The main aim of this thesis  is to investigate the nuclear structure of proton-rich mirror nuclei in some Calcium isotopes like ³⁴Ca-³⁴Si,³⁵Ca-³⁵P, ³⁶Ca-³⁶S, ³⁷Ca-³⁷Cl, ³⁸Ca-³⁸Ar, and ³⁹Ca-³⁹K  were employed a combination of effective theoretical models. The Skyrme interaction within the Hartree–Fock approximation (HF) was utilized to analyze density distributions and symmetry energy parameters, while the Woods–Saxon potential serves as an effective framework to describe the motion of nucleons within the nucleus. The complementary use of these models enables a more accurate determination of charge and matter radii differences in mirror nuclei, as well as the evaluation of proton skin thickness, which plays a central role in constraining the nuclear EoS for asymmetric nuclear matter. The NuShellX@MSU shell model code (Brown & Rae, 2014) was used to get the Skyrme interactions and the model space, which computes the one-body density matrix (OBDM) components for protons and neutrons in the formalism.The employed Skyrme Skzs parameterization and Woods–Saxon potential to calculate the rms of proton, neutron, matter and charge radii, and occupancies numbers for pairs mirror nuclei. Also the  difference of proton radii between mirror nuclei (R_mirr), neutron and proton skin thickness (R_skin) and mirror charges ( radii which are proportional to the derivative of the nuclear equation of state (EoS) at saturation density with ρ0 is 0.16 nucleons per cubic fermi. And then to calculate the symmetry energy (E_sym), and the slope of the symmetry energy (L).The mirror displacement energy (MDE) is also investigation using the Skyrme Skzs set. All thesis values were compared with the existing data.

The thesis included a comparative theoretical study of the nuclear structure properties of proton-rich nuclei and their mirror-image isotopes, highlighting the effect of Coulomb repulsion and isospin refraction on the surface properties of the nucleus. The results showed systematic differences in the mean radical radii of matter, charge, protons, and neutrons, as well as the thickness of the proton shell, depending on the type of nuclear interaction used. It also demonstrated clear correlations between symmetry energy and its slope at saturation density and some nuclear observations of isospin. The calculations of mirror displacement energies showed good agreement with experimental data, confirming their importance as indirect indicators for studying the nuclear structure of unstable nuclei.

The most important recommendations which the study has come up with

1. Extend the present study to other isotopic chains beyond calcium, particularly in fp-shell regions, to generalize the observed correlations between mirror pairs and symmetry energy.
2. Develop optimized Skyrme parametrizations tailored for proton rich systems, given the strong role of Coulomb repulsion in shaping their nuclear structure.
3. Utilize mirror charge radii differences ( ) and skin thickness (R_skin) as experimental observables to constrain the density dependence of symmetry energy in future nuclear structure studies.
4. Representation of the nuclear structure for some exotic mirror nuclei on the nuclear equation of state.
5. Applied the present framework to study other observables such as transition probabilities and exotic decay modes, which could further validate the theoretical predictions for mirror nuclei systems.

the average obtained  Excellent