# On the existence of asymptotically almost periodic solutions for nonlinear systems

- Juan Song
^{1, 2}, - Jianzhi Cao
^{2}Email author and - Xiong Li
^{2}

**2013**:28

https://doi.org/10.1186/1687-1847-2013-28

© Song et al.; licensee Springer 2013

**Received: **13 October 2012

**Accepted: **22 January 2013

**Published: **7 February 2013

## Abstract

In this paper, a nonlinear differential equation ${x}^{\prime}=A(t,x)+f(t)$ is considered. Some new sufficient conditions for the existence of a bounded solution and an asymptotically almost periodic solution, which generalize and improve the previously known results, are established by using a dissipative-type condition for $A(t,x)$. Finally, an example is presented to illustrate the feasibility and effectiveness of the new results.

## 1 Introduction

*etc.*, so they have been widely studied. For example, Medvedev [9] gave a sufficient condition to guarantee the existence of a bounded solution of the following equation:

*n*-space, for $x\in {\mathbb{R}}^{n}$, $\parallel x\parallel $ is any convenient norm of

*x*. Using this result, he also proved the existence of periodic and almost periodic solutions when $A(t,x)$ and $f(t)$ are periodic or almost periodic in

*t*uniformly for

*x*in a bounded subset of ${\mathbb{R}}^{n}$. Shigeo and Masato [4] extended the existence result in [9] by using a dissipative-type condition for $A(t,x)$. Thanh and Nguyen Truong [10] considered the following difference equation:

where ${\mathbb{R}}^{+}=[0,+\mathrm{\infty})$ and $A(t)$ is a linear operator on a Banach space, which is periodic, and $f(t)$ is asymptotically almost periodic. They showed a bounded mild solution *x* is asymptotically almost periodic.

where $A(t,x)\in C({\mathbb{R}}^{+}\times {\mathbb{R}}^{n},{\mathbb{R}}^{n})$ and $f(t)\in C({\mathbb{R}}^{+},{\mathbb{R}}^{n})$. By employing the dissipative-type condition for $A(t,x)$, when $A(t,x)$ and $f(t)$ are asymptotically almost periodic functions, we present some new criteria ensuring the existence of a bounded solution and an asymptotically almost periodic solution of Eq. (1.4). The remaining part is organized as follows. In the next section, we introduce some definitions and lemmas. In Section 3, we obtain two theories, which guarantee the existence of a bounded solution and an asymptotically almost solution of Eq. (1.4). In Section 4, a numerical simulation is carried out to illustrate the main results.

## 2 Preliminaries

Firstly, to establish our main results, it is necessary to make the following assumptions:

where *N* is a positive constant;

*δ*,

*γ*, ${T}_{0}$ such that

where $[\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}]$ is defined as follows (see Definition 2.5).

We now give some definitions which can be found in [3] and [6].

**Definition 2.1**If for any $\u03f5>0$, there exists a positive number $L(\u03f5)$ such that any interval of length $L(\u03f5)$ contains a

*τ*for which

for all $t\in \mathbb{R}$, then $f(t)$ is said to be an almost periodic function.

**Definition 2.2**If for any $\u03f5>0$ and any compact set

*S*in ${\mathbb{R}}^{n}$, there exists a positive number $L(\u03f5,S)$ such that any interval of length $L(\u03f5,S)$ contains a

*τ*for which

for all $t\in \mathbb{R}$ and all $x\in S$, then $A(t,x)$ is said to be almost periodic in *t* uniformly for $x\in {\mathbb{R}}^{n}$.

**Definition 2.3** If $f\in C({\mathbb{R}}^{+},\mathbb{R})$ and $f(t)=g(t)+\alpha (t)$ in ${\mathbb{R}}^{+}$, $g(t)$ is an almost periodic function in ℝ and $\alpha (t)$ is continuous in ${\mathbb{R}}^{+}$, ${lim}_{t\to +\mathrm{\infty}}\alpha (t)=0$, then $f(t)$ is called an asymptotically almost periodic function on ${\mathbb{R}}^{+}$.

**Definition 2.4** If $A(t,x)\in C({\mathbb{R}}^{+}\times {\mathbb{R}}^{n},{\mathbb{R}}^{n})$ and $A(t,x)=B(t,x)+\beta (t,x)$ in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{n}$, and $B(t,x)$ is an almost periodic function in *t* uniformly on $x\in {\mathbb{R}}^{n}$ and $\beta (t,x)$ is continuous in ${\mathbb{R}}^{+}\times {\mathbb{R}}^{n}$, ${lim}_{t\to +\mathrm{\infty}}\beta (t,x)=0$ uniformly on $x\in H\subset \mathrm{\Omega}$, where Ω is an open set on ${\mathbb{R}}^{n}$ and *H* is a compact set, then $A(t,x)$ is said to be an asymptotically almost periodic function in *t*.

