[en] Highlights: • We use the generalized Usadel equations in gauge-invariant form to study the Little–Parks effect in hybrid SC/FM proximity cylinders. • The switching phenomena of the induced superconductivity in the ferromagnet with various vorticities are investigated in detail by solving self-consistently the Usadel equations. • Numerical analysis shows that the vortex states are most energetically favorable than the usual superconducting ground state when the ferromagnetic radius is large. - Abstract: The linearized Usadel equations in a generalized form containing both the spin singlet and the odd frequency spin triplet pair amplitudes in an s-wave state are used to study the oscillatory behavior of the superconducting critical temperature T_c due to the influence of an external magnetic field in proximity coaxial cylinders of a ferromagnetic core surrounded by a superconducting shell. Such geometry modifies the pairing wavefunction periodically with the vortex number L. The exchange field in the ferromagnetic core is chosen to be of the spiral type which rotates in the plane with a spiral wave vector Q. The external magnetic field assumes a constant value and is directed perpendicularly to the plane. The switching phenomena of the induced superconductivity in the ferromagnet with various vortex numbers are investigated in detail by solving self-consistently the Usadel equations to determine the Little–Parks effect i.e., the behavior of T_c versus a magnetic flux threading the ferromagnetic core.