This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations where 1 < α < 2, A is a linear densely defined operator of sectorial type on a complex Banach space X and B is a bounded linear operator defined on X, F is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity F is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

1. Introduction

The almost periodic function introduced seminally by Bohr in 1925 plays an important role in describing the phenomena that are similar to the periodic oscillations which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, and ecosphere. The concept of almost automorphy, which is an important generalization of the classical almost periodicity, was first introduced in the literature [1–4] by Bochner in relation to some aspects of differential geometry. Since then, this pioneer work has attracted more and more attention and has been substantially extended in several different directions. Many authors have made important contributions to this theory (see, for instance, [5–17] and the references therein). Especially, in [5, 6], the authors gave an important overview about the theory of almost automorphic functions and their applications to differential equations.

As a natural extension of almost automorphy, the concept of asymptotic almost automorphy, which is the central issue to be discussed in this paper, was introduced in the literature [18] by N’Guérékata in the early eighties. Since then, this notion has found several developments and has been generalized into different directions. Until now, the asymptotically almost automorphic functions as well as the asymptotically almost automorphic solutions for differential systems have been investigated by many mathematicians; see [19] by Bugajewski and N’Guérékata, [20] by Diagana, Hernández, and dos Santos, and [21] by Ding, Xiao, and Liang for the asymptotically almost automorphic solutions to integrodifferential equations, see [22] by Zhao, Chang, and N’Guérékata for the asymptotically almost automorphic solutions to the nonlinear delay integral equations, and see [23] by Chang and Tang and [24] by Zhao, Chang, and Nieto for the asymptotically almost automorphic solutions to stochastic differential equations, and the existence of asymptotically almost automorphic solutions has become one of the most attractive topics in the qualitative theory of differential equations due to its significance and applications in physics, mathematical biology, control theory, and so on. We refer the reader to the monographs of N’Guérékata [25] for the recently theory and applications of asymptotically almost automorphic functions.

With motivation coming from a wide range of engineering and physical applications, fractional differential equations have recently attracted great attention of mathematicians and scientists. This kind of equations is a generalization of ordinary differential equations to arbitrary noninteger orders. Fractional differential equations find numerous applications in the field of viscoelasticity, feedback amplifiers, electrical circuits, electro analytical chemistry, fractional multipoles, neuron modelling encompassing different branches of physics, chemistry, and biological sciences [26–32]. Many physical processes appear to exhibit fractional order behavior that may vary with time or space. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; we only enumerate here the monographs of Kilbas et al. [26, 27], Diethelm [28], Hilfer [29], Podlubny [30], Miller [31], and Zhou [32] and the papers of Agarwal et al. [33, 34], Benchohra et al. [35, 36], El-Borai [37], Lakshmikantham et al. [38–41], Mophou et al. [42–45], N’Guérékata [46], and Zhou et al. [47–50] and the reference therein.

The study of almost periodic and almost automorphic type solutions to fractional differential equations was initiated by Araya and Lizama [11]. In their work, the authors investigated the existence and uniqueness of an almost automorphic mild solution of the semilinear fractional differential equation

()

when A is a generator of an α-resolvent family and

is the Riemann-Liouville fractional derivative. In [51], Cuevas and Lizama considered the fractional differential equation:

()

where A is a linear operator of sectorial negative type on a complex Banach space X and the fractional derivative is understood in the Riemann-Liouville sense. Under suitable conditions on F(t, x), the authors proved the existence and uniqueness of an almost automorphic mild solution to (2). Cuevas et al. [52, 53] studied, respectively, the pseudo almost periodic and pseudo almost periodic class infinity mild solutions to (2) assuming that

and (t, x)⟶F(t, x) is a pseudo almost periodic and pseudo almost periodic of class infinity function satisfying suitable conditions in x ∈ X. Agarwal et al. [54] studied the existence and uniqueness of a weighted pseudo almost periodic mild solution to equation (2). Ding et al. [55] investigated the existence and uniqueness of almost automorphic solution to (2) assuming that

and (t, x)⟶F(t, x) is Stepanov-like almost automorphic in

satisfying some kind of Lipschitz conditions. Cuevas et al. [56] studied the existence of almost periodic (resp., pseudo almost periodic) mild solutions to equation (2) assuming that

and (t, x)⟶F(t, x) is Stepanov almost (resp., Stepanov-like pseudo almost) periodic in

uniformly for x ∈ X. Chang et al. [57] studied the existence and uniqueness of weighted pseudo almost automorphic solution to equation (2) with Stepanov-like weighted pseudo almost automorphic coefficient. He et al. [58] studied also the existence and uniqueness of weighted Stepanov-like pseudo almost automorphic mild solution to (2). Cao et al. [59] studied the existence and uniqueness of antiperiodic mild solution to (2). In [60], Cuevas et al. showed sufficient conditions to ensure the existence and uniqueness of mild solution for (2) in the following classes of vector-valued function spaces: periodic functions, asymptotically periodic functions, pseudo periodic functions, almost periodic functions, asymptotically almost periodic functions, pseudo almost periodic functions, almost automorphic functions, asymptotically almost automorphic functions, pseudo almost automorphic functions, compact almost automorphic functions, asymptotically compact almost automorphic functions, pseudo compact almost automorphic functions, S-asymptotically ω-periodic functions, decay functions, and mean decay functions.

Recently, Xia et al. [61] established some sufficient criteria for the existence and uniqueness of (μ, ν)-pseudo almost automorphic solution to the semilinear fractional differential equation

()

where 1 < α < 2, A is a sectorial operator of type ω < 0 on a complex Banach space X and B is a bounded linear operator. The fractional derivative is understood in the Riemann-Liouville sense. Their discussion is divided into two cases, i.e.,

, (t, x)⟶F(t, x) is (μ, ν)-pseudo almost automorphic and

, and (t, x)⟶F(t, x) is Stepanov-like (μ, ν)-pseudo almost automorphic. Kavitha et al. [62] studied weighted pseudo almost automorphic solutions of the fractional integrodifferential equation

()

where 1 < α < 2 and

()

A is a linear densely defined sectorial operator on a complex Banach space X,

, and (t, x, y)⟶F(t, x, y) is a weighted pseudo almost automorphic function in

for each x, y ∈ X satisfying suitable conditions. The fractional derivative is understood in the Riemann-Liouville sense. Mophou [63] investigated the existence and uniqueness of weighted pseudo almost automorphic mild solution to the fractional differential equation:

()

where A : D(A) ⊂ X⟶X is a linear densely operator of sectorial type on a complex Banach space X, B : X⟶X is a bounded linear operator and

, and (t, x, y)⟶F(t, x, y) is a weighted pseudo almost automorphic function in

for each x, y ∈ X satisfying suitable conditions. The fractional derivative

is to be understood in Riemann-Liouville sense. Chang et al. [64] investigated some existence results of μ-pseudo almost automorphic mild solutions to (6) assuming that

and (t, x, y)⟶F(t, x, y) is a μ-pseudo almost automorphic function in

for each x, y ∈ X satisfying suitable conditions. For more on the almost periodicity and almost automorphy for fractional differential equations and related issues, we refer the reader to [65–67] and others.

