Boundedness in nonlinear functional differential equations with applications to Volterra integrodifferential equations (Q2567979)

The author considers the system of functional-differential equations \[ x'(t)=G(t, x(\cdot)), \] where \(x\in \mathbb{R}^n\), \(G:\mathbb{R}^+\times \mathbb{R}^n\rightarrow \mathbb{R}^n\) is a given nonlinear continuous function in \(t\) and \(x\). By employing nonnegative definite Lyapunov functions, sufficient conditions are obtained to guarantee the boundedness of solutions to the above system. The presented results are illustrated with several examples for scalar nonlinear Voltera integrodifferential equations and a two-dimensional system of nonlinear Voltera integrodifferential equations.