Weighted Stepanov-like pseudo-almost automorphic mild solutions ...

He et al. Advances in Difference Equations (2015) 2015:74 DOI 10.1186/s13662-015-0410-1

R E S E A R C H Open Access

Weighted Stepanov-like pseudo-almostautomorphic mild solutions for semilinearfractional differential equationsBing He, Junfei Cao* and Bicheng Yang

*Correspondence:[email protected];[email protected] of Mathematics,Guangdong University ofEducation, Xingangzhong Road,Guangzhou, 510310, P.R. China

AbstractThis work is concerned with the existence and uniqueness of weighted Stepanov-likepseudo-almost automorphic mild solutions for a class of semilinear fractionaldifferential equations, Dα

t x(t) = Ax(t) + Dα–1t F(t, x(t)), t ∈R, where 1 < α < 2, A is a linear

densely defined operator of sectorial type of ω < 0 on a complex Banach space X andF is an appropriate function defined on phase space. The fractional derivative isunderstood in the Riemann-Liouville sense. The results obtained are utilized to studythe existence and uniqueness of weighted Stepanov-like pseudo-almostautomorphic mild solutions for a fractional relaxation-oscillation equation.

Keywords: weighted Stepanov-like pseudo-almost automorphic function;semilinear fractional differential equation; fractional relaxation-oscillation equation;solution operator; fractional integral

1 IntroductionIn this paper, we are concerned with the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for the following semilinear fractional dif-ferential equations:

Dαt x(t) = Ax(t) + Dα–

t F(t, x(t)

), t ∈R, ()

where < α < ,

A : D(A) ⊂ X → X

is a linear densely defined operator of sectorial type of ω < on a complex Banach space X,and

F : R× X → X

is an appropriate function. The fractional derivative is understood in the Riemann-Liouville sense.

The almost periodic function was introduced seminally by Bochner in []. It playsan important role in describing the phenomena that are similar to the periodic oscillations

© 2015 He et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly credited.

http://dx.doi.org/10.1186/s13662-015-0410-1

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He et al. Advances in Difference Equations (2015) 2015:74 of 36

Table 1 Historical development of almost periodicity

Function Original reference

Almost periodic (AP) Bochner [1]Asymptotic almost periodic (AAP) Fréchet [16]Pseudo almost periodic (PAP) Zhang [17]Weighted pseudo-almost periodic (WPAP) Diagana [18]Stepanov-like almost periodic (SpAP) Stepanov [19, 20]Stepanov-like pseudo-almost periodic (SpPAP) Diagana [21]Weighted Stepanov-like pseudo-almost periodic (SpWPAP) Diagana et al. [22]

which can be observed frequently in many fields, such as celestial mechanics, nonlinearvibration, electromagnetic theory, plasma physics, engineering, ecosphere, and so on [–]. In mathematics, the almost periodic functions are closely connected with harmonicanalysis, differential equations, dynamical systems, and so on [], they are the generaliza-tion of continuous periodic and quasi-periodic functions. In the last several decades, thebasic theories on the almost periodic functions have been well developed [–], and beenapplied successfully to the investigation of almost periodic dynamics produced by manydifferent kinds of differential equations [–], and they have been some of the most at-tractive topics in the qualitative theory of differential equations for nearly century becauseof their significance and applications in areas such as physics, mathematical biology, con-trol theory, and other related fields. As a result, several concepts were introduced as gen-eralizations or restrictions of almost periodicity, such as asymptotic almost periodicity,pseudo-almost periodicity, weighted pseudo-almost periodicity, Stepanov-like almost pe-riodic, Stepanov-like pseudo-almost periodic and weighted Stepanov-like pseudo-almostperiodic (see, for example, [–]; see Table and the references cited therein for moredetails).

In the earlier s, Bochner introduced the concept of almost automorphic function[–] in relation to some aspects of differential geometry. The notion of almost auto-morphic function was introduced to avoid some assumptions of uniform convergence thatarise when using almost periodic function, it is an important generalization of the classicalalmost periodic function. From that time the theory of almost automorphic function hasbeen studied by numerous authors, and it also has become one of the most attractive topicsin the qualitative theory of differential equations because of its significance and applica-tions. Meanwhile, stimulated by [–], many interesting generalizations of the almostautomorphic function have been introduced, including asymptotic almost automorphyby N’Guérékata [], pseudo-almost automorphy by Xiao et al. [], weighted pseudo-almost automorphy by Blot et al. [], Stepanov-like almost automorphy by Casarino [],Stepanov-like pseudo-almost automorphy by Diagana [] and weighted Stepanov-likepseudo-almost automorphy by Xia and Fan []. The generalizations of almost automor-phy follow closely a historical development very similar to that of almost periodicity andmore and more general types of almost automorphy are developed (see Table and thereferences cited therein for more details). The relationship between the various types ofalmost periodicity and almost automorphy is depicted in Figure .

In recent years, the theory of almost automorphy and its various extensions have at-tracted a great deal of attention of many mathematicians due to their significance andapplications in physics, mathematical biology, control theory, and so on. The existence,uniqueness, and stability of almost automorphic solution have been one of the most at-tractive topics in the context of various kinds of abstract differential equations [, ],


Table 2 Historical development of almost automorphy

Function Original reference

Almost automorphic (AA) Bochner [23]Asymptotic almost automorphic (AAA) N’Guérékata [27]Pseudo almost automorphic (PAA) Xiao et al. [28]Weighted pseudo-almost automorphic (WPAA) Blot [29]Stepanov-like almost automorphic (SpAA) Casarino [30]Stepanov-like pseudo-almost automorphic (SpPAA) Diagana [31]Weighted Stepanov-like pseudo-almost automorphic (SpWPAA) Xia and Fan [32]

Figure 1 Relationship between almost periodic, automorphic functions, and their extensions, where‘→’ denotes the subset relation ‘⊂’.

partial differential equations [, ], functional differential equations [, ], integro-differential equations [] and general dynamic systems []. For more on these studiesand related issues, we refer the reader to the references cited therein. In connection withdifferential equations, the great importance from both the applied and the theoreticalpoint of view of the existence of periodic solutions is well known. However, either be-cause models are only an approximation of reality or due to numerical errors, in practiceit is impossible to verify whether a solution is exactly periodic. The concept of Stepanov-like almost automorphic function allows relaxing some assumptions to obtain solutionsthat have properties similar to those of a periodic function. Meanwhile, the applicationsof the new theory for these generalized functions, especially the Stepanov-like almost au-tomorphic function, to various types of linear, semilinear as well as nonlinear differentialequations were studied extensively (see, e.g., [, , –] and references therein).

In recent years, fractional differential equations have gained considerable interest dueto their applications in various fields of science such as physics, mechanics, chemistryengineering etc. Significant development has been made in ordinary and partial differen-tial equations involving fractional derivatives, we only enumerate here the monographs ofKilbas et al. [, ], Diethelm [], Hilfer [], Podlubny [] and the papers of Agarwalet al. [, ], Benchohra et al. [, ], El-Borai [], Lakshmikantham et al. [–],Mophou et al. [–], N’Guérékata [], and the references therein.

