In the framework of the Authors' research papers devoted to studies of mathematical systems with mixed structures, various extensions of nonlinear integral and integro-differential equations of Volterra and Picone's types play a very important role. The Authors, continuing their previous paper published in the same «Rendiconti Lincei» and concerned with the existence, the unicity and the stability of a new extension of Volterra nonlinear integro-differential equations, deal in what follows with a class of very interesting new extensions of integral and integro-differential Volterra nonlinear equations and prove the existence, the unicity and construct and evaluate different approximate solutions.