Future of the universe in modified gravitational theories: Approaching to the finite-time future singularity

We investigate the future evolution of the dark energy universe in modified gravities including $F(R)$ gravity, string-inspired scalar-Gauss-Bonnet and modified Gauss-Bonnet ones, and ideal fluid with the inhomogeneous equation of state (EoS). Modified Friedmann-Robertson-Walker (FRW) dynamics for all these theories may be presented in universal form by using the effective ideal fluid with an inhomogeneous EoS without specifying its explicit form. We construct several examples of the modified gravity which produces accelerating cosmologies ending at the finite-time future singularity of all four known types by applying the reconstruction program. Some scenarios to resolve the finite-time future singularity are presented. Among these scenarios, the most natural one is related with additional modification of the gravitational action in the early universe. In addition, late-time cosmology in the non-minimal Maxwell-Einstein theory is considered. We investigate the forms of the non-minimal gravitational coupling which generates the finite-time future singularities and the general conditions for this coupling in order that the finite-time future singularities cannot emerge. Furthermore, it is shown that the non-minimal gravitational coupling can remove the finite-time future singularities or make the singularity stronger (or weaker) in modified gravity.