On the variational principle and applications for a class of damped vibration systems with a small forcing term

1.   Introduction

Consider the following forced damped vibration systems:

where $T \gt 0$, $q \in {L^1}(0, T; R)$, $Q(t) = \int_0^t {q(s)ds}$, $f \in {L^1}(0, T; {R^N})$, $A(t) = [{a_{ij}}(t)]$ is an invertible symmetric $N \times N$ matrix-valued function defined in $[0, T]$ with ${a_{ij}} \in C{\text{([}}0, T{\text{]}})$ for all $i, j = 1, 2, \cdots N$ and there exists a positive constant $\lambda$ such that $\lambda |\xi {|^{\text{2}}} \leqslant {\text{(}}A{\text{(}}t{\text{)}}\xi, \xi {\text{), }}$ for all $\xi \in {R^N}$, $a.e.{\text{ }}t \in {\text{[}}0, T{\text{]}}$ and $F:[0, T] \times {R^N} \to R$ satisfies the following assumption:

(A) $F(t, x)$ is measurable in $t$ for every $x \in {R^N}$ and continuously differentiable in $x$ for $a.e.{\text{ }}t \in {\text{[}}0, T{\text{]}}$, and there exist $a \in C{\text{(}}{R^ + }, {R^ + }{\text{)}}$, $b \in {L^1}(0, T; {R^ + })$ such that

When $q(t) \equiv 0$, $A(t) = {I_{N \times N}}$ and $f{\text{(}}t{\text{)}} \equiv 0$, the forced damped vibration systems [i.e., (1.1)] reduce to the following classical second order non-autonomous Hamiltonian systems:

By the variational method, many existence results have been obtained under some suitable conditions in the last two decades. The readers may refer to ^{[1,2,3,4,5,6,7,8,9,10,11,12]} for more relevant results. In particular, Wang and Zhang ^[4] gave the following two existence theorems of periodic solutions of problem (1.2).

Theorem A. Suppose that $F$ satisfies assumption (A) and the following conditions:

${\text{(}}H_1^ * {\text{)}}$ There exist constants ${C_0} \gt 0,$ ${K_1} \geqslant 0,$ ${K_2} \geqslant 0,$ $\alpha \in {\text{[}}0, 1{\text{)}}$ and a non-negative function $h \in C{\text{([0, }} + \infty {\text{); [0, }} + \infty {\text{))}}$ with the properties:

(ⅰ) $(i) h{\text{(}}s{\text{)}} \leqslant h{\text{(}}t{\text{) }}\ \ \ \ \ \ \ \forall s \leqslant t, s, t \in {\text{[}}0, + \infty {\text{), }}$

(ⅱ) $h{\text{(}}s + t{\text{)}} \leqslant {C_0}{\text{(}}h{\text{(}}s{\text{) }} + {\text{ }}h{\text{(}}t{\text{)) }}\ \ \ \ \ \ \ \forall s, t \in {\text{[}}0, + \infty {\text{), }}$

(ⅲ) $0 \leqslant h{\text{(}}t{\text{)}} \leqslant {K_1}{t^\alpha } + {K_2}\ \ \ \ \ \ \ \forall t \in {\text{[}}0, + \infty {\text{), }}$

(ⅳ) $h{\text{(}}t{\text{)}} \to + \infty$ as ${\text{ }}t \to + \infty$.

Moreover, there exist $f \in {L^1}{\text{(}}0, T; {R^ + }{\text{)}}$ and $g \in {L^1}{\text{(}}0, T; {R^ + }{\text{)}}$ such that

${\text{(}}H_2^ * {\text{)}}$ There exists a non-negative function $h \in C{\text{([0, }} + \infty {\text{); [0, }} + \infty {\text{))}}$ which satisfies the conditions (ⅰ)–(ⅳ) and $\frac{1}{{{h^2}{\text{(}}|x|{\text{)}}}}\int_0^T {F{\text{(}}t, x{\text{)}}dt} \to + \infty$, as${\text{ |}}x{\text{|}} \to + \infty.$

Then problem (1.2) has at least one solution which minimizes the functional $\varphi$ on $H_T^1$.

