The long-time limit behavior of velocity correlation functions (VCFs) for an underdamped Brownian particle in an oscillatory viscoelastic shear flow is investigated using the generalized Langevin equation with a power-law memory kernel. The influence of a fluctuating environment is modeled by an additive external fractional Gaussian noise. The exact expressions of the correlation functions of the fluctuating components of velocity for the Brownian particle in the shear plane have been calculated. Also, the particle's angular momentum is found. It is shown that in a certain region of the system parameters an interplay of the shear flow, memory effects, and external noise can induce a bounded long-time behavior of the VCFs, even in the shear flow direction, where in the case of internal noise the velocity process is subdiffusive, i.e., unbounded in time. Moreover, we find resonant behavior of the VCFs and the angular momentum versus the shear oscillation frequency, implying that they can be efficiently excited by oscillatory shear. The role of the initial positional distribution of particles is also discussed.
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