The mysterious acceleration of the expansion of the Universe, discovered in 1998, can be described by the cosmological constant introduced by Einstein in 1917. In the quantum cosmology, the existence of a repulsive force is associated with "dark energy" - a hypothetical substance with a negative pressure. But numerous attempts to obtain from the quantum theory the observed value of the cosmological constant lead to values of 120 orders of magnitude larger than the observed value. In the bouncing cosmology, a collapse of the Universe of the previous cycle is supposed, and then its new divergence. We consider the bouncing cosmology in the light of the latest discovery of gravitational waves and a large number of massive black holes, which are a probable candidate for the role of dark matter. When the Universe collapses, a significant part of the gravitational mass of the merging black holes turns into gravitational waves. During mergers, the size of black holes increases. We call the largest of them the Big Black Hole (BBH). In the stage of expansion, the frequency of mergers of black holes will decrease sharply and the reverse process will begin due to the absorption of gravitational waves: the slow growth of the mass of black holes, first of all the BBH, which has the maximum surface. If we assume that gravitational radiation themselves do not generate a gravitational field, then it can be shown that a sharp decrease in the gravitational mass of the Universe causes strong antigravity, which can be responsible for the Big Bang. The gravitational mass of BBH at the stage of expansion of the Universe will grow and the gravitational influence of BBH will stretch a field of galaxies. This stretching effect is described by the appearance of a cosmological constant in the Friedmann equations. An estimated value of the cosmological constant is in good agreement with the observations. A cyclic model of the Universe can be developed on the basis of periodic transformation of the mass of merging black holes into gravitational waves and absorption of the background gravitational radiation by the BBH.