Dynamical ages of the opposite lobes determined {sl independently} of each other suggest that their ratios are between $sim$1.1 to $sim$1.4. Demanding similar values of the jet power and the radio core density for the same GRS, we look for a {sl self
-consistent} solution for the opposite lobes, which results in different density profiles along them found by the fit. A comparison of the dynamical and spectral ages shows that their ratio is between $sim$1 and $sim$5, i.e. is similar to that found for smaller radio galaxies. Two causes of this effect are pointed out.
The dynamical ages of the opposite lobes of selected giant radio sources are estimated using the DYNAGE algorithm of Machalski et al., and compared with their spectral ages estimated and studied by Jamrozy et al. in Paper II. As expected, the DYNAGE
fits give slightly different dynamical ages and other models parameters for the opposite lobes modelled independently each other, e.g. the age ratios are found between ~1.1 to ~1.4. Demanding similar values of the jet power and the radio core density for the same source, we look for a self-consistent solution for the opposite lobes, which results in different density profiles along them found by the fit. We also show that a departure from the equipartition conditions assumed in the model, justified by X-ray observations of the lobes of some nearby radio galaxies, and a relevant variation of the magnetic-field strengths may provide an equalisation of the lobes ages. A comparison of the dynamical and spectral ages shows that a ratio of the dynamical age to the spectral age of the lobes of investigated giant radio galaxies is between ~1 and ~5, i.e. is similar to that found for smaller radio galaxies (e.g. Parma et al. 1999). Supplementing possible causes for this effect already discussed in the literature, like uncertainty of assumed parameters of the model, an influence of a possible departure from the energy equipartition assumption, etc. Arguments are given to suggest that DYNAGE can better take account of radiative effects at lower frequencies than the spectral-ageing analysis.The DYNAGE algorithm is especially effective for sources at high redshifts, for which an intrinsic spectral curvature is shifted to low frequencies.
A problem of the cosmological evolution of the IGM is recalled and a necessity to find distant (z>0.5) giant radio galaxies (GRGs) with the lobe energy densities lower than about 10^{-14} J m^{-3} to solve this problem is emphasized. Therefore we und
ertake a search for such GRGs on the southern sky hemisphere using the SALT. In this paper we present a selected sample of the GRG candidates and the first deep detections of distant host galaxies, as well as the low-resolution spectra of the galaxies identified on the DSS frames. The data collected during the Performance Verification (P-V) phase show that 21 of 35 galaxies with the spectroscopic redshift have the projected linear size greater than 1 Mpc (for H_{0}=71 kmsMpc). However their redshifts do not exceed the value of 0.4 and the energy density in only two of them is less than 10^{-14} J m^{-3}. A photometric redshift estimate of one of them (J1420-0545) suggests a linear extent larger than 4.8 Mpc, i.e. a larger than that of 3C236, the largest GRG known up to now.
