Existence results for impulsive second-order periodic problems
The authors prove existence results for the nonlinear impulsive periodic boundary value problem \[ u''=f(t,u,u');\; u(t_i^+)=J_i(u(t_i)), \; u'(t_i^+)=M_i(u'(t_i)), \; i=1,2, \ldots,m; \] \[ u(0)=u(T), \; u'(0)=u'(T). \] Here, \(f\) is a Carathéodory function and \(J_i\) and \(M_i \) are continuous functions. They prove that if \(| f| \) is bounded in \(L^1\) and there exists a pair of nonordered lower and upper solutions, then this problem has at least one solution that satisfies additional properties with respect to the lower and the upper solutions.
second-order nonlinear ordinary differential equation with impulses
scientific article; zbMATH DE number 2117566