Infinitely many nontrivial periodic solutions for damped vibration problems with asymptotically linear terms

doi:10.1016/j.amc.2014.07.114

Applied Mathematics and Computation

Volume 245, 15 October 2014, Pages 438-446

https://doi.org/10.1016/j.amc.2014.07.114 Get rights and content

Abstract

In this paper, we study a class of damped vibration problem with asymptotically quadratic terms at infinity. We obtain infinitely many nontrivial periodic solutions by variational method. To the best of our knowledge, there is no published result focusing on this class of damped vibration problem with asymptotically quadratic terms at infinity.

Section snippets

Introduction and main results

We shall study the existence of infinitely many nontrivial periodic solutions for the following damped vibration problem $\{\begin{matrix} \ddot{u} + (q (t) I_{N \times N} + B) \dot{u} + (\frac{1}{2} Bq (t) - A (t)) u + H_{u} (t, u) = 0, & t \in R, \\ u (0) - u (T) = \dot{u} (0) - \dot{u} (T) = 0, & T > 0, \end{matrix}$ where $u = u (t) \in C^{2} (R, R^{N}), I_{N \times N}$ is the $N \times N$ identity matrix, $q (t) \in L^{1} (R; R)$ is T-periodic and satisfies $\int_{0}^{T} q (t) dt = 0, A (t) = [a_{ij} (t)]$ is a T-periodic symmetric $N \times N$ matrix-valued function with $a_{ij} \in L^{\infty} (R; R)$ ( $\forall i, j = 1, 2, \dots, N$ ), $B = [b_{ij}]$ is an antisymmetric $N \times N$ constant matrix, $H (t, u) \in C^{1} (R \times R^{N}, R)$ is T-periodic in t and $H_{u} (t, u)$

Variational frameworks and the proof of our main result

In this section, we shall assume that $(H_{1}) – (H_{4})$ hold and $H (t, u)$ is even in u.

Let $W ≔ H_{T}^{1}$ be defined by $H_{T}^{1} ≔ \{u = u (t) : [0, T] \to R^{N} | u is absolutely continuous, u (0) = u (T), and \dot{u} \in L^{2} ([0, T]; R^{N})\}$ with the inner product ${(u, v)}_{W} ≔ \int_{0}^{T} [(u, v) + (\dot{u}, \dot{v})] dt, \forall u, v \in W,$ where $(\cdot, \cdot)$ has been defined in Notations. The corresponding norm of ${(\cdot, \cdot)}_{W}$ is defined by $‖ u ‖_{W} = {(u, u)}_{W}^{1 / 2}$ . Obviously, W is a Hilbert space.

Let $Q (t) ≔ \int_{0}^{t} q (s) ds$ and $‖ u ‖_{0} ≔ {(\int_{0}^{T} e^{Q (t)} (| u |^{2} + | \dot{u} |^{2}) dt)}^{1 / 2}, u \in W,$ where the function q is defined in problem (1.1). We denote by $〈 \cdot, \cdot 〉_{0}$ the inner

Conclusions

In the case where the nonlinearity H of the damped vibration system (1.1) is asymptotically quadratic at infinity and subquadratic at 0, we obtain the existence of infinitely many nontrivial periodic solutions for the system (1.1) by using variational methods, which unify and sharply improve some recent results in the literature. To the best of our knowledge, there is no published result concerning these cases. Here, we can give two nontrivial examples, which show these assumptions of the

Acknowledgments

The author thanks the referees and the editors for their helpful comments and suggestions. The author also thanks Prof. Shiwang Ma for fruitful discussions of the nontrivial example Ex2.

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Citation Excerpt :
If (SH1)–(SH3) hold, and H(t, u) is even in u, then (1.1) possesses infinitely many nontrivial T-periodic solutions. Here we give some comparisons between this paper and the most related papers [3,4]. First, the methods are different.
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^☆: Research supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).

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Infinitely many nontrivial periodic solutions for damped vibration problems with asymptotically linear terms☆

Abstract

Section snippets

Introduction and main results

Variational frameworks and the proof of our main result

Conclusions

Acknowledgments

J. Math. Anal. Appl.

Nonlinear Anal. TMA

J. Differ. Equ.

Nonlinear Anal. TMA

J. Math. Anal. Appl.

Nonlinear Anal.