Best proximity pair theorems for relatively nonexpansive mappings

V. Sankar Raj, P. Veeramani

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.


Keywords

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

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References

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1. Best Proximity Points: Optimal Solutions
S. Sadiq Basha
Journal of Optimization Theory and Applications  vol: 151  issue: 1  first page: 210  year: 2011  
doi: 10.1007/s10957-011-9869-4



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