We investigate the properties of the Ce isotopes with neutron number N = 60～90 and the propertiesof the heavy nuclei near ~(242)Am within the framework of deformed relativistic mean-field (RMF)theory A systematiccomparison between theoretical results and experimental data is made. The calculated binding energes, two-neutronseparation energies, and two-proton separation energies are in good agreement with experimental ones. The variationtrend of experimental quadrupole deformation parameters on the Ce isotopes can he approximately reproduced by theRMF model. It is found that there exists an abnormally large deformation in the ground state of proton-rich Ce isotopes.This phenomenon can be the general behavior of proton-rich nuclei on the neighboring isotopic chains such as Nd andSm isotopes. For the heavy nuclei near ~(242)Am the properties of the ground state and superdeformed isomeric state canbe approximately reproduced by the RMF model. The mechanism of the appearance of anomalously large deformationor superdeformation is analyzed and its influence on nuclear properties is discussed. Further experiments to study theanomalously large deformation in some proton-rich nuclei are suggested. We investigate the properties of the Ce isotopes with neutron number N = 60-90 and the properties of the heavy nuclei near ~ (242) Am within the framework of deformed relativistic mean-field (RMF) theory A systematic comparison between theoretical results and experimental data is The calculated binding energes, two-proton separation energies are in good agreement with experimental ones. The variationtrend of experimental quadrupole deformation parameters on the Ce isotopes can he approximatelyitarget by the RMF model. It is found that there exists an abnormally large deformation in the ground state of proton-rich Ce isotopes .This phenomenon can be the general behavior of proton-rich nuclei on the various isotopic chains such as Nd andSm isotopes. properties of the ground state and superdeformed isomeric state canbe approximately reproduced by the RMF model. The mechanism of the appearance of anomalousl y large deformationor superdeformation is analyzed and its influence on nuclear properties is discussed. Further experiments to study theanomalously large deformation in some proton-rich nuclei are suggested.