Existence of solutions for damped elastic systems in Banach spaces

Abstract

In this article, we study the existence of mild solutions for damped elastic systems in Banach spaces. The discussion is based on the operator semigroup theory and fixed point theorem. In addition, two examples are given to illustrate our results.

1 Introduction

In this article, we study the existence of mild solutions for the semilinear damped elastic systems in Banach spaces E:

$$ \textstyle\begin{cases} u''(t)+\rho Bu'(t)+Au(t)=f(t,u(t)), \quad 0< t< a, \\ u(0)=u_{0}\in D(A), \qquad u'(0)=u_{1}\in E, \end{cases} $$

(1.1)

where $A:D(A)\subset E\to E$ and $B:D(B)\subset E\rightarrow E$ are densely defined closed (possibly unbounded) linear operators on a complex Banach space E and $f\in C([0,a]\times E,E)$.

In 1982, Chen and Russell [1] investigated the following linear elastic system described by the second order equation:

$$ \textstyle\begin{cases} u''(t)+Bu'(t)+Au(t)=0,\quad t>0, \\ u(0)=x_{0}, \qquad u'(0)=y_{0}, \end{cases} $$

(1.2)

in a Hilbert space H with inner $(\cdot ,\cdot )$, where A (the elastic operator) and B (the damping operator) are positive definite self-adjoint operators in H. They reduced (1.2) to the first order equation in $H\times H$:

$$\frac{d}{dt}\left(\begin{array}{c}{A}^{\frac{1}{2}}u\\ u^{\prime }\end{array}\right)=\left(\begin{array}{cc}0& A^{\frac{1}{2}}\\ -A^{\frac{1}{2}}& -B\end{array}\right)\left(\begin{array}{c}{A}^{\frac{1}{2}}u\\ u^{\prime }\end{array}\right).$$

Let $V=D(A^{\frac{1}{2}})$, $\mathscr{H}=V\times H$ with the naturally induced inner products. Then (1.2) is equivalent to the first order equation in $\mathscr{H}$

$$\frac{d}{dt}\left(\begin{array}{c}{A}^{\frac{1}{2}}u\\ u^{\prime }\end{array}\right)={\mathcal{A}}_B\left(\begin{array}{c}{A}^{\frac{1}{2}}u\\ u^{\prime }\end{array}\right),$$

where

$$\begin{array}{c}{\mathcal{A}}_B=\left(\begin{array}{cc}0& I\\ -A& -B\end{array}\right),\hfill \\ D({\mathcal{A}}_B)=D(A)\times [D\left(A^{\frac{1}{2}}\right)\cap D(B)].\hfill \end{array}$$

Chen and Russell [1] conjectured that $\mathscr{A}_{B}$ is the infinitesimal generator of an analytic semigroup on $\mathscr{H}$ if

$$ D \bigl(A^{\frac{1}{2}} \bigr)\subset D(B) $$

and either of the following two inequalities holds for some $\beta _{1},\beta _{2}>0$:

$$\begin{aligned}& \beta _{1} \bigl(A^{\frac{1}{2}}v,v \bigr)\leq (Bv,v)\leq \beta _{2} \bigl(A^{\frac{1}{2}}v,v \bigr), \quad v \in D \bigl(A^{\frac{1}{2}} \bigr); \\& \beta _{1}(Av,v)\leq \bigl(B^{2}v,v \bigr)\leq \beta _{2}(Av,v), \quad v\in D(A). \end{aligned}$$

The complete proofs of the two conjectures were given by Huang [2, 3]. Then, other sufficient conditions for $\mathscr{A}_{B}$ or its closure $\overline{\mathscr{A}_{B}}$ to generate an analytic or differentiable semigroup on $\mathscr{H}$ were discussed in [4,5,6,7,8,9,10], by choosing B to be an operator comparable with $A^{\alpha }$ for $0 <\alpha \leq 1$, based on an explicit matrix representation of the resolvent operator of $\mathscr{A}_{B}$ or $\overline{\mathscr{A}_{B}}$.

In [11], Fan, Li, and Chen studied the existence of mild solutions for the elastic system with structural damping in Banach spaces:

$$ \textstyle\begin{cases} u''(t)+\rho Au'(t)+A^{2}u(t)=f(t,u(t)), \quad 0< t< a, \\ u(0)=u_{0}\in D(A), \qquad u'(0)=u_{1}\in E, \end{cases} $$

(1.3)

where the damping constant $\rho \geq 2$ and the nonlinearity term f is Lipschitzian in the second variable. In [12], Fan and Li studied the asymptotic stability of solutions and the analyticity and exponential stability of associated semigroups for the following second order semilinear evolution equations:

$$ \textstyle\begin{cases} u''(t)+\rho Au'(t)+A^{2}u(t)=0, \quad t>0, \\ u(0)=u_{0}\in D(A), \qquad u'(0)=u_{1}\in E, \end{cases} $$

(1.4)

where $A:D( A)\subset E\rightarrow E$ is a sectorial linear operator on a complex Banach space E and $\rho >2\cos \alpha$ for a fixed value $\alpha \in (0,\frac{\pi }{2})$.

In [13], Fan and Gao discussed the asymptotic behavior of solutions for the linear elastic system with structural damping:

$$ \textstyle\begin{cases} u''(t)+\rho Au'(t)+A^{2}u(t)=h(t), \quad t>0, \\ u(0)=u_{0}\in D(A), \qquad u'(0)=u_{1}\in E, \end{cases} $$

(1.5)

and the semilinear elastic system with structural damping (1.3) in Banach spaces, where $\rho >2\cos \alpha$, for a fixed value $\alpha \in (0,\frac{\pi }{2})$, A is a sectorial operator, −A generates an analytic and exponentially stable semigroup on E, $h:[0,+\infty )\rightarrow E$ is continuous, and f is Lipschitz continuous in the second variable. The discussion is based on the operator semigroup theory and some fixed point theorem.

