Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects ⎧ u”(t) + g(t)u’(t) + f(t,u(t)) = 0, a.e. t ∈ [0,T ⎨ u(0) = u(T) = 0 ⎩ $\Delta u^{\text{'}}\left(t_j\right)=u^{\text{'}}(t{⁺}_j-u^{\text{'}}\left(t{¯}_j\right)=I_j\left(u\left(t_j\right)\right)$, j = 1,...,p, are established, where $t₀=0<t₁<\cdots <t_p<t_{p+1}=T$, g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and $I_j:ℝ\to ℝ$, j = 1,...,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness of our results....

In this paper, a class of damped vibration problems with impulsive effects is considered. An existence result is obtained by using the variational method and the critical point theorem due to Brezis and Nirenberg. The obtained result is also valid and new for the corresponding second-order impulsive Hamiltonian system. Finally, an example is presented to illustrate the feasibility and effectiveness of the result.

The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(\right|\dot{u}\left(t\right){|}^{p-2}\dot{u}\left(t\right))=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0,\\ \Delta {\dot{u}}^i\left(t_j\right)={\dot{u}}^i\left(t_j^{+}\right)-{\dot{u}}^i\left(t_j^{-}\right)=I_{ij}\left(u^i\left(t_j\right)\right),i=1,2,\cdots ,N;j=1,2,\cdots ,m.\end{array}$$ By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.

If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of the free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed curves. There exists a rigorous geometric way to formulate this sort of problems on smooth manifolds with boundary, which we review here in a friendly self-contained way. As an application,...

We derive both local and global generalized Bianchi identities for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the a priori introduction of a connection. The proof is based on a global decomposition of the variational Lie derivative of the generalized Euler-Lagrange morphism and the representation of the corresponding generalized Jacobi morphism on gauge-natural bundles. In particular, we show that...

We consider a conservative second order Hamiltonian system $$\ddot{q}+\nabla V\left(q\right)=0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ 0 = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.

The existence of infinitely many solutions for a mixed boundary value problem is established. The approach is based on variational methods.

Under no Ambrosetti-Rabinowitz-type growth condition, we study the existence of infinitely many solutions of the p(x)-Laplacian equations by applying the variant fountain theorems due to Zou [Manuscripta Math. 104 (2001), 343-358].