Equation (6) is motivated by physical problems. Indeed, due to their applications in fields of science where characteristics of anomalous diffusion are presented, type (6) equations are attracting increasing interest (cf. [68–70] and references therein). For example, anomalous diffusion in fractals [69] or in macroeconomics [71] has been recently well studied in the setting of fractional Cauchy problems like (6). For this reason, (6) has gotten a considerable attention in recent years (cf. [51–64, 68–71] and the references therein).

To the best of our knowledge, much less is known about the existence of asymptotically almost automorphic mild solutions to (6) when the nonlinearity F(t, x, y) as a whole loses the Lipschitz continuity with respect to x and y. Motivated by the abovementioned works, the purpose of this paper is to establish some new existence results of asymptotically almost automorphic mild solutions to (6). In our results, the nonlinearity

, (t, x, y)⟶F(t, x, y) does not have to satisfy a (locally) Lipschitz condition (see Remark 22). However, in many papers (for instance, [11, 51–64]) on almost periodic type and almost automorphic type solutions to fractional differential equations, to be able to apply the well-known Banach contraction principle, a (locally) Lipschitz condition for the nonlinearity of corresponding fractional differential equations is needed. As can be seen, our results generalize those as well as related research and have more broad applications. In particular, as application and to illustrate our main results, we will examine some sufficient conditions for the existence of asymptotically almost automorphic mild solutions to the fractional relaxation-oscillation equation given by

()

with boundary conditions u(t, 0) = u(t, π) = 0,

, where

is a function and p, μ, and ν are positive constants.

The rest of this paper is organized as follows. In Section 2, some concepts, the related notations, and some useful lemmas are introduced and established. In Section 3, we prove the existence of asymptotically almost automorphic mild solutions to such problems. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation given in Section 4.

2. Preliminaries

This section is concerned with some notations, definitions, lemmas, and preliminary facts which are used in what follows.

From now on, let (X, ‖·‖) and (Y, ‖·‖_Y) be two Banach spaces and

(resp.,

) is the space of all X-valued bounded continuous functions (resp., jointly bounded continuous functions

). Furthermore,

(resp.,

) is the closed subspace of

(resp.,

) consisting of functions vanishing at infinity (vanishing at infinity uniformly in any compact subset of Y × Y, in other words,

()

where

is an any compact subset of Y × Y). Let also

be the Banach space of all bounded linear operators from X into itself endowed with the norm:

()

For a bounded linear operator

, let ρ(A) and D(A) stand for the resolvent and domain of A, respectively.

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

Definition 1 ((Bochner) [1] (N’Guérékata) [6]). A continuous function is said to be almost automorphic if for every sequence of real numbers , there exists a subsequence {s_n} such that

()

is well defined for each

and

()

Denote by the set of all such functions.

Remark 2 (see [6].)By the point-wise convergence, the function Θ(t) in Definition 1 is measurable but not necessarily continuous. Moreover, if Θ(t) is continuous, then F(t) is uniformly continuous (cf., e.g., [17], Theorem 2.6), and if the convergence in Definition 1 is uniform on , one gets almost periodicity (in the sense of Bochner and von Neumann). Almost automorphy is thus a more general concept than almost periodicity. There exists an almost automorphic function which is not almost periodic. The function given by

()

is an example of such functions [72].

Lemma 3 (see [5].) is a Banach space with the norm .

Definition 4 (see [6].)A continuous function is said to be almost automorphic in uniformly for all (x, y) ∈ K, where K is any bounded subset of Y × Y, if for every sequence of real numbers , there exists a subsequence {s_n} such that

()

and

()

The collection of those functions is denoted by

Remark 5. The function given by

()

is almost automorphic in

uniformly for all (x, y) ∈ K, where K is any bounded subset of X × X, X = L²[0, π].

Similar to Lemma 2.2 of [73] and Proposition 3.2 of [63], we have the following result on almost automorphic functions.

Lemma 6. Let be almost automorphic in uniformly for all (x, y) ∈ K, where K is any bounded subset of X × X, and assume that F(t, x, y) is uniformly continuous on K uniformly for , that is, for any ε > 0, there exists δ > 0 such that x₁, x₂, y₁, y₂ ∈ K and ‖x₁ − y₁‖ + ‖x₂ − y₂‖ < δ imply that

()

Let

be almost automorphic. Then the function

defined by Υ(t) = F(t, x(t), y(t)) is almost automorphic.

Proof. Suppose that {s_n} is a sequence of real numbers. Then by the definition of almost automorphic functions, we can extract a subsequence {τ_n} of {s_n} such that

()

Write

()

Then

()

Since x(t) and y(t) are almost automorphic, then x(t), y(t) and

, and

are bounded. Therefore we can choose a bounded subset K ⊂ X × X, such that

()

By (P₁), (P₃), and the uniform continuity of F(t, x, y) in (x(t), y(t)) ∈ K, we have

()

Moreover, by (P₅),

()

so remembering the above triangle inequality, we deduce that

()

Using the same argument we can prove that

()

This proves that Υ(t) is almost automorphic by the definition.

Remark 7. If F(t, x, y) satisfies a Lipschitz condition with respect to x and y uniformly in , i.e., for each pair x₁, x₂, y₁, y₂ ∈ X,

()

uniformly in

, where L > 0 is called the Lipschitz constant for the function F(t, x, y), then F(t, x, y) is uniformly continuous on K uniformly for

, where K is any bounded subset of X × X.

Remark 8. If F(t, x, y) satisfies a local Lipschitz condition with respect to x and y uniformly in , i.e., for each pair x₁, x₂, y₁, y₂ ∈ X, ,

()

where

, then F(t, x, y) is uniformly continuous on K uniformly for

, where K is any bounded subset of X × X.

Definition 9 (see [6].)A continuous function is said to be asymptotically almost automorphic if it can be decomposed as F(t) = G(t) + Φ(t), where

()

Denote by the set of all such functions.

Remark 10. The function defined by

()

is an asymptotically almost automorphic function with

()

Lemma 11 (see [6].) is also a Banach space with the supremum norm ‖·‖_∞.

Definition 12 (see [6].)A continuous function is said to be asymptotically almost automorphic if it can be decomposed as F(t, x, y) = G(t, x, y) + Φ(t, x, y), where

()

Denote by the set of all such functions.

