Behavior of maximally defined solutions of a nonlinear Volterra equation

Abstract:

This paper is concerned with the behavior of solutions of an n-dimensional nonlinear Volterra integral equation \[ x(t) = f(t) + \int _0^t {g(t,s,x(s))ds,\quad t \geqslant 0.} \] In particular, sufficient conditions for a solution $x(t)$ on its maximal interval of existence $[0,T)$ to possess the property that $|x(t)| \to + \infty$ as $t \to {T^ - }$ are obtained. Thus these additional conditions give a positive answer to the problem posed by Miller [3, p. 145]. One can construct examples, satisfying the hypotheses given in [3], which provide a negative answer to this problem, see Artstein [1, Appendix A].