Theorem B. Suppose that ${\text{(}}H_1^ * {\text{)}}$ and assumption (A) hold. Assume that

${\text{(}}H_3^ * {\text{)}}$ $\frac{1}{{{h^2}{\text{(}}|x|{\text{)}}}}\int_0^T {F{\text{(}}t, x{\text{)}}dt} \to - \infty$as${\text{ |}}x{\text{|}} \to + \infty$. Then problem (1.2) has at least one solution in $H_T^1$.

With the increase in research, scholars began to study a more general form of Hamiltonian systems: damped vibration systems. In damped vibration, due to the need of the system overcoming resistance, the displacement and energy of vibration continuously reduced and their decreasing trend is correlated to factors such as the natural frequency and damping coefficient of the system. Therefore, the damped vibration system is greatly based on physics and can be one of the important mathematical models.

Wu ^[13] studied the existence of periodic solutions of the following damped vibration systems:

where $q(t)\dot u(t)$ is called damping term. The systems [i.e., (1.3)] are called damped vibration systems in physics. Wu put forward the variational principle of problem (1.3) for the first time and studied further the existence of periodic solution of problem (1.3). Subsequently, in case $A{\text{(}}t{\text{)}} = 0,$ Wang ^[14] studied the existence of periodic solution of the corresponding system.

In addition, the vibration of a nonlinear vibration system under the action of a periodic dynamic force $f{\text{(}}t{\text{)}}$ is called forced vibration. Take the spring oscillator model as an example. Suppose that the spring oscillator is subjected to both resistance $- \gamma \frac{{{\text{d}}x}}{{{\text{d}}t}}$ and dynamic force ${F_0}{\text{cos}}\omega t$, and then Dynamic equation of the spring oscillator is

This equation is a special forced damped vibration system. Another example is the famous Duffing oscillator model: Assume that it is subjected to both resistance $c\dot x$ and dynamic force $f{\kern 1pt} {\text{cos}}\omega t$, and then the dynamic equation of the forced damped vibration of the Duffing oscillator is

The forced damped vibration systems [i.e., (1.1)] we have studied are more general than the two equations above. Therefore, the systems [i.e., (1.1)] are proved to not only have a very strong physical background but also be a more general class of new systems.

Generally, a nonlinear vibration system is complex and it is difficult to get a strong solution to a differential equation. In recent years, the variational method has been used by many scholars to study the existence of solutions of differential equations, such as the classical second order non-autonomous Hamiltonian systems [i.e., (1.2)] (see ^{[1,2,3,4,5,6,7,8,9,10,11,12]}), the damped vibration systems [i.e., (1.3)] (See ^[13,14]) and the damped random impulsive differential equations under Dirichlet boundary value conditions (See ^[15,16,17]). The variational principle, including the Hamilton principle, is widely used in the nonlinear vibration theory. For the case of Hamiltonian-based frequency formulation for nonlinear oscillators (See ^[18]), its Hamilton principle is established by the semi-inverse method.

Inspired by ^[4,13], we obtain a new class of forced damped vibration systems [i.e., (1.1)] and decide to study the existence of periodic solutions of this problem by the variational method. We explore, in depth, the existence of variational construction for problem (1.1) and study further the existence of periodic solutions of it under some solvability conditions by following the least action principle and the Saddle Point Theorem 4.7 in ^[3], and obtain two new existence theorems.

2.   The variational principle

Let us suppose $H_T^1$ = ${\text{\{ }}u{\text{:[}}0, T{\text{]}} \to {R^N}|u$ is absolutely continuous, $u(0) = u(T)$, $\dot u \in {L^2}({\text{0}}, T{\text{; }} {R^N})$} with the inner product

where $u \in H_T^1$ and $|{\kern 1pt} \; \cdot \; |$ are the usual inner product and norm of ${R^N}$. The corresponding norm is defined by

Then, $H_T^1$ is obviously a Hilbert space.

Set

Obviously, the norm ${\left\| \cdot \right\|_0}$ is equivalent to the usual one $\left\| \cdot \right\|$ on $H_T^1$. The proof is similar to the corresponding parts in ^[19].

Let $\tilde H_T^1 = \{ u \in H_T^1|\int_0^T {udt} = 0\}$, it is easy to know that $\tilde H_T^1$ is a subset of $H_T^1$, and $H_T^1$ = ${R^N} \oplus \tilde H_T^1$. It follows from Proposition 1.3 in ^[3] that

and

Hence,

Define the functional $\varphi {\text{(}}u{\text{)}}$ on $H_T^1$ by

We have the following facts.

