Sufficient conditions for existence and diffeomorphism of a nonlinear vector function

New conditions are derived for existence and uniqueness of solutions of nonlinear algebraic equations arising in nonlinear resistive networks and power networks. It is proved that a continuous differentiable function f(x) = y, x ∈ Rⁿ, y ∈ Rⁿhas at least one solution for every finite y if the Jacobian is positive (or negative) definite for all large x. In addition to this, if the Jacobian is nonsingular for all x, then f is a C´ diffeomorphism.