doi:10.1155/2011/895079
Research Article
Existence of Pseudo-Almost Automorphic Mild
Solutions to Some Nonautonomous Partial
Evolution Equations
Toka Diagana
Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, DC 20059, USA
Correspondence should be addressed to Toka Diagana,[email protected]
Received 15 September 2010; Accepted 29 October 2010
Academic Editor: Jin Liang
Copyrightq2011 Toka Diagana. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We use the Krasnoselskii fixed point principle to obtain the existence of pseudo almost automorphic mild solutions to some classes of nonautonomous partial evolutions equations in a Banach space.
1. Introduction
LetXbe a Banach space. In the recent paper by Diagana1, the existence ofalmost automorphic
mild solutions to the nonautonomous abstract differential equations
ut Atut ft, ut, t∈R, 1.1
whereAtfort∈ Ris a family of closed linear operators with domainsDAtsatisfying Acquistapace-Terreni conditions, and the functionf : R×X → Xis almost automorphic in
t ∈ R uniformly in the second variable, was studied. For that, the author made extensive use of techniques utilized in2, exponential dichotomy tools, and the Schauder fixed point theorem.
In this paper we study the existence of pseudo-almost automorphic mild solutions to the nonautonomous partial evolution equations
d
whereAtfort∈Ris a family of linear operators satisfying Acquistpace-Terreni conditions and F, G are pseudo-almost automorphic functions. For that, we make use of exponential dichotomy tools as well as the well-known Krasnoselskii fixed point principle to obtain some reasonable sufficient conditions, which do guarantee the existence of pseudo-almost automorphic mild solutions to1.2.
The concept of pseudo-almost automorphy is a powerful generalization of both the notion of almost automorphy due to Bochner 3 and that of pseudo-almost periodicity due to Zhang see4, which has recently been introduced in the literature by Liang et al. 5–7. Such a concept, since its introduction in the literature, has recently generated several developments; see, for example,8–12. The question which consists of the existence of pseudo-almost automorphic solutions to abstract partial evolution equations has been made; see for instance10,11,13. However, the use of Krasnoselskii fixed point principle to establish the existence of pseudo-almost automorphic solutions to nonautonomous partial evolution equations in the form1.2is an original untreated problem, which is the main motivation of the paper.
The paper is organized as follows:Section 2is devoted to preliminaries facts related to the existence of an evolution family. Some preliminary results on intermediate spaces are also stated there. Moreover, basic definitions and results on the concept of pseudo-almost automorphy are also given.Section 3is devoted to the proof of the main result of the paper.
2. Preliminaries
LetX,·be a Banach space. IfLis a linear operator on the Banach spaceX, then,DL,ρL,
σL,NL, andRLstand, respectively, for its domain, resolvent, spectrum, null-space or kernel, and range. IfL:DDL⊂X→Xis a linear operator, one setsRλ, L: λI−L−1
for allλ∈ρA.
IfY,Zare Banach spaces, then the spaceBY,Zdenotes the collection of all bounded linear operators fromYintoZequipped with its natural topology. This is simply denoted by
BYwhenYZ. IfPis a projection, we setQI−P.
2.1. Evolution Families
This section is devoted to the basic material on evolution equations as well the dichotomy tools. We follow the same setting as in the studies of Diagana1.
AssumptionH.1given below will be crucial throughout the paper.
H.1The family of closed linear operators At for t ∈ R on X with domain
DAt possibly not densely definedsatisfy the so-called Acquistapace-Terreni conditions, that is, there exist constants ω ≥ 0, θ ∈ π/2, π, K, L ≥ 0, and
μ, ν∈0,1withμν >1 such that
Sθ
{0} ⊂ρAt−ω λ, Rλ, At−ω ≤ K
1|λ|,
At−ωRλ, At−ωRω, At−Rω, As ≤L|t−s|μ|λ|−ν,
2.1
It should mentioned thatH.1 was introduced in the literature by Acquistapace et al. in14,15forω 0. Among other things, it ensures that there exists a unique evolution family
U{Ut, s:t, s∈Rsuch thatt≥s}, 2.2
onXassociated withAtsuch thatUt, sX⊂DAtfor allt, s∈Rwitht≥s, and aUt, sUs, r Ut, rfort, s, r∈Rsuch thatt≥s≥r;
bUt, t Ifort∈RwhereIis the identity operator ofX; c t, s→Ut, s∈BXis continuous fort > s;
dU·, s∈C1s,∞, BX,∂U/∂tt, s AtUt, sand
AtkUt, s≤Kt−s−k, 2.3
for 0< t−s≤1,k0,1;
e ∂Ut, s/∂sx −Ut, sAsxfort > sandx∈DAswithAsx∈DAs. It should also be mentioned that the above-mentioned proprieties were mainly established in16, Theorem 2.3and17, Theorem 2.1; see also15,18. In that case we say thatA·generates the evolution familyU·,·. For some nice works on evolution equations, which make use of evolution families, we refer the reader to, for example,19–29.
