VARIATIONAL APPROACH TO A CLASS OF NONAUTONOMOUS SECOND-ORDER SYSTEMS ON TIME SCALES WITH IMPULSIVE EFFECTS

Abstract

In this paper, we present a recent approach via variational methodsand critical point theory to obtain the existence of solutions forthe nonautonomous second-order system on time scales with impulsive effects \[ \begin{cases} u^{\Delta^2}(t) = \nabla F(\sigma(t),u(\sigma(t))), & \Delta\textrm{-a.e.} \; t \in [0,T]_{\mathbb{T}}^{\kappa}; \\ u(0) - u(T) = \dot{u}(0) - \dot{u}(T) = 0, \\ (u^i)^{\Delta}(t_{j}^{+}) - (u^i)^{\Delta}(t_{j}^{-}) = I_{ij}(u^{i}(t_{j})), & i=1,2,\ldots,N, j=1,2,\ldots,p, \end{cases}\] where $t_{0} = 0 \lt t_{1} \lt t_{2} \lt \ldots \lt t_{p} \lt t_{p+1} = T$, $t_j \in [0,T]_\mathbb{T}$ ($j = 0,1,2,\ldots,p+1$), $u(t) = \big(u^{1}(t), u^{2}(t), \ldots, u^{N}(t)\big) \in \mathbb{R}^N$, $I_{ij} : \mathbb{R} \rightarrow \mathbb{R}$ ($i=1,2,\ldots,N$, $j=1,2,\ldots,p$) are continuous and $F: [0,T]_\mathbb{T} \times \mathbb{R}^{N} \rightarrow \mathbb{R}$. Finally, two examples are presented to illustrate the feasibility andeffectiveness of our results.