Best proximity points of cyclic mappings

In this manuscript, we proved that the existence of best proximity points for the cyclic operators T defined on a union of subsets A , B of a uniformly convex Banach space X with T ( A ) ⊂ B , T ( B ) ⊂ A and satisfying the condition ‖ T x − T y ‖ ≤ α 3 [ ‖ x − y ‖ + ‖ T x − x ‖ + ‖ T y − y ‖ ] + ( 1 − α ) diam ( A , B ) for α ∈ ( 0 , 1 ) and ∀ x ∈ A , ∀ y ∈ B , where diam ( A , B ) = inf { ‖ x − y ‖ : x ∈ A , y ∈ B } .