Meanwhile due to their applications in fields of science where characteristics of anoma-lous diffusion are presented, type () equations are attracting increasing interest (cf. [–] and references therein). For example, anomalous diffusion in fractals [] or in macroe-conomics [] has been recently well studied in the setting of fractional Cauchy problemslike (). While the study of almost automorphic mild solutions to () in the borderlinecase α = was well studied in [, ]. In [] Cuevas and Lizama considered () when < α < and A is a linear operator of sectorial negative type on a complex Banach space,


under suitable conditions on F , the authors proved the existence and uniqueness of an al-most automorphic mild solution to (). Cuevas et al. [] and [] study, respectively, thepseudo-almost periodic and pseudo-almost periodic of class infinity mild solutions to ()assuming that F : R× X → X is a pseudo-almost periodic and pseudo-almost periodic ofclass infinity functions satisfying some appropriate conditions in x ∈ X. See also [, ]where the S-asymptotically ω-periodic solutions to () are studied. Recently, Agarwal etal. [] studied the existence and uniqueness of a weighted pseudo-almost periodic mildsolution to (), and Cao et al. [] studied the existence of anti-periodic mild solutionsto ().

From Figure , we know that the weighted Stepanov-like pseudo-almost automorphicfunction is the most widely used function of the almost periodic type functions, and tothe best of our knowledge, the existence of weighted Stepanov-like pseudo-almost au-tomorphic mild solutions for the semilinear fractional differential equation () is a sub-ject that has not been treated in the literature. Our purpose in this paper is to establishsome results concerning the existence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions for equations that can be modeled in the form ().Upon making some appropriate assumptions, some sufficient conditions for the existenceand uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solutions to() are given. In particular, as application, and to illustrate our main results, we will exam-ine some sufficient conditions for the existence and uniqueness of weighted Stepanov-likepseudo-almost automorphic mild solutions to the fractional relaxation-oscillation equa-tion given by

∂αt u(t, x) = ∂

x u(t, x) – pu(t, x) + ∂α–t F

(t, u(t, x)

), t ∈R, x ∈ [,π ],

with boundary conditions

u(t, ) = u(t,π ) = , t ∈ R,

where F satisfies some additional conditions.The rest of this paper is organized as follows. In Section we recall some concepts and

prove some preliminary results. The section that follows contains the main results of thispaper with four existence and uniqueness theorems. In the last section, we prove the exis-tence and uniqueness of weighted Stepanov-like pseudo-almost automorphic mild solu-tions for a fractional relaxation-oscillation equation as an example to illustrate our mainresults.

2 PreliminariesWe begin this section by giving some notations. Throughout this paper, let p ∈ [,∞), de-note by N, Z and R the set of positive integers, the set of integers and the set of real num-bers, respectively. Let (X,‖·‖), (Y ,‖·‖Y ) be two Banach spaces. Let BC(R, X) (respectively,BC(R× Y , X)) denote the space of bounded continuous functions with supremum norm

‖x‖∞ = sup{∥∥x(t)

∥∥ : t ∈R}


(respectively, the space of jointly bounded continuous functions). By L(Y , X) we denote theBanach space of all bounded linear operators from Y to X. If Y = X, it is simply denotedby L(X).

Now, let us recall some basic definitions and results on almost automorphic functions.

Definition . (Bochner []) A continuous function f : R → X is said to be almost au-tomorphic if for every sequence of real numbers {s′

n}∞n=, one can extract a subsequence{sn}∞n= such that

g(t) = limn→∞ f (t + sn),

is well defined in t ∈R, and

limn→∞ g(t – sn) = f (t),

for each t ∈R.Denote by AA(R, X) the set of all such functions.

Definition . [] A continuous function

f : R× Y → X

is said to be almost automorphic if f (t, x) is almost automorphic in t ∈ R uniformly for allx ∈ K , where K is any bounded subset of Y .

Denote by AA(R× Y , X) the set of all such functions.

Remark . The function g in Definition . is measurable but not necessarily continuous.Moreover, if g is continuous, then f is uniformly continuous (cf., e.g., [], Theorem .). Ifthe convergence in Definition . is uniform in t ∈R, then f is almost periodic. A classicalexample of almost automorphic function (not almost periodic) is (cf. [, ])

f (t) = sin

(

+ cos t + cos√

t

), t ∈ R.

Next, let us recall some definitions and basic results on Stepanov-like almost automor-phic functions (for more details, see []).

Definition . The Bochner transform

f b(t, s), t ∈ R, s ∈ [, ],

of a function f : R → X is defined by

f b(t, s) := f (t + s).

Definition . Let p ∈ [,∞). The space BSp(R, X) of all Stepanov bounded functions,with the exponent p, consists of all measurable functions f : R → X such that

f b ∈ L∞(R, Lp([, ], X

)).


This is a Banach space with the norm

‖f ‖Sp :=∥∥f b∥∥

L∞(R,Lp) = supt∈R

(∫ t+

t

∥∥f (τ )∥∥p dτ

) p

.

Definition . The space SpAA(R, X) of Stepanov-like almost automorphic functionsconsists of all f ∈ BSp(R, X) such that

f b ∈ AA(R, Lp([, ], X

)).

That is, a function f ∈ Lploc(R, X) is said to be Stepanov-like almost automorphic if its

Bochner transform

f b : R → Lp([, ], X)

is almost automorphic in the sense that for every sequence of real numbers {s′n}∞n=, there

exist a subsequence {sn}∞n= and a function g ∈ Lploc(R, X) such that

[∫

∥∥f (t + s + sn) – g(t + s)∥∥p ds

] p

→ ,

and

[∫

∥∥g(t + s – sn) – f (t + s)∥∥p ds

] p

→ ,

as n → ∞ for all t ∈R.

Definition . A function

f : R× Y → X, (t, x) → f (t, x)

with

f (·, x) ∈ Lploc(R, X)

for each x ∈ Y is said to be Stepanov-like almost automorphic in t ∈R uniformly for x ∈ Y ,if t → f (t, x) is Stepanov-like almost automorphic for each x ∈ Y . That is, for every se-quence of real numbers {s′

n}∞n=, there exist a subsequence {sn}∞n= and a function

g(·, x) ∈ Lploc(R, X)

such that

[∫

∥∥f (t + s + sn, x) – g(t + s, x)

∥∥p ds

] p

→ ,


and

[∫

∥∥g(t + s – sn, x) – f (t + s, x)

∥∥p ds

] p

→ ,

as n → ∞ for all t ∈R and x ∈ Y .Denote by SpAA(R× Y , X) the set of all such functions.

Remark . It is clear that, if x : R → X is an almost automorphic function, then x is aStepanov-like almost automorphic function, that is,

AA(R, X) ⊂ SpAA(R, X).

Let U be the set of all functions ρ : R → [,∞) which are positive and locally integrableover R. For a given r > and each ρ ∈ U , set

m(r,ρ) :=∫ r

–rρ(x) dx,

and the notation U∞ stands for the set of weight functions

U∞ :={ρ ∈ U : lim

r→∞ m(r,ρ) = ∞}

.

For ρ ∈ U∞, define the weighted ergodic space

PAA(R, X,ρ) :={ϕ ∈ BC(R, X) : lim

r→∞

m(r,ρ)

∫ r

–r

∥∥ϕ(t)∥∥ρ(t) dt =

},

PAA(R× Y , X,ρ)

:={ϕ ∈ C(R× Y , X) : ϕ(·, x) is bounded for each x ∈ Y and

limr→∞

m(r,ρ)

∫ r

–r

∥∥ϕ(t, x)

∥∥ρ(t) dt = uniformly in compact subset of Y

}.

Definition . [] Let ρ ∈ U∞. A continuous function

f ∈ BSp(R, X)

is said to be weighted Stepanov-like pseudo-almost automorphic (or weighted Sp-pseudo-almost automorphic) if it can be decomposed as

f = g + ϕ,

where

g ∈ SpAA(R, X), ϕ ∈ PAA(R, Lp([, ], X

),ρ

).