Remark 13. The function given by

()

is asymptotically almost automorphic in

uniformly for all (x, y) ∈ K, where K is any bounded subset of X × X, X = L²[0, π] and

()

Next we give some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 14 (see [26].)The fractional integral of order α > 0 with the lower limit t₀ for a function f is defined as

()

provided that the right-hand side is point-wise defined on [t₀, ∞, where Γ is the Gamma function.

Definition 15 (see [26].)Riemann-Liouville derivative of order α > 0 with the lower limit t₀ for a function can be written as

()

The first and maybe the most important property of Riemann-Liouville fractional derivative is that, for t > t₀ and α > 0, one has , which means that Riemann-Liouville fractional differentiation operator is a left inverse to the Riemann-Liouville fractional integration operator of the same order α.

It is important to define sectorial operator for the definition of mild solution of any fractional abstract equations. So, let us now give the definitions of sectorial linear operators and their associated solution operators.

Definition 16 ([74] sectorial operator). A closed and linear operator A is said to be sectorial of type ω and angle θ if there exist 0 < θ < π/2, M > 0, and such that its resolvent ρ(A) exists outside the sector and

()

Sectorial operators are well studied in the literature, usually for the case ω = 0. For a recent reference including several examples and properties we refer the reader to [74]. Note that an operator A is sectorial of type ω if and only if ωI − A is sectorial of type 0.

Definition 17 (see [75].)Let A be a closed and linear operator with domain D(A) defined on a Banach space X. We call A the generator of a solution operator if there are and a strongly continuous function such that {λ^α : Re⁡λ > ω}⊆ρ(A) and

()

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

Note that if A is sectorial of type ω with 0 ≤ θ ≤ π(1 − α/2), then A is the generator of a solution operator given by

()

where γ is a suitable path lying outside the sector ω + Σ_θ (cf. [74]).

Very recently, Cuesta in [74](Theorem 1) has proved that if A is a sectorial operator of type ω < 0 for some M > 0 and 0 ≤ θ < π(1 − α/2), then there exists C > 0 such that

()

In the border case α = 1, this is analogous to saying that A is the generator of a exponentially stable C₀-semigroup. The main difference is that in the case α > 1 the solution family S_α(t) decays like t^−α. Cuesta’s result proves that S_α(t) is, in fact, integrable.

In the following, we present the following compactness criterion, which is a special case of the general compactness result of Theorem 2.1 in [76].

Lemma 18 (see [76].)A set is relatively compact if

(1)
D is equicontinuous;
(2)
lim_|t|⟶∞x(t) = 0 uniformly for x ∈ D;
(3)
the set D(t)≔{x(t) : x ∈ D} is relatively compact in X for every .

The following Krasnoselskii’s fixed point theorem plays a key role in the proofs of our main results, which can be found in many books.

Lemma 19 (see [77].)Let U be a bounded closed and convex subset of X and J₁, J₂ be maps of U into X such that J₁x + J₂y ∈ U for every pair x, y ∈ U. If J₁ is a contraction and J₂ is completely continuous, then J₁x + J₂x = x has a solution on U.

3. Asymptotically Almost Automorphic Mild Solutions

In this section, we study the existence of asymptotically almost automorphic mild solutions for the semilinear fractional differential equations of the form

()

where A : D(A) ⊂ X⟶X is a linear densely defined operator of sectorial type of ω < 0 on a complex Banach space X, B : X⟶X is a bounded linear operator and

, and (t, x, y)⟶F(t, x, y) is a given function to be specified later. The fractional derivative

is to be understood in Riemann-Liouville sense.

We recall the following definition that will be essential for us.

Definition 20 (see [63].)Assume that A generates an integrable solution operator S_α(t). A continuous function satisfying the integral equation

()

is called a mild solution on

to (39).

In the proofs of our results, we need the following auxiliary result.

Lemma 21. Given and , let

()

Then

Proof. Firstly, note that

()

Then

()

which implies that Φ₁(t) is well defined and continuous on

. Since

, then for any ε > 0 and every sequence of real numbers

, there exist a subsequence {s_n}, a function

, and

such that

()

Define

()

Then

()

for each n > N and every

. This implies that

()

is well defined for each

By a similar argument one can obtain

()

Thus

Since , one can choose an N₁ > 0 such that ‖Z(t)‖ < ε for all t > N₁. This enables us to conclude that, for all t > N₁,

()

which implies

()

On the other hand, from

it follows that there exists an N₂ > 0 such that ‖Z(t)‖ < ε for all t < −N₂. This enables us to conclude that, for all t < −N₂,

()

which implies

()

Now we are in position to state and prove our first main result. To prove our main result, let us introduce the following assumptions:

with

()

and there exists a constant L > 0 such that, for all

and x₁, x₂, y₁, y₂ ∈ X,

()

(H₂) There exist a function

and a nondecreasing function

such that, for all

and x, y ∈ X with ‖x‖ + ‖y‖ ≤ r,

()

Remark 22. Assuming that F(t, x, y) satisfies the assumption (H₁), it is noted that F(t, x, y) does not have to meet the Lipschitz continuity with respect to x and y. Such class of asymptotically almost automorphic functions F(t, x, y) are more complicated than those with Lipschitz continuity with respect to x and y and little is known about them.

Let β(t) be the function involved in assumption (H₂). Define

()

Lemma 23. .

Proof. Since , one can choose a T₁ > 0 such that ‖β(t)‖ < ε for all t > T₁. This enables us to conclude that, for all t > T₁,

()

which implies

()

On the other hand, from

it follows that there exists a T₂ > 0 such that ‖β(t)‖ < ε for all t < −T₂. This enables us to conclude that, for all t < −T₂,

()

which implies

()

Theorem 24. Assume that A is sectorial of type ω < 0. Let satisfy the hypotheses (H₁) and (H₂). Put . Then (39) has at least one asymptotically almost automorphic mild solution provided that

()

Proof. The proof is divided into the following five steps.

Step 1. Define a mapping Λ on by

()

and prove Λ has a unique fixed point

Firstly, since the function s⟶F₁(s, v(s), Bv(s)) is bounded in and

()

this implies that (Λv)(t) exists. Moreover from

satisfying (54), together with Lemma 6 and Remark 7, it follows that

()

This, together with Lemma 21, implies that Λ is well defined and maps

into itself.

In the sequel, we verify that Λ is continuous.

Let v_n(t), v(t) be in with v_n(t)⟶v(t) as n⟶∞; then one has

()

Therefore, as n⟶∞ and Λv_n⟶Λv, hence Λ is continuous.

Next, we prove that Λ is a contraction on and has a unique fixed point .

