On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces
This paper studies an initial value problem of Volterra type in a real Banach space. At time zero the values of unkonwn function \(x\) and its derivative are given, and for \(t > 0\) the values of \(x''(t)\) are given as a function of \(t\) and \(x(s)\), \(0 \leq s \leq t\), except at some given time instances \(0 < t_1 < t_2 \cdots \) where \(x\) and \(x'\) have jump discontinuities, with values that depend on \(x(t_k)\) and \(x'(t_k)\). An existence and uniqueness theorem for this equation is developed, and an iterative scheme for the solution is developed, with an error estimate. The results are illustrated with two examples.
nonlinear second-order impulsive integro-differential equation
scientific article; zbMATH DE number 5017405
scientific article