MULTIPLICATIVE CYCLIC CONTRACTION MAPPINGS CLASS AND BEST PROXIMITY POINT THEOREMS

Terrang, A.U, Akinwunmi, S.A, Oyewola, D.O and Bitrus, M.M

Abstract

Let A and B be non-empty subset of a multiplicative metric space (𝑋, 𝑑) and 𝑇: 𝐴 ∪ 𝐵 → 𝐴 ∪ 𝐵 be a multiplicative R-cyclic contraction with respect to 𝛹. Then there exists a sequence {𝑥𝑛}𝑛∈ℕ ⊂ 𝐴 ∪ 𝐵 such that 𝑙𝑖𝑚𝑛→∞ 𝑑(𝑥𝑛, 𝑥𝑛+1) = 𝑖𝑛𝑓𝑛∈ℕ𝑑(𝑥𝑛, 𝑥𝑛+1 ) = 𝑑(𝐴, 𝐵), then 𝑙𝑖𝑚𝑛→∞ 𝑑(𝑥𝑛, 𝑥𝑛+1) = 𝑖𝑛𝑓𝑛∈ℕ𝑑(𝑥𝑛, 𝑥𝑛+1 ) = 𝑑𝑖𝑠𝑡(𝐴, 𝐵). This paper provides solutions to numerous problems in physics, optimization and economics, which can be reduced to finding a common best proximity point of some non-linear operator. We considered the application of cyclic contraction mapping on the multiplicative metric space then we obtain 𝑙𝑖𝑚𝑛→∞ 𝑑(𝑥𝑛, 𝑥𝑛+1) = 𝑖𝑛𝑓𝑛∈ℕ 𝑑(𝑥𝑛, 𝑥𝑛+1 ) = 𝑑𝑖𝑠𝑡(𝐴, 𝐵), 𝑑(𝐴, 𝐵) ≤ 𝑑(𝑣, 𝑇𝑣) ≤ 𝑑(𝐴, 𝐵), 𝑑(𝑣, 𝑇𝑣) = 𝑑(𝐴, 𝐵), 𝑑(𝑇𝑥, 𝑇𝑦) ≤ (𝑑(𝑥, 𝑦)) 𝛹(𝑑(𝑥,𝑦)) ∙ 𝑑(𝐴, 𝐵) 1−𝛹(𝑑(𝑥,𝑦)) ≤ (𝑚𝑎𝑥 {𝑑(𝑥, 𝑦),[𝑑(𝑇𝑥, 𝑥) ∙ 𝑑(𝑇𝑦, 𝑦) ∙ 𝑚𝑖𝑛 {𝑑(𝑥, 𝑇𝑦), 𝑑(𝑦, 𝑇𝑥)}] 1 2) 𝜑(𝑑(𝑥,𝑦)) ∙ 𝑑(𝐴, 𝐵) 1−𝜑(𝑑(𝑥,𝑦)) for all 𝑥 ∈ 𝐴 and 𝑦 ∈ 𝐵. Read full PDF

Keywords: Contraction mapping, Collapse, Height, Waist 2010 Mathematics Subject Classification: 16W2, 10X2, 06F05 and 09F06

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