Abstract
In this paper, we define α -admissible and α - ϕ -fuzzy cone contraction in fuzzy cone metric space to prove some fixed point theorems. Some related sequences with contraction mappings have been discussed. Ultimately, our theoretical results have been utilized to show the existence of the solution to a nonlinear integral equation. This application is also illustrative of how fuzzy metric spaces can be used in other integral type operators.
Highlights
Oner et al [18] introduced the idea of fuzzy cone metric space (FCM space), proved some basic properties, and developed the first version of “Banach contraction principle for fixed point” in FCM spaces which is stated as follows: “let (U, M€ α, ∗) be a complete FCM space in which fuzzy cone contractive sequences are Cauchy and let h: U ⟶ U be a fuzzy cone contractive mapping being a ∈ (0, 1) the contractive constant. en, h has a unique fixed point.”
Ur Rehman et al [19] presented some extended “fuzzy cone Banach contraction results” in FCM spaces for some weaker conditions
The definition of FCM spaces with different contractive conditions has been commonly used
Summary
Let a FM M€ α be triangular in a G-complete FCM space (U, M€ α, ∗) and let h: U ⟶ U be an α-φ-fuzzy cone contractive if the following axioms hold:. Using the right side continuity of a function φ and let the limit l ⟶ + ∞, we obtained the contradiction as follows:. Let a FM M€ α be triangular in a G-complete FCM space (U, M€ α, ∗), and let h: U ⟶ U be an η-admissible. To establish the unique FP of an α-φ-fuzzy cone contraction map, let the hypothesis (H) is given as follows:. ∀l ∈ N, t′ ≫ θ and μl ⟶ z ∈ U as l ⟶ + ∞, α(μl, z, t′) ≤ 1 and α(z, hz, t′) ≤ 1, ∀l ∈ N t′ ≫ θ en, h has a FP in U
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