In fact, let v₁(t), v₂(t) be in , and similar to the above proof of the continuity of Λ, one has

()

which implies

()

Together with (61), this proves that Λ is a contraction on

. Thus, Banach’s fixed point theorem implies that Λ has a unique fixed point

Step 2. Set

()

For the above v(t), define Γ≔Γ¹ + Γ² on

()

and prove that Γ maps

into itself, where k₀ is a given constant.

Firstly, from (54) it follows that, for all and ω(s) ∈ X,

()

which implies that

()

According to (55), one has

()

for all

and ω(s) ∈ X with ‖ω(s)‖ ≤ r; then

()

Those, together with Lemma 21, yield that Γ is well defined and maps

into itself.

On the other hand, in view of (55) and (61) it is not difficult to see that there exists a constant k₀ > 0 such that

()

This enables us to conclude that, for any

and

()

which implies that

. Thus Γ maps

into itself.

Step 3. Show that Γ¹ is a contraction on .

In fact, for any and , from (54) it follows that

()

Thus

()

which implies that

()

Thus, in view of (61), one obtains the conclusion.

Step 4. Show that Γ² is completely continuous on .

Given ε > 0. Let with ω_k⟶ω₀ in as k⟶+∞. Since , one may choose a t₁ > 0 big enough such that, for all t ≥ t₁,

()

Also, in view of (H₁), we have

()

for all s ∈ −∞, t₁] as k⟶+∞ and

()

Hence, by the Lebesgue dominated convergence theorem we deduce that there exists an N > 0 such that

()

whenever k ≥ N. Thus

()

whenever k ≥ N. Accordingly, Γ² is continuous on

In the sequel, we consider the compactness of Γ².

Set B_r(X) for the closed ball with center at 0 and radius r in X, , and z(t) = Γ²(u(t)) for . First, for all and ,

()

and in view of

, which follows from Lemma 23, one concludes that

()

Hence, given ε₀ > 0, one can choose a ξ > 0 such that

()

Thus we get

()

where c(K) denotes the convex hull of K. Using that S_α(·) is strongly continuous, we infer that

()

is a relatively compact set and

, which implies that V is a relatively compact subset of X.

Next, we verify the equicontinuity of the set .

Let k > 0 be small enough and and . Then by (55) we have

()

which implies the equicontinuity of the set

Now an application of Lemma 18 justifies the compactness of Γ².

Step 5. Show that (39) has at least one asymptotically almost automorphic mild solution.

Firstly, the complete continuity of Γ², together with the results of Steps 2 and 3 as well as Lemma 19, yields that Γ has at least one fixed point ; furthermore .

Then, consider the following coupled system of integral equations:

()

From the result of Step 1, together with the above fixed point

, it follows that

()

is a solution to system (91). Thus

()

and it is a solution to the integral equation

()

that is, x(t) is an asymptotically almost automorphic mild solution to (39).

Taking A = −ρ^αI with ρ > 0 in (39), the above theorem gives the following corollary.

Corollary 25. Let satisfy (H₁) and (H₂). Put . Then (39) admits at least one asymptotically almost automorphic mild solution whenever

()

Remark 26. It is interesting to note that the function α⟶αsin(π/α)/ρπ is increasing from 0 to 2/ρπ in the interval 1 < α < 2. Therefore, with respect to condition (61), the class of admissible terms F₁(t, x(t), Bx(t)) is the best in the case α = 2 and the worst in the case α = 1.

Theorem 24 can be extended to the case of F₁(t, x, y) being locally Lipschitz continuous with respect to x and y, where we have the following result.

with

()

and for all x₁, x₂, y₁, y₂ ∈ X,

()

where L(t) is a function on

Theorem 27. Assume that A is sectorial of type ω < 0. Let satisfy the hypotheses () and (H₂) with . Put . Let . Then (39) has at least one asymptotically almost automorphic mild solution provided that

()

Proof. The proof is divided into the following five steps.

Step 1. Define a mapping Λ on by (62) and prove that Λ has a unique fixed point .

Firstly, similar to the proof in Step 1 of Theorem 24, we can prove that (Λv)(t) exists. Moreover from satisfying (97), together with Lemma 6 and Remark 8, it follows that

()

This, together with Lemma 21, implies that Λ is well defined and maps

into itself.

In the sequel, we verify that Λ is continuous.

Let v_n(t), v(t) be in with v_n(t)⟶v(t) as n⟶∞; then one has

()

Therefore, as n⟶∞ and Λv_n⟶Λv, hence Λ is continuous.

Next, we prove that Λ is a contraction on and has a unique fixed point .

In fact, for v₁(t), v₂(t) in , similar to the above proof of the continuity of Λ, one has

()

which implies that

()

Hence, by (98), together with the contraction principle, Λ has a unique fixed point

Step 2. Set

()

For the above v(t), define Γ≔Γ¹ + Γ² on

as (69) and prove that Γ maps

into itself, where k₀ is a given constant.

Firstly, from (97) it follows that, for all , ω(s) ∈ X,

()

which together with

implies that

()

According to (55), one has

()

for all

and ω(s) ∈ X with ‖ω(s)‖ ≤ r; then

()

Those, together with Lemma 21, yield that Γ is well defined and maps

into itself.

On the other hand, in view of (55) and (98) it is not difficult to see that there exists a constant k₀ > 0 such that

()

This enables us to conclude that, for any

and

()

which implies that

. Thus Γ maps

into itself.

Step 3. Show that Γ¹ is a contraction on .

In fact, for any and , from (97) it follows that

()

Thus

()

which implies that

()

Thus, in view of (98), one obtains the conclusion.

Step 4. Show that Γ² is completely continuous on .

The proof is similar to the proof in Step 4 of Theorem 24.

Step 5. Show that (39) has at least one asymptotically almost automorphic mild solution.

The proof is similar to the proof in Step 5 of Theorem 24.

Taking A = −ρ^αI with ρ > 0 in (39), Theorem 27 gives the following corollary.

Corollary 28. Let satisfy and (H₂) with . Put . Let . Then (39) admits at least one asymptotically almost automorphic mild solution whenever

()

Now we consider a more general case of equations introducing a new class of functions L(t). We have the following result.

(

) There exists a function

such that, for all

and x, y ∈ X,

()

Theorem 29. Assume that A is sectorial of type ω < 0. Let satisfy the hypotheses () and () with . Moreover the integral exists for all . Then (39) has at least one asymptotically almost automorphic mild solution.

Proof. The proof is divided into the following five steps.

Step 1. Define a mapping Λ on by (62) and prove that Λ has a unique fixed point .

Firstly, similar to the proof in Step 1 of Theorem 27, we can prove that Λ is well defined and maps into itself; moreover Λ is continuous.

Next, we prove that Λ is a contraction on and has a unique fixed point .

