  We use the correlation relation between the energy emitted by the GRBs in
their prompt phases and the X-ray afterglow fluxes, in an effort to constrain
cosmological parameters and aiming to construct a Hubble diagram at high
redshifts, i.e. beyond those found with Type Ia supernovae.
  We use a sample of 126 \textit{Swift} GRBs, that we have selected among more
than 800 long bursts observed until April 2015. The selection is based on a few
observational constraints: GRB flux higher than 0.4 $photons/cm^2/s$ in the
band 15-150 keV, spectrum fitted with simple power law, redshift accurately
known and given, and X-ray afterglow observed and flux measured.
  The statistical method of maximum likelihood is then used to determine the
best cosmological parameters ($\Omega_M$, $\Omega_{\Lambda}$) that give the
best correlation for two relations: a) the Amati relation (between intrinsic
spectral peak energy $E_{p,i}$ and the equivalent isotropic energy), b) the
Dainotti relation, namely between the X-ray afterglow luminosity $L_X$ and the
break time $T_a$, which is observed in the X-ray flux FX.
  Although the number of GRBs with high redshifts is rather small, and despite
the notable dispersion found in the data, the results we have obtained are
quite encouraging and promising. The results obtained using the Amati relation
are close to those obtained using the Type Ia supernovae, and they appear to
indicate a universe dominated by dark energy. However, those obtained with the
correlation between the break time and the X-ray afterglow luminosity is
consistent with the findings of the WMAP study of the cosmic microwave
background radiation, and they seem to indicate a de Sitter-Einstein universe
dominated by matter.