In [14], Diagana studied the well-posedness and existence of bounded solutions to the linear elastic systems with damping:

$$ \textstyle\begin{cases} u''(t)+\rho Bu'(t)+Au(t)=f(t), \quad t>0, \\ u(0)=u_{0}\in D(A), \qquad u'(0)=u_{1}\in E, \end{cases} $$

where $A:D(A)\subset E\to E$ and $B:D(B)\subset E\rightarrow E$ are densely defined closed (possibly unbounded) linear operators on a complex Banach space E and $f:\mathbb{R}_{+}\rightarrow E$ is a continuous function.

In [15], the authors considered nonlinear evolution equations of second order in Banach spaces:

$$ \textstyle\begin{cases} u''(t)+\rho Au'(t)+A^{2}u(t)=f(t,u(t),u_{t}), \quad t\in I=[0,T], \\ u(s)=\varphi (s), \quad s\leq 0, \\ u'(0)+h(u)=\psi , \end{cases} $$

where u is the unknown function defined on I and, taking values in E, $u_{t}$ is the history state defined by $u_{t}:(-\infty ,0] \rightarrow E$, $u_{t}(s)=u(t+s)$, $t\in I$. By means of the fixed point for condensing maps, they proved the existence and exponential decay of mild solutions.

In [16], the authors studied a class of elastic systems with structural damping:

$$ \textstyle\begin{cases} u''(t)+\rho Au'(t)+A^{2}u(t)=f(t,u(t)), \quad t>0, \\ u(0)+g(u)=x_{0}, \qquad u'(0)+h(u)=y_{0}, \end{cases} $$

where $A:D(A)\subset E\rightarrow E$ is a closed linear operator, $\rho \geq 2$ is a given constant, $x_{0}\in D(A)$, $y_{0}\in E$. By using the measure of noncompactness on the space of continuous functions on the half line, they established the existence of mild solutions with explicit decay rate of exponential type. Although the above discussions have been an interesting subject, the theory still remains to be developed to nonlinear case.

However, motivated by the above works, ideas and methods based on paper [14], in this paper, we investigate the existence of mild solutions for the elastic system (1.1) in complex Banach spaces. We give the expression of the solution of Problem (1.1), which is different from the expression given in article [14]. Our results presented in this paper improve and generalize many classical results [11,12,13,14, 17,18,19,20].

The paper is organized as follows. In Sect. 2, we introduce some notations and recall some basic known results. In Sect. 3 we present the existence of mild solutions to Problem (1.1) in a complex Banach space. In Sect. 4, we give an example to illustrate our results.

2 Preliminaries

Let E be a complex Banach space with the norm $\|\cdot \|$. For any constant $a>0$, denote $J=[0,a]$. Let $C(J,E)$ be the Banach space of all continuous functions from J into E endowed with the supremum norm $\|u\|_{C}=\sup_{t\in J}\|u(t)\|$ for every $u\in C(J,E)$. Let $\mathcal{L}(E)$ be the Banach space of all linear and bounded operators on E. The notations $D(L)$ and $\rho (L)$ stand respectively for the domain and resolvent of L, set $R(\lambda , L)=(\lambda I-L)^{-1}$ for all $\lambda \in \rho (L)$.

Definition 2.1

A $C_{0}$-semigroup $T(t)$ ($t \geq 0$) in E is said to be equicontinuous if $T(t)$ is continuous by the operator norm for every $t > 0$.

Now we introduce some basic definitions and properties about Kuratowski measure of noncompactness that will be used in sequel.

Definition 2.2

([21, 22])

The Kuratowski measure of noncompactness $\alpha (\cdot )$ defined on the bounded set S of a Banach space E is

$$ \alpha (S):=\inf \Biggl\{ \delta >0:S=\bigcup_{i=1}^{m}S_{i} \text{ with } \operatorname{diam}(S_{i})\leq \delta \text{ for }i=1,2, \dots ,m \Biggr\} . $$

In this paper, we denote by $\alpha (\cdot )$, $\alpha _{C}(\cdot )$ and the Kuratowski measure of noncompactness on the bounded set of E, $C(J,E)$, respectively. For any $D\subset C(J,E)$ and $t\in J$, set $D(t)=\{u(t)\mid u\in D\}$, then $D(t)\subset E$. If $D\subset C(J,E)$ is bounded, then $D(t)$ is bounded in E and $\alpha (D(t))\leq \alpha _{C}(D)$. For more details about the properties of the Kuratowski measure of noncompactness, we refer to the monographs [21, 22].

Definition 2.3

([22])

Let E be a Banach space, and let S be a nonempty subset of E. A continuous mapping $Q:S\to E$ is called k-set-contractive if there exists a constant $k\in [0,1)$ such that, for every bounded set $\varOmega \subset S$,

$$ \alpha \bigl(Q(\varOmega ) \bigr)\leq k\alpha (\varOmega ). $$

Lemma 2.1

([22])

Let E be a Banach space. Assume that $\varOmega \subset E$ is a bounded closed and convex set on E, the operator $Q:\varOmega \to \varOmega$ is k-set-contractive. Then Q has at least one fixed point in Ω.

Lemma 2.2

([23])

Let E be a Banach space, and let $D\subset E$ be bounded. Then there exists a countable set $D_{0}\subset D$ such that $\alpha (D)\leq 2 \alpha (D_{0})$.