In fact, for v₁(t), v₂(t) is in and defines a new norm

()

where

and k is a fixed positive constant. Let

; then we have

()

which implies that

()

Hence Λ has a unique fixed point

when k is greater than

Step 2. Set . For the above v(t), define Γ≔Γ¹ + Γ² on as (69) and prove that Γ maps into itself, where k₀ is a given constant.

Firstly, from (97) it follows that, for all , ω(s) ∈ X,

()

which together with

implies that

()

According to (114), one has

()

for all

and ω(s) ∈ X with ‖ω(s)‖ ≤ r; then

()

Those, together with Lemma 21, yield that Γ is well defined and maps

into itself.

On the other hand, it is not difficult to see that there exists a constant k₀ > 0 such that

()

when k is large enough. This enables us to conclude that, for any

and

()

which implies that

. Thus Γ maps

into itself.

Step 3. Show that Γ¹ is a contraction on .

In fact, for any and , from (97) it follows that

()

Thus

()

which implies

()

Thus, when k is greater than

, one obtains the conclusion.

Step 4. Show that Γ² is completely continuous on .

Given ε > 0. Let with ω_n⟶ω₀ in as n⟶+∞. Since , one may choose a t₁ > 0 big enough such that, for all t ≥ t₁,

()

Also, in view of

, we have

()

for all s ∈ (−∞, t₁] as k⟶+∞ and

()

Hence, by the Lebesgue dominated convergence theorem we deduce that there exists an N > 0 such that

()

whenever k ≥ N. Thus

()

whenever k ≥ N. Accordingly, Γ² is continuous on

In the sequel, we consider the compactness of Γ².

Set B_r(X) for the closed ball with center at 0 and radius r in X, , and z(t) = Γ²(u(t)) for . First, for all and ,

()

in view of

which follows from Lemma 23; one concludes that

()

Hence, for given ε₀ > 0, one can choose a ξ > 0 such that

()

Thus we get

()

where c(K) denotes the convex hull of K. Using the fact that S_α(·) is strongly continuous, we infer that

()

is a relatively compact set and

, which implies that V is a relatively compact subset of

Next, we verify the equicontinuity of the set , given ε₁ > 0. In view of (114), together with the continuity of {S_α(t)} _t>0, there exists an η > 0 such that, for all and t₂ ≥ t₁ with t₂ − t₁ < η,

()

Also, one can choose a k > 0 such that

()

which implies that, for all

and t₂ ≥ t₁,

()

Then one has

()

which implies the equicontinuity of the set

Now an application of Lemma 18 justifies the compactness of Γ².

Step 5. Show that (39) has at least one asymptotically almost automorphic mild solution.

The proof is similar to the proof in Step 5 of Theorem 24.

Taking A = −ρ^αI with ρ > 0 in (39), Theorem 29 gives the following corollary.

Corollary 30. Let satisfy and with . Moreover the integral exists for all . Then (39) has at least one asymptotically almost automorphic mild solution.

4. Applications

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

Consider the following fractional relaxation-oscillation equation:

()

where

is a function and p, μ, and ν are positive constants.

Take X = L²([0, π]) and define the operator A by

()

where

()

It is well known that Bu = u^′′ is self-adjoint, with compact resolvent, and is the infinitesimal generator of an analytic semigroup on X. Hence, pI − B is sectorial of type ω = −p < 0. Let

()

Then it is easy to verify that

are continuous and

satisfying

()

that is,

()

furthermore

()

And

()

that is,

()

which implies

. Furthermore

()

Thus, (142) can be reformulated as the abstract problem (39) and the assumptions (H₁) and (H₂) hold with

()

the assumption

holds with L(t) = μa(t), and the assumption

holds.

In consequence, the fractional relaxation-oscillation equation (142) has at least one asymptotically almost automorphic mild solutions if either

()

(Theorem 24) or

()

(Theorem 27), where

or the integral

()

exists for all

(Theorem 29).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the NNSF of China (no. 11561009) and (no. 41665006), the Guangdong Province Natural Science Foundation (no. 2015A030313896), the Characteristic Innovation Project (Natural Science) of Guangdong Province (no. 2016KTSCX094), the Science and Technology Program Project of Guangzhou (no. 201707010230), and the Guangxi Province Natural Science Foundation (no. 2016GXNSFAA380240).

