Existence and control of weighted pseudo almost periodic solutions for abstract nonlinear vibration systems

Abstract

In many vibration problems, it is very important to know precisely the bounds of the stability/instability frequencies and the associated amplitude ranges. This paper investigates the solvability and control of some weighted pseudo almost periodic solutions of abstract nonlinear vibration differential systems. Some sufficient conditions for the solvability and exponential stability of these systems are obtained. Moreover, the precise bound of Lyapunov exponents is estimated.

Introduction

Nonlinear vibration problems arise everywhere in engineering, such as in oil pipelines, nuclear reactors, marine risers, heat exchangers and so on. ¹ Analytical methods for various nonlinear oscillators arising in practical applications have been studied extensively in the past few decades. Very recently, Li and Yang ² studied the vibration analysis of conveying fluid pipe via He’s variational iteration method.^3,4 They transformed the governing equation of the pipe conveying fluid to the following abstract equation

$$u(t)=T(t-t_0)u_0+\lambda {\displaystyle {\int }_{t_0}^{t}T}(t-s)\alpha (s,u(s))\text{d}s$$

(1)

For dynamical systems, the authors^5–7 studied the oscillation/nonoscillation of solutions of different classes of differential systems; the authors^8–11 investigated the stability of neural networks by constructing suitable Lyapunov functionals and combining with fixed-point technique. ¹² Zhang^13–15 firstly studied the pseudo almost periodicity. The authors^16–27 studied the existence, uniqueness and/or stability of the solutions to the abstract differential equations.

In this paper, we shall study some new sufficient conditions for the existence of the solutions of the following evolution equation with a bounded nonlinear term

$$\begin{array}{l}u\prime (t)=Au(t)+\lambda \alpha (t,u(t)),\hfill \\ u(0)=u_0\in R^n,t\ge 0\hfill \end{array}$$

(2)

which is the differential form of (1). Moreover, we also investigate the stability of some weighted pseudo almost periodic (WPAP) solutions of abstract nonlinear vibration differential systems. Some sufficient conditions for the solvability and exponential stability of these systems are obtained. Meanwhile, the precise bound of Lyapunov exponents is estimated.

Preliminaries

Let $(W,||·||),(V,||·|{|}_v)$ be Banach spaces, R real number set. Denote U the collection of weights functions $\theta :R\to (0,\infty )$ , which are locally integrable over R with $\theta >0$ for almost each $u\in R.$ For $\theta \in U$ and for $r>0,$ we set

$$\chi (r,\theta ):={\displaystyle {\int }_{-r}^r\theta }(x)\text{d}x$$

(3)

$$\begin{array}{l}{U}_{\infty }:=\left\{\theta \in U,\underset{r\to \infty }{\lim }\chi (r,\theta )=\infty \right\},\\ U_B:=\left\{\theta \in U_{\infty },\theta \text{is}\text{bounded}\text{ and }\underset{x\in R}{\inf }\theta (x)>0\right\}\end{array}$$

It is obviously that $U_B\mathrm{\subseteq }{U}_{\infty }\mathrm{\subseteq }U$

Let $BC(R,W)$ be the space of all W-valued bounded continuous maps. The space $BC(R,W)$ with the sup norm $\left|\right|u|{|}_{\infty }=\underset{t\in R}{\sup }||u(t)||$ is a Banach space. Let AP(W) be set of Bohr almost periodic maps.Let Let

$$\theta \in U_{\infty }.$$

Define

$$\begin{array}{l}PB{C}_0(W,\theta ):\hfill \\ =\left\{\alpha \in BC(R,W):\underset{r\to \infty }{\lim }\frac{1}{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r|}|\alpha (s)||\theta (s)\text{d}s=0\right\}\hfill \end{array}$$

(4)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\begin{array}{l}PB{C}_0(V,W,\theta ):\hfill \\ =\{F\in C(R\times V,W):F(·,v) \text{is}\text{bounded}\text{ for }\text{each}v\in V\hfill \\ \text{ and }\underset{r\to \infty }{\lim }\frac{1}{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r|}|F(s,\varphi )||\theta (s)\text{d}s=0\text{uniformly}\text{in}\text{compact}\text{subset}\text{of}V\}\hfill \end{array}$$