Theorem 2.1. The functional $\varphi {\text{(}}u{\text{)}}$ is continuously differentiable and weak lower semi-continuous on $H_T^1$.

Proof. Set $L(t, x, y) = {e^{Q(t)}}[\frac{1}{2}(A(t)y, y) + F(t, x) + (f(t), x){\text{]}}$ for all $x, y \in {R^N}$ and $t \in [0, T]$. Then $L(t, x, y)$ satisfies all assumptions of Theorem 1.4 in ^[3]. Hence, by Theorem 1.4 in ^[3], we know that the functional $\varphi {\text{(}}u{\text{)}}$ is continuously differentiable on $H_T^1$ and

for all $u, v \in H_T^1$. Moreover, the proof for the weak lower semi-continuity of $\varphi {\text{(}}u{\text{)}}$ is similar to the corresponding parts in [3, P_12-13].

Theorem 2.2. If $u \in H_T^1$ is a solution of the Euler equation $\varphi '(u) = {\text{0}}$, then $u$ is a solution of problem (1.1).

Proof. Since $\varphi '(u) = {\text{0}}$, then

for all $u, v \in H_T^1$. i.e.,

By the Fundamental Lemma and Remarks in [3, P_6-7], we know that ${e^{Q(t)}}A(t)\dot u(t)$ has a weak-derivative, and

where $C$ is a constant. We identify the equivalence class ${e^{Q(t)}}A(t)\dot u(t)$ and its continuous representation

Then, by (2.4), we have

i.e., $\dot u{\text{(}}0{\text{)}} = 0$.

By (2.4) and (2.5), one has

i.e., ${e^{Q{\text{(}}T{\text{)}}}}\dot u{\text{(}}T{\text{)}} = 0$.

By the existence of $\dot u(t)$, we draw a conclusion similar to (2.5), that is

i.e., $u(0) - u(T) = {\text{0}}$.

Therefore, $u$ satisfies the following periodic boundary condition

Moreover, by (2.3), $u$ satisfies the following forced damped vibration equation

Hence, $u$ is a solution of problem (1.1). This completes the proof.

From the proof of Theorems 2.1 and 2.2, it can be seen that the variational principle of problem (1.1) is indeed the $\varphi (u)$[i.e., (2.2)] we defined above.

In fact, we can also directly derive the variational principle of problem (1.1) by using the semi-inverse method ^[18]. The derivation process is as follows.

In case $q(t) \equiv 0$, we can easily obtain the following variational principle:

In order to obtain the variational principle of problem (1.1), we introduce an integrating factor $g{\text{(}}t{\text{)}}$ which is an unknown function of time, and consider the following integral:

where $G$ is an unknown function of $u$ and/or its derivatives. The semi-inverse method is to identify such $g$ and $G$ that the stationary condition of Eq (2.6) satisfies problem (1.1). The Euler–Lagrange equation of Eq (2.6) reads

where $\frac{{\delta G}}{{\delta u}}$ is called variational derivative ^[20,21,22] defined as

We re-write Eq (2.7) in the form

Comparing Eq (2.8) with problem (1.1), we set

Therefore, we have

Consequently, we obtain the needed variational principle for problem (1.1), which reads

Obviously, ${\varphi _2}{\text{(}}u{\text{)}} = \varphi {\text{(}}u{\text{)}}$.

3.   Existence of solutions for the forced damped vibration systems

In convenience, we set

Theorem 3.1. Let $F(t, x) = {F_1}(t, x) + {F_2}(x)$, suppose that ${F_1}{\text{(}}t, x{\text{)}}$ and ${F_2}{\text{(}}x{\text{)}}$ satisfy assumption (A) and the following conditions:

${\text{(}}{H_1}{\text{)}}$ There exist constants ${C_1} \gt 0,$ ${K_1} \geqslant 0,$ $\alpha \in (\frac{1}{2}, 1)$ and a non-negative function ${h_1} \in C{\text{([0, }} + \infty {\text{); [0, }} + \infty {\text{))}}$ with the properties:

$({\rm{i}})\ {h_1}{\text{(}}s{\text{)}} \leqslant {h_1}{\text{(}}t{\text{) }}\ \ \ \ \ \forall s \leqslant t, \; \; \; s, t \in {\text{[}}0, + \infty {\text{)}},$

$({\rm{ii}})\ {h_1}{\text{(}}s + t{\text{)}} \leqslant {C_1}{\text{(}}{h_1}{\text{(}}s{\text{) }} + {\text{ }}{h_1}{\text{(}}t{\text{)) }}\ \ \ \ \ \forall s, t \in {\text{[}}0, + \infty {\text{)}},$

$({\rm{iii}})\ \mathop {{\text{lim}}}\limits_{s \to + \infty } \frac{{{h_1}{\text{(}}s{\text{)}}}}{{{s^\alpha }}} \leqslant {K_1}.$

Moreover, there exist ${r_1}, {r_2} \in {L^1}{\text{(}}0, T; {R^ + }{\text{)}}$ such that

${\text{(}}{H_2}{\text{)}}$ There exist a constant ${K_2} \gt 0$ and two functions $k \in {L^1}{\text{(0}}, T; {R^ + }{\text{)}}$ with $\int_0^T {k(t)dt} \lt \frac{{3\lambda {d_2}}}{{{d_1}{K_2}T}}$ and ${h_2} \in C{\text{([0, }} + \infty {\text{); [0, }} + \infty {\text{))}}$ which is non-decreasing, such that

and

${\text{(}}{H_3}{\text{)}} \mathop {\lim }\limits_{|x| \to + \infty } \frac{1}{{|x{|^{2\alpha }}}}\int_0^T {{e^{Q(t)}}F(t, x)dt} \gt \frac{{C_1^2d_1^2TK_1^2}}{{3\lambda {d_2}}}{(\int_0^T {{r_1}(t)dt})^2}.$

Then problem (1.1) has at least one solution which minimizes the functional $\varphi {\text{(}}u{\text{)}}$ on $H_T^1$.

Proof. It is clear that

It follows from condition ${\text{(}}{H_1}{\text{)}}$ and Sobolev's inequality that

It follows from the condition ${\text{(}}{H_2}{\text{)}}$ and Sobolev's inequality that

for sufficiently large $\left\| {\tilde u} \right\|_\infty ^{}$. By Sobolev's inequality, we have

Thus, by (3.1)–(3.4), we obtain

Since

and $\frac{{\lambda {d_2}}}{4} - \frac{{{d_1}{K_2}T}}{{12}}\int_0^T {k(t)dt} \gt 0$ as $\int_0^T {k(t)dt} \lt \frac{{3\lambda {d_2}}}{{{d_1}{K_2}T}}$, it follows from ${\text{(}}{H_1}{\text{), }}$ ${\text{(}}{H_3}{\text{), }}$ (3.5) and (3.6) that $\varphi {\text{(}}u{\text{)}} \to + \infty$ as ${F_2}{\text{(}}x{\text{)}} = {\text{0}}$ i.e., $\varphi {\text{(}}u{\text{)}}$ is coercive. By Theorem 1.1 and Corollary 1.1 in ^[3] (i.e., the least action principle), we complete the proof of Theorem 3.1.

Remark 1. The condition ${\text{(}}{H_4}{\text{)}}$ in Theorem 3.1 is weaker than the condition $h{\text{(}}t{\text{)}} \leqslant {K_1}{t^\alpha } + {K_2}$ in Theorem 1.1 in ^[4] (i.e., Theorem A in the present paper), so that Theorem 3.1 generalizes Theorem 1.1 in ^[4] even in the case of ${F_2}{\text{(}}x{\text{)}} = {\text{0}}$, $q{\text{(}}t{\text{)}} \equiv 0$, $A{\text{(}}t{\text{)}} = {I_{N \times N}}$ and $f{\text{(}}t{\text{)}} \equiv 0$.

