The authors consider the following nonlinear Volterra integral equation: \[ (1) H(t) \in u(t) + b*Au(t),\;t \in \mathbb{R}^+ = [0, +\infty), \] where \(b: \mathbb{R}^+ \to \mathbb{R}\), \(H: \mathbb{R}^+ \to X\) are given, \(*\) expresses the convolution \(b*g(t) = \int^t_0 b(t - s) g(s)ds\), and propose a necessary and sufficient condition which ensures the strong convergence and the weak convergence of the solutions of equation (1).