(5)

A map $\alpha \in BC(R,W)$ is called WPAP if $\alpha =\beta +\gamma ,$ in which $\beta \in AP(W)$ and $\gamma \in PB{C}_0(W,\theta ).$ The set of all WPAP maps is denoted by $WPAP(W,\theta ).$ Meanwhile, a map $F\in C(R\times V,W)$ is called by WPAP in $t\in R$ and uniformly in $\varphi \in V$ if $F=G+\Phi ,$ in which $G\in AP(V,W)$ and $\Phi \in PB{C}_0(V,W,\theta ).$ The set of all such maps is denoted by $WPAP(V,W,\theta ).$

Remark 1 Let $\theta \in U_{\infty }.$ If $\alpha ,\beta \in WPAP(W,\theta ), s\in R,$ then $\alpha +s\beta$ is also in $WPAP(W,\theta ).$ Moreover, the decomposition of a WPAP map is unique since $g(R)\subset \overline{\alpha (R)}$ (for more denotes and details, please see literature^{13–15,26,27}).

Lemma 2 Let W be a real Banach space, Ω be a bounded open subset of W, $0\in \Omega ,T:\overline{\Omega }\to W$ is a completely continuous operator. Then, either there exists $w\in \partial \Omega ,\mu >1$ such that $T(w)=\mu w$ or there exists a fixed point $w^{*}\in \overline{\Omega }.$

Theorem 3 Assume that $\sigma \in WPAP(V,W,\theta )$ such that

$$\left|\right|\sigma (t,u)-\sigma (t,v)\left|\right|\le \varsigma (t)\left|\right|u-y|{|}_V\text{ for }\text{all}u,v\in V,t\in R,$$

where the nonnegative $\varsigma \in L^1(R)$ . If $\nu \in WPAP(V,\theta )$ then $\sigma (·,\nu (·))\in WPAP(W,\theta ).$

Proof. The proof is similar with the one given in Zhang, ¹⁵ so we omit it.

Main results

This section, we will devote to the existence of a WPAP solution to the abstract differential equation

$$u\prime (t)=Au(t)+\lambda \alpha (t,u(t)),t\in R$$

(6)

where $\lambda >0$ is a parameter, A is the infinitesimal generator of an exponentially stable C₀-semigroup T(t) on a Banach space W and $\alpha :R\times W\mapsto W$ is a WPAP function.

In the following, we need the condition

(B) $\alpha (t,0)\cancel{\equiv }0$ , and there exists a function $p\in L^1(R), p>0$ such that

$$\left|\right|\alpha (t,u)-\alpha (t,v)\left|\right|\le p(t)\left|\right|u-v\left|\right|,\text{a}.\text{e}.(t,u)\in R\times W$$

and there exists $t_0\in R$ such that $p(t_0)\ne 0$ . Moreover, we assume that $p(t)\le M(M>0)$ for all $t\in R.$

Remark 4 By the results in Zang, ¹⁵ if $\alpha :R\times W\mapsto W$ is a WPAP function, then α is bounded. So, our assumption $p(t)\le M(M>0)$ for all $t\in R$ is reasonable. Moreover, since $\{T(t):t\ge 0\}$ is an exponentially stable C₀-semigroup, we may assume that $\left|\right|T(t)\left|\right|\le L{e}^{-\omega t}$ .

Define a mapping Q by

$$(Qu)(t):=\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)\alpha (s,u(s))\text{d}s$$

Lemma 5 If u(t) is WPAP, then Qu is also WPAP.

