Best proximity pair theorems for relatively nonexpansive mappings
By V. Sankar Raj, P. Veeramani. 2009
95 citations recorded by the source corpus.
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 = O. 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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/d2c2348bb57400a04f89d706a78d5a2d928cd0ef
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In Applied Mathematics
In Computational Theory and Mathematics
In Geometry and Topology
- Naseer Shahzad
- S. Sadiq Basha
- Nawab Hussain
- Poom Kumam
- G. Jungck
- Wutiphol Sintunavarat
- Ancuţa Mihaela Petric
Citations in the corpus
- Best proximity point theorems for generalized proximal contractions (2012, background): “[16] have established a best proximity point theorem for relatively non-expansive mappings, a different treatment to which has been furnished in [17].”
- Best Proximity Point Results for Modified --Proximal Rational Contractions (2013, unknown): “For more details on this approach, we refer the reader to [5, 7–32].”