Problems of a perpetually oscillating universe

A model is constructed where a Friedmann universe, when collapsing, passes the region of Planek's dimensions near the classical singularity in the De Sitter state. The model assumes that the condition of the one-loop approximation $\text{R}{}_{\mathrm{\mu \nu \lambda \delta }}\text{· R}{}^{\mathrm{\mu \nu \lambda \delta }}\text{≤}\text{1}\text{l}{}_{\mathrm{<mtext>P1</mtext>}}{}^4$ is a universal law of nature. In the case of a dust-like matter the law restricts the mass density (?) to Planck's density $\text{?}{}_{\mathrm{<mtext>P1</mtext>}}\text{=}\text{c}{}^5\text{khx}{}^2$; $\text{?}{}^2\text{?}{}_{\mathrm{<mtext>P1</mtext>}}{}^2\text{≤ 1}$. In the dust-like model it is assumed that the gravitation constant χ depends on the density as $\text{χ = χ}{}_0\text{φ(}\text{?}{}^2\text{?}{}_{\mathrm{<mtext>P1</mtext>}}\text{?1)}$, the function φ vanishes at ? = ?_P1 so that the matter tensor in the right-hand side of the Einstein equation χT_μ^ν disappears in this limit, and the Friedmann universe becomes a De Sitter universe whose Λ¹ term is written in the form $\text{Λ}{}^1\text{θ(}\text{?}{}^2\text{?}{}_{\mathrm{<mtext>P1</mtext>}}{}^2\text{)}$ and at ? → ?_P1, θ → 1. As kh → 0, the theory becomes classical. Some difficulties of a perpetually oscillating model, namely, entropy increase, mass increase due to particle production, and increase of metric perturbations (appearance of gravitational waves) in the process of collapse, are considered in the framework of the model. Various possibilities of the mathematical apparatus of the theory that naturally involve limitations on the curvature value and, in particular, nonlinear Born-type lagrangians are discussed.