Best proximity pair theorems for relatively nonexpansive mappings

V. Sankar Raj; P. Veeramani

doi:10.4995/agt.2009.1784

Authors

V. Sankar Raj Indian Institute of Technology Madras
P. Veeramani Indian Institute of Technology Madras

DOI:

https://doi.org/10.4995/agt.2009.1784

Keywords:

Best proximity pair, Relatively nonexpansive map, Cyclic contraction map, Strictly convex space, Uniformly convex Banach space, Fixed point, Metric projection

Abstract

Let A, B be nonempty closed bounded convex subsets of a uniformly convex Banach space and T : AâˆªB â†’ AâˆªB be a map such that T(A) âŠ† B, T(B) âŠ† A and ǁTx âˆ’ Tyǁ ≤ ǁx âˆ’ yǁ, for x in A and y in B. The fixed point equation Tx = x does not possess a solution when A âˆ© B = Ø. In such a situation it is natural to explore to find an element x0 in A satisfying ǁx0 âˆ’ Tx0ǁ = inf{ǁa âˆ’ bǁ : a âˆˆ A, b âˆˆ 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.

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Author Biographies

V. Sankar Raj, Indian Institute of Technology Madras

Department of Mathematics

P. Veeramani, Indian Institute of Technology Madras

Department of Mathematics

References

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