Abstract In this study, we have combined the new concept of generalized momentum operator in quantum mechanics with the framework of position-dependent mass which plays a crucial role in nanomaterials sciences and technologies. We have derived the generalized Schrodinger equation and we have discussed several of its consequences. The generalized Schrodinger equation may describe a large number of quantum mechanical problems related to semiconductors and materials sciences. We have analyzed several problems depending on the forms of the position-dependent mass and the generalized momentum operator. For particular cases, it was observed that free particles behave as oscillators with energy levels less than the standard result which corresponds to the harmonic oscillator. The generalized Schrodinger equation is suitable also to describe quantum damping oscillators. We have also analyzed the dynamics of enormously slowly growing exponentially position-dependent electrons mass moving in a weak periodic potential and we have derived the consequent energy levels in particular for large lattice's length. The corresponding energy levels are enhanced and separated into allowed bands divided by band gaps. This result is in agreement with recent observations which are related to the presence of weak disorder and nonlinearities. Further details are analyzed accordingly.