Proof. Since u(t) is WPAP, we have $u\in WPAP(W,\theta )$ and $\alpha =g+\varphi$ in which $g\in AP(W)$ and $\varphi \in PB{C}_0(W,\theta ).$ Using (B) and Theorem 3, it follows that $\alpha (·,u(·))=g(·,u(·))+\varphi (·,u(·))$ where $g(·,u(·))\in AP(W)$ and $\varphi (·,u(·))\in PB{C}_0(W,\theta ).$ Thus

$$\begin{array}{c}(Qu)(t)=\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)\alpha (s,u(s))\text{d}s=\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)g(s,u(s))\text{d}s+\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)\varphi (s,u(s))\text{d}s.\end{array}$$

Let

$$\begin{array}{l}(Q_{1}u)(t):=\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)g(s,u(s))\text{d}s,\text{ and }\\ (Q_{2}u)(t):=\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)\varphi (s,u(s))\text{d}s.\end{array}$$

Because $g(·,u(·))\in AP(W),$ then for each $\epsilon >0,$ one has $\delta (\epsilon )>0$ such that

$$\left|\right|g(t+\tau ,u(t+\tau ))-g(t,u(t))\left|\right|<\frac{\omega \epsilon }{\lambda M}$$

for every interval of length $\delta (\epsilon )$ containing a τ and for all $t\in R.$ It is easily obtained that

$$\left|\right|(Q_{1}u)(t+\tau )-(Q_{1}u)(t)\left|\right|<\epsilon$$

for all $t\in R,$ and thus $(Q_{1}u)(·)\in AP(W).$ Now, we show that

$$\underset{r\to \infty }{\lim }\frac{1}{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r|}|(Q_{2}u)(s)||\theta (s)\text{d}s=0$$

We write that

$$\underset{r\to \infty }{\lim }\frac{1}{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r|}|(Q_{2}u)(s)||\theta (s)\text{d}s=I(\theta )+J(\theta )$$

where

$$I(\theta )=\underset{r\to \infty }{\lim }\frac{\lambda }{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r\theta }(t)\text{d}t{\displaystyle {\int }_{-r}^t|}|T(t-s)\varphi (s,u(s))||\text{d}s$$

and

$$J(\theta )=\underset{r\to \infty }{\lim }\frac{\lambda }{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r\theta }(t)\text{d}t{\displaystyle {\int }_{-\infty }^{-r}|}|T(t-s)\varphi (s,u(s))||\text{d}s$$

Since $\varphi (·,u(·))\in PB{C}_0(W,\theta )$ , we have

$$\begin{array}{l}J(\theta )\le \underset{r\to \infty }{\lim }\frac{\lambda L}{x(r,\theta )}{\displaystyle {\int }_{-\infty }^{-r}{e}^{\omega s}}\Vert \varphi (s,u(s))\Vert \text{d}s{\displaystyle {\int }_{-r}^r{e}^{-\omega t}}\theta (t)\text{d}t\\ \hspace{1em}\hspace{1em}\le \underset{t\in R}{\text{sup}}\Vert \varphi (t,u(t))\Vert \underset{r\to \infty }{\lim }\frac{\lambda L}{x(r,\theta )}{\displaystyle {\int }_{-r}^r{e}^{-\omega (r+1)}}\theta (t)\text{d}t\\ \hspace{1em}\hspace{1em}\le \underset{t\in R}{\text{sup}}\Vert \varphi (t,u(t))\Vert {\Vert \theta \Vert }_{L_{Loc}^1}\underset{r\to \infty }{\lim }\frac{\lambda L}{x(r,\theta )}\left(-\frac{e^{-2\omega r}}{\omega }+\frac{1}{\omega }\right)=0\end{array}$$

Similarly,

$$I(\theta )\le \underset{r\to \infty }{\lim }\frac{\lambda L}{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r|}|\varphi (t,u(t))||\theta (t)\text{d}t{\displaystyle {\int }_{-r}^t{e}^{-\omega (t-s)}}\text{d}s=\frac{\lambda L}{\omega }\underset{r\to \infty }{\lim }\frac{1}{\chi (r,\theta )}{\displaystyle {\int }_{-r}^r|}|\varphi (t,u(t))\left|\right|\theta (t)\text{d}t=0$$

This completes the proof.

Lemma 6 Assume that (B) holds. Then the operator $Q:WPAP(W,\theta )\to WPAP(W,\theta )$ is completely continuous.