In other words, a function

f ∈ Lploc(R, X)


is said to be weighted Stepanov-like pseudo-almost automorphic relatively to the weightρ ∈ U∞, if its Bochner transform

f b : R → Lp([, ], X)

is weighted pseudo-almost automorphic in the sense that there exist two functions g,ϕ :R → X such that

f = g + ϕ,

where

gb ∈ AA(R, X), ϕ ∈ PAA(R, Lp([, ], X

),ρ

).

We denote by SpWPAA(R, X) the set of all such functions.

Definition . [] Let ρ ∈ U∞. A function

f : R× Y → X, (t, x) → f (t, x)

with

f (·, x) ∈ Lploc(R, X)

for each x ∈ Y is said to be weighted Stepanov-like pseudo-almost automorphic (or Sp-weighted pseudo-almost automorphic) if it can be expressed as

f = g + ϕ,

where

g ∈ SpAA(R× Y , X), ϕ ∈ PAA(R× Y , Lp([, ], X

),ρ

).

We denote by SpWPAA(R× Y , X) the set of all such functions.

Now we give some lemmas for weighted Stepanov-like pseudo-almost automorphicfunctions.

Lemma . [] Let ρ ∈ U∞. Assume that

PAA(R, Lp([, ], X

),ρ

)

is translation invariant. Then the decomposition of a Sp-weighted pseudo-almost automor-phic function is unique.

Lemma . [] SqWPAA(R, X,ρ) ⊂ SpWPAA(R, X,ρ) for ≤ p < q < +∞.


Lemma . [] Assume that

f , f, f ∈ SpWPAA(R, X,ρ), ρ ∈ U∞.

Then(i) f + f ∈ SpWPAA(R, X,ρ).

(ii) λf ∈ SpWPAA(R, X,ρ) for any λ ∈R.(iii) If

lim supt→∞

ρ(t + τ )ρ(t)

and lim supT→∞

m(T + |τ |,ρ)m(T ,ρ)

are finite for τ ∈R, then

f (t – τ ) ∈ SpWPAA(R, X,ρ).

Lemma . [] Let ρ ∈ U∞. The space SpWPAA(R, X,ρ) equipped with the norm ‖ · ‖Sp

is a Banach space.

Lemma . [] Assume that ρ ∈ U∞,

f = g + ϕ ∈ SpWPAA(R× X, X,ρ)

with

gb ∈ AA(R× X, Lp([, ], X

)), ϕ ∈ PAA

(R× X, Lp([, ], X

)),

and:(i) There exist constants Lf , Lg > such that

∥∥f (t, x)– f (t, y)∥∥ ≤ Lf ‖x–y‖,

∥∥g(t, x)–g(t, y)∥∥ ≤ Lg‖x–y‖, x, y ∈ X, t ∈R.

(ii) h = α + β ∈ SpWPAA(R, X,ρ) with

αb ∈ AA(R, Lp([, ], X

)), ϕ ∈ PAA

(R, Lp([, ], X

)),

and

K ={α(t) : t ∈R

}

is compact in X .Then

f(·, h(·)) ∈ SpWPAA(R, X,ρ).

Lemma . [] Assume that ρ ∈ U∞,

f = g + ϕ ∈ SpWPAA(R× X, X,ρ)


with

gb ∈ AA(R× X, Lp([, ], X

)), ϕ ∈ PAA

(R× X, Lp([, ], X

)),

and:(i) There exist nonnegative functions

Lf , Lg ∈ SrAA(R,R)

with

r ≥ max

{p,

pp –

}

such that

∥∥f (t, x)– f (t, y)

∥∥ ≤ Lf (t)‖x–y‖,

∥∥g(t, x)–g(t, y)

∥∥ ≤ Lg(t)‖x–y‖, x, y ∈ X, t ∈R.

(ii) h = α + β ∈ SpWPAA(R, X,ρ) with

αb ∈ AA(R, Lp([, ], X

)), ϕ ∈ PAA

(R, Lp([, ], X

)),

and

K ={α(t) : t ∈R

}

is compact in X .Then there exists q ∈ [, p) such that

f(·, h(·)) ∈ SpWPAA(R, X,ρ).

Now we give a lemma.

Lemma . Let {xn(t)}n∈N be a sequence of Stepanov-like pseudo-almost automorphicfunctions such that

∫

∥∥xn(t + s) – x(t + s)∥∥p ds → , ()

as n → ∞ for each t ∈R, then

x ∈ SpAA(R, X).

Proof For any i ∈N fixed, since

xi(t) ∈ SpAA(R, X),


for every sequence of real numbers {s′n}n∈N, there exist a subsequence {sn}n∈N and a func-

tion yi ∈ Lploc(R, X) such that

[∫

∥∥xi(t + sn + s) – yi(t + s)∥∥p ds

] p

→ , ()

[∫

∥∥yi(t – sn + s) – xi(t + s)

∥∥p ds

] p

→ , ()

as n → ∞ for all t ∈R. On the other hand, from (), one can easily deduce that {xn(t)}n∈Nis a Cauchy sequence with respect to ‖ · ‖Sp . Observe that, for each t ∈R, the sequence yi

is also a Cauchy sequence in Lploc(R, X). Indeed, if we write

yi(t) – yj(t) = yi(t) – xi(t + sn) + xi(t + sn) – xj(t + sn) + xj(t + sn) – yj(t),

then for a sufficiently large n, one gets

[∫

∥∥yi(t + s) – yj(t + s)

∥∥p ds

] p

≤[∫

(∥∥yi(t + s) – xi(t + s + sn)∥∥ +

∥∥xi(t + s + sn) – xj(t + s + sn)∥∥

+∥∥xj(t + s + sn) – yj(t + s)

∥∥)p ds

] p

≤ [∫

(∥∥yi(t + s) – xi(t + s + sn)∥∥p +

∥∥xi(t + s + sn) – xj(t + s + sn)∥∥p

+∥∥xj(t + s + sn) – yj(t + s)

∥∥p)ds

] p

.

By (), (), and (), the sequence of yi is a Cauchy sequence in Lploc(R, X).

Using the completeness of Lploc(R, X), we denote by y(t) the pointwise limit of yi(t). Now

let us prove that

x(t) ∈ SpAA(R, X).

Note that the inequality below holds for any index i and any t ∈R,

[∫

∥∥x(t + s + sn) – y(t + s)∥∥p ds

] p

≤[∫

(∥∥x(t + s + sn) – xi(t + s + sn)∥∥

+∥∥xi(t + s + sn) – yi(t + s)

∥∥ +

∥∥yi(t + s) – y(t + s)

∥∥)p ds

] p

≤ [∫

(∥∥x(t + s + sn) – xi(t + s + sn)∥∥p +

∥∥xi(t + s + sn) – yi(t + s)∥∥p

+∥∥yi(t + s) – y(t + s)

∥∥p)ds

] p

.


So, from () and the fact that y(t) is the pointwise limit of yi(t), for any sufficiently smallε > there exists a sufficiently large i, such that for each t ∈R,

∫

∥∥xi(t + s + sn) – yi(t + s)

∥∥p ds <

εp

p+ ,

∫

∥∥x(t + s + sn) – xi(t + s + sn)∥∥p ds <

εp

p+ .

Now for this sufficiently large i, from () and (), there exists a sufficient N such that forany n > N one has

∫

∥∥yi(t + s) – y(t + s)

∥∥p ds <

εp

p+ .

Thus

[∫

∥∥x(t + s + sn) – y(t + s)

∥∥p ds

] p

< ε, for n > N ,

which implies

[∫

∥∥x(t + s + sn) – y(t + s)

∥∥p ds

] p

→ ,

as n → ∞ pointwise on R. One can use the same steps to prove that

[∫

∥∥y(t + s – sn) – x(t + s)∥∥p ds

] p

→ ,

as n → ∞ pointwise on R. That is,

x(t) ∈ SpAA(R, X).

