Abstract
This paper aims to introduce the new concept of rational type fuzzy-contraction mappings in fuzzy metric spaces. We prove some fixed point results under the rational type fuzzy-contraction conditions in fuzzy metric spaces with illustrative examples to support our results. This new concept will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces. Moreover, we present an application of a nonlinear integral type equation to get the existing result for a unique solution to support our work.
Highlights
In 1965, the theory of fuzzy set was introduced by Zadeh [14]
One direction is the evaluation of test results which is the application of fuzzy logic in the processing of students evaluation; the application is expected to represent the mechanisms of human thought processes capable of resolving the problem of evaluation of students, which can be directly monitored by the teacher
We use the concept of Gregory and Sapena [32] and the “triangular property of fuzzy metric” presented by Bari and Vetro [33] and prove some unique fixed point Journal of Mathematics theorems under the rational type fuzzy-contraction conditions in G-complete FM-spaces with some illustrative examples. is new theory will play a very important role in the theory of fuzzy fixed point results and can be generalized for different contractive type mappings in the context of fuzzy metric spaces
Summary
Definition 1 (see [47]). An operation ∗: [0, 1]2 ⟶ [0, 1] is called a continuous t-norm, if (i) ∗ is commutative, associative, and continuous. (ii) 1 ∗ ξ1 ξ1 and ξ1 ∗ ξ2 ≤ ξ3 ∗ ξ4, whenever ξ1 ≤ ξ3 and ξ2 ≤ ξ4, ∀ξ1, ξ2, ξ3, ξ4 ∈ [0, 1]. Let (U, Mr, ∗ ) be a FM-space, v1 ∈ U, and a sequence (μj) in U is (i) Converges to v1 if ξ ∈ (0, 1) and t > 0∃j1 ∈ N, such that Mr(μj, v1, t) > 1 − ξ, ∀j ≥ j1. (iii) (U, Mr, ∗ ) is complete if every Cauchy sequence is convergent in U. A FM-space (U, Mr, ∗ ) is called G-complete if every G-Cauchy sequence is convergent. Let (U, Mr, ∗ ) be a FM-space and l: U ⟶ U. en, l is said to be fuzzy-contractive if. We present some rational type fixed point results under the rational type fuzzy-contraction conditions in G-complete FM-spaces by using the “triangular property of fuzzy metric.”. In the last section of this paper, we present an integral type application for a unique solution to support our work
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