Proof. Let $\left\{x_n\right\}\in WPAP(W,\theta )$ such that $x_n\to x$ in $WPAP(W,\theta )$ as $n\to \infty .$ Then there exists a bounded set $K\subset W$ s.t. $x_n(t),x(t)\in K$ for all $t\in R, n=1,2,\dots .$ By (B), given $\epsilon >0,$ there is $\delta =\frac{\omega \epsilon }{\lambda LM}>0$ such that $x,y\in K$ and $\left|\right|x-y\left|\right|<\delta$ . Then

$$\left|\right|\alpha (t,x)-\alpha (t,y)\left|\right|<\frac{\omega \epsilon }{\lambda L}\text{ for }\text{all}t\in R$$

For the above δ, there exists N s.t. $\left|\right|x_n(t)-x(t)\left|\right|<\delta$ for all $n>N$ and $t\in R.$ Then $\left|\right|\alpha (t,x_n(t))-\alpha (t,x(t))\left|\right|<\frac{\omega \epsilon }{\lambda L}$ . Hence

$$\begin{array}{c}\left|\right|(Q{x}_n)(t)-(Qx)(t)\left|\right|=\lambda \Vert {\displaystyle {\int }_{-\infty }^{t}T}(t-s)(\alpha (s,x_n(s))-\alpha (s,x_n(s)))\text{d}s\Vert \\ \le \lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)\left|\right|\alpha (s,x_n(s))-\alpha (s,x_n(s))\left|\right|\text{d}s\le \frac{\omega \epsilon }{L}{\displaystyle {\int }_{-\infty }^{t}L}{e}^{-\omega (t-s)}\text{d}s=\epsilon \end{array}$$

for all $n\in N$ and any $t\in R.$ It follows that Q is continuous.

Let $\Omega \subset WPAP(W)$ be bounded, i.e. there exists a constant C > 0 such that $\left|\right|u\left|\right|<C$ for all $u\in \Omega .$ Then, for $u\in \Omega ,$ we have

$$\begin{array}{l}\left|\right|Qu(t)\left|\right|=\lambda \left|\right|{\displaystyle {\int }_{-\infty }^{t}T}(t-s)\alpha (s,u(s))\text{d}s\left|\right|\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\le \lambda {\displaystyle {\int }_{-\infty }^t|}|T(t-s)\alpha (s,u(s))||\text{d}s\le \lambda {\displaystyle {\int }_{-\infty }^t|}|T(t-s)\left|\right|(p(s)|\left|u\right||+||\alpha (s,0)||)\text{d}s\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\le \lambda (MC+D){\displaystyle {\int }_{-\infty }^{t}L}{e}^{-\omega (t-s)}\text{d}s=\frac{\lambda L(MC+D)}{\omega }<+\infty \end{array}$$

Hence, $Q(\Omega )$ is bounded.

Let $u\in \Omega , -\infty <t_1<t_2<\infty , \epsilon >0$ and $\rho =\frac{\epsilon }{6L(MC+D)}$ , setting $(Qu)(t_2)-(Qu)(t_1)=I_1+I_2+I_3$ , where

$$\begin{array}{c}{I}_1=\lambda {\displaystyle {\int }_{-\infty }^{t_1-\rho }(T(}{t}_2-s)-T(t_1-s))\alpha (s,u(s))\text{d}s\\ I_2=\lambda {\displaystyle {\int }_{t_1-\rho }^{t_1}(T(}{t}_2-s)-T(t_1-s))\alpha (s,u(s))\text{d}s\end{array}$$

and

$$\begin{array}{l}{I}_3=\lambda {\displaystyle {\int }_{t_1}^{t_2}T}(t_2-s)\alpha (s,u(s))\text{d}s\hfill \end{array}$$

Since $\{T(t):t\ge 0\}$ is a C₀-semigroup and $\left|\right|T(t)\left|\right|\le L{e}^{-\omega t}$ , there exists $\delta =\delta (\epsilon ,\rho )<\rho ,$ s.t. $t_2-t_1<\delta$ implies $\left|\right|T\left(\frac{t}{2}\right)-T\left(\frac{t}{2}+t_2-t_1\right)\left|\right|\le \frac{\omega \epsilon }{4\lambda L(MC+D)}$ Then, one has