Lemma 2.3

([24])

Let E be a Banach space, and let $D=\{u_{n}\}\subset C([b_{1},b_{2}],E)$ be a bounded and countable set for constants $-\infty < b_{1}< b_{2}<+\infty$. Then $\alpha (D(t))$ is Lebesgue integral on $[b_{1},b_{2}]$, and

$$ \alpha \biggl( \biggl\{ \int _{b_{1}}^{b_{2}} u_{n}(t)\,dt: n\in \mathbb{N} \biggr\} \biggr) \leq 2 \int _{b_{1}}^{b_{2}}\alpha \bigl(D(t) \bigr)\,dt. $$

Lemma 2.4

([21])

Let E be a Banach space, and let $D\subset C([b_{1},b_{2}],E)$ be bounded and equicontinuous. Then $\alpha (D(t))$ is continuous on $[b_{1},b_{2}]$, and

$$ \alpha _{C}(D)=\max_{t\in [b_{1},b_{2}]}\alpha \bigl(D(t) \bigr). $$

Lemma 2.5

([25])

Assume $f\in C(J,E)$ and that A is the infinitesimal generator of $C_{0}$-semigroup $(T(t))_{t\geq 0}$. Then the inhomogeneous Cauchy problem

$$ \textstyle\begin{cases} u'(t)=Au(t)+f(t), \quad t\in J, \\ u(0)=u_{0}\in D(A) \end{cases} $$

(2.1)

has a mild solution u given by

$$ u(t)=T(t)u_{0}+ \int _{0}^{t} T(t-s)f(s)\,ds, \quad t\in J. $$

Thoughts and methods based on paper [13]. We consider the following linear damped elastic system:

$$ \textstyle\begin{cases} u''(t)+\rho Bu'(t)+Au(t)=h(t), \quad t\in J, \\ u(0)=u_{0}\in D(A), \qquad u'(0)=u_{1}\in E, \end{cases} $$

(2.2)

where $A:D(A)\subset E\to E$ and $B:D(B)\subset E\rightarrow E$ are densely defined closed (possibly unbounded) linear operators on a complex Banach space E and $h:J \rightarrow E$.

For the second order evolution equation

$$ u''(t)+\rho Bu'(t)+Au(t)=h(t), $$

(2.3)

it has the following decomposition:

$$ \biggl(\frac{d}{dt}+E_{1}(\rho ) \biggr) \biggl( \frac{d}{dt}+E_{2}(\rho ) \biggr)u=h(t), \quad t>0. $$

(2.4)

That is,

$$ \frac{d^{2}u}{dt^{2}}+ \bigl(E_{1}(\rho )+E_{2}( \rho ) \bigr) \frac{du}{dt}+E_{1}(\rho )E_{2}(\rho )u=h(t). $$

(2.5)

It follows from (2.3) and (2.5) that

$$ E_{1}(\rho )+E_{2}(\rho )=\rho B, \qquad E_{1}( \rho )E_{2}(\rho )=A. $$

(2.6)

By (2.6), we have

• (i) if $C(\rho )=\rho ^{2}B^{2}-4A=L^{2}(\rho )>0$, then

  $$ \begin{gathered} E_{1}(\rho )=\frac{\rho B-\sqrt{\rho ^{2}B^{2}-4A}}{2}= \frac{\rho B-L( \rho )}{2}, \\ E_{2}(\rho )=\frac{\rho B+\sqrt{\rho ^{2}B^{2}-4A}}{2}=\frac{\rho B+L( \rho )}{2}; \end{gathered} $$

  (2.7)

• (ii) if $C(\rho )=\rho ^{2}B^{2}-4A=L^{2}(\rho )=0$, then

  $$ E_{1}(\rho )=E_{2}(\rho )=\frac{\rho B}{2}; $$

• (iii) if $C(\rho )=\rho ^{2}B^{2}-4A=-L^{2}(\rho )<0$, then

  $$ \begin{gathered} E_{1}(\rho )=\frac{\rho B-\sqrt{\rho ^{2}B^{2}-4A}}{2}= \frac{\rho B-iL( \rho )}{2}, \\ E_{2}(\rho )=\frac{\rho B+\sqrt{\rho ^{2}B^{2}-4A}}{2}=\frac{\rho B+iL( \rho )}{2}. \end{gathered} $$

  (2.8)

Remark 2.1

In order to study the existence to Eq. (1.1), we will make use of the above linear operator which links both A and B: $C(\rho )=\rho ^{2}B^{2}-4A=L^{2}(\rho )$ with $D(C(\rho ))=D(B^{2})\cap D(A)$. In the following discussion, we will focus on the following cases: $C(\rho )=L^{2}(\rho )>0$ and $C(\rho )=L^{2}(\rho )=0$ for densely closed linear operator $L(\rho ):D(L(\rho ))\subset E\rightarrow E$. Obviously, $C(\rho )=0$ corresponds to the case studied in papers [11, 12]. For more details, see [14].

Lemma 2.6

Assume that there exists a densely defined closed linear operator $L(\rho ):D(L(\rho ))\subset E\rightarrow E$ such that $u_{0}\in D(L( \rho ))\cap D(B)$ and $C(\rho )=\rho ^{2}B^{2}-4A=L^{2}(\rho )$ and $BL(\rho )=L(\rho )B$. Let $h\in C(J,E)$, $-E_{1}(\rho )$ and $-E_{2}(\rho )$ be respectively the infinitesimal generators of $C_{0}$-semigroups $T_{1}(t)$ ($t\geq 0$) and $T_{2}(t)$ ($t\geq 0$). Then Problem (2.2) has a unique solution given by

$$\begin{aligned} u(t)&= T_{2}(t)u_{0}+ \int _{0}^{t}T_{2}(t-s)T_{1}(s) \bigl(u_{1}+E_{2}(\rho )u _{0} \bigr)\,ds \\ &\quad{}+ \int _{0}^{t} \int _{0}^{s}T_{2}(t-s)T_{1}(s- \tau )h(\tau )\,d\tau \,ds, \end{aligned}$$

where $E_{1}(\rho )$, $E_{2}(\rho )$ were defined in (2.7).

Proof

Let

$$ \frac{du}{dt}+E_{2}(\rho )u=v(t), \quad t\in J, $$

which means

$$ v_{0}:=v(0)=u_{1}+E_{2}(\rho )u_{0}. $$

So we reduce the linear elastic system (2.2) to the following two abstract Cauchy problems in a Banach space E:

$$ \textstyle\begin{cases} \frac{dv}{dt}+E_{1}(\rho )v=h(t), \quad t\in J, \\ v(0)=v_{0}, \end{cases} $$

(2.9)

and

$$ \textstyle\begin{cases} \frac{du}{dt}+E_{2}(\rho )u=v(t), \quad t\in J, \\ u(0)=u_{0}. \end{cases} $$

(2.10)

It is clear that (2.9) and (2.10) are linear inhomogeneous initial value problems for $-E_{1}(\rho )$ and $-E_{2}(\rho )$, respectively. Thus, by operator semigroups theory [11], $-E_{1}(\rho )$ and $-E_{2}(\rho )$ are infinitesimal generators of $C_{0}$-semigroups, which implies that initial value problems (2.9) and (2.10) are well posed.

