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

: This paper is dedicated to studying the existence of periodic solutions to a new class of forced damped vibration systems by the variational method. The advantage of this kind of system is that the coefficient of its second order term is a symmetric N N  matrix valued function rather than the identity matrix previously studied. The variational principle of this problem is obtained by using two methods: the direct method of the calculus of variations and the semi-inverse method. New existence conditions of periodic solutions are created through several auxiliary functions so that two existence theorems of periodic solutions of the forced damped vibration systems are obtained by using the least action principle and the saddle point theorem in the critical point theory. Our results improve and extend many previously known results


Introduction
Consider the following forced damped vibration systems: , 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: with the properties: Moreover, there exist  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: 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.

The variational principle
Let us suppose 1 is absolutely continuous, and ||  are the usual inner product and norm of N R . The corresponding norm is defined by Then, 1 T H is obviously a Hilbert space. Set Obviously, the norm 0  is equivalent to the usual one  on 1 T H . The proof is similar to the corresponding parts in [19]. Let Hence, We have the following facts.
Moreover, the proof for the weak lower semi-continuity of () u  is similar to the corresponding parts in [3, P12-13].
By the Fundamental Lemma and Remarks in [3, P6-7], we know that has a weak-derivative, and and its continuous representation Then, by (2.4), we have i.e., By (2.4) and (2.5), one has By the existence of ) (t u  , we draw a conclusion similar to (2.5), that is i.e., 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) 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.
, we can easily obtain the following variational principle: In order to obtain the variational principle of problem (1.1), we introduce an integrating factor ) (t g 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 u G   is called variational derivative [20][21][22] defined as We re-write Eq (2.7) in the form (2.8) 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

Existence of solutions for the forced damped vibration systems
In convenience, we set satisfy assumption (A) and the following conditions： with the properties: (3.1) It follows from condition ) ( 1 H and Sobolev's inequality that

r t C h u h u t u t dt u e r t dt
It follows from the condition ) ( 2 H and Sobolev's inequality that Thus, by (3.1)-(3.4), we obtain (3.5) , and suppose that a.e  .
Then problem (1.1) has at least one solution on 1 T H . Proof. We will use the Saddle Point Theorem 4.7 in [3] Hence, which contradicts the boundedness of () n u  . Therefore, } { | u | n is bounded, and then } { n u is bounded by (3.7). We conclude that the (PS) condition is satisfied.
Next, we only need to prove the following conditions: In fact, by ) which implies that

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