$$\begin{array}{l}\left|\right|I_1\left|\right|=\lambda \Vert {\displaystyle {\int }_{\rho }^{+\infty }(T(}{t}_2-t_1+t)-T(t))\alpha (t_1-t,u(t_1-t))\text{d}t\Vert =\lambda \Vert {\displaystyle {\int }_{\rho }^{+\infty }T}\left(\frac{t}{2}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \left(T\left(t_2-t_1+\frac{t}{2}\right)-T\left(\frac{t}{2}\right)\right)\alpha \left(t_1-t,u(t_1-t)\right)dt\Vert \le \lambda (MC+D)\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {\displaystyle {\int }_{\rho }^{+\infty }L}{e}^{-\omega t/2}\Vert T\left(t_2-t_1+\frac{t}{2}\right)-T\left(\frac{t}{2}\right)\Vert \text{d}t\\ \hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{0.5em}\le \lambda (MC+D){\displaystyle {\int }_{\rho }^{+\infty }\frac{\omega \epsilon L{e}^{-\omega t/2}}{4\lambda L(MC+D)}}\text{d}t\le \frac{\epsilon }{2}.\end{array}$$

By a direct calculation, we have

$$\left|\right|I_2\left|\right|\le \lambda (MC+D){\displaystyle {\int }_{t_1-\rho }^{t_1}2}L\text{d}s=2\lambda L(MC+D)·\rho =\frac{\epsilon }{3}$$

and

$$\left|\right|I_3\left|\right|\le \lambda (MC+D){\displaystyle {\int }_{t_1}^{t_2}L}{e}^{-\omega (t_2-s)}\text{d}s=\lambda L(MC+D)·\rho =\frac{\epsilon }{6}.$$

Then, for $x\in \Omega$ and $t_2-t_1<\delta$

$$\left|\right|(Qx)(t_2)-(Qx)(t_1)\left|\right|\le \left|\right|I_1\left|\right|+\left|\right|I_2\left|\right|+\left|\right|I_3\left|\right|\le \frac{\epsilon }{2}+\frac{\epsilon }{3}+\frac{\epsilon }{6}=\epsilon .$$

That is to say, $Q(\Omega )$ is equicontinuity.

By the Arzela-Ascoli theorem, we have

$$Q:WPAP(W,\theta )\to WPAP(W,\theta )$$

is completely continuous. The proof is complete.

Theorem 7 Suppose that (B) holds. Then there exists a constant ${\lambda }^{*}>0$ such that for any $0<\lambda \le {\lambda }^{*}$ , equation (6) has at least one nontrivial WPAP solution.

Proof. It is well-known that problem (6) has a solution $u=u(t)$ if and only if u solves the operator equation

$$u(t)=(Qu)(t):=\lambda {\displaystyle {\int }_{-\infty }^{t}T}(t-s)\alpha (s,u(s))\text{d}s$$

in W. So we need to seek a fixed point of Q in $WPAP(W,\theta )$ . By Lemma 6, the operator $Q:WPAP(W,\theta )\to WPAP(W,\theta )$ is a completely continuous operator.

Since $\left|\right|\alpha (t,u)-\alpha (t,0)\left|\right|\le p(t)\left|\right|u-0\left|\right|,a.e.t\in R$ , we know $\left|\right|\alpha (t,u)\left|\right|\le p(t)\left|\right|u\left|\right|+\left|\right|\alpha (t,0)\left|\right|$ . Let