References

1 Bochner S., Continuous mappings of almost automorphic and almost periodic functions, Proceedings of the National Acadamy of Sciences of the United States of America. (1964) 52, 907–910, https://doi.org/10.1073/pnas.52.4.907, MR0168997.
10.1073/pnas.52.4.907
CAS PubMed Web of Science® Google Scholar
2 Bochner S., Uniform convergence of monotone sequences of functions, Proceedings of the National Acadamy of Sciences of the United States of America. (1961) 47, 582–585, https://doi.org/10.1073/pnas.47.4.582, MR0126094.
10.1073/pnas.47.4.582
CAS PubMed Web of Science® Google Scholar
3 Bochner S., A new approach to almost periodicity, Proceedings of the National Acadamy of Sciences of the United States of America. (1962) 48, 2039–2043, https://doi.org/10.1073/pnas.48.12.2039, MR0145283.
10.1073/pnas.48.12.2039
CAS PubMed Web of Science® Google Scholar
4 Bochner S. and Von Neumann J., On compact solutions of operational-differential equations, I. Annals of Mathematics: Second Series. (1935) 36, no. 1, 255–291, https://doi.org/10.2307/1968678, MR1503222.
10.2307/1968678
Google Scholar
5 N′Guérékata G. M., Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, 2001, Kluwer Academic/Plenum Publishers, New York, London, Moscow, MR1880351.
10.1007/978-1-4757-4482-8
Google Scholar
6 N’Guerekata G. M., Topics in Almost Automorphy, 2005, Springer, New York, NY, USA, MR2107829, Zbl1073.43004.
Google Scholar
7 Veech W. A., Almost automorphic functions, Proceedings of the National Acadamy of Sciences of the United States of America. (1963) 49, 462–464, MR0152830, https://doi.org/10.1073/pnas.49.4.462, Zbl0173.33402.
10.1073/pnas.49.4.462
CAS PubMed Web of Science® Google Scholar
8 Campos J. and Tarallo M., Almost automorphic linear dynamics by Favard theory, Journal of Differential Equations. (2014) 256, no. 4, 1350–1367, https://doi.org/10.1016/j.jde.2013.10.018, MR3145760, Zbl1343.34035, 2-s2.0-84890876406.
10.1016/j.jde.2013.10.018
Web of Science® Google Scholar
9 Caraballo T. and Cheban D., Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard′s separation condition, I. Journal of Differential Equations. (2009) 246, no. 1, 108–128, https://doi.org/10.1016/j.jde.2008.04.001, MR2467017.
10.1016/j.jde.2008.04.001
Google Scholar
10 Mahto L. and Abbas S., PC-almost automorphic solution of impulsive fractional differential equations, Mediterranean Journal of Mathematics. (2015) 12, no. 3, 771–790, https://doi.org/10.1007/s00009-014-0449-3, MR3376811, Zbl1325.34014, 2-s2.0-84938740608.
10.1007/s00009-014-0449-3
Web of Science® Google Scholar
11 Araya D. and Lizama C., Almost automorphic mild solutions to fractional differential equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2008) 69, no. 11, 3692–3705, https://doi.org/10.1016/j.na.2007.10.004, MR2463325.
10.1016/j.na.2007.10.004
Web of Science® Google Scholar
12 Mophou G. M. and N’Guérékata G. M., On some classes of almost automorphic functions and applications to fractional differential equations, Computers & Mathematics with Applications. An International Journal. (2010) 59, no. 3, 1310–1317, https://doi.org/10.1016/j.camwa.2009.05.008, MR2579493.
10.1016/j.camwa.2009.05.008
Web of Science® Google Scholar
13 Abadias L. and Lizama C., Almost automorphic mild solutions to fractional partial difference-differential equations, Applicable Analysis: An International Journal. (2016) 95, no. 6, 1347–1369, https://doi.org/10.1080/00036811.2015.1064521, MR3479007, Zbl1381.35205.
10.1080/00036811.2015.1064521
Web of Science® Google Scholar
14 Fu M. and Liu Z., Square-mean almost automorphic solutions for some stochastic differential equations, Proceedings of the American Mathematical Society. (2010) 138, no. 10, 3689–3701, https://doi.org/10.1090/S0002-9939-10-10377-3, MR2661567, Zbl1202.60109, 2-s2.0-77955628630.
10.1090/S0002-9939-10-10377-3
Web of Science® Google Scholar
15 Cao J., Yang Q., and Huang Z., Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations, Stochastics. An International Journal of Probability and Stochastic Processes. (2011) 83, no. 3, 259–275, https://doi.org/10.1080/17442508.2010.533375, MR2810592, Zbl1221.60078.
10.1080/17442508.2010.533375
Google Scholar
16 Liu Z. and Sun K., Almost automorphic solutions for stochastic differential equations driven by Lévy noise, Journal of Functional Analysis. (2014) 266, no. 3, 1115–1149, https://doi.org/10.1016/j.jfa.2013.11.011, MR3146813.
10.1016/j.jfa.2013.11.011
Web of Science® Google Scholar
17 N’guérékata G. M., Comments on almost automorphic and almost periodic functions in Banach spaces, Far East Journal of Mathematical Sciences (FJMS). (2005) 17, no. 3, 337–344, MR2183191.
Google Scholar
18 N’Guérékata G. M., Sur les solutions presqu automorphes d’équations différentielles abstraites, Annales des Sciences Mathématiques du Québec. (1981) no. 1, 69–79, MR626577.
Google Scholar
19 Bugajewski D. and N′Guérékata G. M., On the topological structure of almost automorphic and asymptotically almost automorphic solutions of differential and integral equations in abstract spaces, Nonlinear Analysis: Theory, Methods & Applications. (2004) 59, no. 8, 1333–1345, https://doi.org/10.1016/j.na.2003.08.012, Zbl1071.34055, 2-s2.0-9244234347.
10.1016/j.na.2003.08.012
Web of Science® Google Scholar
20 Diagana T., Hernández E. M., and dos Santos J., Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2009) 71, no. 1-2, 248–257, https://doi.org/10.1016/j.na.2008.10.046, MR2518032.
10.1016/j.na.2008.10.046
Web of Science® Google Scholar
21 Ding H.-S., Xiao T.-J., and Liang J., Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, Journal of Mathematical Analysis and Applications. (2008) 338, no. 1, 141–151, https://doi.org/10.1016/j.jmaa.2007.05.014, MR2386405, Zbl1142.45005, 2-s2.0-34548847768.
10.1016/j.jmaa.2007.05.014
Web of Science® Google Scholar
22 Zhao J.-Q., Chang Y.-K., and N’guérékata G. M., Existence of asymptotically almost automorphic solutions to nonlinear delay integral equations, Dynamic Systems and Applications. (2012) 21, no. 2-3, 339–349, MR2918384.
CAS Web of Science® Google Scholar
23 Chang Y.-K. and Tang C., Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process, Stochastics. An International Journal of Probability and Stochastic Processes. (2016) 88, no. 7, 980–1011, https://doi.org/10.1080/17442508.2016.1178748, MR3529857.
10.1080/17442508.2016.1178748
Web of Science® Google Scholar
24 Zhao Z.-H., Chang Y.-K., and Nieto J. J., Square-mean asymptotically almost automorphic process and its application to stochastic integro-differential equations, Dynamic Systems and Applications. (2013) 22, no. 2-3, 269–284, MR3100203, Zbl1300.34182, 2-s2.0-84880503996.
Web of Science® Google Scholar
25 N’Guérékata G. M., Spectral Theory for Bounded Functions and Applications to Evolution Equations, 2017, Nova Science Publishers, NY, USA, MR3676744.
