  We present a study of the contribution of antishadowing effects on the gluon
distribution functions $G(x,Q^2)$ in light of the
Gribov-Levin-Ryskin-Mueller-Qiu, Zhu-Ruan-Shen (GLR-MQ-ZRS) nonlinear equation
at small-$x$, where $x$ is the momentum fraction or Bjorken variable and $Q^2$
is the four momentum transfer squared or photon virtuality. In this work, we
have solved the GLR-MQ-ZRS nonlinear equation using Regge like the behavior of
gluons in the kinematic range of $10^{-2}\leq x \leq 10^{-6}$ and $5\,GeV^2\,
\leq Q^2\leq 100\, GeV^2$ respectively. We have obtained the solution of
$G(x,Q^2)$ by considering two particular cases: (a) $\alpha_s$ fixed; and (b)
the leading order QCD dependency of $\alpha_{s}$ on $Q^2$. A comparative
analysis is also performed where we compare the gluon distribution function due
to inclusion of the antishadowing effect with that of the gluon distribution
without including the antishadowing effect. Our obtained results of $G(x,Q^2)$
are compared with NNPDF3.0, CT14 and PDF4LHC. We also compare our results with
the result obtained from the IMParton C++ package. Using the solutions of
$G(x,Q^2)$, we have also predicted $x$ and $Q^2$ evolution of the logarithmic
derivative of proton's $F_2$ structure function i.e. $dF_2 (x,Q^2)/d\ln Q^2$.
We incorporated both the leading order(LO) and next-to-leading order (NLO) QCD
contributions of the gluon-quark splitting kernels, in $dF_2 (x,Q^2)/d\ln Q^2$.
Our result of $dF_2 (x,Q^2)/d\ln Q^2$ agrees reasonably well with the
experimental data recorded by HERA's H1 detector.