The proof is finished. �

We need some basic definitions and properties of the fractional calculus theory whichare used further in this paper.

Definition . [] The fractional Riemann-Liouville integral of order α > with thelower limit t for a function f is defined as

Iαf (t) =

�(α)

∫ t

t

(t – s)α–f (s) ds, t > t,α > ,

provided the right-hand side is pointwise defined on [t,∞), where � is the Gamma func-tion.


Definition . [] The Riemann-Liouville derivative of order α > with the lower limitt for a function f : [t,∞) →R can be written as

Dαt f (t) =

�(n – α)

dn

dtn

∫ t

t

(t – s)–αf (s) ds, t > t, n – < α < n.

Remark . The first and maybe the most important property of the Riemann-Liouvillefractional derivative is that, for t > t and α > , one has

Dαt(Iαf (t)

)= f (t),

which means that the Riemann-Liouville fractional differentiation operator is a left inverseto the Riemann-Liouville fractional integration operator of the same order α.

In the following, we give the definitions of sectorial linear operators and their associatedsolution operators.

Recall that a closed and linear operator A is said to be sectorial of type ω and angle θ ifthere exist

< θ <π

, M > , ω ∈R,

such that its resolvent exists outside the sector

ω + Sθ :={ω + λ : λ ∈ C,

∣∣arg(–λ)

∣∣ < θ

},

and

∥∥(λ – A)–∥∥ ≤ M|λ – ω| , λ /∈ ω + Sθ .

Sectorial operators are well studied in the literature, usually for the case ω = . For a recentreference including several examples and properties we refer the reader to []. Note thatan operator A is sectorial of type ω if and only if ωI – A is sectorial of type .

Definition . [] Let A be a closed and linear operator with domain D(A) defined ona Banach space X. We call A is the generator of a solution operator if there are ω ∈R anda strongly continuous function

Sα : R+ → L(X)

such that

{λα : Reλ > ω

} ⊆ ρ(A)

and

λα–(λα – A)–x =

∫ ∞

e–λtSα(t)x dt, Reλ > ω, x ∈ X.

In this case, Sα(t) is called the solution operator generated by A.


We note that if A is sectorial of type ω with

≤ θ ≤ π

( –

α

),

then A is the generator of a solution operator given by

Sα(t) :=

π i

∫

γ

e–λtλα–(λα – A)– dλ, ()

where γ is a suitable path lying outside the sector ω + �θ (cf. []).Very recently, Cuesta in [], Theorem , has proved that if A is a sectorial operator of

type ω < for some M > and

≤ θ < π

( –

α

),

then there exists C > such that

∥∥Sα(t)∥∥

L(X) ≤ CM + |ω|tα

, ()

for t ≥ . In the border case α = , this is analogous to saying that A is the generator ofan exponentially stable C-semigroup. The main difference is that in the case α > thesolution family Sα(t) decays like t–α . Cuesta’s result proves that Sα(t) is, in fact, integrable.

Now we give another lemma.

Lemma . Assume that () is true. Given a function

F(t) ∈ SpWPAA(R, X).

Let

[�F](t) :=∫ t

–∞Sα(t – s)F(s) ds.

Then [�F](t) is weighted Stepanov-like pseudo-almost automorphic.

Proof Firstly, note that

∫ ∞

+ |ω|sα

ds =ω–

α π

α sin πα

for < α < . ()

By condition (), one has

[∫ t+

t

∥∥∥∥

∫ σ

–∞Sα(σ – s)F(s) ds

∥∥∥∥

p

dσ

] p

=[∫ t+

t

∥∥∥∥

∫ ∞

Sα(τ )F(σ – τ ) dτ

∥∥∥∥

p

dσ

] p


≤ CM[∫ t+

t

∫ ∞

(

+ |ω|τα

)p∥∥F(σ – τ )∥∥p dτ dσ

] p

≤ CM‖F‖Sp

(∫ ∞

(

+ |ω|τα

)p

dτ

) p

≤ CM‖F‖Sp

(∫ ∞

+ |ω|τα

dτ

) p

= CM‖F‖Sp

[ω–

α π

α sin πα

] p

.

Thus, � is well defined and �F is bounded. On the other hand, for any t, h ∈R,

(∫ t+

t

∥∥[�F](σ + h) – [�F](σ )

∥∥p dσ

) p

=(∫ t+

t

∥∥∥∥

∫ σ+h

–∞Sα(σ + h – s)F(s) ds –

∫ σ

–∞Sα(σ – s)F(s) ds

∥∥∥∥

p

dσ

) p

=(∫ t+

t

∥∥∥∥

∫ σ

–∞Sα(σ – s)

[F(s + h) – F(s)

]ds

∥∥∥∥

p

dσ

) p

=(∫ t+

t

∥∥∥∥

∫ ∞

Sα(τ )

[F(σ – τ + h) – F(σ – τ )

]dτ

∥∥∥∥

p

dσ

) p

≤ CM∥∥F(t + h) – F(t)

∥∥

Sp

(∫ ∞

[

+ |ω|τα

]p

dτ

) p

≤ CM∥∥F(t + h) – F(t)

∥∥Sp

(∫ ∞

+ |ω|τα

dτ

) p

= CM[

ω– α π

α sin πα

] p ∥∥F(t + h) – F(t)

∥∥

Sp ,

which shows that �F is continuous. Since

F ∈ SpWPAA(R, X),

there exist

G ∈ SpAA(R, X) and � ∈ PAA(R, Lp([, ], X

),ρ

),

such that F = G + �. So

x(t) =∫ t

–∞Sα(t – σ )F(σ ) dσ =

∫ t

–∞Sα(t – σ )G(σ ) dσ +

∫ t

–∞Sα(t – σ )�(σ ) dσ

= �(t) + �(t).

We only need to verify

�(t) ∈ SpAA(R, X), �(t) ∈ PAA(R, X,ρ).


First we prove that

�(t) ∈ SpAA(R, X).

Let {s′m}m∈N be a sequence of real numbers. Since

G ∈ SpAA(R, X),

there exist a subsequence {sm}m∈N of {s′m}m∈N and a function G̃ such that

[∫

∥∥G(t + sm + s) – G̃(t + s)

∥∥p ds

] p

→ , ()

[∫

∥∥G̃(t – sm + s) – G(t + s)∥∥p ds

] p

→ , ()

as m → ∞ pointwise on R for each x ∈ X. Let

[�G̃](t) :=∫ t

–∞Sα(t – s)G̃(s) ds.

Thus

(∫

∥∥[�G](t + s + sm) – [�G̃](t + s)∥∥p ds

) p

=(∫

∥∥∥∥

∫ t

–∞Sα(σ )G(t + s + sm – σ ) dσ –

∫ t

–∞Sα(σ )G̃(t + s – σ ) dσ

∥∥∥∥

p

ds)

p

=(∫

∥∥∥∥

∫ t

–∞Sα(σ )

[G(t + s + sm – σ ) – G̃(t + s – σ )

]dσ

∥∥∥∥

p

ds)

p

≤(∫

(∫ t

–∞

∥∥Sα(σ )

∥∥∥∥G(t + s + sm – σ ) – G̃(t + s – σ )

∥∥dσ

)p

ds)

p

≤(∫

∫ t

–∞

∥∥Sα(σ )∥∥p∥∥G(t + s + sm – σ ) – G̃(t + s – σ )

∥∥p dσ ds)

p

≤(∫

∫ ∞

[CM

+ |ω|σα

]p∥∥G(t + s + sm – σ ) – G̃(t + s – σ )∥∥p dσ ds

) p

=(∫ ∞

[CM

+ |ω|σα

]p ∫

∥∥G(t + s + sm – σ ) – G̃(t + s – σ )∥∥p ds dσ

) p

≤(∫ ∞

[CpMp

+ |ω|σα

]∫

∥∥G(t + s + sm – σ ) – G̃(t + s – σ )∥∥p ds dσ

) p

.

