∈AA(ℝ,𝕏β) and η∈AA0(ℝ,𝕏β),
then(4.5)(Λ1u)(t)=∫−∞tAT(t−s)Pξ(s)ds+∫−∞tAT(t−s)Pη(s)ds+∫+∞tAT(t−s)Qξ(s)ds+∫+∞tAT(t−s)Qη(s)ds.Set(4.6)Ξ(t)=∫−∞tAT(t−s)Pξ(s)ds+∫+∞tAT(t−s)Qξ(s)ds,Π(t)=∫−∞tAT(t−s)Pη(s)ds+∫+∞tAT(t−s)Qη(s)ds. Next, we show that Ξ∈AA(ℝ,𝕏α) and Π∈AA0(ℝ,𝕏α).
Now, to prove thatΞ∈AA(ℝ,𝕏α).
Let us take a sequence (σn)n∈ℕ and show that there exists a subsequence (sn)n∈ℕ such that(4.7)limm→+∞limn→+∞∥Ξ(t+sn−sm)−Ξ(t)∥α=0foreacht∈ℝ.Since ξ∈AA(ℝ,𝕏β),
there exists a subsequence (sn)n∈ℕ such that(4.8)limm→+∞limn→+∞∥ξ(t+sn−sm)−ξ(t)∥β=0foreacht∈ℝ.On the other hand, we
have(4.9)Ξ(t+sn−sm)−Ξ(t)=∫−∞(t+sn−sm)AT(t+sn−sm−s)Pξ(s)ds−∫−∞tAT(t−s)Pξ(s)ds+∫+∞t+sn−smAT(t+sn−sm−s)Qξ(s)ds−∫+∞tAT(t−s)Qξ(s)ds=∫0+∞AT(s)Pξ(t+sn−sm−s)ds−∫0+∞AT(s)Pξ(t−s)ds−(∫−∞0AT(s)Qξ(t+sn−sm−s)ds−∫−∞0AT(s)Qξ(t−s)ds)=∫0+∞AT(s)P[ξ(t+sn−sm−s)−ξ(t−s)]ds−∫−∞0AT(s)Q[ξ(t+sn−sm−s)−ξ(t−s)]ds.Passing to the norm ∥⋅∥α,0<α<1, remembering the triangle inequalities,  (2.7), then we obtain that(4.10)∥Ξ(t+sn−sm)−Ξ(t)∥α≤∫0+∞∥AT(s)P[ξ(t+sn−sm−s)−ξ(t−s)]∥αds+∫−∞0∥AT(s)Q[ξ(t+sn−sm−s)−ξ(t−s)]∥αds≤c∫0+∞sβ−α−1e−γs∥ξ(t+sn−sm−s)−ξ(t−s)∥βds+c∫−∞0eδs∥ξ(t+sn−sm−s)−ξ(t−s)∥βds.Thus, (4.8) and Lebesgue
dominated convergence theorem lead to (4.7), therefore, to Ξ∈AA(ℝ,𝕏α).
To finish the proof, we will prove that Π∈AA0(ℝ,𝕏α).
It is obvious that Π∈BC(ℝ,𝕏α),
the left task is to show that(4.11)limr→∞12r∫−rr∥Π(t)∥αdt=0.Using (2.7), we have(4.12)0≤limr→∞12r∫−rr∥Π(t)∥αdt≤I+J,where(4.13)I:=limr→∞c2r∫−rrdt∫−∞t(t−s)β−α−1e−γ(t−s)∥η(s)∥βds=limr→∞c2r∫−rrdt∫0+∞σβ−α−1e−γσ∥η(t−σ)∥βdσ=limr→∞c∫0+∞σβ−α−1e−γσ[12r∫−rr∥η(t−σ)∥βdt]dσ,J:=limr→∞c2r∫−rrdt∫t+∞eδ(t−s)∥η(s)∥βds=limr→∞c2r∫−rrdt∫−∞0eδσ∥η(t−σ)∥βdσ=limr→∞c∫−∞0eδσ[12r∫−rr∥η(t−σ)∥βdt]dσ.Then, by the Lebesgue dominated
convergence theorem and the fact that η∈AA0(ℝ,𝕏β),
one has I=J=0. Hence, Π∈AA0(ℝ,𝕏α) and we end the proof.Lemma 4.3.
Let assumptions (H2)–(H6) hold. Define the nonlinear operatorΛ2 by(4.14)(Λ2u)(t)=∫−∞tT(t−s)Pg(s,Cu(s))ds−∫t+∞T(t−s)Qg(s,Cu(s))ds.Then, Λ2 maps PAA(ℝ,𝕏α) into itself.Proof.
The proof is similar to that of Lemma4.2, so we omit it.Theorem 4.4.
Under the assumptions (H1)–(H6), partial hyperbolic evolution equation
(1.1) admits a unique pseudo almost automorphic mild solution if(4.15)Θ=Lω¯[k(α)+cδ+C(α)δ+cΓ(β−α)γβ−α+M(α)Γ(1−α)γ1−α]<1.Proof.
Firstly, define the nonlinear operatorΛ on BC(ℝ,𝕏α) by(4.16)(Λu)(t)=−f(t,Bu(t))−∫−∞tAT(t−s)Pf(s,Bu(s))ds+∫t+∞AT(t−s)Qf(s,Bu(s))ds+∫−∞tT(t−s)Pg(s,Cu(s))ds−∫t+∞T(t−s)Qg(s,Cu(s))ds=−f(t,Bu(t))−(Λ1u)(t)+(Λ2u)(t).Let u∈PAA(ℝ,𝕏α),
then f(⋅,Bu(⋅))∈PAA(ℝ,𝕏β)⊂PAA(ℝ,𝕏α),
as proved in Lemma 4.2. Together with Lemmas 4.2 and 4.3, it follows that the
operator Λ maps PAA(ℝ,𝕏α) into itself. Secondly, we will show that Λ admits a unique fixed point in PAA(ℝ,𝕏α).
Letv,w∈PAA(ℝ,𝕏α),
then the triangle inequality reads(4.17)∥(Λv)(t)−(Λw)(t)∥α≤∥f(t,Bv(t))−f(t,Bw(t))∥α+∥(Λ1v)(t)−(Λ1w)(t)∥α+∥(Λ2v)(t)−(Λ2w)(t)∥α.By (H1), (H5), and (H6), we
obtain(4.18)∥f(t,Bv(t))−f(t,Bw(t))∥α≤k(α)L∥Bv(t)−Bw(t)∥≤k(α)Lω¯∥v−w∥∞,α.By (H2), Lemma 2.6, (H5), and
(H6), we obtain(4.19)∥(Λ1v)(t)−(Λ1w)(t)∥α≤∫−∞t∥AT(t−s)P[f(s,Bv(s))−f(s,Bw(s))]∥αds+∫t+∞∥AT(t−s)Q[f(s,Bv(s))−f(s,Bw(s))]∥αds≤cLω¯∥v−w∥∞,α∫−∞t(t−s)β−α−1e−γ(t−s)ds+cLω¯∥v−w∥∞,α∫t+∞eδ(t−s)ds≤cLω¯∥v−w∥∞,α[Γ(β−α)γβ−α+1δ].Similarly, by (H2), Lemma 2.5,
(H5), and (H6), we obtain(4.20)∥(Λ2v)(t)−(Λ2w)(t)∥α≤∫−∞t∥T(t−s)P[g(s,Cv(s))−g(s,Cw(s))]∥αds+∫t+∞∥T(t−s)Q[g(s,Cv(s))−g(s,Cw(s))]∥αds≤M(α)Lω¯∥v−w∥∞,α∫−∞t(t−s)−αe−γ(t−s)ds+C(α)Lω¯∥v−w∥∞,α∫t+∞eδ(t−s)ds≤Lω¯∥v−w∥∞,α[M(α)Γ(1−α)γ1−α+C(α)δ].Combining the above inequality
together, we obtain ∥Λv−Λw∥∞,α≤Θ∥v−w∥∞,α,
where(4.21)Θ=Lω¯[k(α)+cδ+C(α)δ+cΓ(β−α)γβ−α+M(α)Γ(1−α)γ1−α].Clearly, if Θ<1,
then the operator Λ becomes a strict contraction on PAA(ℝ,𝕏α).
Remembering that PAA(ℝ,𝕏α) equipped with the α-sup norm: ∥u∥∞,α=supt∈ℝ∥u(t)∥α is a Banach space by Remark 3.9, the classical Banach fixed-point theorem leads to
the desired conclusion.