Theorem 3.2. Let $F(t, x) = {F_1}(t, x) + {F_2}(x)$, and suppose that ${F_1}{\text{(}}t, x{\text{)}}$ and ${F_2}{\text{(}}x{\text{)}}$ satisfy assumption (A) and${\text{(}}{H_2}{\text{)}}$. If the following conditions hold:

${\text{(}}{H_4}{\text{)}}$ There exist constants ${\text{(}}{H_5}{\text{)}}$ ${K_1} \geqslant 0,$ $\alpha \in {\text{[}}0, 1{\text{)}}$ and a non-negative function ${h_1} \in C{\text{([0, }} + \infty {\text{); [0, }} + \infty {\text{))}}$ with the properties:

$({\rm{i}})\ {h_1}{\text{(}}s{\text{)}} \leqslant {h_1}{\text{(}}t{\text{) }}\ \ \ \ \forall s \leqslant t, s, t \in {\text{[}}0, + \infty {\text{)}},$

$({\rm{ii}})\ {h_1}{\text{(}}s + t{\text{)}} \leqslant {C_1}{\text{(}}{h_1}{\text{(}}s{\text{) }} + {\text{ }}{h_1}{\text{(}}t{\text{)) }}\ \ \ \ \forall s, t \in {\text{[}}0, + \infty {\text{)}},$

$({\rm{iii}}) \ \mathop {{\text{lim}}}\limits_{s \to + \infty } \frac{{{h_1}{\text{(}}s{\text{)}}}}{{{s^\alpha }}} \leqslant {K_1}.$

Moreover, there exist $\varphi {\text{(}}{u_n}{\text{)}}$ such that

${\text{(}}{H_5}{\text{)}}\ \mathop {{\text{lim}}}\limits_{|x| \to + \infty } \frac{1}{{h_1^2{\text{(}}|x|{\text{)}}}}\int_0^T {{e^{Q{\text{(}}t{\text{)}}}}{\text{(}}F{\text{(}}t, x{\text{)}} + |f{\text{(}}t{\text{)}}||x|{\text{)}}dt} = - \infty.$

Then problem (1.1) has at least one solution on$H_T^1$.

Proof. We will use the Saddle Point Theorem 4.7 in ^[3] to prove Theorem 3.2. First, we prove that the functional $\varphi {\text{(}}u{\text{)}}$ satisfies the (PS) condition. Suppose that $\{ {u_n}\}$ is a (PS) sequence for $\varphi {\text{(}}u{\text{)}}$, that is, $\mathop {\lim }\limits_{n \to \infty } \varphi '({u_n}) = 0$ and $\varphi {\text{(}}{u_n}{\text{)}}$ is bounded. By a way similar to (3.2)–(3.4), we have

and

Hence,

It follows from (2.1) that

Thus,

By (3.2)–(3.4) and ${\text{(}}{H_5}{\text{)}}$, we have

which contradicts the boundedness of $\varphi {\text{(}}{u_n}{\text{)}}$. Therefore, ${\text{\{ }}|{\bar u_n}|{\text{\} }}$ is bounded, and then ${\text{\{ }}{u_n}{\text{\} }}$ is bounded by (3.7). We conclude that the (PS) condition is satisfied.

Next, we only need to prove the following conditions:

${\text{(}}{l_1}{\text{)}}$ $\varphi {\text{(}}u{\text{)}} \to - \infty$ as $\left| u \right| \to + \infty$ in ${R^N}$;

${\text{(}}{l_2}{\text{)}}$ $\varphi {\text{(}}u{\text{)}} \to + \infty$ as $\left\| u \right\| \to + \infty$ in $\tilde H_T^1$.

In fact, by ${\text{(}}{H_5}{\text{)}}$, we obtain

Thus ${\text{(}}{l_1}{\text{)}}$ is proved.

For $u \in \tilde H_T^1$, as we have argued in (3.2) and (3.3), we have

and

which implies that

for all $u \in \tilde H_T^1$.

By Wirtinger's inequality, one has

We know that $\frac{{\lambda {d_2}}}{4} - \frac{{{d_1}{K_2}T}}{{12}}\int_0^T {k{\text{(}}t{\text{)}}dt} \gt 0$ as $\int_0^T {k{\text{(}}t{\text{)}}dt} \lt \frac{{3\lambda {d_2}}}{{{d_1}{K_2}T}}$. It follows from (3.8) that $\varphi (u) \to + \infty$ as $\lambda \gt 0.$ in ${F_1}{\text{(}}t, x{\text{)}} = {\text{(}}\frac{{{d_2}}}{{3{d_1}}}T - t{\text{)l}}{{\text{n}}^{\frac{{\text{3}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$, that is, ${F_2}{\text{(}}x{\text{)}} = \frac{{\lambda {d_2}}}{{{d_1}{K_2}{T^2}}}|x{|^2}$ is proved.

