  In this work, our focus is on exploring the potential of current GRB
measurements to provide reliable constraints on cosmological model parameters
at high redshift. This work is divided into two parts. First, we calibrate the
Amati relation in a model-independent way by using Hubble parameter
measurements obtained from the differential ages of the galaxies. We further
check if the Amati relation parameters evolve with the GRBs' redshift or not,
using the data of Old Astrophysical Objects. The results indicate that GRBs do
seem to evolve with redshift. In the second part, we test different
cosmological models with the calibrated GRB data obtained by using constant and
dynamical Amati relation. Our results indicate that the present quality of GRB
data is not good enough to put tight constraints on the cosmological
parameters. Hence we perform a joint analysis with the combined data of GRBs
and Type Ia Supernovae (SNe) and find that this can considerably enhance
cosmological constraints in contrast to solely relying on GRBs.
