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We study the cosmology of a generalized Galileon field with five covariant Lagrangians in which is replaced by general scalar functions f₈ () (i=1, , 5). For these theories, the equations of motion remain at second order in time derivatives. We constrain the forms of the functions f₈ () from the requirement to possess de Sitter solutions responsible for dark energy. There are two possible choices for power-law functions f₈ (), depending on whether the coupling F () with the Ricci scalar R is independent of or depends on. The former corresponds to the covariant Galileon theory that respects the Galilean symmetry in the Minkowski space-time. For generalized Galileon theories we derive the conditions for the avoidance of ghosts and Laplacian instabilities associated with scalar and tensor perturbations as well as the condition for the stability of de Sitter solutions. We also carry out a detailed analytic and numerical study for the cosmological dynamics in those theories.
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