Existence of solutions for a class of second-order differential equations with impulsive effects (Q2795267)

The author considers the boundary value problem for an impulsive differential system of second order NEWLINE\[NEWLINE \begin{aligned} &-(\rho(x)u')' + A(x)u = \nabla F(x,u(x)) \quad \text{a.e.}\;x \in (0,T), \\ &\rho(x_j+) {u'}^{i}(x_j^+) - \rho(x_j^-) {u'}^{i}(x_j^-) = I_{ij}(u^i(x_j)),\;i=1,\ldots,N,\;j=1,\ldots,l,\\ &\alpha_1u'(0) - \alpha_2u(0) = 0, \quad \beta_1u'(T) + \beta_2u(T) = 0, \end{aligned} NEWLINE\]NEWLINE where \(l \in \mathbb{N}\), \(0 = x_0 < x_1 < x_2 < \ldots < x_l < x_{l+1}=T\), \(A : [0,T] \to \mathbb{R}^{N\times N}\) is continuous, symetric matrix-valued mapping, \(\rho\) is essentially bounded on \([0,T]\), \(\alpha_i,\beta_i\), \(i=1,2\) are positive constants, \(I_{ij} : \mathbb{R} \to \mathbb{R}\) are continuous and \(\nabla F\) is the gradient of \(F : [0,T]\times\mathbb{R}^N \to \mathbb{R}\) with respect to the second variable.NEWLINENEWLINEUsing a variational approach existence results for the IBVP are obtained. Sufficient conditions for the existence of at least two nonzero weak solutions are found.