Multiple solutions of nonlinear impulsive differential equations with Dirichlet boundary conditions via variational method
Using a variational formulation of the second order nonlinear impulsive (Dirichlet) boundary value problem \[ -u''(t) + \lambda u(t) = f(t, u(t)), \;a.e. \; t\in [0,T], \] \[ \Delta u'(t_j) = I_j(u(t_j)), j = 1,2,\cdots,p, u(0) = u(T) = 0, \] where \(T > 0, f: [0,T]\times \mathbb{R} \rightarrow \mathbb{R}\) is continuous, \(0= t_0 < t_1 < \cdots < t_p < t_{p+1} = T\), \(\Delta u'(t_i) = u'(t_j^+) - u'(t_j^-), I_j: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, some existence and multiplicity theorems for the solutions are obtained. The impulsive functions \(I_j\) are assumed to be superlinear whereas \(f\) may be superlinear or sublinear. The results presented here extend and improve some recent results obtained in this direction.
scientific article; zbMATH DE number 6135270
scientific article