Multiplicative Cyclic Contraction Mappings Class and Best Proximity Point Theorems

Abstract

Let A and B be non-empty subset of a multiplicative metric space
(????, ????) and ????: ???? ∪ ???? → ???? ∪ ???? be a multiplicative R-cyclic contraction
with respect to ????. Then there exists a sequence {????????}????∈ℕ ⊂ ???? ∪ ???? such
that lim????→∞
????(????????, ????????+1) = ????????????????∈ℕ????(????????, ????????+1
) = ????(????, ????), then
lim????→∞
????(????????, ????????+1) = ????????????????∈ℕ????(????????, ????????+1
) = ????????????????(????, ????). 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 lim????→∞
????(????????, ????????+1) =
????????????????∈ℕ ????(????????, ????????+1
) = ????????????????(????, ????), ????(????, ????) ≤ ????(????, ????????) ≤ ????(????, ????),
????(????, ????????) = ????(????, ????), ????(????????, ????????) ≤ (????(????, ????))
????(????(????,????))
∙
????(????, ????)
1−????(????(????,????)) ≤ (max {????(????, ????),[????(????????, ????) ∙ ????(????????, ????) ∙
min {????(????, ????????), ????(????, ????????)}]
1
2)
????(????(????,????))
∙ ????(????, ????)
1−????(????(????,????))
for all ???? ∈ ????
and ???? ∈ ????.