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 ????????????????→∞
????(????????, ????????+1) = ????????????????∈ℕ????(????????, ????????+1
) =
????(????, ????), then ????????????????→∞
????(????????, ????????+1) = ????????????????∈ℕ????(????????, ????????+1
) =
????????????????(????, ????). 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 ????????????????→∞
????(????????, ????????+1) =
????????????????∈ℕ ????(????????, ????????+1
) = ????????????????(????, ????), ????(????, ????) ≤ ????(????, ????????) ≤ ????(????, ????),
????(????, ????????) = ????(????, ????), ????(????????, ????????) ≤ (????(????, ????))
????(????(????,????))
∙
????(????, ????)
1−????(????(????,????)) ≤ (???????????? {????(????, ????),[????(????????, ????) ∙ ????(????????, ????) ∙
???????????? {????(????, ????????), ????(????, ????????)}]
1
2)
????(????(????,????))
∙ ????(????, ????)
1−????(????(????,????))
for all ???? ∈
???? and ???? ∈ ????.