Boundedness of solutions of a system of integro-differential equations (Q1066045)

The paper deals with the integro-differential equation \[ (E)\quad \dot x(t)=-\int^{0}_{-1}b(s)g(x(t+s))ds, \] where \(x(t)\in R^ n\), \(g: R^ n\to R^ n\) is locally Lipschitzian and possesses a potential \(G: R^ n\to R\) \((g=\text{grad} G)\), \(b\in C([-1,0],R)\) is nondecreasing, strictly convex and \(b(-1)=0\). The authors give several geometric criteria for the boundedness of solutions to (E) or some of its components which complements the classical boundedness criterion: \(\lim_{\| x\| \to \infty}G(x)=\infty\). (They are applicable e.g. to the cases like: \(g(x)=x \sin x\), \(G(x)=\| x\|^ 2\cos \| x\|^ 2\), \(g(x)=\sin x\) or \(g(x,y)=2(\sin x \cos y\), sin y cos x).)