Best proximity pair theorems for relatively nonexpansive mappings

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 : 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.

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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

Handbook of metric fixed point theory, Edited by W. A. Kirk and Brailey Sims, Kluwer Acad. Publ., Dordrecht, 2001. MR1904271 (2003b:47002)

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S. Sadiq Basha and P. Veeramani, Best proximity pair theorems for multifunctions with open fiber, J. Approx. Theory. 103 (2000), 119–129. (2000)

S. Singh, B. Watson and P. Srivastava, Fixed point theory and best approximation: the KKM-map principle, Kluwer Acad. Publ., Dordrecht, 1997. MR1483076 (99a:47087)

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How to Cite

[1]
V. Sankar Raj and P. Veeramani, “Best proximity pair theorems for relatively nonexpansive mappings”, Appl. Gen. Topol., vol. 10, no. 1, pp. 21–28, Apr. 2009.

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