On the existence of solutions for impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions (Q610858)

The authors study the following impulsive Duffing dynamic equations on time scales \[ u^{\Delta \Delta}(t)+Cu^{\Delta}(\sigma(t))-r(t)u(\sigma(t))+f(\sigma(t),u(\sigma(t)))=h(t), \text{ a.e. } t \in [0,\sigma(T)]^{\kappa^2}_{\mathbb{T}}, \] \[ \Delta u^{\Delta}(t_j)=u^{\Delta}(t_j^+)-u^{\Delta}(t_j^-)= I_{j}(u(t_j)),\quad j=1,\dots,p, \] \[ u(0)=0=u(\sigma(T)) \] with \(T>0\), \(C\) a regressive constant, \(\{t_j\}_{j=1}^{p}\) a finite increasing sequence in \((0,\sigma(T))\), \(t_0=0\), \(t_{p+1}=\sigma(T)\), \(r \in L^{\infty}[0,\sigma(T)]_{\mathbb{T}}\), \(h \in L^{2}[0,\sigma(T)]_{\mathbb{T}}\), \(f\) continuous and \(I_j\) continuous, for \(j=1,\dots,p\). They establish the variational framework for the problem and, hence, sufficient conditions for the existence of weak solutions by using critical point theorems, providing some examples of application.