$$m=\frac{D}{M},\Omega =\{u\in C[R]:|\left|u\right||<m\}.$$

Suppose $u\in \partial \Omega ,\mu >1$ such that $Qu=\mu u$ . Then

$$\mu m=\mu \left|\right|u\left|\right|=\left|\right|Qu\left|\right|.$$

Since

$$\begin{array}{l}\left|\right|Qu(t)\left|\right|=\lambda \left|\right|{\displaystyle {\int }_{-\infty }^{t}T}(t-s)\alpha (s,u(s))\text{d}s\left|\right|\le \lambda {\displaystyle {\int }_{-\infty }^t|}|T(t-s)\alpha (s,u(s))||\text{d}s\\ \hspace{1em}\hspace{1em}\hspace{1em} \le \lambda {\displaystyle {\int }_{-\infty }^t|}|T(t-s)|\left|\right||\alpha (s,u(s))||\text{d}s\le \lambda {\displaystyle {\int }_{-\infty }^t|}|T(t-s)\left|\right|(p(s)|\left|u\right||+||\alpha (s,0)||)\text{d}s\\ \hspace{1em}\hspace{1em}\hspace{1em} \le \lambda L\left|\right|u\left|\right|{\displaystyle {\int }_{-\infty }^t{e}^{-\omega (t-s)}}p(s)\text{d}s+\lambda L{\displaystyle {\int }_{-\infty }^t{e}^{-\omega (t-s)}}\left|\right|\alpha (s,0)\left|\right|\text{d}s\le \frac{\lambda LM}{\omega }\left|\right|u\left|\right|+\frac{\lambda DL}{\omega }\end{array}$$

Choose ${\lambda }^{*}=\frac{\omega }{2LM}.$ Then when $0<\lambda \le {\lambda }^{*}$ , we have

$$\mu \left|\right|u\left|\right|\le \frac{1}{2}\left|\right|u\left|\right|+\frac{D}{2M}.$$

Consequently

$$\mu \le \frac{1}{2}+\frac{D}{2mM}=1$$

This contradicts $\mu >1$ . By Lemma 2, Q has a fixed point $u^{*}\in \overline{\Omega }$ . Then when $0<\lambda \le {\lambda }^{*}$ , equation (6) has a nontrivial WPAP solution. ◊

Example 8 Consider the following nonlinear oscillator system which generalizes the one in He et al. ²⁸

$$\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\prime =\lambda \left[\begin{array}{cc}5& 3\\ -4& -2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{\sqrt{1-t}}\left(x_1(t)-\cos x_1(t)\right)\end{array}\right]$$

with the initial value $x_1(0)=1, x_2(0)=0.$ If we choose $p(t)=\frac{1}{\sqrt{1-t}}$ , by a simple computation, we have

$$\left|\right|T(t)\left|\right|\le e^{-t},{\lambda }^{*}=\frac{\omega }{2LM}=\frac{1}{2}$$

Then, by Theorem 7, equation (6) has one nontrivial WPAP solution for any $\lambda \in (0,\frac{1}{2}]$ .

Theorem 9 Suppose that $\alpha (t,0)\cancel{\equiv }0$ , and

$$0\le H=\underset{\left|u\right|\to +\infty }{\lim \sup }\underset{t\in R}{\max }\frac{|\alpha (t,u)|}{\left|u\right|}<+\infty$$

(7)

Then, there exists a constant ${\lambda }^{*}>0$ such that for any $0<\lambda \le {\lambda }^{*}$ , equation (6) has at least one nontrivial WPAP solution.

Proof. Let $\epsilon >0$ such that $H+1-\epsilon >0$ . By (7), there exists $\varsigma >0$ such that

$$|\alpha (t,u)|\le (H+1-\epsilon )\left|u\right|,\left|u\right|\ge \varsigma ,t\in R$$

Let $\mathrm{ϵ}={\max }_{t\in R,\left|u\right|\le \varsigma }|\alpha (t,u)|$ . Then for any $(t,u)\in R\times R$ , we have

$$|\alpha (t,u)|\le (H+1-\epsilon )\left|u\right|+\mathrm{ϵ}.$$

From Theorem 7 we know that equation (6) has at least one nontrivial WPAP solution. ◊

In this end, we consider the exponential stability of the solution for the following system with nonlinearity

$$\begin{array}{l}x\prime (t)=Ax(t)+\lambda \alpha (t,x(t)),\hfill \\ x(0)=x_0\in R^n,t\ge 0\hfill \end{array}$$

(8)

If there exist constants $m\ge 1$ and $\eta >0$ such that any solution x(t) of equation (8) satisfies the inequality

$$\text{E}\left|\right|x(t)\left|\right|\le m\mathrm{\hspace{0.17em}exp}[-\eta (t-t_0)]\left|\right|x_0\left|\right|$$

for any $0\le t_0\le t$ . In this paper, the infimum of $\mu =-\eta$ for which the above inequality exists is called the Lyapunov exponent.