Thus using Lemma 2.6, if $h\in C(J,E)$, Problem (2.9) has a mild solution v given by

$$ v(t)=T_{1}(t)v_{0}+ \int _{0}^{t}T_{1}(t-s)h(s)\,ds. $$

(2.11)

Similarly, if $v\in C(J,E)$, then the mild solution of Problem (2.10) is expressed by

$$ u(t)=T_{2}(t)u_{0}+ \int _{0}^{t}T_{2}(t-s)v(s)\,ds. $$

(2.12)

Substituting (2.11) into (2.12), we get

$$\begin{aligned} u(t)&= T_{2}(t)u_{0}+ \int _{0}^{t}T_{2}(t-s)T_{1}(s) \bigl(u_{1}+E_{2}(\rho )u _{0} \bigr)\,ds \\ &\quad {}+ \int _{0}^{t} \int _{0}^{s}T_{2}(t-s)T_{1}(s- \tau )h(\tau )\,d\tau \,ds. \end{aligned}$$

 □

Throughout this paper, we assume that $-E_{1}(\rho )$ and $-E_{2}( \rho )$ generate $C_{0}$-semigroups $T_{1}(t)$ ($t\geq 0$) and $T_{2}(t)$ ($t\geq 0$) on E, respectively.

Based on the above discussion, motivated by the definition of mild solutions in [13], we give the definition of a mild solution of Problem (1.1) as follows.

Definition 2.4

Let $f\in C(J\times E,E)$, $-E_{1}(\rho )$ and $-E_{2}(\rho )$ be respectively the infinitesimal generators of $C_{0}$-semigroups $T_{1}(t)$ ($t\geq 0$) and $T_{2}(t)$ ($t\geq 0$). A function $u:J\rightarrow E$ is said to be a mild solution of Problem (1.1) if $u(0)=u_{0}$ and

$$\begin{aligned} u(t)&= T_{2}(t)u_{0}+ \int _{0}^{t}T_{2}(t-s)T_{1}(s) \bigl(u_{1}+E_{2}(\rho )u _{0} \bigr)\,ds \\ &\quad{}+ \int _{0}^{t} \int _{0}^{s}T_{2}(t-s)T_{1}(s- \tau )f \bigl(\tau ,u(\tau ) \bigr)\,d \tau \,ds, \end{aligned}$$

where $E_{1}(\rho )$, $E_{2}(\rho )$ were defined in (2.7).

3 Main results

To obtain the existence of a mild solution for Problem (1.1), we introduce the following hypotheses:

• (H1) Assume that there exists a densely defined closed linear operator $L(\rho ):D(L(\rho ))\subset E\rightarrow E$ such that $u_{0}\in D(L(\rho ))\cap D(B)$ and

  $$ C(\rho )=\rho ^{2}B^{2}-4A=L^{2}(\rho ), \qquad BL(\rho )=L(\rho )B. $$

• (H2) The nonlinear function $f:J \times E\to E$ is continuous, there exist a Lebesgue integrable function $\varphi \in L^{\infty }(J, \mathbb{R}^{+})$ and a nondecreasing continuous function $\varPsi :[0,+ \infty )\to (0,+\infty )$ such that

  $$ \bigl\Vert f(t,u) \bigr\Vert \leq \varphi (t) \varPsi \bigl( \Vert u \Vert \bigr) $$

  for all $u\in C(J,E)$.

• (H3) There exists a positive constant L such that $L<\frac{1}{4a ^{2} M_{1} M_{2} }$ for any countable set $D\subset E$,

  $$ \alpha \bigl(f(t,D) \bigr)\leq L\alpha (D),\quad t\in J. $$

• (H4) The $C_{0}$-semigroups $T_{2}(t)$ ($t\geq 0$) and $T_{1}(t)$ ($t \geq 0$) are equicontinuous for $t>0$.

Theorem 3.1

Assume that $-E_{1}(\rho )$ and $-E_{2}(\rho )$ are respectively the infinitesimal generators of $C_{0}$-semigroups $T_{1}(t)$ ($t\geq 0$) and $T_{2}(t)$ ($t\geq 0$). If conditions (H1)–(H4) hold and

$$ a^{2}M_{1}M_{2} \Vert \varphi \Vert _{L^{\infty }(J,\mathbb{R}^{+})} \liminf_{n\rightarrow +\infty }\frac{\varPsi (n)}{n}< 1, $$

(3.1)

then, for every $u_{0}\in D(L(\rho ))\cap D(B)$, $u_{1}\in E$, Problem (1.1) has at least one mild solution $u\in C(J,E)$.

Proof

Define the operator $\mathcal{F}:C(J,E)\rightarrow C(J,E)$ by

$$ \begin{aligned}[b] \mathcal{F}u(t)&= T_{2}(t)u_{0}+ \int _{0}^{t}T_{2}(t-s)T_{1}(s) \bigl(u_{1}+E _{2}(\rho )u_{0} \bigr)\,ds \\ &\quad {}+ \int _{0}^{t} \int _{0}^{s}T_{2}(t-s)T_{1}(s- \tau )f \bigl(\tau ,u(\tau ) \bigr)\,d \tau \,ds. \end{aligned} $$

(3.2)

It is easy to see that the operator $\mathcal{F}$ is well defined on $C(J,E)$. From Definition 2.4, one can easily see that the mild solution of Problem (1.1) is equivalent to a fixed point of the operator $\mathcal{F}$ defined by (3.2). Next, we will prove that the operator $\mathcal{F}$ has at least one fixed point.