From (), (), and (), obviously, the last inequality goes to as m → ∞ pointwise on R.Similarly one can prove that

[∫

∥∥[�G̃](t + s – sm) – [�G](t + s)

∥∥p ds

] p

→ ,


as m → ∞ pointwise on R. Thus we conclude that

[�G] ∈ SpAA(R, X).

In the following, we prove that

�(t) ∈ PAA(R, X,ρ).

To complete the proof, consider for each n = , , . . . , the integrals

�n(t) =∫ t–n+

t–nSα(t – σ )�(σ ) dσ ,

for each t ∈R. Note that

�n(t) =∫ t–n+

t–nSα(t – σ )�(σ ) dσ =

∫ n

n–Sα(σ )�(t – σ ) dσ ,

and by using the Hölder inequality, one gets

∥∥�n(t)

∥∥

Sp = supt∈R

(∫ t+

t

∥∥�n(τ )

∥∥p dτ

) p

= supt∈R

(∫

∥∥�n(t + s)∥∥p ds

) p

= supt∈R

(∫

∥∥�n(t + s)∥∥p ds

) p

= supt∈R

(∫

∥∥∥∥

∫ n

n–Sα(σ )�(t + s – σ ) dσ

∥∥∥∥

p

ds)

p

≤ supt∈R

(∫

(∫ n

n–

∥∥Sα(σ )

∥∥∥∥�(t + s – σ )

∥∥dσ

)p

ds)

p

≤ supt∈R

(∫

∫ n

n–

∥∥Sα(σ )

∥∥p∥∥�(t + s – σ )

∥∥p dσ ds

) p

≤ supt∈R

(∫

∫ n

n–

[CM

+ |ω|σα

]p∥∥�(t + s – σ )∥∥p dσ ds

) p

= supt∈R

(∫ n

n–

[CM

+ |ω|σα

]p ∫

∥∥�(t + s – σ )∥∥p ds dσ

) p

= supt∈R

(∫ n

n–

[CM

+ |ω|σα

]p

‖�‖pSp dσ

) p

= CM‖�‖Sp

(∫ n

n–

[

+ |ω|σα

]p

dσ

) p

≤ CM‖�‖Sp

+ |ω|(n – )α

≤ CM‖�‖Sp

|ω|

(n – )α.


From < α < , it follows that

CM‖�‖pSp

|ω|∞∑

n=

(n – )α

< ∞,

one can deduce from the well-known Weierstrass test that the series∞∑

n=

�n(t)

is convergent in the sense of the norm ‖ · ‖Sp uniformly on R. Now let

�(t) :=∞∑

n=

�n(t), for each t ∈R.

Observe that

�(t) =∫ t

–∞Sα(t – σ )G(σ ) dσ , for each t ∈ R.

Clearly, for any t, h ∈R,

(∫ t+

t

∥∥�(σ + h) – �(σ )

∥∥p dσ

) p

=(∫ t+

t

∥∥∥∥

∫ σ+h

–∞Sα(σ + h – s)�(s) ds –

∫ σ

–∞Sα(σ – s)�(s) ds

∥∥∥∥

p

dσ

) p

=(∫ t+

t

∥∥∥∥

∫ σ

–∞Sα(σ – s)

[�(s + h) – �(s)

]ds

∥∥∥∥

p

dσ

) p

=(∫ t+

t

∥∥∥∥

∫ ∞

Sα(τ )

[�(σ – τ + h) – �(σ – τ )

]dτ

∥∥∥∥

p

dσ

) p

≤ CM∥∥�(t + h) – �(t)

∥∥

Sp

(∫ ∞

[

+ |ω|τα

]p

dτ

) p

≤ CM∥∥�(t + h) – �(t)

∥∥Sp

(∫ ∞

+ |ω|τα

dτ

) p

= CM[

ω– α π

α sin πα

] p ∥∥�(t + h) – �(t)

∥∥

Sp ,

which shows that � is continuous. So, we only need to show that

limT→∞

m(T ,ρ)

∫ T

–T

∥∥�(t)∥∥ρ(t) dt = .

In fact, one has

∥∥�n(t)

∥∥ =

∥∥∥∥

∫ t–n+

t–nSα(t – σ )�(σ ) dσ

∥∥∥∥

≤∫ t–n+

t–n

∥∥Sα(t – σ )∥∥∥∥�(σ )

∥∥dσ


≤∫ t–n+

t–n

CM + |ω|(t – σ )α

∥∥�(σ )∥∥dσ

≤(∫ t–n+

t–n

[CM

+ |ω|(t – σ )α

]q) q(∫ t–n+

t–n

∥∥�(σ )∥∥p dσ

) p

≤ CM + |ω|(n – )α

(∫ t–n+

t–n

∥∥�(σ )∥∥p dσ

) p

≤ CM|ω|

(n – )α

(∫ t–n+

t–n

∥∥�(σ )∥∥p dσ

) p

,

where q = p/(p – ). Then

m(T ,ρ)

∫ T

–T

∥∥�n(t)

∥∥ρ(t) dt

≤ CM|ω|

(n – )α

m(T ,ρ)

∫ T

–T

(∫ t–n+

t–n

∥∥�(σ )

∥∥p dσ

) pρ(t) dt,

and hence

�n(t) ∈ PAA(R, X,ρ)

since

� ∈ PAA(R, Lp([, ], X

),ρ

).

From


and

m(T ,ρ)

∫ T

–T

∥∥�(t)

∥∥ρ(t) dt

≤ CM|ω|

(n – )α

m(T ,ρ)

∫ T

–T

∥∥∥∥∥�(t) –

N∑

n=

�n(t)

∥∥∥∥∥ρ(t) dt

+N∑

n=

CM|ω|

(n – )α

m(T ,ρ)

∫ T

–T

∥∥�n(t)∥∥ρ(t) dt,

it follows that

�(t) ∈ PAA(R, X,ρ).

Therefore,

x(t) ∈ SpWPAA(R, X).

The proof is now complete. �


3 Stepanov-like almost automorphic mild solutionsLet < α < . We first consider the linear version for (), that is

Dαt x(t) = Ax(t) + Dα–

t F(t), t ∈R. ()

Observe that () can be viewed as the limiting equation for the equation

y′(t) =∫ t

(t – s)α–

�(α – )Ay(s) + F(t), t ≥ , y() = x ∈ X, ()

in the sense that the solutions x(t) of () and y(t) of () are asymptotic to each other ast → ∞. In fact, if we assume that A is sectorial of type ω with

≤ θ < π

( –

α

),

then () is well posed (cf. []) and the variation of parameters formula allows us to writethe solution of () as

y(t) = Sα(t)x +∫ t

Sα(t – s)F(s) ds, t ≥ ,

where the family of operators Sα(t) is given by (). On the other hand, if Sα(t) is integrable,then the solution of () is given by

x(t) =∫ t

–∞Sα(t – s)F(s) ds. ()

Hence

y(t) – x(t) = Sα(t)x –∫ ∞

tSα(s)F(t – s) ds,

which shows that

y(t) – x(t) → , as t → ∞

whenever F ∈ Lp(R+, X) for some p ∈ [, +∞).From Cuesta’s result, it follows that Sα(t) is integrable. Thus the above considerations

motivate the following definition.

