alpha-optimal best proximity point result involving proximal contraction mappings in fuzzy metric spaces

Abstract

In this paper, we introduce α-proximal fuzzy contraction of type−I and II in complete fuzzy metric space and obtain some fuzzy proximal and optimal coincidence point results. The obtained results further unify, extend and generalize some already existing results in literature. We also provide some examples which show the validity of obtained results and a comparison is also given which shows that contractive mappings and obtained results further generalizes already existing results in literature. c 2017 all rights reserved.Keywords: Fuzzy metric space, α-proximal fuzzy contraction of type−I, α-proximal fuzzy contraction of type−II, fuzzy expansive mapping, optimal coincidence best proximity point, t-norm. 2010 MSC: 47H10, 47H04, 47H07.

Optimal Approximate Solution of Coincidence Point Equations in Fuzzy Metric Spaces

“…, which is an α-proximal fuzzy contraction of type-I defined in [25]. A similar explanation exist for α-proximal fuzzy contraction of type-II.…”

“…Remark 1. Latif et al [25] defined α-proximal fuzzy contraction of type-I and type-II. If we define η(t) = 1 − ψ(t α ), where ψ ∈ Ψ (as defined in [25]) and α ∈ [0, 1], then η ∈ H. Then, α f -proximal contraction of first and second kind will reduce to α-proximal fuzzy contraction of type-I and type-II in [25].…”

“…Latif et al [25] defined α-proximal fuzzy contraction of type-I and type-II. If we define η(t) = 1 − ψ(t α ), where ψ ∈ Ψ (as defined in [25]) and α ∈ [0, 1], then η ∈ H. Then, α f -proximal contraction of first and second kind will reduce to α-proximal fuzzy contraction of type-I and type-II in [25]. If we take α(x, y, t) = 1 for all x, y ∈ A, t > 0 and η(t) = 1 − ψ(t α ) in Theorem (1), (2) and simplify our results along with some minor conditions on involved mappings, we obtain Theorems 2.2, 3.2, 3.5, and 3.8 in [25].…”

“…Explanation: Take η(t) = 1 − ψ(t α ), where ψ ∈ Ψ (as defined in [25]) and α ∈ (0, 1) in α f -proximal contraction of first kind defined in Equation 2 We have…”

Coincidence Point Results for Multivalued Suzuki Type Mappings Using θ-Contraction in b-Metric Spaces

“…It gets more exposure due to the vast applications of fuzzy metric spaces in controlling the noise in data, smoothing the data, and decision-making, but the authors did not pay attention to study the best proximity point theory in fuzzy metric spaces. In 2012, N. Saleem et al investigated best proximity and coincidence point results in fuzzy metric spaces [10][11][12][13][14][15].…”