1 Introduction

This work is mainly concerned with the existence of asymptotically almost automorphic mild solutions to fractional order neutral integro-differential equation

$$\begin{aligned}&\displaystyle D_t^{\alpha }[x(t)-k_1(t,x(t))]=A[x(t)-k_1(t,x(t))]+D_t^{\alpha -1}f(t,x(t),Kx(t)),\qquad \end{aligned}$$
(1.1)
$$\begin{aligned}&\displaystyle \text{ K }x(t)=\int _{-\infty }^tk(t-s)h(s,x(s))ds \nonumber \\&\displaystyle x(0)=x_0 \quad t\in R, \end{aligned}$$
(1.2)

where \(1<\alpha <2\) and \(A{:}D(A)\subset X\rightarrow X\) is a linear densely defined operator of sectorial type on a complex Banach space \((X,\Vert .\Vert ),\, k\) satisfy \(|k(t)|\le c_ke^{-bt}\) for \(t\ge 0\) and \(c_k, b\) are positive constants, \(f{:}R\times X\times X\rightarrow X\), \(h{:}R\times X\rightarrow X\) and \(k_1{:}R\times X\rightarrow X\) are asymptotically almost automorphic functions in t for each \(x,y\in X\) satisfying suitable conditions. The fractional derivative \(D_t^{\alpha }\) is to be understood in Riemann–Liouville sense.

Neutral differential equations arise in many areas of applied mathematics and for this reason, this type of equation has received much attention in recent years see [14, 15, 23, 24, 26, 31]. Due to their applications in several fields of science [5, 17, 18], fractional differential equations are attracting increasing interest, because of their numerical treatment. Properties of the solutions have been studied in several contexts see [1–4, 7–10, 32] and references therein.

The concept of asymptotically almost automorphy was introduced by N’Guérékata [19]. Since then, these functions have generated several developments and applications, and we refer the reader to [6, 12, 13, 15, 16, 20, 21, 25, 28, 29, 33–35] and the references therein.

Recently, Kavitha et al. [27] studied weighted pseudo almost automorphic solution of the following fractional integro-differential equation

$$\begin{aligned}&\displaystyle D_t^{\alpha }x(t)=Ax(t)+D_t^{\alpha -1}f(t,x(t),Kx(t)) \quad t\in R, \quad \text{ where } \, \, 1<\alpha <2 \, \, \text {and} \\&\displaystyle Kx(t)=\int _{-\infty }^tk(t-s)h(s,x(s))ds, \end{aligned}$$

where A is linear densely defined sectorial operator.

Motivated by the above work, in this paper we study the existence of asymptotically almost automorphic solutions to (1.1)–(1.2). The organization of the paper is as follows. In Sect. 2, we give some basic definitions and results. In Sect. 3, we establish the existence of asymptotic almost automorphic solution of equations (1.1)–(1.2). In Sect. 4, examples are given to support the theory.

2 Preliminaries and Basic Results

In this section, we introduce notations, definitions, lemmas and preliminary facts which are used throughout this work.

Let \((X,\Vert \cdot \Vert )\) and \((Y,\Vert \cdot \Vert _Y)\) be two complex Banach spaces. The notation C(R, X) (respectively \(C(R\times X,X)\)) denotes the collection of all continuous functions from R to X. Let \(\mathcal {BC}(R,X)\) (respectively \(\mathcal {BC}(R\times X,X)\)) denote the collection of all X-valued bounded continuous functions (respectively, the class of jointly bounded continuous functions \(f:R\times X\rightarrow X\)). The space \(\mathcal {BC}(R,X)\) equipped with the sup norm defined by

$$\begin{aligned} \Vert f\Vert _{\infty }=\sup _{t\in R}\Vert f(t)\Vert \end{aligned}$$

is a Banach space. The notation \(\mathcal L(X,Y)\) stands for the space of bounded linear operators from X into Y endowed with the uniform operator topology and we abbreviate it into \(\mathcal L(X)\) whenever \(X=Y\).

Definition 2.1

[21, 22]. Let \(f{:}R\rightarrow X\) be a bounded continuous function. We say that f is almost automorphic if for every sequence of real numbers \((s_n)_{n\in \mathcal {N}}\), there exists a subsequence \((\tau _n)_{n\in \mathcal {N}}\) such that

$$\begin{aligned} g(t)=\lim _{n\rightarrow \infty }f(t+\tau _n) \end{aligned}$$

is well-defined for each \(t\in R\) and

$$\begin{aligned} \lim _{n\rightarrow \infty }g(t-\tau _n)=f(t) \end{aligned}$$

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

Definition 2.2

[21, 22]. A continuous function \(f{:}R\times X\rightarrow X\) is called almost automorphic in t uniformly for x in compact subsets of X if for every compact subset \(\mathcal {K}\) of X and every real sequence \((s_n)_{n\in \mathcal {N}}\) there exists a subsequence \((\tau _n)_{n\in \mathcal {N}}\) such that

$$\begin{aligned} g(t,x)=\lim _{n\rightarrow \infty }f(t+\tau _n,x) \end{aligned}$$

is well-defined for each \(t\in R, \, x\in \mathcal {K}\) and

$$\begin{aligned} \lim _{n\rightarrow \infty }g(t-\tau _n,x)=f(t,x) \end{aligned}$$

for each \(t\in R, \, x\in \mathcal {K}\). Denote by \(AA(R\times X, X)\) the set of all such functions.