Definition . A function x : R → X is said to be a mild solution to () if the function

s → Sα(t – s)F(s)

is integrable on (–∞, t) for each t ∈R and

x(t) =∫ t

–∞Sα(t – σ )F(σ ) dσ .


Similarly, a function x : R → X is said to be a mild solution to () if the function

s → Sα(t – s)F(s, x(s)

)

is integrable on (–∞, t) for each t ∈R and

x(t) =∫ t

–∞Sα(t – σ )F

(σ , x(σ )

)dσ .

To study the existence and uniqueness of weighted Stepanov-like pseudo-almost auto-morphic mild solutions to (), we first consider the existence and uniqueness of weightedStepanov-like pseudo-almost automorphic mild solutions to the linear fractional differ-ential equation () with < α < ,

A : D(A) ⊂ X → X

is a linear densely defined operator of sectorial type of ω < on a complex Banach spaceX and

F : R → X

is a weighted Stepanov-like pseudo-almost automorphic function. The fractional deriva-tive is understood in the Riemann-Liouville sense.

The following are the main results for the linear fractional differential equations ().

Theorem . Assume that A is sectorial of type ω < . Then () admits a weightedStepanov-like pseudo-almost automorphic mild solution.

Proof Since

F ∈ SpWPAA(R, X),

there exist

G ∈ SpAA(R, X) and � ∈ PAA(R, Lp([, ], X

),ρ

),

such that F = G + �. So

x(t) =∫ t

–∞Sα(t – σ )F(σ ) dσ

=∫ t

–∞Sα(t – σ )G(σ ) dσ +

∫ t

–∞Sα(t – σ )�(σ ) dσ

= �(t) + �(t).

We only need to verify

�(t) ∈ SpAA(R, X), �(t) ∈ PAA(R, X,ρ).


First we prove that

�(t) ∈ SpAA(R, X).

Consider for each n = , , . . . , the integrals

ϒn(t) =∫ t–n+

t–nSα(t – σ )G(σ ) dσ ,


ϒn(t) =∫ t–n+

t–nSα(t – σ )G(σ ) dσ =

∫ n

n–Sα(σ )G(t – σ ) dσ ,

and by using the Hölder inequality, one gets

∥∥ϒn(t)

∥∥

Sp = supt∈R

(∫ t+

t

∥∥ϒn(τ )

∥∥p dτ

) p

= supt∈R

(∫

∥∥ϒn(t + s)∥∥p ds

) p

= supt∈R

(∫

∥∥ϒn(t + s)∥∥p ds

) p

= supt∈R

(∫

∥∥∥∥

∫ n

n–Sα(σ )G(t + s – σ ) dσ

∥∥∥∥

p

ds)

p

≤ supt∈R

(∫

(∫ n

n–

∥∥Sα(σ )

∥∥∥∥G(t + s – σ )

∥∥dσ

)p

ds)

p

≤ supt∈R

(∫

∫ n

n–

∥∥Sα(σ )

∥∥p∥∥G(t + s – σ )

∥∥p dσ ds

) p

≤ supt∈R

(∫

∫ n

n–

[CM

+ |ω|σα

]p∥∥G(t + s – σ )∥∥p dσ ds

) p

= supt∈R

(∫ n

n–

[CM

+ |ω|σα

]p ∫

∥∥G(t + s – σ )∥∥p ds dσ

) p

= supt∈R

(∫ n

n–

[CM

+ |ω|σα

]p

‖G‖pSp dσ

) p

= CM‖G‖Sp

(∫ n

n–

[

+ |ω|σα

]p

dσ

) p

≤ CM‖G‖Sp

+ |ω|(n – )α

≤ CM‖G‖Sp

|ω|

(n – )α.


From < α < , it follows that

CM‖G‖pSp

|ω|∞∑

n=

(n – )α

< ∞,

one can deduce from the well-known Weierstrass test that the series

∞∑

n=

ϒn(t)

is convergent in the sense of the norm ‖ · ‖Sp uniformly on R. Now let

�(t) :=∞∑

n=

ϒn(t), for each t ∈R.

Observe that

�(t) =∫ t

–∞Sα(t – σ )G(σ ) dσ , for each t ∈R.

Clearly, for any t, h ∈R,

(∫ t+

t

∥∥�(σ + h) – �(σ )∥∥p dσ

) p

=(∫ t+

t

∥∥∥∥

∫ σ+h

–∞Sα(σ + h – s)G(s) ds –

∫ σ

–∞Sα(σ – s)G(s) ds

∥∥∥∥

p

dσ

) p

=(∫ t+

t

∥∥∥∥

∫ σ

–∞Sα(σ – s)

[G(s + h) – G(s)

]ds

∥∥∥∥

p

dσ

) p

=(∫ t+

t

∥∥∥∥

∫ ∞

Sα(τ )

[G(σ – τ + h) – G(σ – τ )

]dτ

∥∥∥∥

p

dσ

) p

≤ CM∥∥G(t + h) – G(t)

∥∥Sp

(∫ ∞

[

+ |ω|τα

]p

dτ

) p

≤ CM∥∥G(t + h) – G(t)

∥∥Sp

(∫ ∞

+ |ω|τα

dτ

) p

= CM[

ω– α π

α sin πα

] p ∥∥G(t + h) – G(t)

∥∥Sp ,

which shows that � is continuous.Now let us show that each

ϒn ∈ SpAA(R, X).

Indeed, let {s′m}m∈N be a sequence of real numbers. Since

G ∈ SpAA(R, X),


there exist a subsequence {sm}m∈N of {s′m}m∈N and a function G̃ such that

[∫

∥∥G(t + sn + s) – G̃(t + s)∥∥p ds

] p

→ ,

[∫

∥∥G̃(t – sn + s) – G(t + s)

∥∥p ds

] p

→ ,

as m → ∞ pointwise on R. Moreover, if we let

ϒ̃n(t) =∫ n

n–Sα(σ )G̃(t – σ ) dσ ,

one has

(∫

∥∥xn(t + s + sm) – x̃n(t + s)

∥∥p ds

) p

=(∫

∥∥∥∥

∫ n

n–Sα(σ )G(t + s + sm – σ ) dσ –

∫ n

n–Sα(σ )G̃(t + s – σ ) dσ

∥∥∥∥

p

ds)

p

=(∫

∥∥∥∥

∫ n

n–Sα(σ )

[G(t + s + sm – σ ) – G̃(t + s – σ )

]dσ

∥∥∥∥

p

ds)

p

≤(∫

(∫ n

n–

∥∥Sα(σ )∥∥∥∥G(t + s + sm – σ ) – G̃(t + s – σ )

∥∥dσ

)p

ds)

p

≤(∫

∫ n

n–

∥∥Sα(σ )

∥∥p∥∥G(t + s + sm – σ ) – G̃(t + s – σ )

∥∥p dσ ds

) p

≤(∫

∫ n

n–

[CM

+ |ω|σα

]p∥∥G(t + s + sm – σ ) – G̃(t + s – σ )∥∥p dσ ds

) p

=(∫ n

n–

[CM

+ |ω|σα

]p ∫

∥∥G(t + s + sm – σ ) – G̃(t + s – σ )

∥∥p ds dσ

) p

.