Definition 2.1. One says that an evolution family U has an exponential dichotomy or is
hyperbolic if there are projections Pt t ∈ R that are uniformly bounded and strongly continuous intand constantsδ >0 andN≥1 such that
fUt, sPs PtUt, s;
gthe restriction UQt, s : QsX → QtX of Ut, s is invertible we then set
UQs, t:UQt, s−1;
hUt, sPs ≤Ne−δt−sandU
Qs, tQt ≤Ne−δt−sfort≥sandt, s∈R.
Under Acquistpace-Terreni conditions, the family of operators defined by
Γt, s
⎧ ⎨ ⎩
Ut, sPs, ift≥s, t, s∈R,
−UQt, sQs ift < s, t,∈R
2.4
are called Green function corresponding toUandP·.
This setting requires some estimates related to Ut, s. For that, we introduce the interpolation spaces forAt. We refer the reader to the following excellent books30–32
for proofs and further information on theses interpolation spaces.
LetAbe a sectorial operator onXfor that, in assumptionH.1, replaceAtwithA
and letα∈0,1. Define the real interpolation space
XA α :
x∈X:xAα :sup
r>0
which, by the way, is a Banach space when endowed with the norm · Aα. For convenience we further write
XA
0 :X, xA0 :x, XA1 :DA,
xA1 :ω−Ax.
2.6
Moreover, letXA:DAofX. In particular, we have the following continuous embedding:
DA→XAβ →Dω−Aα→XAα →XA→X, 2.7
for all 0< α < β <1, where the fractional powers are defined in the usual way. In general,DAis not dense in the spacesXA
α andX. However, we have the following
continuous injection:
XA
β →DA
·A α
2.8
for 0< α < β <1.
Given the family of linear operatorsAtfort∈R, satisfyingH.1, we set
Xt
α:XAαt, Xt:XAt 2.9
for 0≤α≤1 andt∈R, with the corresponding norms.
Now the embedding in 2.7 holds with constants independent of t ∈ R. These interpolation spaces are of classJα 32, Definition 1.1.1, and hence there is a constantcα
such that
ytα≤cαy1−αAtyα, y∈DAt. 2.10
We have the following fundamental estimates for the evolution familyUt, s.
Proposition 2.2 see 33. Suppose that the evolution family U Ut, s has exponential dichotomy. Forx∈X,0≤α≤1, andt > s, the following hold.
iThere is a constantcα, such that
Ut, sPsxtα≤cαe−δ/2t−st−s−αx. 2.11
iiThere is a constantmα, such that U
Qs, tQtx s
α≤mαe
−δt−sx. 2.12
H.2The domain DAt D is constant in t ∈ R. Moreover, the evolution family
U Ut, st≥s generated byA· has an exponential dichotomy with constants
N, δ >0 and dichotomy projectionsPtfort∈R.
2.2. Pseudo-Almost Automorphic Functions
Let BCR,X denote the collection of all X-valued bounded continuous functions. The space BCR,X equipped with its natural norm, that is, the sup norm is a Banach space. Furthermore,CR,Ydenotes the class of continuous functions fromRintoY.
Definition 2.3. A functionf ∈CR,Xis said to be almost automorphic if, for every sequence of real numberssnn∈N, there exists a subsequencesnn∈Nsuch that
gt: lim
n→ ∞ftsn 2.13
is well defined for eacht∈R, and
lim
n→ ∞gt−sn ft 2.14
for eacht∈R.
If the convergence above is uniform int∈R, thenfis almost periodic in the classical Bochner’s sense. Denote byAAXthe collection of all almost automorphic functionsR→X. Note thatAAXequipped with the sup-norm · ∞turns out to be a Banach space.