The space of all continuous functions \(m{:}R^+\rightarrow X\) such that \(\lim _{t\rightarrow \infty }m(t)=0\) is denoted by \(C_0(R^+,X)\). Moreover, we denote \(C_0(R^+\times X,X),\) the space of all continuous functions from \(R^+\times X\) to X satisfying \(\lim _{t\rightarrow \infty }m(t,x)=0\) in t and uniformly in \(x\in X\).

Definition 2.3

A continuous function \(f{:}R^+\rightarrow X\) is called asymptotically almost automorphic iff it can be written as \(f=g+\phi \), where \(g\in AA(R,X)\) and \(\phi \in C_0(R^+,X)\). This kind of functions is denoted by \(AAA(R^+,X)\).

Definition 2.4

A continuous function \(f{:}R^+\times X\rightarrow X\) is called asymptotically almost automorphic iff it can be written as \(f=g+\phi \), where \(g\in AA(R\times X,X)\) and \(\phi \in C_0(R^+\times X,X)\). This kind of functions is denoted by \(AAA(R^+\times X,X)\).

We state a Lemma by Liang et al. [28] about the composition result.

Lemma 2.1

Let \(f(t,x)=g(t,x)+\phi (t,x)\) is an asymptotically almost automorphic function with \(g(t,x)\in AA(R\times X,X)\) and \(\phi (t,x)\in C_0(R^+\times X,X)\) and f(t, x) is uniformly continuous on any bounded subset \(\Omega \subset X\) uniformly in t. Then for \(x(\cdot )\in AAA(R^+,X)\), the function \(f(\cdot , x(\cdot ))\in AAA(R^+\times X,X)\).

Definition 2.5

[8]. A closed linear operator (A, D(A)) with dense domain D(A) in a Banach space X is said to be sectorial of type \(\omega \) and angle \(\theta \) if there are constants \(\omega \in R, \, \theta \in (0,\frac{\pi }{2}),\, M>0\) such that its resolvent exists outside the sector

$$\begin{aligned}&\omega +\Sigma _{\theta }:=\{\lambda +\omega :\lambda \in \mathcal {C}, |\arg (-\lambda )|<\theta \}, \end{aligned}$$
(2.1)
$$\begin{aligned}&\quad \Vert (\lambda -A)^{-1}\Vert \le \frac{M}{|\lambda -\omega |}, \quad \lambda \notin \omega +\Sigma _{\theta }. \end{aligned}$$
(2.2)

Definition 2.6

Let \(1<\alpha <2\). Let A be a closed and linear operator with domain D(A) defined on a Banach space X. We say that A is the generator of a solution operator if there exist \(\omega \in R\) and a strongly continuous function \(S_{\alpha }:R_{+}\rightarrow \mathcal {L}(X)\) such that \(\{\lambda ^{\alpha }:Re\lambda >\omega \}\subset \rho (A)\) and

$$\begin{aligned} \lambda ^{\alpha -1}(\lambda ^{\alpha }I-A)^{-1}x=\int _0^{\infty }e^{-\lambda t}S_{\alpha }(t)xdt, \quad Re\lambda >\omega , \quad x\in X. \end{aligned}$$

In [8], Cuesta proves that if A is sectorial of type \(\omega \in R\) with \(0\le \theta <\pi (1-\alpha /2)\), then A is a generator of a solution operator given by

$$\begin{aligned} S_{\alpha }(t)=\frac{1}{2\pi i}\int _{\mathbb {G}}e^{\lambda t}\lambda ^{\alpha -1}(\lambda ^{\alpha }-A)^{-1}d\lambda , \quad t\ge 0 \end{aligned}$$

with \(\mathbb {G}\) a suitable path lying outside the sector \(\omega +\Sigma _0\). Furthermore he shows that the following Lemma holds.

Lemma 2.2

[8][Theorem 1]. Let \(A{:}D(A)\subset X\rightarrow X\) be a sectorial operator in a complex Banach space X, satisfying hypothesis (2.1) and (2.2), for some \(M>0, \omega <0\) and \(0\le \theta <\pi (1-\alpha /2)\). Then there exists \(C(\theta ,\alpha )>0\) depending solely on \(\theta \) and \(\alpha \), such that

$$\begin{aligned} \Vert S_{\alpha }(t)\Vert _{\mathcal {L}(X)}\le \frac{C(\theta ,\alpha )M}{1+|\omega |t^{\alpha }}, \quad t\ge 0. \end{aligned}$$
(2.3)

Now, we recall a useful compactness criterion.

Let \(h: R^+\rightarrow [1,\infty )\) be a continuous function such that \(h(t)\rightarrow \infty \) as \(t\rightarrow \infty \). We consider the space

$$\begin{aligned} C_{h}(X)=\left\{ u\in C(R^+,X): \lim _{t\rightarrow \infty }\frac{u(t)}{h(t)}=0\right\} . \end{aligned}$$

The space \(C_h(X)\) is a Banach space equipped with the norm

\(\Vert u\Vert _{h}=\sup _{t\in R^+}\frac{\Vert u(t)\Vert }{h(t)}\). (see [11]).