Obviously, the last inequality goes to as m → ∞ pointwise on R. Similarly one can provethat

[∫

∥∥ϒ̃n(t + s – sm) – ϒn(t + s)∥∥p ds

] p

→ ,

as m → ∞ pointwise on R. Thus we conclude that each

ϒn ∈ SpAA(R, X)

and consequently their uniform limit

�(t) ∈ SpAA(R, X),

by using Lemma ..


In the following, we prove that

�(t) ∈ PAA(R, X,ρ).

To complete the proof, consider for each n = , , . . . , the integrals

�n(t) =∫ t–n+

t–nSα(t – σ )�(σ ) dσ ,


�n(t) =∫ t–n+

t–nSα(t – σ )�(σ ) dσ =

∫ n

n–Sα(σ )�(t – σ ) dσ .

By carrying out similar arguments as above, we know that �n(t) is bounded and continu-ous, and

∞∑

n=

�n(t)

is uniformly convergent on R. Let

�(t) :=∞∑

n=

�n(t), for each t ∈R,

then

�(t) =∫ t

–∞Sα(t – σ )�(σ ) dσ , for each t ∈R.

It is obvious that �(t) is bounded and continuous. So, we only need to show that

limT→∞

m(T ,ρ)

∫ T

–T

∥∥�(t)

∥∥ρ(t) dt = .

In fact, one has

∥∥�n(t)∥∥ =

∥∥∥∥

∫ t–n+

t–nSα(t – σ )�(σ ) dσ

∥∥∥∥

≤∫ t–n+

t–n

∥∥Sα(t – σ )

∥∥∥∥�(σ )

∥∥dσ

≤∫ t–n+

t–n

CM + |ω|(t – σ )α

∥∥�(σ )

∥∥dσ

≤(∫ t–n+

t–n

[CM

+ |ω|(t – σ )α

]q) q(∫ t–n+

t–n

∥∥�(σ )

∥∥p dσ

) p

≤ CM + |ω|(n – )α

(∫ t–n+

t–n

∥∥�(σ )∥∥p dσ

) p

≤ CM|ω|

(n – )α

(∫ t–n+

t–n

∥∥�(σ )

∥∥p dσ

) p

,


where q = p/(p – ). Then

m(T ,ρ)

∫ T

–T

∥∥�n(t)∥∥ρ(t) dt

≤ CM|ω|

(n – )α

m(T ,ρ)

∫ T

–T

(∫ t–n+

t–n

∥∥�(σ )∥∥p dσ

) pρ(t) dt,

and hence


since

� ∈ PAA(R, Lp([, ], X

),ρ

).

From


and

m(T ,ρ)

∫ T

–T

∥∥�(t)∥∥ρ(t) dt

≤ CM|ω|

(n – )α

m(T ,ρ)

∫ T

–T

∥∥∥∥∥�(t) –

N∑

n=

�n(t)

∥∥∥∥∥ρ(t) dt

+N∑

n=

CM|ω|

(n – )α

m(T ,ρ)

∫ T

–T

∥∥�n(t)

∥∥ρ(t) dt,

it follows that

�(t) ∈ PAA(R, X,ρ).

Therefore,

x(t) ∈ SpWPAA(R, X).

In view of the above, it follows that x(t) is the bounded weighted Stepanov-like pseudo-almost automorphic mild solution to (). The proof is now complete. �

Now we investigate the Stepanov-like almost automorphic mild solutions to the nonlin-ear fractional differential equation (), the following are the main results.

Theorem . Assume that A is sectorial of type ω < and ρ ∈ U∞. Let

F = G + � ∈ SpWPAA(R× X, X,ρ)


with

Gb ∈ AA(R× X, Lp([, ], X

)), � ∈ PAA

(R× X, Lp([, ], X

)),

and there exist nonnegative functions

LF , LG ∈ SrAA(R,R)

with

r ≥ max

{p,

pp –

}

such that

∥∥F(t, x) – F(t, y)∥∥ ≤ LF (t)‖x – y‖, ()

∥∥G(t, x) – G(t, y)

∥∥ ≤ LG(t)‖x – y‖, x, y ∈ X, t ∈R, ()

where

LF (t) ∈ Lp(R).

Then () admits a unique weighed Stepanov-like pseudo-almost automorphic mild solution.

Proof Define the operator � on SpWPAA(R, X) by

�x(t) =∫ t

–∞Sα(t – σ )F

(σ , x(σ )

)dσ .

From Lemma ., it follows that

F(·) = F(·, x(·)) ∈ SpWPAA(R, X).

From the function

+ |ω|tα

being integrable on R+ (α > ) and the proof of Lemma ., one can easily see that �x

is well defined and continuous. Then by using the proof of Theorem . with the aboveLemma ., one has

�x ∈ SpWPAA(R, X)

whenever

x ∈ SpWPAA(R, X).


Thus � maps SpWPAA(R, X) into itself. It suffices now to show that this operator � has aunique fixed point in SpWPAA(R, X). For this, let x, y be in SpWPAA(R, X) and define

Cα := supt∈R

∥∥Sα(t)∥∥,

one has

∥∥�x(t) – �y(t)

∥∥

Sp

= supt∈R

(∫ t+

t

∥∥∥∥

∫ τ

–∞Sα(τ – σ )

[F(σ , x(σ )

)– F

(σ , y(σ )

)]dσ

∥∥∥∥

p

dτ

) p

≤ supt∈R

(∫ t+

t

∫ τ

–∞Lp(σ )

∥∥S(τ – σ )

∥∥p∥∥x(σ ) – y(σ )

∥∥p dσ dτ

) p

≤ Cα‖L‖p‖x – y‖Sp .

In general we get

∥∥[

�nx](t) –

[�ny

](t)

∥∥

Sp

≤ Cnα

(n – )!

(∫ t

–∞Lp(σ )

(∫ σ

–∞Lp(τ ) dτ

)n–

dσ

) p‖x – y‖Sp

≤ Cnα

n!

((∫ t

–∞L(σ ) dσ

) p)n

‖x – y‖Sp

≤ (Cα‖L‖p)n

n!‖x – y‖Sp .

Hence, since

(Cα‖L‖p)n

n!<

for n sufficiently large, by the contraction principle � has a unique fixed point

x ∈ SpWPAA(R, X).

We note that conditions of type () have been previously considered in the literature foralmost automorphic functions []. Our motivation comes from their use in the study ofpseudo-almost periodic solutions of semilinear Cauchy problems []. Now we considerthe more general case of equations introducing a new class of functions L which do notnecessarily belong to Lp(R). We have the following result. �



with

Gb ∈ AA(R× X, Lp([, ], X

)), � ∈ PAA

(R× X, Lp([, ], X

)),


and there exist nonnegative functions

LF , LG ∈ SrAA(R,R)

with

r ≥ max

{p,

pp –

}

such that

∥∥F(t, x) – F(t, y)∥∥ ≤ LF (t)‖x – y‖,

∥∥G(t, x) – G(t, y)∥∥ ≤ LG(t)‖x – y‖, x, y ∈ X, t ∈R,

where the integral

∫ t

–∞LF (σ ) dσ

exists for all t ∈ R. Then () admits a unique weighed Stepanov-like pseudo-almost auto-morphic mild solution.

Proof Define a new norm

∣∣‖x‖∣∣ := sup

t∈R

{v(t)

∥∥x(t)

∥∥

Sp}

,

where

v(t) :=[e–k

∫ t–∞ L(σ ) dσ

] p ,

and k is a fixed positive constant greater than

Cα := supt∈R

∥∥Sα(t)∥∥.

