A comparison of microscopic theories for superconductivity in the limit of strong electron correlations is presented. We consider the results for the two-dimensional t-J model obtained within a projection technique for the Green functions in terms of the Hubbard operators and a slave-fermion representation for the RVB state. It is argued that the latter approach resulting in an odd-symmetry p-wave pairing for fermions is inadequate.

The theory of the superconductivity mediated by kinematic and exchange interactions in t−J and two-band Hubbard models in a paramagnetic state is formulated. The Dyson equations for the matrix Green functions in terms of the Hubbard operators are obtained in the non-crossing approximation. To calculate superconducting T c a numerical solution of self-consistent Eliashberg equations is proposed.

A general theory of superconductivity is formulated within the thermodynamic Green function method for various types of pairing mediated by phonons, spin fluctuations, and strong Coulomb correlations in the Hubbard and t-J models. A rigorous Dyson equation for matrix Green functions is derived in terms of a self-energy as a many-particle Green function. By applying the noncrossing approximation for the self-energy, a closed selfconsistent system of equations is obtained, similar to the conventional Eliashberg equations. A brief discussion of superconductivity mediated by kinematic interaction with an estimation of a superconducting transition temperature in the Hubbard model is given.

The pairing of quasiparticles in a CuO2 plane is studied within a spin polaron formulation of the t-t ′ -J model. Our numerical solution of the Eliashberg equations unambiguously shows d-wave pairing between spin polarons on different sublattices mediated by the exchange of spin-fluctuations, and a strong doping dependence of the quasiparticle bandwidth. The transition temperature Tc is an increasing function of J/t and crosses a maximum at an optimal doping concentration δopt. For the t-J model with J/t = 0.4 we obtain Tc ≃ 0.013t at δopt ≃ 0.2.

The theory of superconducting pairing due to the exchange and kinematical interactions in the t − J model in a paramagnetic state is developed. The Dyson equation for the matrix Green functions in terms of the Hubbard operators is obtained in the noncrossing approximation. The linearized selfconsistent system of Eliashberg equations is proposed to study the temperature and doping dependence of the quasi-particle hole spectrum in the normal state and to calculate the temperature of the superconducting phase transition and the symmetry of the gap function.