Lemma 2.3

[11]. A subset \(K^{^\prime }\subset C_h(X)\) is a relatively compact set if it verifies the following conditions:

  1. (c-1)

    The set \(K^{^\prime }_b=\{u_{[0,b]}:u\in K^{^\prime }\}\) is relatively compact in C([0, b], X) for all \(b\ge 0\).

  2. (c-2)

    \(\lim _{t\rightarrow \infty }\frac{\Vert u(t)\Vert }{h(t)}=0\) uniformly for all \(u\in K^{^\prime }\).

3 Asymptotically Almost Automorphic Mild Solutions

Before starting our main results in this section, we recall the definition of the mild solution to (1.1)–(1.2).

Definition 3.1

[3]. A continuous function \(x:R^+\rightarrow X\) satisfying the integral equation

$$\begin{aligned} x(t)=S_{\alpha }(t)[x_0-k_1(0,x_0)]+k_1(t,x(t))+\int _0^tS_{\alpha }(t-s)f\big (s,x(s),Kx(s)\big )ds, \end{aligned}$$

is called the mild solution of the problem (1.1)–(1.2).

We only need integrability of function f so that the right-hand expression is well defined and therefore it is called mild solution. If we put the condition \(f \in \mathfrak {C}_{\mu }, \ 1< \mu <2,\) where \(\mathfrak {C}_{\mu }\) is the space of all functions such that \(t^{\mu }f\) is continuous, then the solution is called classical solution.

We make the following assumptions:

  1. (H1)

    A is a sectorial operator of type \(\omega <0\).

  2. (H2)

    \(k_1\in AAA(R^+\times X,X)\) and \(f\in AAA(R^+\times X\times X,X)\) and there exist positive constants \(L_1, L_2, L_3\) such that

    $$\begin{aligned} (i)\quad \Vert k_1(t,x)-k_1(t,y)\Vert \le L_1\Vert x-y\Vert , \quad x,y\in X \end{aligned}$$
    $$\begin{aligned} (ii)\quad \Vert f(t,x_1,y_1)-f(t,x_2,y_2)\Vert \le L_2\Vert x_1-x_2\Vert +L_3\Vert y_1-y_2\Vert , \end{aligned}$$

    where \(x_i,y_i\in X, \, i=1,2\) and \(t\in R^+\).

  3. (H3)

    The function \(h:R^+\times X\rightarrow X\) is an asymptotically almost automorphic in t uniformly in \(x\in X\) and satisfies

    $$\begin{aligned} \Vert h(t,x)-h(t,y)\Vert \le L_4\Vert x-y\Vert \quad \text {for each}\ x,y\in X. \end{aligned}$$

The following lemmas are from [16].

Lemma 3.1

Let \(f=g+\phi \in AAA(R^+\times X,X)\) with \(g\in AA(R\times X,X),\phi \in C_0(R^+\times X,X)\) satisfying the Hypothesis (H2)(ii). If \(x(t)\in AAA(R^+,X)\) then \(f(\cdot ,x(\cdot ))\in AAA(R^+\times X,X)\).

Lemma 3.2

Let \(f=g+\phi \in AAA(R^+\times X\times X\rightarrow X)\) with \(g\in AA(R,X),\ \phi \in C_0(R^+,X)\). Then \(Q(t):=\int _0^tS_{\alpha }(t-s)f(s)ds\in AAA(R^+,X)\).

Proof

We observe that

$$\begin{aligned} Q(t)&=\int _0^tS_{\alpha }(t-s)g(s)ds+\int _0^tS_{\alpha }(t-s)\phi (s)ds \\&=\int _{-\infty }^tS_{\alpha }(t-s)g(s)ds-\int _{-\infty }^0S_{\alpha }(t-s)g(s)ds+\int _0^tS_{\alpha }(t-s)\phi (s)ds. \end{aligned}$$

Let \(Q(t)=R(t)+S(t)\), where

$$\begin{aligned}&\displaystyle R(t):=\int _{-\infty }^tS_{\alpha }(t-s)g(s)ds \\&\displaystyle S(t):=\int _0^tS_{\alpha }(t-s)\phi (s)ds-\int _{-\infty }^0S_{\alpha }(t-s)g(s)ds. \end{aligned}$$

Now, let \((s_n^\prime )\) be an arbitrary sequence of real numbers. Since \(g\in AA(R,X)\) there exists a subsequence \(s_n\) of \((s_n^\prime )\) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }g(t+s_n)=\overline{g}(t),\quad \text {for all}\, t\in R \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\overline{g}(t-s_n)=g(t),\quad \text {for all}\, t\in R. \end{aligned}$$

We define \(\overline{R}(t):=\int _{-\infty }^tS_{\alpha }(t-s)\overline{g}(s)ds\).

