We study the existence of stop bands for left-handed Alfv¿nic ion cyclotron waves propagating in the direction of magnetic field in a multicomponent plasma. These effects are discussed: finite ion temperature, inhomogenous magnetic field, and nonlinear wave amplitude. All of them affect the existence of stop bands and set critical bounds on the relevant physical parameters, particularly on the density of heavy ions. Using the model of a linearly varing, longitudinally inhomogeneous magnetic field, we calculate the critical lower bound on heavy ion density as a function of longitudinal inhomogeneity, and we discuss the difference between the truly inhomogeneous and essentially homogeneous situations. Typical values for the wave vector in the resonance region are derived. We also find the upper bound on the wave amplitude for the linear theory to be applicable, and we derive the condition for stop band formation in the nonlinear case. The obtained results are applied to the magnetospheric environment and are shown to lead to relevant modifications. For example, the formation of stop bands by thermal heavy ions is strongly dependent on the local inhomogeneity of the magnetic field. ¿American Geophysical Union 1990