MULTIPLICATIVE CYCLIC CONTRACTION MAPPINGS CLASS AND BEST PROXIMITY POINT THEOREMS

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

Abstract

Let A and B be non-empty subset of a multiplicative metric space (𝑋, 𝑑) and 𝑇: 𝐴 βˆͺ 𝐡 β†’ 𝐴 βˆͺ 𝐡 be a multiplicative R-cyclic contraction with respect to 𝛹. Then there exists a sequence {π‘₯𝑛}π‘›βˆˆβ„• βŠ‚ 𝐴 βˆͺ 𝐡 such that limπ‘›β†’βˆž 𝑑(π‘₯𝑛, π‘₯𝑛+1) = π‘–π‘›π‘“π‘›βˆˆβ„•π‘‘(π‘₯𝑛, π‘₯𝑛+1 ) = 𝑑(𝐴, 𝐡), then limπ‘›β†’βˆž 𝑑(π‘₯𝑛, π‘₯𝑛+1) = π‘–π‘›π‘“π‘›βˆˆβ„•π‘‘(π‘₯𝑛, π‘₯𝑛+1 ) = 𝑑𝑖𝑠𝑑(𝐴, 𝐡). The current article 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 limπ‘›β†’βˆž 𝑑(π‘₯𝑛, π‘₯𝑛+1) = π‘–π‘›π‘“π‘›βˆˆβ„• 𝑑(π‘₯𝑛, π‘₯𝑛+1 ) = 𝑑𝑖𝑠𝑑(𝐴, 𝐡), 𝑑(𝐴, 𝐡) ≀ 𝑑(𝑣, 𝑇𝑣) ≀ 𝑑(𝐴, 𝐡), 𝑑(𝑣, 𝑇𝑣) = 𝑑(𝐴, 𝐡), 𝑑(𝑇π‘₯, 𝑇𝑦) ≀ (𝑑(π‘₯, 𝑦)) 𝛹(𝑑(π‘₯,𝑦)) βˆ™ 𝑑(𝐴, 𝐡) 1βˆ’π›Ή(𝑑(π‘₯,𝑦)) ≀ (max {𝑑(π‘₯, 𝑦),[𝑑(𝑇π‘₯, π‘₯) βˆ™ 𝑑(𝑇𝑦, 𝑦) βˆ™ min {𝑑(π‘₯, 𝑇𝑦), 𝑑(𝑦, 𝑇π‘₯)}] 1 2) πœ‘(𝑑(π‘₯,𝑦)) βˆ™ 𝑑(𝐴, 𝐡) 1βˆ’πœ‘(𝑑(π‘₯,𝑦)) for all π‘₯ ∈ 𝐴 and 𝑦 ∈ 𝐡. Read full PDF

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

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