**Definition 2.5**Functional $[\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}]:{\mathbb{R}}^{n}\times {\mathbb{R}}^{n}\u27f6\mathbb{R}$:

The following lemma on the functional $[\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}]$ is well known (see [6]).

**Lemma 2.1** [6]

*Let*

*x*,

*y*

*and*

*z*

*be in*${\mathbb{R}}^{n}$.

*Then the functional*$[\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}]$

*has the following properties*:

- (1)
$[x,y]={inf}_{h>0}{h}^{-1}(\parallel x+hy\parallel -\parallel x\parallel )$;

- (2)
$|[x,y]|\le \parallel y\parallel $;

- (3)
$[x,y+z]\le [x,y]+[x,z]$;

- (4)
*Let**u**be a function from a real interval**J**into*${\mathbb{R}}^{n}$*such that*${u}^{\prime}({t}_{0})$*exists for an interior point of**J*.*Then*${D}_{+}\parallel u({t}_{0})\parallel $*exists and*${D}_{+}\parallel u({t}_{0})\parallel =[u({t}_{0}),{u}^{\prime}({t}_{0})],$

*where* ${D}_{+}\parallel u({t}_{0})\parallel $ *denotes the right derivative of* $\parallel u(t)\parallel $ *at* ${t}_{0}$.

**Lemma 2.2**$f(t)\in C({\mathbb{R}}^{+},{\mathbb{R}}^{n})$

*is an asymptotically almost periodic function if and only if for any*$\u03f5>0$,

*there exist positive numbers*$L(\u03f5)$

*and*$T(\u03f5)$

*such that any interval of length*$L(\u03f5)$

*contains an*

*ω*

*such that when*$t\ge T(\u03f5)$,

**Lemma 2.3** [4]

*Suppose that*(${C}_{2}$)

*is satisfied*.

*Let*$u(t)$

*and*$v(t)$

*be solutions of*(1.1)

*on an interval*$[a,b)$.

*Then*

*for all* $t\in [a,b)$.

In order to obtain our main results, we should prove the following lemma.

**Lemma 2.4**

*Suppose that*(${C}_{2}$)

*is satisfied*.

*Then*

*and*

*Proof*It follows from (${C}_{2}$) that there exists a ${T}_{1}>{T}_{0}$ such that

This completes the proof. □

## 3 Existence of bounded solutions and asymptotically periodic solutions

In this section, it will be shown that, under certain conditions, the system (1.4) has a bounded solution and an asymptotically periodic solution.

**Theorem 3.1**

*Suppose that conditions*(${C}_{1}$), (${C}_{2}$)

*are satisfied*.

*Let*

*r*

*be defined as*

*where*

*and*

*Then Eq*. (1.4) *has a bounded solution* $u(t)$ *on* ${\mathbb{R}}^{+}$ *such that* $\parallel u(t)\parallel \le r$ *for* $t\in {\mathbb{R}}^{+}$. *Furthermore*, *if* $v(t)$ *is any solution of Eq*. (1.4), *then* $\parallel u(t)-v(t)\parallel \to 0$ *as* $t\to +\mathrm{\infty}$.

*Proof*If $A(t,0)\not\equiv 0$ for $t\in {\mathbb{R}}^{+}$, we replace $A(t,x)$ and $f(t)$ by $A(t,x)-A(t,0)$ and $f(t)+A(t,0)$, respectively. We assume, henceforth, that $A(t,0)\equiv 0$ and $\parallel f(t)\parallel \le N$ for all $t\in {\mathbb{R}}^{+}$ and fix a vector ${u}_{0}\in {\mathbb{R}}^{n}$ with $\parallel {u}_{0}\parallel ={r}_{0}$. For each positive integer

*n*with $n>{T}_{0}>\frac{1}{n}$, we consider the following Cauchy problem:

for $0<h\le {h}_{0}$. Since ${r}_{1}>{r}_{0}=\frac{N}{\delta}$, $p(\tau ){r}_{1}\le -\delta {r}_{1}<-N$, $p(\tau ){r}_{1}+N<0$. Thus, for *ϵ* with $0<\u03f5<-(p(\tau ){r}_{1}+N)$, there exists a sufficiently small $h>0$ such that $\parallel {u}_{n}(\tau +h)\parallel <{r}_{1}$. This contradicts the definition of *τ*. Then $\parallel {u}_{n}(t)\parallel \le {r}_{0}$ for all $t\in [\frac{1}{n},{T}_{0}]$.

