  The generalized Langevin equation describes anomalous dynamics. Noise is not
only the origin of uncertainty but also plays a positive role in helping to
detect signal with information, termed stochastic resonance (SR). This paper
analyzes the anomalous resonant behaviors of the generalized Langevin system
with a multiplicative dichotomous noise and an internal tempered Mittag-Leffler
noise. For the system with fluctuating harmonic potential, we obtain the exact
expressions of several SR, such as, the first moment, the amplitude and the
autocorrelation function for the output signal as well as the signal-noise
ratio. We analyze the influence of the tempering parameter and memory exponent
on the bona fide SR and the general SR. Moreover, it is detected that the
critical memory exponent changes regularly with the increase of tempering
parameter. Almost all the theoretical results are validated by numerical
simulations.
