  Newton method is one of the most powerful methods for finding solutions of
nonlinear equations and for proving their existence. In its "pure" form it has
fast convergence near the solution, but small convergence domain. On the other
hand damped Newton method has slower convergence rate, but weaker conditions on
the initial point. We provide new versions of Newton-like algorithms, resulting
in combinations of Newton and damped Newton method with special step-size
choice, and estimate its convergence domain. Under some assumptions the
convergence is global. Explicit complexity results are also addressed. The
adaptive version of the algorithm (with no a priori constants knowledge) is
presented. The method is applicable for under-determined equations (with $m<n$,
$m$ being the number of equations and $n$ being the number of variables). The
results are specified for systems of quadratic equations, for composite
mappings and for one-dimensional equations and inequalities.