Now, consider

$$\begin{aligned} R(t+s_n)&=\int _{-\infty }^{t+s_n}S_{\alpha }(t+s_n-s)g(s)ds\\&=\int _{-\infty }^{t}S_{\alpha }(t-\sigma )g(\sigma +s_n)d\sigma \\&=\int _{-\infty }^{t}S_{\alpha }(t-\sigma )g_n(\sigma )d\sigma , \end{aligned}$$

where \(g_n(\sigma )=g(\sigma +s_n),\ n=1,2,\ldots \)

$$\begin{aligned} R(t+s_n)=\int _0^{\infty }S_{\alpha }(\sigma )g_n(t-\sigma )d\sigma \end{aligned}$$

Now, by inequality (2.3)

$$\begin{aligned} \Vert R(t+s_n)\Vert\le & {} \int _0^{\infty }\frac{C(\theta ,\alpha )M}{1+|\omega |\sigma ^{\alpha }}\Vert g_n(t-\sigma )\Vert d\sigma \\\le & {} C(\theta ,\alpha )M\frac{|w|^{-1/\alpha }\pi }{\alpha \sin (\pi /\alpha )}\Vert g\Vert _{\infty } \end{aligned}$$

and by continuity of \(S_{\alpha }(\cdot )x\) we have \(S_{\alpha }(t-\sigma )g_n(\sigma )\rightarrow S_{\alpha }(t-\sigma )\overline{g}(\sigma )\) as \(n\rightarrow \infty \) for each \(\sigma \in R\) fixed and any \(t\ge \sigma \). Then by the Lebesgue dominated convergence theorem,

$$\begin{aligned} R(t+s_n)\rightarrow \overline{R}(t)\quad \text{ as }\quad n\rightarrow \infty \quad \text{ for } \text{ all } \quad t\in R. \end{aligned}$$

In similar way we can show that

$$\begin{aligned} \overline{R}(t-s_n)\rightarrow R(t)\quad \text{ as }\quad n\rightarrow \infty \quad \text{ for } \text{ all } \quad t\in R. \end{aligned}$$

This shows that \(R(t)\in AA(R,X)\).

Now let us show that \(S(t)\in C_0(R^+,X)\). Since \(\phi \in C_0(R^+,X)\), for each \(\epsilon >0\) there exists a constant \(T>0\) such that \(\Vert \phi (s)\Vert \le \epsilon \) for all \(s\ge T\). Then for all \(t\ge T\), we deduce,

$$\begin{aligned} \Vert S(t)\Vert&\le C(\theta ,\alpha )M\Vert \phi \Vert _{\infty }\int _0^{t/2}\frac{1}{1+|\omega |(t-s)^{\alpha }}ds+\epsilon C(\theta ,\alpha )M\int _{t/2}^t\frac{1}{1+|\omega |(t-s)^{\alpha }}ds\\&\quad +C(\theta ,\alpha )M\Vert g\Vert _{\infty }\int _{-\infty }^0\frac{1}{1+|\omega |(t-s)^{\alpha }}ds\\&\le C(\theta ,\alpha )M[\Vert \phi \Vert _{\infty }+\Vert g\Vert _{\infty }]\int _t^{\infty }\frac{1}{1+|\omega |s^{\alpha }}ds\\&\quad +\epsilon C(\theta ,\alpha )M\Vert g\Vert _{\infty }\int _0^{\infty }\frac{1}{1+|\omega |s^{\alpha }}ds\\&\le C(\theta ,\alpha )M[\Vert \phi \Vert _{\infty }+\Vert g\Vert _{\infty }]\int _t^{\infty }\frac{1}{1+|\omega |s^{\alpha }}ds+\frac{\epsilon C(\theta ,\alpha )M|\omega |^{-1/\alpha }\pi }{\alpha \sin (\pi /\alpha )}. \end{aligned}$$

Therefore, \(\lim _{t\rightarrow \infty }S(t)=0\), that is, \(S(t)\in C_0(R^+,X)\). This completes the proof. \(\square \)

The first existence and uniqueness result is based on Banach’s contraction principle.

Theorem 3.1

Let \(f=g+\phi \in AAA(R^+\times X\times X,X)\) with \(g\in AA(R\times X\times X,X)\) and \(\phi \in C_0(R^+\times X\times X,X)\). Assume that (H1)-(H3) hold. Then (1.1)-(1.2) has a unique asymptotically almost automorphic mild solution provided

$$\begin{aligned} L_1+\Big (L_2+L_3L_4\frac{c_k}{b}\Big )C(\theta ,\alpha )M\frac{|w|^{-1/\alpha }\pi }{\alpha \sin (\pi /\alpha )}<1. \end{aligned}$$
(3.1)

Proof

Consider the operator \(\Gamma :AAA(R^+,X)\rightarrow AAA(R^+,X)\) such that

$$\begin{aligned} (\Gamma x)(t)=S_{\alpha }(t)[x_0-k_1(0,x_0)]+k_1(t,x(t))+\int _0^tS_{\alpha }(t-s)f(s,x(s),Kx(s))ds. \end{aligned}$$