## 5. Application

Example 5.1.
Take𝕏:=C[0,π] equipped with the sup norm. Define the
operator A by(5.1)Aφ(ξ):=φ′′(ξ)+aφ(ξ),∀ξ∈(0,π),φ∈D(A),where D(A):={φ∈C2[0,π],φ(0)=φ(π)=0}⊂C[0,π] and a∈ℝ is a constant.
Clearly,A is sectorial, and hence is the generator of an
analytic semigroup. If a≠n2,
then A generates a hyperbolic analytic semigroup (T(t))t≥0 on 𝕏.
Let(5.2)𝕏α=DA(α,∞)={C2α[0,π],if0<α<12,Cν2α[0,π],if12<α<1,where Cν2α[0,π]={φ∈C2α[0,π],φ(0)=φ(π)=0}  (see [10] for more
details).
Define the operatorC by(5.3)Cφ(ξ):=q(ξ)φ(ξ),∀ξ∈(0,π),φ∈D(C),where D(C):={φ∈C[0,π],φ(0)=φ(π)=0}⊂C[0,π],
then 𝕏α⊂D(C) and ∥C∥B(𝕏α,𝕏)=∥q∥∞=supξ∈[0,π]q(ξ).
Let(5.4)g(t,u)=usin12+cost+cos2t+maxk∈ℤ{e−(t±k2)2}cosu,t∈ℝ,u∈𝕏,then g∈PAA(ℝ×𝕏,𝕏), g is Lipschitz with respect to the second
argument with L=2,𝒩(t,u)=usin(1/(2+cost+cos2t)) is uniformly continuous in any bounded subset K⊂𝕏 uniformly for t∈ℝ.Now the above discussion can be formulated as
follows.Proposition 5.2.
Letg(t,u)=usin(1/(2+cost+cos2t))+maxk∈ℤ{e−(t±k2)2}cosu,t∈ℝ,u∈𝕏. Suppose that the constant a≠n2 for n∈ℕ.
Then, nonlinear heat equation (1.2) admits a unique 𝕏α-valued pseudo almost automorphic mild solution
if(5.5)2∥q∥∞[C(α)δ+M(α)Γ(1−α)γ1−α]<1.



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*Source: 405092-2008-11-06.xml*