Let $\varOmega _{R}=\{u\in C(J,E):\|u\|_{C}\leq R\}$, then $\varOmega _{R}$ is a bounded closed and convex set in $C(J,E)$. Note that $T_{1}(t)$ ($t \geq 0$) and $T_{2}(t)$ ($t\geq 0$) are $C_{0}$-semigroups on E, then there exist $M_{1}\geq 1$ and $M_{2}\geq 1$ such that

$$ M_{1}=\sup_{t\in J} \bigl\Vert T_{1}(t) \bigr\Vert _{\mathcal{L}(E)}, \qquad M_{2}=\sup_{t\in J} \bigl\Vert T_{2}(t) \bigr\Vert _{\mathcal{L}(E)}. $$

Firstly, we prove that there exists a constant $R>0$ such that $\mathcal{F}(\varOmega _{R})\subset \varOmega _{R}$. Assume to the contrary that, for each $n\in \mathbb{N}$, there exists a sequence $\{u_{n}\} _{n=1}^{\infty }\subset \varOmega _{n}$ with $\|u_{n}\|_{C}\leq n$ but $\|\mathcal{F}(u_{n})\|_{C}>n$. Then, by (3.2) and assumptions (H1), (H2), we have

$$\begin{aligned} \bigl\Vert (\mathcal{F}u_{n}) (t) \bigr\Vert & \leq M_{2} \Vert u_{0} \Vert +M_{1}M_{2}a \bigl\Vert u_{1}+E _{2}(\rho )u_{0} \bigr\Vert +M_{1}M_{2} \int _{0}^{t} \int _{0}^{s} \bigl\Vert f \bigl(\tau ,u_{n}( \tau ) \bigr) \bigr\Vert \,d\tau \,ds \\ &\leq M_{2} \Vert u_{0} \Vert +M_{1}M_{2}a \bigl\Vert u_{1}+E_{2}(\rho )u_{0} \bigr\Vert +M_{1}M _{2}a^{2}\varPsi \bigl( \Vert u_{n} \Vert \bigr) \Vert \varphi \Vert _{L^{\infty }(J,\mathbb{R}^{+})}. \end{aligned}$$

(3.3)

Hence,

$$ \begin{aligned}[b] 1&< \frac{1}{n} \bigl\Vert (\mathcal{F}u_{n}) (t) \bigr\Vert \\ &\leq \frac{1}{n} \bigl(M_{2} \Vert u _{0} \Vert +M_{1}M_{2}a \bigl\Vert u_{1}+E_{2}( \rho )u_{0} \bigr\Vert \bigr)+M_{1}M_{2}a^{2} \Vert \varphi \Vert _{L^{\infty }(J,\mathbb{R}^{+})}\frac{\varPsi (n)}{n}. \end{aligned} $$

(3.4)

Passing to the limit in (3.4), we get a contradiction. Hence, for some positive number $R, \mathcal{F}(\varOmega _{R})\subset \varOmega _{R}$.

Next, we prove that $\mathcal{F}$ is continuous in $\varOmega _{R}$. To this end, let $u_{n}\in \varOmega _{R}$ be a sequence such that $u_{n}\rightarrow u$ in $\varOmega _{R}$. By the continuity of nonlinear term f with respect to the second variable, for each $s\in J$, we have

$$ f \bigl(s,u_{n}(s) \bigr)\rightarrow f \bigl(s,u(s) \bigr), \quad n \rightarrow \infty , $$

(3.5)

that is, for all $\epsilon >0$, there exists N, when $n>N$, we have

$$ \bigl\Vert f \bigl(s,u_{n}(s) \bigr)-f \bigl(s,u(s) \bigr) \bigr\Vert \leq \epsilon . $$

(3.6)

Now, we have

$$ \begin{aligned} \bigl\Vert (\mathcal{F}u_{n}) (t)-( \mathcal{F}u) (t) \bigr\Vert &\leq M_{1}M_{2} \int _{0} ^{t} \int _{0}^{s} \bigl\Vert f \bigl(\tau ,u_{n}(\tau ) \bigr)-f \bigl(\tau ,u(\tau ) \bigr) \bigr\Vert \,d\tau \,ds \\ &\leq M_{1}M_{2}a^{2} \bigl\Vert f \bigl(\tau ,u_{n}(\tau ) \bigr)-f \bigl(\tau ,u(\tau ) \bigr) \bigr\Vert . \end{aligned} $$

So, when $n>N$, we have

$$ \Vert \mathcal{F}u_{n}-\mathcal{F}u \Vert _{C}\leq M_{1}M_{2}a^{2}\epsilon , $$

which means that $\mathcal{F}$ defined by (3.2) is continuous in $\varOmega _{R}$.

In the following, we demonstrate that the operator $\mathcal{F}:\varOmega _{R}\to \varOmega _{R}$ is equicontinuous. For any $u\in \varOmega _{R}$ and $0< t' < t''\leq a$, we obtain that

$$\begin{aligned} &\bigl\Vert (\mathcal{F}_{2}u) \bigl(t'' \bigr)-(\mathcal{F}_{2}u) \bigl(t' \bigr) \bigr\Vert \\ &\quad \leq \bigl\Vert T_{2} \bigl(t'' \bigr)u _{0}-T_{2} \bigl(t' \bigr)u_{0} \bigr\Vert \\ &\qquad{}+ \biggl\Vert \int _{0}^{t'} \bigl[T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr]T_{1}(s) \bigl(u _{1}+E_{2}(\rho )u_{0} \bigr)\,ds \biggr\Vert \\ &\qquad{}+ \biggl\Vert \int _{t'}^{t''} T_{2} \bigl(t''-s \bigr)T_{1}(s) \bigl(u_{1}+E_{2}(\rho )u _{0} \bigr)\,ds \biggr\Vert \\ &\qquad{}+ \biggl\Vert \int _{0}^{t'} \int _{0}^{s} \bigl[T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr]\times T _{1}(s-\tau )f \bigl(\tau ,u(\tau ) \bigr)\,d\tau \,ds \biggr\Vert \\ &\qquad{}+ \biggl\Vert \int _{t'}^{t''} \int _{0}^{s}T_{2} \bigl(t''-s \bigr)\times T_{1}(s- \tau )f \bigl(\tau ,u(\tau ) \bigr)\,d\tau \,ds \biggr\Vert \\ &\quad := I_{1}+I_{2}+I_{3}+I_{4}+I_{5}, \end{aligned}$$

