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Some aspects of generalized modified gravity models

In the present work, we review some general aspects of modified gravity theories, investigating mathematical and physical properties and, more specifically, the feature of viable and realistic models able to reproduce the dark energy epoch and the early-time inflation. We will discuss the black hole solutions in generalized theories of gravity: it is of fundamental interest to understand how properties and laws of black holes in General Relativity can be addressed in the framework of modified theories. In particular, we will discuss the energy issue and the possibility to derive the First Law of thermodynamics from the field equations. Then, in the analysis of cosmological solutions, we will pay a particular attention to the occurrence of finite-time future singularities and to the possibility to avoid them in $\mathcal F(R,G)$-gravity. Furthermore, realistic models of $F(R)$-gravity will be analyzed in the detail. A general feature occurring in matter era will be shown, namely the high derivatives of Hubble parameter may be influenced by the high frequency oscillation of the dark energy and some correction term is required in order to stabilize the theory at high redshift. The inflationary scenario is also carefully analyzed and an unified description of the universe is risen. In the final part of the work, we will have a look at the last developments in modified gravity, namely we will investigate cosmological and black hole solutions in a covariant field theory of gravity and we will introduce the extended "teleparallel" $F(T)$-gravity theories. A nice application to the dark matter problem will be presented.