On Almost b-Metric Spaces and Related Fixed Point Results

Abstract

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b-metrics to the corresponding b-metrics. Later, we show that this approach can not work for all kinds of contractions. To confirm this, we present a proof in which the contraction condition is such that it cannot be reduced to corresponding b-metrics.

Some New Observations and Results for Convex Contractions of Istratescu’s Type

“…Since the result is similar to Lemma 2 and is valid in r-almost b-metric spaces (see [12]), the r-almost version of Lemma 4 is as follows: let k ∈ N and {ς n } be a sequence in the r-almost b-metric space (Υ, η, s) so that…”

“…Definition 6. ( [12]) Let Υ be a nonempty set and s ≥ 1. Let η ab : Υ × Υ → [0, +∞) and ς, ω, σ, ς n ∈ Υ so that:…”

A Characterization of Quasi-Metric Completeness in Terms of α–ψ-Contractive Mappings Having Fixed Points

“…In their interesting and germinal paper [1], Samet, Vetro, and Vetro obtained various fixed point theorems in terms of α-ψ contractions which allowed them to deduce, in an elegant and direct way, several important and well-known fixed point results from [2][3][4][5]. Many authors have continued the research of this type of contractions and their generalizations in different contexts (see e.g., [6][7][8][9][10][11][12]). Recently, Fulsa and Taş [13] have presented a careful and extensive study for several generalized α-ψ contractions in the realm of quasi-metric spaces.…”

On Fixed-Point Results in Controlled Partial Metric Type Spaces with a Graph

“…[1][2][3]. Many researchers focused on developing new metric spaces and several contractions [4][5][6][7][8][9][10]. There is a large body of work that contributes to the development of this theory.…”