mathematics Article Some New Observations on Geraghty and ´ Ciri´ c Type Results in b-Metric Spaces Nabil Mlaiki 1 *, Nebo ˇ jša Dedovic 2 , Hassen Aydi 3,4 *, Milanka Gardaševi´ c-Filipovi´ c 5 , Bandar Bin-Mohsin 6 , and Stojan Radenovi´ c 6 1 Department of Mathematics and General Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia 2 Faculty of Agriculture, University of Novi Sad, Trg Dositeja Obradovi´ ca 8, 21000 Novi Sad, Serbia 3 Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse 4000, Tunisia 4 China Medical University Hospital, China Medical University, Taichung 40402, Taiwan 5 School of Computing, Union University, 11000 Belgrade, Serbia 6 Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia * Correspondence: nmlaiki@psu.edu.sa (N.M); hassen.aydi@isima.rnu.tn (H.A) Received: 10 June 2019; Accepted: 15 July 2019; Published: 18 July 2019   Abstract: We discuss recent fixed point results in b-metric spaces given by Pant and Panicker (2016). Our results are with shorter proofs. In addition, for ε ∈ (1, 3], our results are genuine generalizations of ones from Pant and Panicker. Keywords: (Ω, ω)−admissible mappings; b-metric spaces; b-completeness; b-Cauchyness; generalized quasi-contraction MSC: 47H10; 54H25 1. Introduction We start with the following. Definition 1 ([1,2]). Let f be a self-mapping on a metric space ( X, d). For μ ∈ X, take O (μ, n)= {μ, f μ, ..., f n μ} and O (μ, ∞)= {μ, f μ, ..., f n μ, ...} , where n ∈ N. The set O (μ, ∞) is called an orbit of f . Such ( X, d) is said to be f -orbitally complete if each Cauchy sequence in O (μ, ∞) converges in ( X, d) . It is well known that every complete metric space is f -orbitally complete for each self-mapping f on X. Its converse does not hold (see [1,2]). Two very known and important generalizations of the Banach contraction principle [3] obtained by ´ Ciri´ c[1] and Geraghty [4] as follows: Theorem 1 ([1]). Let ( X, d) be an f -orbitally complete metric space and f : X → X be a quasi-contraction, i.e., there is λ ∈ [0, 1) so that d ( f μ, f τ) ≤ λ · max {d (μ, τ) , d (μ, f μ) , d (τ, f τ) , d (μ, f τ) , d ( f μ, τ)} , (1) for all μ, τ ∈ X. Then, f possesses a unique fixed point. Mathematics 2019, 7, 643; doi:10.3390/math7070643 www.mdpi.com/journal/mathematics