Among other things, almost automorphic functions satisfy the following properties.
Theorem 2.4see34. Iff, f1, f2∈AAX, then
if1f2 ∈AAX,
iiλf ∈AAXfor any scalarλ,
iiifα∈AAX, wherefα:R → Xis defined byfα· f·α,
ivthe rangeRf :{ft:t∈R}is relatively compact inX, thusfis bounded in norm, viffn → funiformly onR, where eachfn∈AAX, thenf∈AAXtoo.
LetY, · Ybe another Banach space.
Definition 2.5. A jointly continuous functionF :R×Y→Xis said to be almost automorphic int∈Rift→Ft, xis almost automorphic for allx∈KK⊂Ybeing any bounded subset. Equivalently, for every sequence of real numberssnn∈N, there exists a subsequencesnn∈N
such that
Gt, x: lim
is well defined int∈Rand for eachx∈K, and
lim
n→ ∞Gt−sn, x Ft, x 2.16
for allt∈Randx∈K.
The collection of such functions will be denoted byAAY,X.
For more on almost automorphic functions and related issues, we refer the reader to, for example,1,4,9,13,34–39.
Define
P AP0R,X:
f∈BCR,X: lim
r→ ∞
1 2r
r
−r
fsds0
. 2.17
Similarly,P AP0Y,Xwill denote the collection of all bounded continuous functions
F:R×Y→Xsuch that
lim
T→ ∞
1 2r
r
−r
Fs, xds0 2.18
uniformly inx∈K, whereK⊂Yis any bounded subset.
Definition 2.6 see Liang et al. 5, 6. A function f ∈ BCR,X is called pseudo-almost automorphic if it can be expressed asf gφ, whereg ∈ AAXandφ ∈P AP0X. The collection of such functions will be denoted byP AAX.
The functionsg andφappearing in Definition 2.6are, respectively, called thealmost automorphicand theergodic perturbationcomponents off.
Definition 2.7. A bounded continuous functionF :R×Y→Xbelongs toAAY,Xwhenever it can be expressed asFG Φ, whereG∈AAY,XandΦ∈P AP0Y,X. The collection of such functions will be denoted byP AAY,X.
An important result is the next theorem, which is due to Xiao et al.6.
Theorem 2.8see6. The spaceP AAXequipped with the sup norm · ∞is a Banach space.
The next composition result, that isTheorem 2.9, is a consequence of12, Theorem 2.4.
Theorem 2.9. Suppose thatf:R×Y→Xbelongs toP AAY,X;fgh, withx→gt, xbeing uniformly continuous on any bounded subsetKofYuniformly int∈R. Furthermore, one supposes
that there existsL >0such that
ft, x−f
t, y≤Lx−yY 2.19
for allx, y∈Yandt∈R.
We also have the following.
Theorem 2.10see6. Iff :R×Y→Xbelongs toP AAY,Xand ifx→ft, xis uniformly continuous on any bounded subsetKofYfor eacht∈R, then the function defined byht ft, ϕt
belongs toP AAXprovided thatϕ∈P AAY.
3. Main Results
Throughout the rest of the paper we fixα, β, real numbers, satisfying 0 < α < β < 1 with 2β > α1.
To study the existence of pseudo-almost automorphic solutions to1.2, in addition to the previous assumptions, we suppose that the injection
Xα→X 3.1
is compact, and that the following additional assumptions hold:
H.3Rω, A· ∈ AABX,Xα. Moreover, for any sequence of real numbersτnn∈N there exist a subsequenceτnn∈Nand a well-defined functionRt, ssuch that for eachε >0, one can findN0, N1∈Nsuch that
Rt, s−Γtτn, sτnBX,Xα≤εH0t−s 3.2
whenevern > N0fort, s∈R, and
Γt, s−Rt−τn, s−τnBX,Xα≤εH1t−s 3.3
whenevern > N1 for allt, s ∈ R, whereH0, H1 : 0,∞ → 0,∞withH0, H1 ∈
L10,∞.
H.4 aThe functionF:R×Xα→Xis pseudo-almost automorphic in the first variable
uniformly in the second one. The functionu→Ft, uis uniformly continuous on any bounded subsetKofXαfor eacht∈R. Finally,
Ft, u∞≤ Muα,∞, 3.4
whereuα,∞ supt∈Rutα andM : R → R is a continuous, monotone increasing function satisfying
lim
r→ ∞
Mr
bThe functionG:R×X→Xβis pseudo-almost automorphic in the first variable
uniformly in the second one. Moreover,Gis globally Lipschitz in the following sense: there existsL >0 for which
Gt, u−Gt, vβ≤Lu−v 3.6
for allu, v∈Xandt∈R.
