ISSN 1686-0209 Thai Journal of Mathematics Volume 18 Number 4 (2020) Pages 1961–1977 http://thaijmath.in.cmu.ac.th Fixed Point Theorems Concerning Rational Geraghty Contraction in a b-Metric Space Samira Rahrovi and Hossein Piri * Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, 5551761167, Iran e-mail : s.rahrovi@ubonab.ac.ir (S. Rahrovi); h.piri@ubonab.ac.ir (H. Piri) Abstract In this paper, we introduce some new type of rational Geraghty contractive mappings in the setup of complete b-metric spaces and investigate the existence of fixed points for such mappings. We also provide an example to illustrate the presented results. MSC: 74H10; 54H25 Keywords: fixed point; metric space; rational geraghty contraction Submission date: 25.04.2018 / Acceptance date: 26.02.2019 1. Introduction Banach contraction principle has been extended and generalized by many researchers by using different forms of contractive conditions in various spaces (see [1–6] and the references therein) . Some of such generalizations are obtained by contraction conditions containing rational expressions. In this direction, in 1973, Geraghty [7] introduced a contraction in which the contraction constant was replaced by a function having some specific properties. Since then, several papers dealt with fixed point theory for rational Geraghty contractive mappings (see, e.g., [8–13]). Czerwik in [14] introduced the concept of a b-metric space. Since then many math- ematicians done several work on involving fixed point for single-valued and multivalued operators in b-metric spaces (see [14–21] and the references therein). Definition 1.1 ([14]). Let X be a nonempty set and s ≥ 1 be a given real number. A mapping d : X × X → R + is said to be a b-metric if for all x, y, z ∈ X the following conditions are satisfied: (bM 1 ) d(x, y) = 0 if and only if x = y; (bM 2 ) d(x, y)= d(y,x); (bM 3 ) d(x, z) ≤ s[d(x, y)+ d(y,z)]. In this case, the pair (X, d) is called a b-metric space (with constant s). *Corresponding author. Published by The Mathematical Association of Thailand. Copyright c  2020 by TJM. All rights reserved.