By Lemma 2.4, we obtain ${e}^{{\int}_{{T}_{0}}^{t}p(\sigma )\phantom{\rule{0.2em}{0ex}}d\sigma}\to 0$, when $t\to +\mathrm{\infty}$, ${\int}_{{T}_{0}}^{t}p(\sigma )\phantom{\rule{0.2em}{0ex}}d\sigma \to -\mathrm{\infty}$, and then $\parallel u(t)-v(t)\parallel \to 0$ as $t\to +\mathrm{\infty}$. This completes the proof. □

**Theorem 3.2** *Suppose that* $A(t,x)$ *is asymptotically almost periodic in* *t* *uniformly for* $x\in Sr(0)$, *where* *r* *is a positive number defined by Eq*. (3.1) *and* $Sr(0)=\{x\in {\mathbb{R}}^{n};\parallel x\parallel \le r\}$, *and* $f(t)$ *is an asymptotically almost periodic function*. *Suppose*, *furthermore*, *that the condition* (${C}_{2}$) *is satisfied*. *Then Eq*. (1.4) *has an asymptotically almost periodic solution on* ${\mathbb{R}}^{+}$.

*Proof* First, we prove that $f(t)$ is bounded. $f(t)=g(t)+\alpha (t)$ in ${\mathbb{R}}^{+}$, and $g(t)$ is an almost periodic function in ℝ. For any $\epsilon \le 1$, there is an $l(\epsilon )>0$, when $t\in [0,l(\epsilon )]$, there is an $M>0$, $\parallel g(t)\parallel \le M$. For any $t\in \mathbb{R}$, choose $\tau \in [-t,-t+l(\epsilon )]$, then $t+\tau \in [0,l(\epsilon )]$, $\parallel g(t+\tau )\parallel <M$ and $\parallel g(t+\tau )-g(t)\parallel <1$, so for any *t*, $\parallel g(t)\parallel <M+1$. While $\alpha (t)\to 0$ $(t\to \mathrm{\infty})$, we have a positive $N>0$ such that $\parallel f(t)\parallel <N$, then the condition (${C}_{1}$) is satisfied. Conditions (${C}_{1}$) and (${C}_{2}$) are satisfied, let $u(t)$ be a bounded solution of (1.4) on ${\mathbb{R}}^{+}$ obtained in Theorem 3.1. Note that $\parallel u(t)\parallel \le r$ for all $t\in {\mathbb{R}}^{+}$, where *r* is a number defined by Eq. (3.1).

*t*uniformly for $x\in Sr(0)$. For each $\u03f5>0$, there exist a positive number ${t}_{1}(\u03f5,Sr(0))$ and a positive number $L(\u03f5,Sr(0))$ such that any interval of length $L(\u03f5,Sr(0))$ contains an

*ω*,

*η*is a positive number to be chosen later appropriately, and $t-\eta \ge 0$. We show that

We will show that $\parallel u(t+\omega )-u(t)\parallel \le {K}_{0}\u03f5$, where ${K}_{0}$ is a positive constant independent of *ϵ* and *ω*.

We must estimate ${e}^{{\int}_{t-\eta}^{t}p(\sigma )\phantom{\rule{0.2em}{0ex}}d\sigma}$ for *t* large enough.

*ω*, when $t\ge T=max\{{t}_{1}(\u03f5,Sr(0)),4{T}_{1}+{T}_{0}\}$, $\eta =4{T}_{1}$, ${K}_{0}=\gamma +K$,

From Lemma 2.2, $u(t)$ is an asymptotically almost periodic solution of Eq. (1.4). This completes the proof. □

**Remark 3.1** In [4], employing the dissipative-type condition for $A(t,x)$, the authors gave some sufficient conditions to prove the existence of a bounded solution, a periodic or almost periodic solution of the equation ${x}^{\prime}=A(t,x)+f(t)$. Extension of this result has been obtained in one direction: from periodic and almost periodic to asymptotically almost periodic forcing. The equation can be more widely used with asymptotically almost periodic functions.

**Remark 3.2** The condition (${C}_{2}$) implies the following hypothesis.

*δ*, ${\delta}_{1}$, ${T}_{0}$ and ${T}_{1}$ such that

We know that (${C}_{2}^{\prime}$) can also be used to prove the lemmas in Section 2 and the theorems in Section 3 leaving the conclusion unchanged. (${C}_{2}^{\prime}$) as well as (${C}_{2}$) yields the existence of a bounded solution, and the process of the proof is similar to the proof before, and we need not necessarily do it again.

## 4 The example

where $A(t,x)=-x(sint+sin\sqrt{2}t+3)+sin\frac{1}{{(1+t)}^{2}}$ is an asymptotically almost periodic function in $t\in {\mathbb{R}}^{+}$ uniformly on *x* which belongs to a compact set and $f(t)=-(sint+sin\sqrt{2}t+3)+sin\frac{1}{{(1+t)}^{2}}$ is an asymptotically almost periodic function on ${\mathbb{R}}^{+}$.

we know that (${C}_{1}$) is satisfied.

(${C}_{2}$) is satisfied too.

## Declarations

### Acknowledgements

The authors would like to thank the two referees for their valuable suggestions and comments concerning improvement of the work.

## Authors’ Affiliations

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