A general class of f (R) gravity models with minimally coupling a nonlocal scalar field is considered. The Ostrogradski representation for nonlocal gravitational models with a quadratic potential and the way of its localization are proposed. We study the action with an arbitrary analytic function F( g ), which has both simple and double roots. The way of localization allows to find particular solutions of nonlocal equations of gravity.
The stability of isotropic cosmological solutions for two-field models in the Bianchi I metrics is considered. We have proved that the sufficient conditions for Lyapunov stability in the Friedmann-Robertson-Walker metric are sufficient for the stability under anisotropic perturbations in the Bianchi I metric as well. The standard way to construct cosmological models with exact solutions in the Friedmann-Robertson-Walker metric is the superpotential method. We have used the superpotential method to construct stable kink-type solutions and obtained conditions on superpotential, which are sufficient conditions for the Lyapunov stability. We analyze the stability of isotropic kink-type solutions for quintom models related to the string field theory.
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