Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces

Abstract

This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation $u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u\left(t\right))$ in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if $A\left(t\right)$ is time independent.

Introduction

In 1962, Bochner [1] introduced the concept of almost automorphy, which is an important generalization of almost periodicity. Afterwards, Zaki [2] extended the concept to almost automorphic vector-valued functions, paving the road to many applications to differential equations. In the last decade, the almost automorphy and almost periodicity of the solutions to various evolution equations have been widely investigated (see, e.g., [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]).

On the other hand, in the 1920s, Stepanov [16] introduced a generalization of almost periodic functions—Stepanov almost periodic functions, which are not necessarily continuous. Recently, N’Guérékata and Pankov [17] studied a generalization of almost automorphic functions—Stepanov almost automorphic functions, which are also not necessarily continuous.

Most recently, [18], [13] introduced and initiated the study of pseudo-almost automorphy, which is a meaningful generalization of almost automorphy as well as pseudo-almost periodicity. Diagana [4] started the research on Stepanov-like pseudo-almost automorphy. Moreover, [14] introduced and investigated firstly the concept of bi-almost automorphy. On the basis of these works, in this paper, we investigate the existence of almost automorphic solutions to the following semilinear evolution equation:

$$u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f(t,u\left(t\right))$$

where the nonlinear term is Stepanov-like almost automorphic. We first establish a composition theorem for Stepanov-like almost automorphic functions, and then we study the existence and uniqueness of almost automorphic solutions to the linear equation

$$u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)+f\left(t\right)$$

with Stepanov-like almost automorphic $f$. Finally, combining the composition theorem and the existence and uniqueness theorem for equation (1.2), we explore the existence of an almost automorphic solution to equation (1.1) with a Stepanov almost automorphic nonlinear term.

As we will see, the composition theorem plays a key role in this paper, and the establishment of a composition theorem is more difficult than for the almost automorphic case due to the complexity of Stepanov almost automorphic functions. Moreover, our Lipschitz assumption on $f(t,u)$ (see (A0)) is weaker than those in many earlier works, and is more natural in the case of $f(\cdot ,u)$ being Stepanov almost automorphic. In addition, in some earlier works, $A\left(t\right)$ is assumed to be periodic. Here, we do not make this assumption.

Throughout this paper, we denote by $\mathbb{N}$ the set of positive integers, by $\mathbb{R}$ the set of real numbers, by $X$ a Banach space, and by $mesE$ the Lebesgue measure for any subset $E\subset \mathbb{R}$. Moreover, we assume $1\le p<+\infty$ if there is no special statement.

First, let us recall some definitions and basic results for almost automorphic functions (for more details, see [12]).

Definition 1.1

A continuous function $f:\mathbb{R}\to X$ is called almost automorphic if for every real sequence $\left(s_m\right)$, there exists a subsequence $\left(s_n\right)$ such that

$$g\left(t\right)≔\underset{n\to \infty }{lim}f(t+s_n)$$

is well defined for each $t\in \mathbb{R}$ and

$$\underset{n\to \infty }{lim}g(t-s_n)=f\left(t\right)$$

for each $t\in \mathbb{R}$. Denote by $AA\left(X\right)$ the set of all such functions.

A classical example of an almost automorphic function (not almost periodic) is

$$f\left(t\right)=sin\frac{1}{2+cost+cos\sqrt{2}t}\text{,}t\in \mathbb{R}\text{.}$$

Lemma 1.2

The following hold true:

• Assume that $f\in AA\left(X\right)$ . Then the range ${\mathcal{R}}_f=\{f\left(t\right):t\in \mathbb{R}\}$ is precompact in $X$, and so $f$ is bounded.

• Assume that $f$, $g\in AA\left(X\right)$ . Then $f+g\in AA\left(X\right)$.

• Assume that $f_n\in AA\left(X\right)$ and $f_n\to f$ uniformly on $\mathbb{R}$ . Then $f\in AA\left(X\right)$.

• Equipped with the sup norm

  $$\Vert f\Vert =\underset{t\in \mathbb{R}}{sup}\Vert f\left(t\right)\Vert \text{,}$$

  $AA\left(X\right)$ turns out to be a Banach space.

Proof

See [12].  □

Definition 1.3

A continuous function $f:\mathbb{R}\times X\to X$ is called almost automorphic in $t$ for each $x\in X$ if for every real sequence $\left(s_n^{\prime }\right)$, there exists a subsequence $\left(s_n\right)$ such that

$$g(t,x)≔\underset{n\to \infty }{lim}f(t+s_n,x)$$

is well defined for each $t\in \mathbb{R}$, $x\in X$ and

$$\underset{n\to \infty }{lim}g(t-s_n,x)=f(t,x)$$

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

Next, let us recall the definition of bi-almost automorphic functions, which was introduced originally by Xiao, Zhu, and Liang [14].

