X-ray absorption spectra of the oxide systems Ba_{1 - x} K_x BiO₃ and BaPbO₃ above the Bi and Pb L₃ absorption edges were investigated. It was shown that oxygen ions move in a double-well potential and their oscillations are correlated with the charge-carrier movement. The observed breathing-like oxygen vibration in the double-well potential with large amplitude and low frequency causes the strong electron-phonon coupling and high T_c -values for doped BaBiO₃ . Based on the experimental data, a model of the relationship of the electronic and local crystal structures is proposed that is in good agreement with the results from transport measurements, and inelastic neutron and electron scattering, Raman scattering, and photoemission spectroscopy. In the framework of the model, the possible reasons for the superconductivity in perovskite-like oxides are discussed.