Abstract
In this study, we introduce fuzzy weak ϕ -contraction and Suzuki-type fuzzy weak ϕ -contraction and employ these to prove some fuzzy fixed point results for fuzzy mappings in the setting of metric spaces, which is followed by an example to support our claim. Next, we deduce some corollaries and fixed point results for multivalued mappings from our main result. Finally, as an application of our result, we provide the existence of a solution for a Fredholm integral inclusion.
Highlights
Introduction and Preliminaries e idea of fuzzy mapping was inspired by the fuzzy set theory given by Zadeh [1]
It was initiated by Heilpern [2] in 1981, defined to be a mapping from an arbitrary set to a subfamily of fuzzy sets in metric linear spaces
He established a fuzzy expansion of Banach contraction principle
Summary
We define the same class of mappings used in [20, 21]. Let Φ denotes the set of all mappings φ: [0, ∞) ⟶ [0, ∞) satisfying the following:. (b) a Suzuki-type fuzzy weak φ-contraction mapping if the following condition is satisfied: 12pα(μ, Sμ) ≤ d(μ, ]) ⇒ Dα(Sμ, S]) ≤ d(μ, ]) − φ Dα(Sμ, S]), (7). Otherwise, we continue this process and obtain a sequence μn satisfying the following conditions: μn+1 ∈ Sμnα, μn+2 ∈ Sμn+1α,. Let (M, d) be a complete metric space and S: M ⟶ Wα(M) a fuzzy weak φ-contraction, such that for every μ ∈ M, (Sμ)α is closed. En, there exists μ∗ ∈ M, such that μ∗α is a fuzzy fixed point of S, i.e., μ∗α ⊂ Sμ∗. If the fuzzy mapping S is a Suzuki-type fuzzy weak φ-contraction, it immediately satisfies the following (37) contraction condition:. Let (M, d) be a complete metric space and S: M ⟶ Wα(M) a fuzzy mapping satisfying (43). We can obtain the results corresponding to eorem 2
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