As is well known, the solution x(t) of equation (8) can be written as

$$x(t)=T(t-t_0)x_0+\lambda {\displaystyle {\int }_{t_0}^{t}T}(t-s)\alpha (s,x(s))\text{d}s$$

(9)

where the exponentially stable C₀-semigroup T(t) is generated by operator A. Suppose that there exist positive constants m and β such that

$$\left|\right|T(t-s)\left|\right|\le m\mathrm{\hspace{0.17em}exp}\left[-\beta (t-s)\right]$$

by Theorem 7, we have the following theorem.

Theorem 10 Suppose that $\left|\right|\alpha (t,x)\left|\right|\le l\left|\right|x\left|\right|,$ then there exists a constant ${\lambda }^{*}>0$ such that for any $0<\lambda \le {\lambda }^{*}$ , equation (8) has at least one nontrivial WPAP solution. Moreover, the general exponent μ of (8) satisfies the inequality,

$$\mu \le \lambda lm-\beta$$

(10)

Thus, for such $m,\beta ,$ equation (8) is exponentially stable $(\text{i}\text{.e}. \mu <0)$ if

$$l<l_1=\frac{\beta }{\lambda m}$$

Proof. From

$$x(t)=T(t-t_0)x_0+\lambda {\displaystyle {\int }_{t_0}^{t}T}(t-s)\alpha (s,x(s))\text{d}s$$

and

$$T(t-s)\le m\mathrm{\hspace{0.17em}exp}[-\beta (t-s)]$$

we have

$$\begin{array}{l}\left|\right|x(t)\left|\right|\le m\mathrm{\hspace{0.17em}exp}[-\beta (t-t_0)]\left|\right|x_0\left|\right|+\lambda m{\displaystyle {\int }_{t_0}^t\mathrm{\hspace{0.17em}exp}}[-\beta (t-s)]\left|\right|\alpha (s,x(s))\left|\right|\text{d}s\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le m\mathrm{\hspace{0.17em}exp}[-\beta (t-t_0)]\left|\right|x_0\left|\right|+\lambda m{\displaystyle {\int }_{t_0}^t\mathrm{\hspace{0.17em}exp}}[-\beta (t-s)]l\left|\right|x(s)\left|\right|\text{d}s\hfill \end{array}$$

(11)

The substitution

$$x(t)=\mathrm{\hspace{0.17em}exp}(-\beta t)y(t)$$

(12)

reduces (11) to the form

$$\left|\right|y(t)\left|\right|\le C+\lambda ml{\displaystyle {\int }_{t_0}^t|}|y(s)||\text{d}s$$

(13)

where

$$C=m\mathrm{\hspace{0.17em}exp}(\beta t_0)\left|\right|x(t_0)\left|\right|.$$

Thus, it follows from the known Gronwall inequality that

$$\left|\right|y(t)\left|\right|\le C\mathrm{\hspace{0.17em}exp}[\lambda ml(t-t_0)]\left|\right|y_0\left|\right|$$

which along with (12) imply that

$$\mathbb{E}|x(t)||\le m\mathrm{\hspace{0.17em}exp}[(\lambda ml-\beta )(t-t_0)]\mathbb{E}|\left|x_0\right||$$

Then, (10) is obtained and the proof is complete.

Remark 11 Noting that m and β become very more complex when $A=A(t)$ , and the technique here does not work.

Remark 12. Even slight nonlinearities in vibrating dynamics can cause instability bands and unstable amplitudes. By the works of Peleg and Hing, ²⁴ in many vibration dynamics or vibration problems, it is vital to know precisely the bounds of the instability/stability frequencies and the associated amplitude rang. In this paper, we calculated the precise Lyapunov exponents of vibrating dynamics with bounded or unbounded nonlinearities. Thus, our results can be used for vibration dynamics or vibration problems.

Conclusion

In this paper, we considered some WPAP solutions for vibrating dynamics. Firstly, we obtained some sufficient conditions for at least one nontrivial WPAP solution. Secondly, we studied the exponential stability and the precise bound of Lyapunov exponents for the systems with bounded or unbounded nonlinearities.

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.