Definition 1.4 [14]

A continuous function $f(t,s):\mathbb{R}\times \mathbb{R}\to X$ is called bi-almost automorphic if for every sequence of real numbers $\left(s_m\right)$, we can extract a subsequence $\left(s_n\right)$ such that $g(t,s)={lim}_{n\to \infty }f(t+s_n,s+s_n)$ is well defined for each $t,s\in \mathbb{R}$, and ${lim}_{n\to \infty }g(t-s_n,s-s_n)=f(t,s)$ for each $t,s\in \mathbb{R}$. $bAA(\mathbb{R}\times \mathbb{R},X)$ stands for the set of all such functions.

Example 1.5 [14]

(i) If $f(t,s)=g(t-s)$ for some $g\in C(\mathbb{R},X)$, then $f\in bAA(\mathbb{R}\times \mathbb{R},X)$. (ii) Let $h(t,s)=sintcoss,t,s\in \mathbb{R}$. Then $h\in bAA(\mathbb{R}\times \mathbb{R},\mathbb{R})$.

Moreover, we recall some definitions and basic results for Stepanov-like almost automorphic functions (for more details, see [17]).

Definition 1.6

The Bochner transform $f^b(t,s)$, $t\in \mathbb{R}$, $s\in [0,1]$, of a function $f\left(t\right)$ on $\mathbb{R}$, with values in $X$, is defined by

$$f^b(t,s)≔f(t+s)\text{.}$$

Definition 1.7

The space $B{S}^p\left(X\right)$ of all Stepanov bounded functions, with the exponent $p$, consists of all measurable functions $f$ on $\mathbb{R}$ with values in $X$ such that

$${\Vert f\Vert }_{S^p}≔\underset{t\in \mathbb{R}}{sup}{\left({\int }_t^{t+1}{\Vert f\left(\tau \right)\Vert }^{p}d\tau \right)}^{1/p}<+\infty \text{.}$$

It is obvious that $L^p(\mathbb{R};X)\subset B{S}^p\left(X\right)\subset L_{loc}^p(\mathbb{R};X)$ and $B{S}^p\left(X\right)\subset B{S}^q\left(X\right)$ whenever $p\ge q\ge 1$.

Definition 1.8

The space $A{S}^p\left(X\right)$ of $S^p$-almost automorphic functions ($S^p$-a.a. for short) consists of all $f\in B{S}^p\left(X\right)$ such that $f^b\in AA\left(L^p(0,1;X)\right)$. In other words, a function $f\in L_{\mathrm{loc}}^p(\mathbb{R};X)$ is said to be $S^p$-almost automorphic if its Bochner transform $f^b:\mathbb{R}\to L^p(0,1;X)$ is almost automorphic in the sense that for every sequence of real numbers $\left(s_n^{\prime }\right)$, there exists a subsequence $\left(s_n\right)$ and a function $g\in L_{\mathrm{loc}}^p(\mathbb{R};X)$ such that

$$\underset{n\to \infty }{lim}{\left({\int }_0^1{\Vert f(t+s_n+s)-g(t+s)\Vert }^{p}ds\right)}^{1/p}=0\text{,}$$

and

$$\underset{n\to \infty }{lim}{\left({\int }_0^1{\Vert g(t-s_n+s)-f(t+s)\Vert }^{p}ds\right)}^{1/p}=0\text{,}$$

for each $t\in \mathbb{R}$.

Remark 1.9

It is clear that if $1\le p<q<\infty$ and $f\in L_{\mathrm{loc}}^q(\mathbb{R};X)$ is $S^q$-almost automorphic, then $f$ is $S^p$-almost automorphic. Also if $f\in AA\left(X\right)$, then $f$ is $S^p$-almost automorphic for any $1\le p<\infty$.