H.5The operator Atis invertible for eacht ∈ R, that is, 0 ∈ ρAtfor eacht ∈ R. Moreover, there existsc0>0 such that
sup
t,s∈R
AsAt−1
BX,Xβ
< c0. 3.7
To study the existence and uniqueness of pseudo-almost automorphic solutions to 1.2we first introduce the notion of a mild solution, which has been adapted to the one given in the studies of Diagana et al.35, Definition 3.1.
Definition 3.1. A continuous function u : R → Xα is said to be a mild solution to 1.2
provided that the functions → AsUt, sPsGs, usis integrable ons, t, the function
s → AsUQt, sQsGs, usis integrable ont, sand
ut −Gt, ut Ut, sus Gs, us
−
t
s
AsUt, sPsGs, usds
s
t
AsUQt, sQsGs, usds
t
s
Ut, sPsFs, usds−
s
t
UQt, sQsFs, usds,
3.8
fort≥sand for allt, s∈R.
Under assumptionsH.1,H.2, andH.5, it can be readily shown that1.2has a mild solution given by
ut −Gt, ut−
t
−∞AsUt, sPsGs, usds
∞
t
AsUQt, sQsGs, usds t
−∞Ut, sPsFs, usds
−
∞
t
UQt, sQsFs, usds
3.9
We denote bySandTthe nonlinear integral operators defined by
Sut
t
−∞Ut, sPsFs, usds−
∞
t
UQt, sQsFs, usds,
Tut −Gt, ut−
t
−∞AsUt, sPsGs, usds
∞
t
AsUQt, sQsGs, usds.
3.10
The main result of the present paper will be based upon the use of the well-known fixed point theorem of Krasnoselskii given as follows.
Theorem 3.2. LetCbe a closed bounded convex subset of a Banach spaceX. Suppose the (possibly nonlinear) operatorsT andSmapCintoXsatisfying
1for allu, v∈C, thenSuTv∈C;
2the operatorTis a contraction;
3the operatorSis continuous andSCis contained in a compact set.
Then there existsu∈Csuch thatuTuSu.
We need the following new technical lemma.
Lemma 3.3. For eachx∈X, suppose that assumptions (H.1), (H.2) hold, and letα, βbe real numbers such that0< α < β <1with2β > α1. Then there are two constantsrα, β, dβ>0such that
AtUt, sPsxβ≤rα, βe−δ/4t−st−s−βx, t > s, 3.11
AtUQt, sQsx β≤d
βe−δs−tx, t≤s. 3.12
Proof. Letx∈X. First of all, note thatAtUt, sBX,Xβ≤Kt−s−1−βfor allt, ssuch that 0< t−s≤1 andβ∈0,1.
Lettingt−s≥1 and usingH.2and the above-mentioned approximate, we obtain
AtUt, sxβAtUt, t−1Ut−1, sxβ
≤ AtUt, t−1BX,XβUt−1, sx
≤MKeδe−δt−sx
K1e−δt−sx
K1e−3δ/4t−st−sβt−s−βe−δ/4t−sx.
A straightforward consequence ofLemma 3.3is the following.
Corollary 3.4. For eachx∈X, suppose that assumptions (H.1), (H.2), and (H.5) hold, and letα, βbe real numbers such that0< α < β <1with2β > α1. Then there are two constantsrα, β, dβ>0
such that
AsUt, sPsxβ≤rα, βe−δ/4t−st−s−βx, t > s, 3.18
AsUQt, sQsx β≤d
βe−δs−tx, t≤s. 3.19
Proof. We make use ofH.5andLemma 3.3. Indeed, for eachx∈X,
AsUt, sPsxβAsA−1tAtUt, sPsx
β
≤AsA−1t
BX,Xβ
AtUt, sPsx
≤c0kAtUt, sPsxβ
≤c0kr
α, βe−δ/4t−st−s−βx
rα, βe−δ/4t−st−s−βx, t > s.
3.20
Equation3.19has already been provedsee the proof of3.12.