Applying Lemma 3.1, we infer that \(k_1(\cdot ,x(\cdot ))\) and \(f(\cdot ,x(\cdot ))\) belong to \(AAA(R^+,X)\). By Lemma 3.2, we obtain that \(\Gamma \) is \(AAA(R^+,X)\)-valued. Furthermore, we have the estimate

$$\begin{aligned} \Vert (\Gamma x)(t)-(\Gamma y)(t)\Vert&=\Big \Vert [k_1(t,x(t))-k_1(t,y(t))] \nonumber \\&\quad +\int _0^tS_{\alpha }(t-s) \big [f(s,x(s),Kx(s)) -f(s,y(s),Ky(s))\big ]ds\Big \Vert \nonumber \\&\le \Vert k_1(t,x(t))-k_1(t,y(t))\Vert \nonumber \\ {}&\quad +\int _0^t\Vert S_{\alpha }(t-s)\Vert _{L(X)}\big \Vert f(s,x(s),Kx(s))\nonumber \\&\quad -f(s,y(s),Ky(s))\big \Vert ds\nonumber \\&\le L_1\Vert x(t)-y(t)\Vert +\int _0^t\frac{C(\theta ,\alpha )M}{1+|\omega |(t-s)^{\alpha }}\big [L_2\Vert x(s)-y(s)\Vert \nonumber \\ {}&\quad +L_3\Vert Kx(s)-Ky(s)\Vert \big ]ds. \end{aligned}$$
(3.2)

Consider

$$\begin{aligned} \Vert Kx(s)-Ky(s)\Vert&\le \int _0^t|k(t-s)|\Vert h(s,x(s))-h(s,y(s))\Vert ds\\&\le \int _0^t|k(t-s)|L_4\Vert x(s)-y(s)\Vert ds\\&\le \sup _{t\in R^+}\Vert x(t)-y(t)\Vert L_4\Big (\int _0^t|k(t-s)|ds\Big )\\&\le \sup _{t\in R^+}\Vert x(t)-y(t)\Vert L_4\int _0^t|k(s)|ds\\&\le \sup _{t\in R^+}\Vert x(t)-y(t)\Vert L_4\int _0^tc_ke^{-bs}ds\\&\le c_k\frac{(1-e^{-bt})}{b}L_4\sup _{t\in R^+}\Vert x(t)-y(t)\Vert . \end{aligned}$$

Using the above estimate, inequality (3.2) becomes

$$\begin{aligned}&\Vert (\Gamma x)(t)-(\Gamma y)(t)\Vert \\&\quad \le L_1\sup _{t\in R^+}\Vert x(t)-y(t)\Vert \\&\quad +\Big [L_2+L_3L_4c_k\Big (\frac{1-e^{-bt}}{b}\Big )\Big ]\sup _{t\in R^+}\Vert x(t)-y(t)\Vert \int _0^t\frac{C(\theta ,\alpha )M}{1+|\omega |s^{\alpha }}ds \\&\quad \le \Big [L_1+\Big [L_2+L_3L_4c_k\Big (\frac{1-e^{-bt}}{b}\Big )\Big ]C(\theta ,\alpha )M\frac{|w|^{-1/\alpha }\pi }{\alpha \sin (\pi /\alpha )}\Big ]\Vert x-y\Vert _{\infty }. \end{aligned}$$

This implies

$$\begin{aligned} \Vert \Gamma x-\Gamma y\Vert _{\infty }\le \Big [L_1+\Big [L_2+L_3L_4c_k\Big (\frac{1-e^{-bt}}{b}\Big )\Big ]C(\theta ,\alpha )M\frac{|w|^{-1/\alpha }\pi }{\alpha \sin (\pi /\alpha )}\Big ]\Vert x-y\Vert _{\infty }, \end{aligned}$$

which proves that \(\Gamma \) is a contraction and we conclude that \(\Gamma \) has a unique fixed point in \(AAA(R^+,X)\). This completes the proof. \(\square \)

We next study the existence of asymptotically almost automorphic mild solutions of (1.1)–(1.2) when the perturbation f is not necessarily Lipschitz continuous. For that, we require the following assumptions:

  1. (H4)

    There exists a continuous nondecreasing function \(W:[0,\infty )\rightarrow (0,\infty )\) such that

    $$\begin{aligned} \Vert f(t,x,y)\Vert \le W(\Vert x\Vert +\Vert y\Vert ) \quad \text {for all} \quad t\ge 0 \,\, \text {and} \quad x\in X. \end{aligned}$$
  2. (H5)

    The functions \(f:R^+\times X\times X\rightarrow X\), \(h:R^+\times X\rightarrow X\) and \(k_1:R^+\times X\rightarrow X\) are asymptotically almost automorphic in t and uniformly for x in compact subsets of X and uniformly continuous on bounded sets of X uniformly in \(t\ge 0\).

