A study of the J/ψππ and J/ψKK¯ systems, treating them as coupled channels, has been made by solving the Faddeev equations, with the purpose of investigating the possibility of generation of the JPC=1−−, Y(4260) resonance due to the interaction between these three mesons. In order to do this, we start by solving the Bethe-Salpeter equation for the two pseudoscalar and for the vector-pseudocalar meson systems using the amplitudes obtained from the lowest order chiral Lagrangians as potentials. With the t-matrices generated from these potentials, which contain the poles of the σ, f0(980) and a0(980) resonances for the pseudoscalar-pseudoscalar system and the pole of the X(3872), alongwith other new charmed resonant states, for the vector-pseudoscalar system, we solve the Faddeev equations. As a result, we get a peak around 4150 MeV with a width ∼ 90 MeV when the invariant mass of the two pseudoscalars is close to that of the f0(980).