Best proximity pair theorems for relatively nonexpansive mappings

[EN] Let A, B be nonempty closed bounded convex subsets of a uniformly convex Banach space and T : A∪B → A∪B be a map such that T(A) ⊆ B, T(B) ⊆ A and ǁTx − Tyǁ ≤ ǁx − yǁ, for x in A and y in B. The fixed point equation Tx = x does not possess a solution when A ∩ B = Ø. In such a situation it is natural to explore to find an element x0 in A satisfying ǁx0 − Tx0ǁ = inf{ǁa − bǁ : a ∈ A, b ∈ B} = dist(A,B). Using Zorn’s lemma, Eldred et.al proved that such a point x0 exists in a uniformly convex Banach space settings under the conditions stated above. In this paper, by using a convergence theorem we attempt to prove the existence of such a point x0 (called best proximity point) without invoking Zorn’s lemma.The authors would like to thank the referee for useful comments and suggestions for the improvement of the paper. The first author acknowledges the Council of Scientific and Industrial Research(India) for the financial support provided in the form of a Junior Research Fellowship to carry out this research work.Sankar Raj, V.; Veeramani, P. (2009). Best proximity pair theorems for relatively nonexpansive mappings. Applied General Topology. 10(1):21-28. doi:10.4995/agt.2009.1784.212810