  This paper presents a nonlinear dynamical model which consists the system of
differential and operator equations. Here differential equation contains a
nonlinear operator acting in Banach space, a nonlinear operator equation with
respect to two elements from different Banach spaces. This system is assumed to
enjoy the stationary state (rest points or equilibrium). The Cauchy problem
with the initial condition with respect to one of the desired functions is
formulated. The second function controls the corresponding nonlinear dynamic
process, the initial conditions are not set. The sufficient conditions of the
global classical solution's existence and stabilization at infinity to the rest
point are formulated. It is demonstrated that a solution can be constructed by
the method of successive approximations under the suitable sufficient
conditions. If the conditions of the main theorem are not satisfied, then
several solutions may exist. Some of solutions can blow-up in a finite time,
while others stabilize to a rest point. The special case of considered
dynamical models are nonlinear differential-algebraic equation (DAE) have
successfully modeled various phenomena in circuit analysis, power systems,
chemical process simulations and many other nonlinear processes. Three examples
illustrate the constructed theory and the main theorem. Generalization on the
non-autonomous dynamical systems concludes the article.