Google Scholar
26 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, 24, Elsevier Science B.V., Amsterdam, North-Holland Mathematics Studies, MR2218073.
10.1016/S0304-0208(06)80001-0
Google Scholar
27 Samko S. G., Kilbas A. A., and Marichev O. I., Fractional integral and derivatives: Theory and applications, 1993, Gordon and Breach Science Publishers, Switzerland, MR1347689.
Google Scholar
28 Diethelm K., The Analysis of Fractional Differential Equations, 2010, Springer Verlag, Berlin, Heidelberg, Germany, Lecture Notes in Mathematics, https://doi.org/10.1007/978-3-642-14574-2_8.
10.1007/978-3-642-14574-2
Web of Science® Google Scholar
29 Hilfer R., Applications of Fractional Calculus in Physics, 2000, World Scientific, Singapore, https://doi.org/10.1142/9789812817747, MR1890104, Zbl0998.26002.
10.1142/3779
Google Scholar
30 Podlubny I., Fractional Differential Equations, 1999, Academic Press, CA, USA, MR1658022.
Web of Science® Google Scholar
31 Miller K. S. and Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, 1993, John Wiley & Sons, NY, USA, A Wiley Interscience Publication, MR1219954.
Google Scholar
32 Zhou Y., Basicheory of Fractional Diferential Equations, 2014, Singapore, World Scientiic.
10.1142/9069
Google Scholar
33 Agarwal R. P., Belmekki M., and Benchohra M., A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Advances in Difference Equations. (2009) 2009, 47, https://doi.org/10.1155/2009/981728, 981728, MR2505633, Zbl1182.34103.
10.1155/2009/981728
Web of Science® Google Scholar
34 Agarwal R. P., Lakshmikantham V., and Nieto J. J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2010) 72, no. 6, 2859–2862, https://doi.org/10.1016/j.na.2009.11.029, MR2580143.
10.1016/j.na.2009.11.029
Web of Science® Google Scholar
35 Benchohra M., Henderson J., Ntouyas S. K., and Ouahab A., Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications. (2008) 338, no. 2, 1340–1350, https://doi.org/10.1016/j.jmaa.2007.06.021, MR2386501, Zbl1209.34096, 2-s2.0-34848916710.
10.1016/j.jmaa.2007.06.021
Web of Science® Google Scholar
36 Agarwal R. P., Benchohra M., and Hamani S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Applicandae Mathematicae. (2010) 109, no. 3, 973–1033, https://doi.org/10.1007/s10440-008-9356-6, MR2596185, Zbl1198.26004, 2-s2.0-77949264980.
10.1007/s10440-008-9356-6
Web of Science® Google Scholar
37 El-Borai M. M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos, Solitons & Fractals. (2002) 14, no. 3, 433–440, https://doi.org/10.1016/S0960-0779(01)00208-9, MR1903295.
10.1016/S0960-0779(01)00208-9
Web of Science® Google Scholar
38 Lakshmikantham V., Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods & Applications. (2008) 60, 3337–3343, https://doi.org/10.1016/j.na.2007.09.025, MR2450543.
10.1016/j.na.2007.09.025
Google Scholar
39 Lakshmikantham V. and Vatsala A. S., Basic theory of fractional differential equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2008) 69, no. 8, 2677–2682, https://doi.org/10.1016/j.na.2007.08.042, MR2446361.
10.1016/j.na.2007.08.042
Web of Science® Google Scholar
40 Lakshmikantham V. and Vatsala A. S., Theory of fractional differential inequalities and applications, Communications in Applied Analysis. (2007) 11, no. 3-4, 395–402, MR2368191, Zbl1159.34006, 2-s2.0-37349014877.
Google Scholar
41 Lakshmikantham V. and Devi J. V., Theory of fractional differential equations in a Banach space, European Journal of Pure and Applied Mathematics. (2008) 1, no. 1, 38–45, MR2379647, Zbl1146.34042.
Google Scholar
42 Mophou G., Nakoulima O., and N’Guérékata G. M., Existence results for some fractional differential equations with nonlocal conditions, Nonlinear Studies. The International Journal. (2010) 17, no. 1, 15–21, MR2647799.
Google Scholar
43 Mophou G. M. and N’Guérékata G. M., Existence of the mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum. (2009) 79, no. 2, 315–322, https://doi.org/10.1007/s00233-008-9117-x, MR2538728, 2-s2.0-71549132795, Zbl1180.34006.
10.1007/s00233-008-9117-x
Web of Science® Google Scholar
44 Mophou G. M. and N’Guérékata G. M., On integral solutions of some nonlocal fractional differential equations with nondense domain, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2009) 71, no. 10, 4668–4675, https://doi.org/10.1016/j.na.2009.03.029, MR2548700.
10.1016/j.na.2009.03.029
Web of Science® Google Scholar
45 Mophou G. M., Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2010) 72, no. 3-4, 1604–1615, https://doi.org/10.1016/j.na.2009.08.046, MR2577561.
10.1016/j.na.2009.08.046
Web of Science® Google Scholar
46 N’Guérékata G. M., A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. (2009) 70, no. 5, 1873–1876, https://doi.org/10.1016/j.na.2008.02.087, MR2492125.
10.1016/j.na.2008.02.087
Web of Science® Google Scholar
47 Zhou Y. and Peng L., On the time-fractional Navier-Stokes equations, Computers & Mathematics with Applications. An International Journal. (2017) 73, no. 6, 874–891, https://doi.org/10.1016/j.camwa.2016.03.026, MR3623092.
10.1016/j.camwa.2016.03.026
Web of Science® Google Scholar
48 Zhou Y. and Peng L., Weak solutions of the time-fractional Navier-Stokes equations and optimal control, Computers & Mathematics with Applications. An International Journal. (2017) 73, no. 6, 1016–1027, https://doi.org/10.1016/j.camwa.2016.07.007, MR3623103.
10.1016/j.camwa.2016.07.007
Web of Science® Google Scholar
49 Zhou Y. and Zhang L., Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems, Computers & Mathematics with Applications. An International Journal. (2017) 73, no. 6, 1325–1345, https://doi.org/10.1016/j.camwa.2016.04.041, MR3623125.
10.1016/j.camwa.2016.04.041
Web of Science® Google Scholar
50 Zhou Y., Vijayakumar V., and Murugesu R., Controllability for fractional evolution inclusions without compactness, Evolution Equations and Control Theory. (2015) 4, no. 4, 507–524, https://doi.org/10.3934/eect.2015.4.507, MR3461697, Zbl1335.34096, 2-s2.0-84957558534.
10.3934/eect.2015.4.507
Web of Science® Google Scholar
51 Cuevas C. and Lizama C., Almost automorphic solutions to a class of semilinear fractional differential equations, Applied Mathematics Letters. (2008) 21, no. 12, 1315–1319, https://doi.org/10.1016/j.aml.2008.02.001, MR2464387, Zbl1192.34006.
10.1016/j.aml.2008.02.001
Web of Science® Google Scholar
52 Agarwal R. P., de Andrade B., and Cuevas C., On type of periodicity and ergodicity to a class of fractional order differential equations, Advances in Difference Equations. (2010) 179750, MR2595646.
10.1186/1687-1847-2010-179750
Web of Science® Google Scholar
53 Agarwal R. P., Cuevas C., and Soto H., Pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Applied Mathematics and Computation. (2011) 37, no. 1-2, 625–634, https://doi.org/10.1007/s12190-010-0455-y, MR2831559, Zbl1368.34008, 2-s2.0-80052610250.