By making use of the Saddle Point Theorem 4.7 in ^[3], we prove that problem (1.1) has at least one solution on ${h_1}{\text{(|}}x|{\text{)}} = {\text{l}}{{\text{n}}^{\frac{{\text{1}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$.

Remark 2. The condition $= {r_1}{\text{(}}t{\text{)}}{h_1}{\text{(|}}x{\text{|)}} + {r_2}{\text{(}}t{\text{)}}$ in Theorem 3.2 is weaker than the condition ${\text{(}}\nabla {F_2}{\text{(}}x{\text{)}} - \nabla {F_2}{\text{(}}y{\text{), }}x - y{\text{)}} = \frac{{2\lambda {d_2}}}{{{d_1}{K_2}{T^2}}}|x - y{|^2} = k{\text{(}}t{\text{)}}{h_2}{\text{(}}|x - y|{\text{)}}$ in Theorem 1.2 in ^[4] (i.e., Theorem B in the present paper), so that Theorem 3.2 generalizes Theorem 1.2 in ^[4] even in the case of $F_2(x)=0, q(t) \equiv 0, A(t)=I_{N \times N} \text { and } f(t) \equiv 0 \text {. }$

4.   Examples

In this section, we give two examples to illustrate the feasibility and effectiveness of our main conclusions.

Example 4.1. Let $A{\text{(}}t{\text{)}} = {\left({\begin{array}{*{20}{c}} {\lambda + t} & 0 & 0 & \cdots & 0 \\ 0 & {\lambda + t} & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & {\lambda + t} \end{array}} \right)_{N \times N}}$, where $\lambda \gt 0.$

Let ${F_1}{\text{(}}t, x{\text{)}} = {\text{(}}\frac{{{d_2}}}{{3{d_1}}}T - t{\text{)l}}{{\text{n}}^{\frac{{\text{3}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$, ${F_2}{\text{(}}x{\text{)}} = \frac{{\lambda {d_2}}}{{{d_1}{K_2}{T^2}}}|x{|^2}$, ${h_1}{\text{(|}}x|{\text{)}} = {\text{l}}{{\text{n}}^{\frac{{\text{1}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$.

Then,

i.e.,

and

Moreover, we obtained the following results through simple derivation:

① $|\nabla {F_1}{\text{(}}t, x{\text{)|}} = \frac{3}{2}{\text{|}}\frac{{{d_2}}}{{3{d_1}}}T - t|{\text{l}}{{\text{n}}^{\frac{{\text{1}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}\frac{{\text{1}}}{{{\text{1}} + |x{|^2}}} \cdot 2|x{\text{|}}$

   $\leqslant \frac{3}{2}{\text{|}}\frac{{{d_2}}}{{3{d_1}}}T - t|{\text{l}}{{\text{n}}^{\frac{{\text{1}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$

    $= {r_1}{\text{(}}t{\text{)}}{h_1}{\text{(|}}x{\text{|)}} + {r_2}{\text{(}}t{\text{)}},$

② ${\text{(}}\nabla {F_2}{\text{(}}x{\text{)}} - \nabla {F_2}{\text{(}}y{\text{), }}x - y{\text{)}} = \frac{{2\lambda {d_2}}}{{{d_1}{K_2}{T^2}}}|x - y{|^2} = k{\text{(}}t{\text{)}}{h_2}{\text{(}}|x - y|{\text{)}}$, ${\text{(}}\int_0^T {k{\text{(}}t{\text{)}}dt} \lt \frac{{3\lambda {d_2}}}{{{d_1}{K_2}T}}{\text{)}}$,

③ $\mathop {{\text{limsup}}}\limits_{s \to + \infty } \frac{{{h_2}{\text{(}}s{\text{)}}}}{{{s^2}}} = 1 \leqslant {K_2}$,

④ $\mathop {{\text{lim}}}\limits_{|x| \to + \infty } \frac{1}{{|x{|^{2\alpha }}}}\int_0^T {{e^{Q{\text{(}}t{\text{)}}}}F{\text{(}}t, x{\text{)}}dt}$ $\varphi {\text{(}}u{\text{)}}$