Lemma 3.5. Under assumptions (H.1), (H.2), (H.3), and (H.4), the mapping S : BCR,Xα → BCR,Xαis well defined and continuous.
Proof. We first show thatSBCR,Xα ⊂ BCR,Xα. For that, letS1 and S2 be the integral operators defined, respectively, by
S1ut
t
−∞Ut, sPsFs, usds,
S2ut
∞
t
UQt, sQsFs, usds.
Now, using2.11it follows that for allv∈BCR,Xα,
It remains to prove that S1 is continuous. For that consider an arbitrary sequence of functions un ∈ BCR,Xα which converges uniformly to some u ∈ BCR,Xα, that is,
Now, using the continuity of F and the Lebesgue Dominated Convergence Theorem we conclude that
Lemma 3.6. Under assumptions (H.1), (H.2), (H.3), and (H.4), the integral operatorSdefined above mapsP AAXαinto itself.
Proof. Let u ∈ P AAXα. Settingφt Ft, ut and using Theorem 2.10 it follows that
thatS1u1 ∈AAXα. Indeed, sinceu1 ∈ AAX, for every sequence of real numbersτnn∈N
there exists a subsequenceτnn∈Nsuch that
v1t: lim
n→ ∞u1tτn 3.26
is well defined for eacht∈Rand
lim
n→ ∞v1t−τn u1t 3.27
for eacht∈R.
SetMt −∞t Ut, sPsu1sdsandNt
t
−∞Ut, sPsv1sdsfor allt∈R. Now
Mtτn−Nt
tτn
−∞ Utτn, sPsu1sds−
t
−∞Ut, sPsv1sds
t
−∞Utτ, sτnPsτnu1sτnds−
t
−∞Ut, sPsv1sds
t
−∞Utτn, sτnPsτnu1sτn−v1sds
t
−∞Utτn, sτnPsτn−Ut, sPsv1sds.
3.28
Using2.11and the Lebesgue Dominated Convergence Theorem, one can easily see that
t
−∞Utτn, sτnPsτnu1sτn−v1sds
α
−→0 as n → ∞, t∈R. 3.29
Similarly, usingH.3and40it follows that
t
−∞Utτn, sτnPsτn−Ut, sPsv1sds
α
−→0 asn → ∞, t∈R. 3.30
Therefore,
Nt lim
n→ ∞Mtτn, t∈R. 3.31
Using similar ideas as the previous ones, one can easily see that
Mt lim
Again using2.11it follows that
lim
r→ ∞
1 2r
r
−r
S1u2tαdt≤rlim→ ∞
cα
2r
r
−r ∞
0
s−αe−δ/2su2t−sds dt
≤ lim
r→ ∞cα
∞
0
s−αe−δ/2s 1
2r
r
−r
u2t−sdt ds.
3.33
Set
Γsr
1 2r
r
−r
u2t−sdt. 3.34
SinceP AP0Xis translation invariant it follows thatt → u2t−sbelongs toP AP0Xfor eachs∈R, and hence
lim
r→∞
1 2r
r
−r
u2t−sdt0 3.35
for eachs∈R.
One completes the proof by using the well-known Lebesgue dominated convergence theorem and the factΓsr→0 asr → ∞for eachs∈R.
The proof forS2 is similar to that ofS1and hence omitted. ForS2, one makes use of 2.12rather than2.11.
Letγ∈0,1, and let BCγR,Xα {u∈BCR,Xα:uα,γ <∞}, where
uα,γ suputαγ sup
t,s∈R, t /s
ut−usα
|t−s|γ . 3.36
Clearly, the space BCγR,Xαequipped with the norm · α,γ is a Banach space, which is
the Banach space of all bounded continuous H ¨older functions fromRtoXα whose H ¨older
exponent isγ.
