  In this work we study non-singular{ bounce cosmology} in the context of the
Lagrange multiplier generalized $F(R)$ gravity theory of gravity. We specify
our study by using a specific variant form of the well known matter{ bounce
cosmology}, with scale factor $a(t) = \left(a_0t^2 + 1 \right)^n$, and we
demonstrate that for $n < 1/2$, the primordial curvature perturbations are
generated deeply in the contraction era. Particularly, we show explicitly that
the perturbation modes exit the horizon at a large negative time during the
contraction era, which in turn makes the ``low-curvature'' regime, the era for
which the calculations of observational indices related to the primordial power
spectrum can be considered reliable. Using the reconstruction techniques for
the Lagrange multiplier $F(R)$ gravity, we construct the form of effective
$F(R)$ gravity that can realize such a cosmological evolution, and we determine
the power spectrum of the primordial curvature perturbations. Accordingly, we
calculate the spectral index of the primordial curvature perturbations and the
tensor-to-scalar ratio, and we confront these with the latest observational
data. We also address the issue of stability of the primordial metric
perturbations, and to this end, we determine the form of $F(R)$ which realizes
the non-singular cosmology for the whole range of cosmic time $-\infty < t <
\infty$, by solving the Friedmann equations without the ``low-curvature''
approximation. This study is performed numerically though, due to the high
complexity of the resulting differential equations. We also investigate the
energy conditions in the present context. The phenomenology of the non-singular
bounce is also studied in the context of a standard $F(R)$ gravity.