Theorem 3.2

Assume that the conditions (H1) and (H4)–(H5) hold. Let inequality (2.3) be satisfied. In addition, suppose the following properties hold:

  1. (i)

    For each \(C\ge 0\)

    $$\begin{aligned} \lim _{t\rightarrow \infty }\frac{1}{h(t)}\int _0^t\frac{W\big ((1+K)Ch(s)\big )}{1+|\omega |(t-s)^{\alpha }}ds=0, \end{aligned}$$

    where h is the function given in Lemma 2.3. We set

    $$\begin{aligned}&\beta (C):=\frac{1}{h(t)}\Big (\Vert S_{\alpha }(t)(x_0-k_1(0,x_0)\Vert +\Vert k_1(t,x(t))\Vert \\&\qquad +C(\theta ,\alpha )M\int _0^t\frac{W\big ((1+K)Ch(s)\big )}{1+|\omega |(t-s)^{\alpha }}ds\Big ), \end{aligned}$$

    where \(C(\theta ,\alpha )\) and M are constants given in (2.3).

  2. (ii)

    There is a constant \(L_1>0\) such that \(\Vert k_1(t,h(t)x)-k_1(t,h(t)y)\Vert \le L_1\Vert x-y\Vert \) for all \(t\ge 0\) and \(x,y\in X\). We set

    $$\begin{aligned} \Omega (C):=\frac{C(\theta ,\alpha )M}{h(t)}\int _0^t\frac{W\big ((1+K)Ch(s)\big )}{1+|\omega |(t-s)^{\alpha }}ds, \end{aligned}$$

    where \(C(\theta ,\alpha )\) and M are the constants given in (2.3) and h is given in Lemma 2.3.

  3. (iii)

    For each \(\epsilon >0\) there is \(\delta >0\) such that for every \(u,v\in C_h(X),\, \Vert u-v\Vert _h\le \delta \) implies that

    $$\begin{aligned} C(\theta ,\alpha )M\int _0^t\frac{\Vert f(s,u(s),Ku(s))-f(s,v(s),Kv(s))\Vert }{1+|\omega |(t-s)^{\alpha }}ds\le \epsilon , \end{aligned}$$

    for all \(t\in R\).

  4. (iv)

    \(L_1+\displaystyle {\liminf }_{\displaystyle r\rightarrow \infty }\displaystyle \frac{\Omega (r)}{r}<1\).

  5. (v)

    \(\displaystyle \lim \inf _{\xi \rightarrow \infty }\frac{\xi }{\beta (\xi )}>1\).

  6. (vi)

    For all \(a,b\in R,\, a<b\) and \(r>0\), the set \(\{f(s,h(s)x,K(h(s)x)):a\le s\le b, \, x\in C_h(X), \, \Vert x\Vert _h\le r\}\) is relatively compact in X.

Then equation (1.1)–(1.2) has an asymptotically almost automorphic mild solution.

Proof

We define the operator \(\Gamma :C_h(X)\rightarrow C_h(X)\) by

$$\begin{aligned} (\Gamma x)(t)&=S_{\alpha }(t)[x_0-k_1(0,x_0)]+k_1(t,x(t)) \\&\quad +\,\int _0^tS_{\alpha }(t-s)f(s,x(s),Kx(s))ds, \, t\ge 0. \end{aligned}$$

Now, we decompose \(\Gamma \) as \(\Gamma =\Gamma _1+\Gamma _2\), where

$$\begin{aligned} (\Gamma _1x)(t)&=S_{\alpha }(t)[x_0-k_1(0,x_0)]+k_1(t,x(t)) \\ (\Gamma _2x)(t)&= \int _0^tS_{\alpha }(t-s)f(s,x(s),Kx(s))ds. \end{aligned}$$

Now, we will show that the operator \(\Gamma _1\) is contraction and \(\Gamma _2\) is completely continuous. For better readability, we break the proof into sequence of steps.

Step 1: We show that \( \Gamma _1\) is contraction on \(C_h(X)\).

Let \(x\in C_h(X)\), we have that

$$\begin{aligned} \frac{\Vert (\Gamma _1x)(t)\Vert }{h(t)}&\le \frac{1}{h(t)}\Big [\Vert S_{\alpha }(t)\Vert [\Vert x_0\Vert +\Vert k_1(0,x_0)\Vert ]\\&\quad +\Vert k_1(t,x(t))-k_1(t,0)\Vert +\Vert k_1(t,0)\Vert \Big ] \\&\le \frac{1}{h(t)}\Big [C(\theta ,\alpha )M[\Vert x_0\Vert +\Vert k_1(0,x_0)\Vert ]+L_1\Vert x\Vert _h+\Vert k_1(\cdot ,0)\Vert _{\infty }\Big ]. \end{aligned}$$

Hence, \(\Gamma _1\) is \(C_h(X)\)-valued. On the other hand, \(\Gamma _1\) is an \(L_1\)-contraction.

Next we show that \(\Gamma _2\) is completely continuous.

Step 2: The operator \(\Gamma _2\) is continuous.

In fact, for any \(\epsilon >0\), we take \(\delta >0\) involved in condition (iii). If \(x,y\in C_h(X)\) and \(\Vert x-y\Vert _h\le \delta \) then

$$\begin{aligned} \Vert (\Gamma _2 x)(t)-(\Gamma _2 y)(t)\Vert&\le C(\theta ,\alpha )M\int _0^t\frac{\Vert f(s,x(s),Kx(s))-f(s,y(s),Ky(s))\Vert }{1+|\omega |(t-s)^{\alpha }}ds\\&\le \epsilon , \end{aligned}$$

which shows the assertion.

