  Based on the self-consistent solution of a nonlinear system of
one-dimensional GL-equations, the onset and destruction of superconductivity,
the phase transitions and hysteresis phenomena are discussed for a cylinder
(radius R) in an axial magnetic field (H) for arbitrary R,kappa,H,m (kappa$ is
the GL-parameter, m is the total vorticity of the system). The edge-suppressed
solutions (which are connected with the jumps of magnetization in the states
with fixed vorticity m), the depressed solutions (responsible for the
hysteresis in type-II superconductors), and the precursor solutions (which
describe the onset of superconductivity in type-I superconductors) are also
studied. The limits of applicability of the so-called linear equation
approximation are discussed.
