Abstract
The aim of this work is to establish results in fixed point theory for a pair of fuzzy dominated mappings which forms a rational fuzzy dominated V-contraction in modular-like metric spaces. Some results via a partial order and using the graph concept are also developed. We apply our results to ensure the existence of a solution of nonlinear Volterra-type integral equations.
Highlights
1 Introduction and preliminaries Fixed point theory has a basic role in analysis
Chistyakov [12] developed the idea of modular metric spaces and discussed briefly modular convergence, convex modular, equivalent metrics, abstract convex cones, and metric semigroups
The main idea behind this new concept is the physical interpretation of the modular. We look at these spaces as the nonlinear version of the classical modular spaces
Summary
Definition 1.15 Let A be a nonempty set, ξ : A → W (A) be a fuzzy mapping, M ⊆ A, and α : A × A → [0, ∞). Ξ is called fuzzy α∗-dominated on M, if for all a ∈ M and 0 < β ≤ 1, we have α∗(a, [ξ a]β) = inf{α(a, l) : l ∈ [ξ a]β } ≥ 1. Let θ0 ∈ , α : × → [0, ∞), and S, T : → W ( ) be two fuzzy α∗-dominated mappings on {TS(θn)}. The pair (S, T) is called a rational fuzzy dominated V -contraction, if there exist τ > 0, γ (θ), β(g) ∈ (0, 1] and V ∈ such that τ + V Hu1 [Sθ ]γ (θ), [Tg]β(g). Up(θn, θm) ≤ u1(θn, θn+1) + u1(θn+1, θn+2) + · · · + u1(θm–1, θm)
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