   $= \lim _{|x| \rightarrow \infty} \frac{1}{|x|^{2 \alpha}} \int_0^{\mathrm{T}} e^{Q(t)}\left(\frac{d_2}{3 d_1} T-t\right) \ln ^{\frac{3}{2}}\left(1+|x|^2\right) d t+\lim _{|x| \rightarrow \infty} \frac{1}{|x|^{2 \alpha}} \int_0^{\mathrm{T}} e^{Q(t)} \frac{\lambda d_2}{d_1 K_2 T^2}|x|^2 d t$

   $\geq \int_0^T e^{Q(t)}\left(\frac{d_2}{3 d_1} T-t\right) d t \cdot \lim _{|x| \rightarrow \infty} \frac{\ln ^{\frac{3}{2}}\left(1+|x|^2\right)}{|x|^{2 \alpha}}+\frac{\lambda d_2^2}{d_1 K_2 T} \lim _{|x| \rightarrow \infty}|x|^{2-2 \alpha}$

  $= +\infty \text {, }$

⑤ $\lambda |x{|^{\text{2}}} \leqslant {\text{(}}\lambda + t{\text{)}}|x{|^2} = {\text{(}}A{\text{(}}t{\text{)}}x, x{\text{)}}$, for all $x \in {R^N}$ and $a.e.{\text{ }}t \in {\text{[}}0, T{\text{]}}$.

It is easy to see that $A{\text{(}}t{\text{)}}$, ${F_1}{\text{(}}t, x{\text{)}}$, ${F_2}{\text{(}}x{\text{)}}$ and ${h_1}{\text{(|}}x|{\text{)}}$ satisfy all conditions of Theorem 3.1. Hence, problem (1.1) has at least one solution on $H_T^1$.

Example 4.2. Let $A{\text{(}}t{\text{)}} = {\left({\begin{array}{*{20}{c}} {\lambda + t} & 0 & 0 & \cdots & 0 \\ 0 & {\lambda + t} & 0 & \cdots & 0 \\ 0 & 0 & \ddots & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & {\lambda + t} \end{array}} \right)_{N \times N}}$, where $\lambda \gt 0.$

Let ${F_1}{\text{(}}t, x{\text{)}} = {\text{(}}\frac{{{d_2}}}{{3{d_1}}}T - t{\text{)l}}{{\text{n}}^{\frac{{\text{3}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$, ${F_2}{\text{(}}x{\text{)}} = - \frac{{\lambda {d_2}}}{{{d_1}{K_2}{T^2}}}|x{|^2}$, ${h_1}{\text{(|}}x|{\text{)}} = {\text{l}}{{\text{n}}^{\frac{{\text{1}}}{{\text{2}}}}}{\text{(}}1 + |x{|^2}{\text{)}}$.

Then,

The derivation of other conditions for Theorem 3.2 is the same as Example 4.1.

It is clear that $A{\text{(}}t{\text{)}}$, ${F_1}{\text{(}}t, x{\text{)}}$, ${F_2}{\text{(}}x{\text{)}}$ and ${h_1}{\text{(|}}x|{\text{)}}$ satisfy all conditions of Theorem 3.2. Therefore, problem (1.1) has at least one solution on $H_T^1$.

5.   Conclusions

In this paper, we study the existence of periodic solutions of the forced damped vibration systems [i.e., (1.1)] by using the variational method and the critical point theory.

First, we provide an expression for functional $\varphi {\text{(}}u{\text{)}}$ and further prove that the functional $\varphi {\text{(}}u{\text{)}}$ is continuously differentiable and weak lower semi-continuous.

Then, we prove that the critical point of $\varphi {\text{(}}u{\text{)}}$ is a solution of problem (1.1) in the sense of weak-derivative. Moreover, we directly derive the variational principle of problem (1.1) via the semi-inverse method.

Finally, it is proved that problem (1.1) has at least one solution under the given sufficient conditions through the least action principle and the saddle point theorem.

In the future, we can continue to study this problem by looking for new sufficient conditions.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (No.11861005), the Joint Special Fund Project for Basic Research of Local Undergraduate Universities in Yunnan Province (No.202101BA070001-219) and Foundation of Dali University (No.KY2319101540).

Conflict of interest

There is no conflict of interest.