  We present the effects of nonlinear corrections to the single differential
cross section d\sigma/dQ^2 and the reduced cross section \sigma_r (x,Q^2) for
the neutral current (NC) e-p scattering at the leading order (LO) and the
next-to-leading order (NLO) approximations in perturbative quantum chromo
dynamics (QCD). Technically, based on the double Laplace transform method, we
first derive the effects of the nonlinear corrections to the proton structure
functions F2(x,Q^2) and FL(x,Q^2) and consequently obtain the corresponding
single differential and reduced cross sections. Our results clearly indicate
the consistency of the nonlinear behavior of the quark and gluon distributions
at low x values. Our numerical results ( obtained in a range of the virtuality
8.5 < Q2 < 5000 GeV ^2 and the Bjorken scale 10^(-5)< x < 1) show that the
effects of these nonlinear corrections to the proton structure functions are
more noticeable at x < 0.001 and, to some extent, control the incremental trend
of these functions at low x values. Moreover, a comparison of our numerical
results of the single differential and reduced cross sections at the NLO
approximations with those of H1 collaboration data shows that the nonlinear
corrections increase the accuracy of calculations rather than the linear
calculations at low to moderate Q^2 values for low x values.