Step 3: We next show that \(\Gamma _2\) is completely continuous.

Let \(V^{^\prime }(t)=\Gamma _2(B_r(C_h(X)))\) and \(v^{\prime }=\Gamma _2(x)\) for \(x\in B_r(C_h(X))\). Initially, we can infer that \(V^{^\prime }_b(t)\) is a relatively compact subset of X for each \(t\in [0,b]\). Infact, using condition (vi) we get that \(N=\{S_{\alpha }(s)f(\xi ,h(\xi )x,K(h(\xi )x)):0\le s\le t,0\le \xi \le t, \Vert x\Vert \le r\}\) is relatively compact. It is each to see that \(V^{^\prime }_b(t)\subset S_{\alpha }(t)[x_0-k_1(0,x_0)]+k_1(t,x(t))+t\overline{C(N)}\), which establishes our assertion.

From the decomposition of

$$\begin{aligned} v^\prime (t+s)-v^\prime (t)&=[S_{\alpha }(t+s)-S_{\alpha }(t)][x_0-k_1(0,x_0)]+k_1(t+s,x(t+s)) \\&\quad -k_1(t,x(t)) \\&\quad +\int _t^{t+s}S_{\alpha }(t+s-\xi )f(\xi ,x(\xi ),Kx(\xi ))d\xi \\&\quad +\int _0^t[S_{\alpha }(\xi +s)-S_{\alpha }(\xi )]f(t-\xi ,x(t-\xi ),Kx(t-\xi ))d\xi , \end{aligned}$$

it follows that the set \(V^{^\prime }_b\) is equicontinuous.

From the condition (i), we have

$$\begin{aligned} \frac{\Vert v^\prime (t)\Vert }{h(t)}&\le \frac{1}{h(t)}\Big [S_{\alpha }(t)[x_0-k_1(0,x_0)]+k_1(t,x(t))\Big ] \\&\quad +\,\frac{C(\theta ,\alpha )M}{h(t)}\int _0^t\frac{W((1+\Vert K\Vert )r h(s)}{1+|\omega |(t-s)^{\alpha }}ds \\&\quad \rightarrow \, 0 \qquad \text{ as }\qquad t\rightarrow \infty . \end{aligned}$$

From Lemma 2.3, we deduce that, \(V^{^\prime }\) is relatively compact set in \(C_h(X)\).

Let us denote \(x^{\lambda }(\cdot )\) be a solution of equation \(x^{\lambda }=\lambda \Gamma (x^{\lambda })\) for some \(\lambda \in (0,1)\). Now using the estimate,

$$\begin{aligned} \Vert x^{\lambda }\Vert _h&\le \Vert S_{\alpha }(t)[x_0-k_1(0,x_0)\Vert +\Vert k_1(t,\Vert x^{\lambda }(t)\Vert _h)\Vert \\&\quad +C(\theta ,\alpha )M\int _0^t\frac{W((1+\Vert K\Vert )r \Vert x^{\lambda }\Vert _hh(s)}{1+|\omega |(t-s)^{\alpha }}ds\\&\le \beta (\Vert x^{\lambda }\Vert _h), \end{aligned}$$

we get \(\frac{\Vert x^{\lambda }\Vert _h}{\beta (\Vert x^{\lambda }\Vert _h)}\le 1\). Using the condition (v) of Theorem 3.2, we have \(\{x^{\lambda }:x^{\lambda }=\lambda \Gamma (x^{\lambda })\}, \lambda \in (0,1)\) is bounded. From Lemmas 3.1 and 3.2, we have that

$$\begin{aligned} \Gamma _i(AAA(R^+\times X,X))\subset AAA((R^+\times X,X)), i=1,2. \end{aligned}$$

Hence, \(\Gamma (AAA(R^+\times X,X))\subset AAA((R^+\times X,X))\) and \(\Gamma _2:(AAA(R^+\times X,X))\rightarrow AAA((R^+\times X,X))\) is completely continuous.

Putting \(B_r:=B_r(AAA(R^+\times X,X))\), we claim that there is \(r>0\) such that \(\Gamma (B_r)\subset B_r\). In fact, if we assume that this assertion is false, then for all \(r>0\) we can choose \(x^r\in B_r\) and \(t^r\ge 0\) such that \(\Vert \Gamma x^r(t^r)\Vert /h(t^r)>r\). We observe that

$$\begin{aligned} \Vert \Gamma x^r(t^r)\Vert&\le C(\theta ,\alpha )M(\Vert x_0\Vert +\Vert k_1(0,x_0)\Vert )+L_1r+\Vert k_1(\cdot ,0)\Vert _{\infty } \\&\quad +C(\theta ,\alpha )M\int _0^{t^r}\frac{W((1+\Vert K\Vert )r h(s)}{1+|\omega |(t^r-s)^{\alpha }}ds. \end{aligned}$$

Thus, \(1\le L_1+\liminf _{r\rightarrow \infty }\frac{\Omega (r)}{r}\), which is contrary to assumption (iv). We have that \(\Gamma _1\) is a contraction on \(B_r\) and \(\Gamma _2(B_r)\) is a compact set. It follows from [30] [Corollary 4.3.2] that \(\Gamma \) has a fixed point \(x\in AAA(R^+\times X,X)\). More precisely, \(x\in AAA(R^+\times X,X)\) and this finishes the proof. \(\square \)