Definition 1.10 [4]

A function $f:\mathbb{R}\times X\to X,(t,u)\mapsto f(t,u)$ with $f(\cdot ,u)\in L_{loc}^p(\mathbb{R},X)$ for each $u\in X$ is said to be $S^p$-almost automorphic in $t\in \mathbb{R}$ uniformly for $u\in X$ if for every sequence of real numbers $\left(s_n^{\prime }\right)$, there exists a subsequence $\left(s_n\right)$ and a function $g:\mathbb{R}\times X\to X$ with $g(\cdot ,u)\in L_{loc}^p(\mathbb{R},X)$ such that

$$\underset{n\to \infty }{lim}{\left({\int }_0^1{\Vert f(t+s_n+s,u)-g(t+s,u)\Vert }^{p}ds\right)}^{1/p}=0\text{,}$$

and

$$\underset{n\to \infty }{lim}{\left({\int }_0^1{\Vert g(t-s_n+s,u)-f(t+s,u)\Vert }^{p}ds\right)}^{1/p}=0\text{,}$$

for each $t\in \mathbb{R}$ and for each $u\in X$. We denote by $A{S}^p(\mathbb{R}\times X,X)$ the set of all such functions.

In this paper, for each compact subset $K\subset X$, we denote by $A{S}_K^p(\mathbb{R}\times X,X)$ the space of all $f\in A{S}^p(\mathbb{R}\times X,X)$ satisfying that for every sequence of real numbers $\left(s_n^{\prime }\right)$, there exists a subsequence $\left(s_n\right)$ and a function $g:\mathbb{R}\times X\to X$ with $g(\cdot ,u)\in L_{loc}^p(\mathbb{R},X)$ such that

$$\underset{n\to \infty }{lim}{\left[{\int }_0^1{\left(\underset{u\in K}{sup}\Vert f(t+s_n+s,u)-g(t+s,u)\Vert \right)}^{p}ds\right]}^{1/p}=0\text{,}$$

and

$$\underset{n\to \infty }{lim}{\left[{\int }_0^1{\left(\underset{u\in K}{sup}\Vert g(t-s_n+s,u)-f(t+s,u)\Vert \right)}^{p}ds\right]}^{1/p}=0\text{,}$$

for each $t\in \mathbb{R}$. It is obvious that

$$A{S}_K^p(\mathbb{R}\times X,X)\subset A{S}^p(\mathbb{R}\times X,X)\subset AA(\mathbb{R}\times X,X)\text{.}$$

We also need to recall some notation for the evolution family and exponential dichotomy (cf., e.g., [19], [20]).

Definition 1.11

A set $\{U(t,s):t\ge s,t,s\in \mathbb{R}\}$ of bounded linear operators on $X$ is called an evolution family if

• $U(s,s)=I$, $U(t,s)=U(t,r)U(r,s)$ for $t\ge r\ge s$ and $t,r,s\in \mathbb{R}$,

• $\{(\tau ,\sigma )\in {\mathbb{R}}^2:\tau \ge \sigma \}\ni (t,s)⟼U(t,s)$ is strongly continuous.

Suppose that the initial value problem

$$u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)\text{,}u\left(s\right)=x\in X\text{,}$$

has an associated evolution family $U(t,s)$, i.e., $\left\{A\left(t\right)\right\}$ generates an evolution family $U(t,s)$.

Definition 1.12

An evolution family $U(t,s)$ is called hyperbolic (or has exponential dichotomy) if there are projections $P\left(t\right)$, $t\in \mathbb{R}$, uniformly bounded and strongly continuous in $t$, and constants $M$, $\omega >0$ such that

• $U(t,s)P\left(s\right)=P\left(t\right)U(t,s)$ for all $t\ge s$,

• the restriction $U_Q(t,s):Q\left(s\right)X\to Q\left(t\right)X$ is invertible for all $t\ge s$ (and we set $U_Q(s,t)=U_Q{(t,s)}^{-1}$),

• $\Vert U(t,s)P\left(s\right)\Vert \le M{e}^{-\omega (t-s)}$ and $\Vert U_Q(s,t)Q\left(t\right)\Vert \le M{e}^{-\omega (t-s)}$ for all $t\ge s$.

Here and below $Q≔I-P$.

Exponential dichotomy is a classical concept in the study of long-term behaviour of evolution equations; see e.g. [20]. If $U(t,s)$ is hyperbolic, then

$$\Gamma (t,s)≔\{\begin{array}{c}U(t,s)P\left(s\right)\text{,}t\ge s,t,s\in \mathbb{R}\text{,}\hfill \\ -U_Q(t,s)Q\left(s\right)\text{,}t<s,t,s\in \mathbb{R}\text{,}\hfill \end{array}$$

is called the Green’s function corresponding to $U(t,s)$ and $P(\cdot )$, and

$$\Vert \Gamma (t,s)\Vert \le \{\begin{array}{c}M{e}^{-\omega (t-s)}\text{,}t\ge s,t,s\in \mathbb{R}\text{,}\hfill \\ M{e}^{-\omega (s-t)}\text{,}t<s,t,s\in \mathbb{R}\text{.}\hfill \end{array}$$