Abstract
The purpose of this work is to introduce a new class of implicit relation and implicit type contractive condition in metric spaces under w -distance functional. Further, we derive fixed point results under a new class of contractive condition followed by three suitable examples. Next, we discuss results about weak well-posed property, weak limit shadowing property, and generalized w -Ulam-Hyers stability of the mappings of a given type. Finally, we obtain sufficient conditions for the existence of solutions for fractional differential equations as an application of the main result.
Highlights
Introduction and PreliminariesIn 1996, Kada et al [1] introduced the concept of a w-distance on a metric space and proved a generalized Caristi fixed point theorem, Ekeland’s ε-variational principle, and the nonconvex minimization theorem according to Mizoguchi and Takahashi [2].Definition 1
A function ω : Ξ × Ξ⟶1⁄20,∞Þ is called a w-distance on Ξ if it satisfies the following properties: (W1) ωðθ, μÞ ≤ ωðθ, νÞ + ωðν, μÞ for any θ, ν, μ ∈ Ξ (W2) ω is lower semicontinuous in its second variable, i.e., if θ ∈ Ξ and νn⟶ν ∈ Ξ, ωðθ, νÞ ≤ liminf n⟶∞ωðθ, νnÞ (W3) For each ε > 0, there exists a δ > 0 such that ωðμ, θÞ ≤ δ and ωðμ, νÞ ≤ δ imply dðθ, νÞ ≤ ε
Let ω be a w-distance on a metric space ðΞ, dÞ and fθng be a sequence in Ξ such that for each ε > 0 there exists Nε ∈ N such that m > n > Nε implies ωðθn, θmÞ < ε, i.e., limm,n?∞ωðθn, θmÞ = 0
Summary
In 1996, Kada et al [1] introduced the concept of a w-distance on a metric space and proved a generalized Caristi fixed point theorem, Ekeland’s ε-variational principle, and the nonconvex minimization theorem according to Mizoguchi and Takahashi [2]. Let ω be a w-distance on a metric space ðΞ, dÞ and fθng be a sequence in Ξ such that for each ε > 0 there exists Nε ∈ N such that m > n > Nε implies ωðθn, θmÞ < ε, i.e., limm,n?∞ωðθn, θmÞ = 0. (2) a mapping I is called a Picard operator if there exists u ∈ Ξ such that FixðIÞ = fug and fInxg converges to u, for all x ∈ Ξ (3) [4] a metric space Ξ is said to be I-orbitally complete if every Cauchy sequence contained in Oðx ; IÞ (for some x in Ξ) converges in Ξ (4) a mapping I is said to be orbitally U-continuous if, for some U ⊂ Ξ × Ξ, the following condition holds: for any x ∈ Ξ and a strictly increasing sequence fnig of positive integers lim Ini x = z ∈ Ξ, ð2Þ i⟶∞.
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