Best proximity points of cyclic mappings

In this manuscript, we proved that the existence of best proximity points for the cyclic operators TT defined on a union of subsets A,BA,B of a uniformly convex Banach space XX with T(A)⊂BT(A)⊂B, T(B)⊂AT(B)⊂A and satisfying the condition ‖Tx−Ty‖≤α3[‖x−y‖+‖Tx−x‖+‖Ty−y‖]+(1−α)diam(A,B) for α∈(0,1)α∈(0,1) and ∀x∈A,∀y∈B, where diam(A,B)=inf{‖x−y‖:x∈A,y∈B}.