We propose a new model of nonlinear electrodynamics with two dimensional parameters. The phenomenon of vacuum birefringence, the principles of causality and unitarity were studied... It was shown that there is no a singularity of the electric field in the center of point-like charges. Corrections to the Coulomb law as $r\rightarrow\infty$ were obtained. Dyonic and magnetized black holes are considered. We show that in the self-dual case, when the electric charge equals the magnetic charge, corrections to Coulomb's law and Reissner$-$Nordstr\"{o}m solutions are absent. The metric function and its asymptotic as $r\rightarrow\infty$ for magnetic black holes were evaluated. We calculate the magnetic mass of the black hole which is finite. The thermodynamic properties and thermal stability of black holes were investigated. We calculated the Hawking temperature, the heat capacity and the Helmholtz free energy of black holes and shown that at some parameters there are second-order phase transitions. It was demonstrated that at some range of parameters the black holes are stable. (read more)