Lemma 3.7. Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5),V S1−S2 maps bounded
sets of BCR,Xαinto bounded sets of BCγR,Xαfor some0< γ < 1, whereS1, S2 are the integral
Proof. Letu∈BCR,Xα, and letgt Ft, utfor eacht∈R. Then we have
S1utα≤kαS1utβ
≤kα
t
−∞
Ut, sPsgsβds
≤kαcβ
t
−∞e
−δ/2t−st−s−βgsds
≤ Muα,∞
kαcβ
∞
0
e−σ
2σ δ
−β2dσ
δ
≤ Muα,∞kαcβ2−1δ1−βΓ1−β,
3.37
and hence
S1uα,∞≤
kαcβ2−1δ1−βΓ1−βMuα,∞. 3.38
Similarly,
S2utα≤kαS2utβ
≤kα
∞
t
UQt, sQsgsβds
≤kαmβ ∞
t
e−δs−tgsds
≤ Muα,∞kαmβδ−1,
3.39
and hence
Now
S2ut2−S2ut1α≤mα t2
t1
e−δs−t1gsds
mα
∞
t2 t2
t1
e−δs−τgsτ
ds
≤Nα, δt2−t1M
uα,∞,
3.44
whereNα, δis a positive constant.
Consequently, lettingγ1−βit follows that
V ut2−V ut1α≤s
α, β, δMuα,∞|t2−t1|γ, 3.45
wheresα, β, δis a positive constant.
Therefore, for eachu∈BCR,Xαsuch that
utα≤R 3.46
for allt∈R, thenV ubelongs to BCγR,Xαwith
V utα≤R 3.47
for allt∈R, whereRdepends onR.
The proof of the next lemma follows along the same lines as that ofLemma 3.6and hence omitted.
Lemma 3.8. The integral operatorV S1−S2maps bounded sets ofAAXαinto bounded sets of BC1−βR,Xα∩AAXα.
Similarly, the next lemma is a consequence of2, Proposition 3.3.
Lemma 3.9. The set BC1−βR,Xα is compactly contained in BCR,X, that is, the canonical injectionid:BC1−βR,Xα→BCR,Xis compact, which yields
id:BC1−βR,X α
AAXα−→AAXα 3.48
is compact, too.
Proof. The proof follows along the same lines as that of2, Proposition 3.4. Recalling that in view ofLemma 3.7, we have
V uα,∞≤pα, β, δMuα,∞,
V ut2−Vt1α≤s
α, β, δMuα,∞|t2−t1|,
3.49
for allu∈BCR,Xα,t1, t2 ∈Rwitht1/t2, wherepα, β, δ, sα, β, δare positive constants. Consequently,u∈BCR,Xαanduα,∞< RyieldV u∈BC1−βR,Xαand
V uα< R1, 3.50
whereR1cα, β, δMR.
Therefore, there existsr >0 such that for allR≥r, the following hold:
VBAAXα0, R
⊂BBC1−βR,X α0, R
BAAXα0, R. 3.51
In view of the above, it follows thatV :D → Dis continuous and compact, whereDis the ball inAAXαof radiusRwithR≥r.
Define
W1ut
t
−∞AsUt, sPsGs, usds,
W2ut
s
t
AsUQt, sQsGs, usds
3.52
for allt∈R.
Lemma 3.11. Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5), the integral operators W1
andW2defined above mapP AAXαinto itself.
Proof. Let u ∈ P AAXα. Again, using the composition of pseudo-almost automorphic
functions Theorem 2.10 it follows that ψ· G·, u· is in P AAXβ whenever u ∈
P AAXα. In particular,
ψ
β,∞sup
t∈RGt, utβ<∞. 3.53
Now writeψ φz, whereφ∈AAXβandz∈P AP0Xβ, that is,W1ψ Ξφ Ξz where
Ξφt:
t
−∞AsUt, sPsφsds,
Ξzt:
t
−∞AsUt, sPszsds.
Clearly,Ξφ∈AAXα. Indeed, sinceφ∈AAXβ, for every sequence of real numbers
Using3.18and the Lebesgue Dominated Convergence Theorem, one can easily see that
Similarly, usingH.3it follows that
Using similar ideas as the previous ones, one can easily see that
Theorem 3.12. Under assumptions (H.1), (H.2), (H.3), (H.4), and (H.5) and ifLis small enough, then1.2has at least one pseudo-almost automorphic solution.
To complete the proof we have to show thatT is a strict contraction. Indeed, for all
u, v∈Xα
Tu−Tvα,∞≤Lkα
1rα, β4 δ
1−β
Γ1−βd
β δ
u−vα,∞ 3.67
and henceT is a strict contraction wheneverLis small enough.
Using the Krasnoselskii fixed point theoremTheorem 3.2it follows that there exists at least one pseudo-almost automorphic mild solution to1.2.
Acknowledgment
The author would like to express his thanks to the referees for careful reading of the paper and insightful comments.
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