where

$$\begin{aligned}& I_{1}= \bigl\Vert T_{2} \bigl(t'' \bigr)u_{0}-T_{2} \bigl(t' \bigr)u_{0} \bigr\Vert , \\& I_{2}= \biggl\Vert \int _{0}^{t'} \bigl[T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr]T_{1}(s) \bigl(u _{1}+E_{2}(\rho )u_{0} \bigr)\,ds \biggr\Vert , \\& I_{3}= \biggl\Vert \int _{t'}^{t''} T_{2} \bigl(t''-s \bigr)T_{1}(s) \bigl(u_{1}+E_{2}(\rho )u _{0} \bigr)\,ds \biggr\Vert , \\& I_{4}= \biggl\Vert \int _{0}^{t'} \int _{0}^{s} \bigl[T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr] \times T_{1}(s-\tau )f \bigl(\tau ,u(\tau ) \bigr)\,d\tau \,ds \biggr\Vert \\& I_{5}= \biggl\Vert \int _{t'}^{t''} \int _{0}^{s}T_{2} \bigl(t''-s \bigr)\times T_{1}(s- \tau )f \bigl(\tau ,u(\tau ) \bigr)\,d\tau \,ds \biggr\Vert . \end{aligned}$$

In fact, we only need to check whether $I_{1}$, $I_{2}$, $I_{3}$, $I_{4}$, and $I_{5}$ tend to 0 independently of $u\in \varOmega _{R}$ when $t''-t'\to 0$.

Note that the function $T_{2}(t)u_{0}$ is continuous for $t\geq 0$. Thus, $T_{2}(t)u_{0}$ is uniformly continuous on J and thus $\lim_{t''\rightarrow t'}I_{1}=0$.

For $I_{2}$, we have

$$\begin{aligned} I_{2} &\leq \int _{0}^{t'} \bigl\Vert T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr\Vert _{ \mathcal{L}(E)}\times \bigl\Vert T_{1}(s) \bigr\Vert _{\mathcal{L}(E)} \bigl\Vert u_{1}+E_{2}( \rho )u_{0} \bigr\Vert \,ds \\ &\leq M_{1} \bigl\Vert u_{1}+E_{2}(\rho )u_{0} \bigr\Vert \int _{0}^{t'} \bigl\Vert T_{2} \bigl(t''-s \bigr)-T _{2} \bigl(t'-s \bigr) \bigr\Vert _{\mathcal{L}(E)}\,ds. \end{aligned}$$

Therefore the continuity of the functions $t\mapsto \|T_{1}(t)\|$ and $t\mapsto \|T_{2}(t)\|$ for $t\in J$ allows us to conclude that $\lim_{t''\rightarrow t'}I_{2}=0$.

For $I_{4}$, we have

$$\begin{aligned} I_{4} &\leq \int _{0}^{t'} \int _{0}^{s} \bigl\Vert T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr\Vert _{\mathcal{L}(E)}\times \bigl\Vert T_{1}(s-\tau ) \bigr\Vert _{\mathcal{L}(E)} \bigl\Vert f \bigl( \tau ,u(\tau ) \bigr) \bigr\Vert \,d\tau \,ds \\ &\leq M_{1}a\varPsi (R) \Vert u \Vert _{L^{\infty }(J,E)}\times \int _{0}^{t'} \bigl\Vert T_{2} \bigl(t''-s \bigr)-T_{2} \bigl(t'-s \bigr) \bigr\Vert _{\mathcal{L}(E)}\,ds. \end{aligned}$$

Consequently, $\lim_{t''\rightarrow t'}I_{4}=0$.

For $I_{3}$, $I_{5}$, we ha

$$\begin{aligned}& I_{3}\leq M_{1}M_{2} \bigl\Vert u_{1}+E_{2}(\rho )u_{0} \bigr\Vert \cdot \bigl\vert t''-t' \bigr\vert , \\& I_{5}\leq M_{1}M_{2}\varPsi (R) \Vert u \Vert _{L^{\infty }(J,E)} \bigl\vert t''-t' \bigr\vert . \end{aligned}$$

Hence, $\lim_{t''\rightarrow t'}I_{3}=\lim_{t''\rightarrow t'}I_{5}=0$.

As a result, $\|(\mathcal{F}u)(t'')-(\mathcal{F}u)(t')\|$ tends to 0 independently of $u\in \varOmega _{R}$ as $t''-t'\to 0$, which means that $\mathcal{F}:\varOmega _{R}\to \varOmega _{R}$ is equicontinuous.

Now, we show that the operator $\mathcal{F}$ is k-set-contractive. For any bounded $D\subset \varOmega _{R}$, by Lemma 2.2, we know that there exists a countable set $D_{0}=\{u_{n}\}\subset D$ such that

$$ \alpha \bigl(\mathcal{F}(D) \bigr)_{C}\leq 2\alpha \bigl( \mathcal{F}(D_{0}) \bigr)_{C}. $$

(3.7)

Since $\mathcal{F}(D_{0})\subset \mathcal{F}(\varOmega _{R})$ is bounded and equicontinuous, we know from Lemma 2.4 that

$$ \alpha \bigl(\mathcal{F}(D_{0}) \bigr)_{C}=\max _{t\in J} \alpha \bigl(\mathcal{F}(D _{0}) (t) \bigr). $$

(3.8)

For every $t\in J$, by Lemma 2.3, assumption (H3), and (3.2), we have

$$ \begin{aligned}[b] \alpha \bigl(\mathcal{F}(D_{0}) (t) \bigr) &= \alpha \biggl( \biggl\{ T_{2}(t)u_{0}+ \int _{0}^{t}T_{2}(t-s)T_{1}(s) \bigl(u_{1}+E_{2}(\rho )u_{0} \bigr)\,ds \\ &\quad {}+ \int _{0}^{t} \int _{0}^{s}T_{2}(t-s)T_{1}(s- \tau )f \bigl(\tau ,u_{n}(\tau ) \bigr)\,d \tau \,ds \biggr\} \biggr) \\ &\leq 2M_{1}M_{2}a \int _{0}^{t} \alpha \bigl( \bigl\{ f \bigl( \tau ,u_{n}(\tau ) \bigr) \bigr\} \bigr)\,d\tau \\ &\leq 2M_{1}M_{2}a \int _{0}^{t} L\alpha \bigl(D_{0}(s) \bigr)\,ds \\ &\leq 2M_{1}M_{2}La^{2}\alpha (D)_{C}. \end{aligned} $$