Let x, y be in SpWPAA(R, X), then one has

v(t)∥∥�x(t) – �y(t)

∥∥

Sp

= v(t) supt∈R

(∫ t+

t

∥∥∥∥

∫ τ

–∞S(τ – σ )

[F(σ , x(σ )

)– F

(σ , y(σ )

)]dσ

∥∥∥∥

p

dτ

) p

≤ Cα supt∈R

(∫ t+

t

∫ τ

–∞vp(τ )Lp(σ )

∥∥x(σ ) – y(σ )

∥∥p dσ dτ

) p

= Cα supt∈R

(∫ t+

t

∫ τ

–∞vp(τ )vp(σ )Lp(σ )

(vp(σ )

)–∥∥x(σ ) – y(σ )∥∥p dσ dτ

) p

≤ Cα

∣∣‖x – y‖∣∣ supt∈R

(∫ t+

t

∫ τ

–∞vp(τ )

(vp(σ )

)–Lp(σ ) dσ dτ

) p


=Cα

k∣∣‖x – y‖∣∣ sup

t∈R

(∫ t+

t

∫ τ

–∞kek

∫ στ L(τ ) dτ L(σ ) dσ dτ

) p

=Cα

k∣∣‖x – y‖∣∣ sup

t∈R

(∫ t+

t

∫ τ

–∞d

dσ

(ek

∫ στ L(τ ) dτ

)dσ dτ

) p

=Cα

ksupt∈R

(∫ t+

t

( – e–k

∫ τ–∞ L(τ ) dτ

)dτ

) p ∣∣‖x – y‖∣∣

≤ Cα

k∣∣‖x – y‖∣∣.

Hence, since

Cα

k< ,

� has a unique fixed point

x ∈ SpWPAA(R, X).

Note that the above result does not include the cases where LF and LG are constants.�



with

Gb ∈ AA(R× X, Lp([, ], X

)), � ∈ PAA

(R× X, Lp([, ], X

)),

and there exist constants LF , LG such that

∥∥F(t, x) – F(t, y)∥∥ ≤ LF‖x – y‖,

∥∥G(t, x) – G(t, y)

∥∥ ≤ LG‖x – y‖, x, y ∈ X, t ∈R,

Then () admits a unique Stepanov-like pseudo-almost automorphic mild solution when-ever

CMLFω– α π < α sin

π

α.

Proof For x, y ∈ SpWPAA(R, X), one has

∥∥�x(t) – �y(t)

∥∥

Sp

= supt∈R

(∫ t+

t

∥∥∥∥

∫ s

–∞Sα(s – σ )

[F(σ , x(σ )

)– F

(σ , y(σ )

)]dσ

∥∥∥∥

p

ds)

p

= supt∈R

(∫ t+

t

∥∥∥∥

∫ ∞

Sα(τ )

[F(s – τ , x(s – τ )

)– F

(s – τ , y(s – τ )

)]dτ

∥∥∥∥

p

ds)

p


≤ L supt∈R

(∫ t+

t

∫ ∞

∥∥Sα(τ )∥∥p∥∥x(s – τ ) – y(s – τ )

∥∥p dτ ds)

p

≤ CML‖x – y‖Sp

(∫ ∞

(

+ |ω|τα

)p

dτ ds)

p

≤ CML‖x – y‖Sp

(∫ ∞

+ |ω|τα

dτ ds)

p

= CML[

ω– α π

α sin πα

] p‖x – y‖Sp .

This proves that � is a strict contraction, so it follows from the Banach contraction map-ping principle that � admits a unique fixed point

x ∈ SpWPAA(R, X),

which is the unique weighed Stepanov-like pseudo-almost automorphic mild solutionto (). �

Taking

A = –ραI, with ρ >

in (), the above theorem gives the following corollary.

Corollary . Let ρ ∈ U∞,


with

Gb ∈ AA(R× X, Lp([, ], X

)), � ∈ PAA

(R× X, Lp([, ], X

)),

and there exist constants LF , LG such that

∥∥F(t, x) – F(t, y)∥∥ ≤ LF‖x – y‖, ()

∥∥G(t, x) – G(t, y)

∥∥ ≤ LG‖x – y‖, x, y ∈ X, t ∈R. ()

Then () admits a unique weighted Stepanov-like pseudo-almost automorphic mild solu-tion whenever

CLF <α sin( π

α)

ρπ.

Remark . It is interesting to note that the function

α → α sin( πα

)ρπ


is increasing from to ρπ

in the interval < α < . Therefore, with respect to the Lipschitzcondition (), the class of admissible semilinear terms F(t, x(t)) is the best in the case α = and the worst in the case α = . Note the direct relation with the term

(t – s)α–

�(α – )

in (), where the singularity becomes better (smooth) when α goes from to .

4 ApplicationsIn this section we give an example to illustrate the above results.

Consider the following fractional relaxation-oscillation equation:

∂αt u(t, x) = ∂

x u(t, x) – μu(t, x) + ∂α–t F

(t, u(t, x)

), t ∈ R, x ∈ [,π ],

u(t, ) = u(t,π ) = , t ∈ R, ()

where μ > , F : R×R→ R is a given function.Take X = L([,π ]) and define the operator A by

Aϕ := ϕ′′ – μϕ, ϕ ∈ D(A),

where

D(A) :={ϕ ∈ L[,π ] : ϕ′′ ∈ L[,π ],ϕ() = ϕ(π )

} ⊂ L[,π ].

It is well known that

Au = u′′

is the generator of an analytic semigroup on L[,π ]. Hence,

μI – A

is sectorial of type

ω = –μ < .

Equation () can be formulated by the inhomogeneous problem (), where

u(t) = u(t, ·).

Example . Let us consider the nonlinearity

F(t, x)(s) = βe–t[

sin

(

+ cos t + cos√

t

)sin

(x(s)

)+ max

{e–(t±k)}

],

for all x ∈ X and s ∈ [,π ], t ∈R. Thus one has

F(t, x) ∈ SpWPAA(R× X, X)


and

∥∥F(t, x) – F(t, y)∥∥

≤∫ π

βe–t∣∣sin

(x(s)

)– sin

(y(s)

)∣∣ds ≤ βe–t∥∥x(s) – y(s)∥∥

.

In consequence, from Theorem ., it follows that the fractional differential equation ()has a unique weighed Stepanov-like pseudo-almost automorphic mild solution.


F(t, x)(s) = βe–|t|[

sin

(

+ cos t + cos√

t

)sin

(x(s)

)+ max

{e–(t±k)}

],



and

∥∥F(t, x) – F(t, y)

∥∥

≤∫ π

βe–|t|∣∣sin

(x(s)

)– sin

(y(s)

)∣∣ds ≤ βe–|t|∥∥x(s) – y(s)∥∥

.

In consequence, from Theorem ., it follows that the fractional differential equation ()has a unique weighed Stepanov-like pseudo-almost automorphic mild solution.


F(t, x)(s) = βe– t

[sin

(

+ cos t + cos√

t

)sin

(x(s)

)+ max

{e–(t±k)}

],



and

∥∥F(t, x) – F(t, y)∥∥

≤∫ π

βe– t

∣∣sin

(x(s)

)– sin

(y(s)

)∣∣ds ≤ βe– t∥∥x(s) – y(s)

∥∥.

In consequence, from Theorem . it follows that the fractional differential equation ()has a unique weighed Stepanov-like pseudo-almost automorphic mild solution whenever

|β| <α sin( π

α)

πCM|μ|– α

.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally to the writing of this paper. All authors read and approved the final version of themanuscript.


AcknowledgementsThis work is supported by the National Natural Science Foundation of China (No. 11301090), Appropriative ResearchingFund for Professors and Doctors, Guangdong University of Education (No. 2013ARF02).

Received: 28 October 2014 Accepted: 9 February 2015

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