10.1007/s12190-010-0455-y
Google Scholar
54 Agarwal R. P., de Andrade B., and Cuevas C., Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations, Nonlinear Analysis: Real World Applications. (2010) 11, no. 5, 3532–3554, https://doi.org/10.1016/j.nonrwa.2010.01.002, MR2683811, Zbl1248.34004.
10.1016/j.nonrwa.2010.01.002
Web of Science® Google Scholar
55 Ding H.-S., Liang J., and Xiao T.-J., Almost automorphic solutions to abstract fractional differential equations, Advances in Difference Equations. (2010) 508374, MR2595645.
10.1186/1687-1847-2010-508374
Web of Science® Google Scholar
56 Cuevas C., Sepúlveda A., and Soto H., Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equations, Applied Mathematics and Computation. (2011) 218, no. 5, 1735–1745, https://doi.org/10.1016/j.amc.2011.06.054, MR2831397.
10.1016/j.amc.2011.06.054
Web of Science® Google Scholar
57 Chang Y.-K., Zhang R., and N’Guérékata G. M., Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Computers & Mathematics with Applications. An International Journal. (2012) 64, no. 10, 3160–3170, https://doi.org/10.1016/j.camwa.2012.02.039, MR2989344.
10.1016/j.camwa.2012.02.039
Web of Science® Google Scholar
58 He B., Cao J., and Yang B., Weighted Stepanov-like pseudo-almost automorphic mild solutions for semilinear fractional differential equations, Advances in Difference Equations. (2015) 2015, 74, https://doi.org/10.1186/s13662-015-0410-1, MR3317932.
10.1186/s13662-015-0410-1
PubMed Google Scholar
59 Cao J., Yang Q., and Huang Z., Existence of anti-periodic mild solutions for a class of semilinear fractional differential equations, Communications in Nonlinear Science and Numerical Simulation. (2012) 17, no. 1, 277–283, https://doi.org/10.1016/j.cnsns.2011.05.005, MR2826006, Zbl1248.35218, 2-s2.0-79961027513.
10.1016/j.cnsns.2011.05.005
Web of Science® Google Scholar
60 Lizama C. and Poblete F., Regularity of mild solutions for a class of fractional order differential equations, Applied Mathematics and Computation. (2013) 224, 803–816, https://doi.org/10.1016/j.amc.2013.09.009, MR3127665, Zbl1334.34016, 2-s2.0-84885361950.
10.1016/j.amc.2013.09.009
Web of Science® Google Scholar
61 Xia Z., Fan M., and Agarwal R. P., Pseudo almost automorphy of semilinear fractional differential equations in Banach spaces, Fractional Calculus and Applied Analysis. (2016) 19, no. 3, 741–764, https://doi.org/10.1515/fca-2016-0039, MR3563608, Zbl1348.34029, 2-s2.0-84979586882.
10.1515/fca-2016-0039
Web of Science® Google Scholar
62 Abbas S., Kavitha V., and Murugesu R., Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations, Journal of Fractional Calculus and Applications. (2013) 4, 1–19, https://doi.org/10.1007/s12044-015-0235-6, Zbl1327.35391.
10.1007/s12044-015-0235-6
Google Scholar
63 Mophou G. M., Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations, Applied Mathematics and Computation. (2011) 217, no. 19, 7579–7587, https://doi.org/10.1016/j.amc.2011.02.048, MR2799772, Zbl1221.34015, 2-s2.0-79955856737.
10.1016/j.amc.2011.02.048
Web of Science® Google Scholar
64 Chang Y.-K. and Luo X.-X., Pseudo almost automorphic behavior of solutions to a semi-linear fractional differential equation, Mathematical Communications. (2015) 20, no. 1, 53–68, MR3352474, Zbl1335.34093, 2-s2.0-84934757221.
Web of Science® Google Scholar
65 Kavitha V., Abbas S., and Murugesu R., (μ1,μ2)-pseudo almost automorphic solutions of fractional order neutral integro-differential equations, Nonlinear Stud. (2017) 24, 669–685, MR3701881.
Google Scholar
66 Kavitha V., Abbas S., and Murugesu R., Asymptotically almost automorphic solutions of fractional order neutral integro-differential equations, Bulletin of the Malaysian Mathematical Sciences Society. (2016) 39, no. 3, 1075–1088, https://doi.org/10.1007/s40840-015-0205-2, MR3515067, Zbl1347.34114, 2-s2.0-84976354253.
10.1007/s40840-015-0205-2
Web of Science® Google Scholar
67 Abbas S., Weighted pseudo almost automorphic solutions of fractional functional differential equations, Cubo (Temuco). (2014) 16, 21–35, MR3185785.
10.4067/S0719-06462014000100003
Google Scholar
68 Bazhlekova E., Fractional Evolution Equations in Banach Spaces, 2001, Eindhoven University of Technology, MR1868564.
Google Scholar
69 Eidelman S. D. and Kochubei A. N., Cauchy problem for fractional diffusion equations, Journal of Differential Equations. (2004) 199, no. 2, 211–255, https://doi.org/10.1016/j.jde.2003.12.002, MR2047909, Zbl1068.35037, 2-s2.0-1942542248.
10.1016/j.jde.2003.12.002
Web of Science® Google Scholar
70 Gorenflo R. and Mainardi F., A. Carpinteri and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics, 1997, Springer-Verlag, NY, USA, Vienna, Austria, 223–276, MR1611585.
10.1007/978-3-7091-2664-6_5
Google Scholar
71 Anh V. V. and Mcvinish R., Fractional differential equations driven by Lévy noise, Journal of Applied Mathematics and Stochastic Analysis. (2003) 16, no. 2, 97–119, https://doi.org/10.1155/S1048953303000078, MR1989577.
10.1155/S1048953303000078
Google Scholar
72 Ding H.-S., Liang J., and Xiao T.-J., Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations, Applicable Analysis: An International Journal. (2009) 88, no. 7, 1079–1091, https://doi.org/10.1080/00036810903156164, MR2561476, Zbl1187.43004.
10.1080/00036810903156164
Web of Science® Google Scholar
73 Liang J., Zhang J., and Xiao T.-J., Composition of pseudo almost automorphic and asymptotically almost automorphic functions, Journal of Mathematical Analysis and Applications. (2008) 340, no. 2, 1493–1499, https://doi.org/10.1016/j.jmaa.2007.09.065, MR2390946, Zbl1134.43001, 2-s2.0-38049089402.
10.1016/j.jmaa.2007.09.065
Web of Science® Google Scholar
74 Haase M., The functional calculus for sectorial operators, Operator Theory: Advances and Applications, 2006, 169, Birkhuser Verlag, Basel, Switzerland.
Google Scholar
75 Cuesta E., Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete and Continuous Dynamical Systems - Series A. (2007) 277–285, MR2409222, Zbl1163.45306.
Google Scholar
76 Ruess W. M. and Summers W. H., Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity, Mathematische Nachrichten. (1988) 135, 7–33, https://doi.org/10.1002/mana.19881350102, MR944213, Zbl0666.46007, 2-s2.0-0001831555.
10.1002/mana.19881350102
Web of Science® Google Scholar
77 Smart D. R., Fixed Point Theorems, 1980, Cambridge University Press, London, UK, MR0467717.
Google Scholar

Citing Literature

All articles

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Asymptotically Almost Automorphic Mild Solutions

4. Applications

Conflicts of Interest

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Asymptotically Almost Automorphic Mild Solutions

4. Applications

Conflicts of Interest

Acknowledgments

References

Citing Literature

References

Related

Information