(3.9)

Therefore, from (3.7) and (3.9) we know that

$$ \alpha \bigl(\mathcal{F}(D) \bigr)_{C}\leq 4M_{1}M_{2}La^{2} \alpha (D)_{C}. $$

And from Definition 2.3 we know that the operator $\mathcal{F}:\varOmega _{R}\to \varOmega _{R}$ is k-set-contractive. It follows from Lemma 2.1 that $\mathcal{F}$ has at least one fixed point $u\in \varOmega _{R}$, which is just a mild solution of Problem (1.1). This completes the proof. □

Remark 3.1

The analytic semigroup and differentiable semigroup are equicontinuous semigroups [25]. In the application of partial differential equations, such as parabolic and strongly damped wave equations, the corresponding solution semigroup is analytic semigroup. Therefore, Theorem 3.1 has a broad applicability.

Remark 3.2

In the case $C(\rho )=-L^{2}(\rho )$, the expression of a mild solution for Problem (1.1) and the conclusion of Theorem 3.1 are correct and meaningful in complex Banach spaces.

4 Examples

Let $\varOmega \subset \mathbb{R}^{N}$ be an open bounded set with sufficiently smooth boundary ∂Ω, and let $E=L^{p}( \varOmega )$. Then E is a Banach space equipped with the $L^{p}$-norm $\|\cdot \|_{p}$.

Example 4.1

Let $p=2$. We consider the following damping elastic system:

$$ \textstyle\begin{cases} \frac{\partial ^{2}u(t,x)}{\partial t^{2}}-2\gamma \Delta \frac{\partial u(t,x)}{\partial t}+\Delta ^{2} u(t,x)=\frac{1}{5}\sin u(t,x), \quad (t,x)\in J\times \varOmega , \\ \Delta u(t,x)=u(t,x)=0, \quad (t,x)\in J\times \partial \varOmega , \\ u(0,x)=u_{0}(x), \qquad \frac{\partial }{\partial t}u(0,x)=u_{1}(x), \quad x\in \varOmega , \end{cases} $$

(4.1)

where $\gamma =\rho \geq 1$ is constant, Δ stands for the Laplace operator in the space variable x, $J=[0,1]$. We define the linear operators A and B in E by

$$\begin{aligned}& Au=\Delta ^{2} u, \quad u\in D(A)=D \bigl(\Delta ^{2} \bigr)= \bigl\{ u\in H^{4}(\varOmega ):\Delta u=u=0 \text{ on } \partial \varOmega \bigr\} , \\& Bu=-2\Delta u, \quad u\in D(B)=H_{0}^{1}(\varOmega )\cap H^{2}_{0}(\varOmega ). \end{aligned}$$

Clearly, $C(\rho )=\rho ^{2}B^{2}-4A=4\Delta ^{2}(\rho ^{2}-1)=L^{2}$, where $L=2\Delta (\rho ^{2}-1)^{\frac{1}{2}}$. It is clear that $BL=LB$. Further,

$$ E_{1}(\rho )=- \bigl(\rho + \bigl(\rho ^{2}-1 \bigr)^{\frac{1}{2}} \bigr)\Delta =-\sigma _{1} \Delta , \qquad E_{2}(\rho )=- \bigl(\rho - \bigl(\rho ^{2}-1 \bigr)^{\frac{1}{2}} \bigr)\Delta =-\sigma _{2} \Delta , $$

(4.2)

where $\sigma _{1}=(\rho +(\rho ^{2}-1)^{\frac{1}{2}})$, $\sigma _{2}=( \rho -(\rho ^{2}-1)^{\frac{1}{2}})$, $E_{1}(\rho )$ and $E_{2}(\rho )$ are invertible bounded linear operators on $L^{2}(\varOmega )$ for all $\rho >0$.

Since Δ is the infinitesimal generator of $C_{0}$-semigroup $T(t)_{t\geq 0}$, furthermore, for any $\rho \geq 1$, (4.2) yields $\sigma _{1}>0$, $\sigma _{2}>0$. Thus, by operator semigroup theory [25], $-E_{1}(\rho )=\sigma _{1}\Delta$ and $-E_{2}(\rho )=\sigma _{2}\Delta$ are the infinitesimal generators of equicontinuous $C_{0}$-semigroups $T_{1}(t)_{t\geq 0}$ and $T_{2}(t)_{t\geq 0}$ on $L^{2}(\varOmega )$, respectively. It follows that

$$ T_{1}(t)=T(\sigma _{1}t), \qquad T_{2}(t)=T( \sigma _{2}t), \quad t\geq 0, $$

which is exponentially stable, i.e.,

$$ \bigl\Vert T_{1}(t) \bigr\Vert \leq e^{-\lambda _{1}\sigma _{1} t}, \qquad \bigl\Vert T_{2}(t) \bigr\Vert \leq e^{-\lambda _{1}\sigma _{2} t} $$

with $\lambda _{1}$ being the first eigenvalue of Δ.

Let $u(t)=u(t,\cdot )$, $f(t,u(t))=\frac{1}{5}\sin u(t,x)$, then Problem (4.1) can be reformulated as the following abstract second order evolution equation in E:

$$ \textstyle\begin{cases} u''(t)+\rho Bu'(t)+Au(t)=f(t,u(t)), \quad t\in J, \\ u(0)=u_{0}, \qquad u'(0)=u_{1}. \end{cases} $$

(4.3)

In order to solve Problem (4.1), we also need the following assumptions:

• (1) $u_{0}\in D(L)\cap D(B)$, $u_{1}\in L^{2}(\varOmega )$.