4 Example

Example 1

Consider the following example for the Theorem 3.1.

$$\begin{aligned} \partial _t^{\alpha }[w(t,x)-k_1(t,w(t,x))]&=\partial _x^2[w(t,x)-k_1(t,w(t,x))]-\mu w(t,x) \\&\quad +\partial _t^{\alpha -1}\Big [\beta w(t,x)(\cos t+\cos \sqrt{2} t)+\beta e^{-t}\sin (w(t,x))\\&\quad +\sin \Big (\int _0^te^{t-s}h(s,w(t,s))ds\Big )\Big ], \quad t\ge 0,\, x\in [0,\pi ],\\ w(t,0)&=w(t,\pi )=0, \quad t\ge 0,\, \mu >0, \end{aligned}$$

where \(1 < \alpha < 2\) and \(w_0\in L^2[0,\pi ]\). Define the linear operator A on \(X=(L^2([0,\pi ]),|\cdot |_2)\) by \(Aw=w^{\prime \prime }-\mu w\) with the domain

$$\begin{aligned} D(A)=\{w\in X:w^{\prime \prime }\in X, w(0)=w(\pi )=0\}. \end{aligned}$$

It is known that \(\Delta w=w^{\prime \prime }\) is the infinitesimal generator of analytic semigroup on \(L^2[0,\pi ]\) and thus A is sectorial of type \(w=-\mu <0\). Denote \(w(t)x=w(t,x)\) and

$$\begin{aligned} f(t,w,Kw)(x)=\beta w(x)(\cos t+\cos \sqrt{2} t)+\beta e^{-|t|}\sin (w(x))+\sin (Kw(x)) \end{aligned}$$

for each \(w\in X\). One can easily see that the function f(t, x, Kx) is asymptotically almost automorphic in t for each \(x\in X\). Now under the condition

$$\begin{aligned} \beta +1<\frac{\alpha \sin (\pi /\alpha )}{3C(\theta ,\alpha )M\mu ^{-1/\alpha }\pi }-L_1, \end{aligned}$$

there exists an unique asymptotically almost automorphic mild solution.

Example 2

One can also consider the following fractional order delay relaxation oscillation equation for \(\alpha \in (1,2),\)

$$\begin{aligned} \frac{\partial ^{\alpha }}{\partial t^{\alpha }}(u(t,x)-k_1(t,u(t,x)))= & {} \frac{\partial ^2}{\partial x^2}((u(t,x)-k_1(t,u(t,x))))-pu(t,x) \nonumber \\&+\frac{\partial ^{\alpha -1}}{\partial t^{\alpha -1}}(f(t,u(t,x),u(t-\tau ,x))), \ \tau >0, \nonumber \\&\quad \quad t \in R, \ x \in (0,\pi ) \nonumber \\ u(t,0)= & {} u(t, \pi ) = 0, \quad t \in R, \nonumber \\ u(t,x)= & {} \phi (t,x) \quad t \in [-\tau ,0], \end{aligned}$$
(4.1)

where \(p>0\) and f is a asymptotic almost automorphic function in t. Also assume that f satisfies Lipschitz condition in both variable with Lipschitz constants \(L_2, L_3.\) Note that \(\displaystyle \int _{-\infty }^tk(t-s)h(s,u(s))ds=\displaystyle \int _{-\infty }^tk(-s)h(s,u_t(s))ds=J(u_t),\) which can be thought like function of \(u_t\) and hence can be considered as functional differential equations. Using the transformation \(u(t)x=u(t,x)\) and define \(Au=\frac{\partial ^2 u}{\partial x^2}-pu, \ u \in D(A),\) where

$$\begin{aligned} D(A)= & {} \Big \{u \in L^2((0,\pi ),R), u^\prime \\&\in L^2((0,\pi ),R), u^{\prime \prime } \in L^2((0,\pi ),R),u(0)=u(\pi )=0 \Big \}, \end{aligned}$$

the above equation can be transformed into

$$\begin{aligned} \frac{d^{\alpha }}{dt^{\alpha }}(u(t)-k_1(t,u(t)))=A(u(t)-k_1(t,u(t)))+\frac{d^{\alpha -1}}{dt^{\alpha -1}}g(t,u(t),u_t(-\tau )),\qquad \, \end{aligned}$$
(4.2)

\(t \in R\) and \(u(t)=\phi (t) \, t \in [-\tau ,0].\) It is to note that A generates an analytic semigroup \(\{T(t) :t\ge 0\) on X,  where \(X=L^2((0,\pi ),R).\) Hence \(pI-A\) is sectorial of type \(\omega =-p<0.\) Further A has discrete spectrum with eigenvalues of the form \(-k^2; k \in N,\) and corresponding normalized eigenfunctions given by \(z_k(x)=(\frac{2}{\pi })^{\frac{1}{2}}\sin (kx).\) As A is analytic. Thus under all the required assumption on f,  the existence of asymptotic almost automorphic solutions is ensured accordingly.