• (2) The partial derivative $f'_{u}(t,x,u)$ is continuous.

Theorem 4.1

If assumptions (1) and (2) are satisfied, then Problem (4.1) has a mild solution $u\in C(J,L^{2}(\varOmega ))$.

Proof

Since $f(t,x,u(t,x))=\frac{1}{5}\sin u(t,x)$ is continuous on $[0,1]\times [0,+\infty )\times E$ and satisfying

$$ \begin{gathered} \bigl\vert f'_{u}(t,x,u) \bigr\vert =\frac{1}{5} \bigl\vert \cos u(t,x) \bigr\vert \leq \frac{1}{5}, \quad (t,x,u)\in [0,1]\times [0,+\infty )\times E; \\ f(t,x,0)=\sin 0=0, \quad (t,x)\in [0,1]\times [0,+\infty ). \end{gathered} $$

(4.4)

From (4.4), for $u\in E$, we have

$$\begin{aligned}& \bigl\Vert f(t,x,u) \bigr\Vert \leq \frac{1}{5} \Vert u \Vert , \quad (t,x)\in [0,1]\times [0,+\infty ), \\& \alpha \bigl(f(t,D) \bigr)\leq \frac{1}{5}\alpha (D),\quad t\in J. \end{aligned}$$

Now take $M_{1}=M_{2}=1$, we calculate

$$\begin{aligned}& a^{2}M_{1}M_{2} \Vert \varphi \Vert _{L^{\infty }(J,\mathbb{R}^{+})} \liminf_{n\rightarrow +\infty }\frac{\varPsi (n)}{n}= \frac{1}{5}< 1, \\& L< \frac{1}{4a^{2} M_{1} M_{2} }=\frac{1}{4}. \end{aligned}$$

From all the assumptions, it is easily seen that the conditions in Theorem 3.1 are satisfied. Hence, by Theorem 3.1, Problem (4.3) has a mild solution $u\in C(J,E)$, which means u is a mild solution for Problem (4.1). □

Example 4.2

Let $p\in [2,\infty )$. Consider the following damping elastic system:

$$ \textstyle\begin{cases} \frac{\partial ^{2}u(t,x)}{\partial t^{2}}+2\rho \Delta \frac{\partial u(t,x)}{\partial t}+\Delta u(t,x)=\frac{1}{5}\sin u(t,x), \quad (t,x)\in J\times \varOmega , \\ u(t,x)=0, \quad (t,x)\in J\times \partial \varOmega , \\ u(0,x)=u_{0}(x), \qquad \frac{\partial }{\partial t}u(0,x)=u_{1}(x), \quad x\in \varOmega , \end{cases} $$

(4.5)

where $\rho >0$ is constant, the function $f:J\times \varOmega \times E \rightarrow E$ is continuous, and Δ stands for the Laplace operator in the space variable x, $J=[0,1]$. We define the linear operators A and B in E by

$$\begin{aligned}& Au=\Delta u, \quad u\in D(A)=W^{2,p}(\varOmega )\cap W^{1,p}_{0}(\varOmega ), \\& Bu=2\Delta u, \quad u\in D(B)=W^{2,p}(\varOmega )\cap W^{1,p}_{0}(\varOmega ). \end{aligned}$$

Clearly, $C(\rho )=\rho ^{2}B^{2}-4A=4(\rho ^{2}\Delta ^{2}-\Delta )=L ^{2}$, where $L=2(\rho ^{2}\Delta ^{2}-\Delta )^{\frac{1}{2}}$. It is clear that $BL=LB$. Further,

$$ E_{1}(\rho )=R_{1}(\rho )\Delta , \qquad E_{2}( \rho )=R_{2}(\rho )\Delta , $$

where $R_{1}(\rho )=[\rho I-(\rho ^{2}I+(-\Delta )^{-1})^{\frac{1}{2}}]$ and $R_{2}(\rho )=[\rho I+(\rho ^{2}I+(-\Delta )^{-1})^{\frac{1}{2}}]$ are invertible bounded linear operators on $L^{p}(\varOmega )$ for all $\rho >0$.

Moreover,

$$ -R_{1}^{-1}(\rho ) \bigl(-E_{1}(\rho ) \bigr)=-R_{2}^{-1}(\rho ) \bigl(-E_{2}(\rho ) \bigr)= \Delta $$

is the infinitesimal generator of a $C_{0}$-semigroup as $T(t)$ ($t \geq 0$) on $L^{p}(\varOmega )$ and $-R_{1}^{-1}(\rho )$, $-R_{2}^{-1}( \rho )$ are invertible. Thus, by operator semigroup theory [25], $-E_{1}(\rho )=-R_{1}(\rho )\Delta$ and $-E_{2}(\rho )=-R_{2}(\rho ) \Delta$ generate positive $C_{0}$-semigroups $T_{1}(t)$ ($t\geq 0$) and $T_{2}(t)$ ($t\geq 0$) on $L^{p}(\varOmega )$.

Let $u(t)=u(t,\cdot )$, $f(t,u(t))=\frac{1}{5}\sin u(t,x)$, then Problem (4.5) can be reformulated as Eqs. (4.3). In order to solve Problem (4.5), we also need the following assumptions:

• (3) $u_{0}\in D(L)\cap D(B)$, $u_{1}\in L^{p}(\varOmega )$.

• (4) The partial derivative $f'_{t}(t,x,u)$ is continuous.

Theorem 4.2

If assumptions (3) and (4) are satisfied, then Problem (4.5) has a mild solution $u\in C(J,L^{p}(\varOmega ))$.

Proof

The proof is similar to that of Theorem 4.1. Here, we omit it. □

5 Conclusions

This paper investigates the existence of mild solutions for damped elastic systems in Banach spaces by the fixed point theorem and operator semigroup theory. Our results presented in this paper improve and generalize many classical results [11,12,13,14]. For future work we will focus on investigating the asymptotic stability of solutions and the analyticity and exponential stability of the associated semigroup for a damping elastic system in Banach spaces.