symmetry S S Article Fixed Points of Proinov E-Contractions Maryam A. Alghamdi 1 , Selma Gulyaz-Ozyurt 2 and Andreea Fulga 3, *   Citation: Alghamdi, M.A.; Gulyaz-Ozyurt S.; Fulga, A. Fixed Points of Proinov E-Contractions. Symmetry 2021, 13, 962. https:// doi.org/10.3390/sym13060962 Academic Editors: Olivia Ana Florea and Ileana Constan¸ ta Ro¸ sca Received: 04 May 2021 Accepted: 25 May 2021 Published: 28 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, College of Science, University of Jeddah, Jeddah 21589, Saudi Arabia; maalghamdi@uj.edu.sa 2 Department of Mathematics, Sivas Cumhuriyet University, Sivas 58140, Turkey; sgulyaz@cumhuriyet.edu.tr or selmagulyaz@gmail.com 3 Department of Mathematics and Computer Sciences, Transilvania University of Brasov, B-dul Eroilor nr.29, 500036 Brasov, Romania * Correspondence: afulga@unitbv.ro Abstract: In this paper, we consider a new type of Proinov contraction on the setting of a symmetrical abstract structure, more precisely, the metric space. Our goal is to expand on some results from the literature using admissible mappings and the concept of E-contraction. The considered examples indicate the validity of the obtained results. Keywords: Proinov type contraction; E-contraction; alpha-admissible 1. Introduction and Preliminaries Fixed point theory is one of the most dynamic research topics of the last two decades. New and interesting results are obtained, following especially two directions: changing the frame (the structure of the abstract space—e.g., b-metric, delta symmetric quasi-metric or non-symmetric metric space, etc.) or changing the property of the operators. The notion of E-contraction was introduced by Fulga and Proca [1]. Later, this concept has been improved by several authors, e.g., [2–4]. Undoubtedly, one of the most interesting, most original, most impressive fixed point theorem published in the last two decades is the result of Proinov [5]. By using certain auxiliary functions, Proinov [5] obtained interesting fixed point theorems that generalize, extend, and unify several recent fixed point results in the literature. In this paper, we shall propose a new type of contraction, namely, Proinov type E-contraction, which combines the Proinov approach and the E-contraction setting. First, we recall the basic results and definitions. Definition 1. Let (X, d) be a metric space and the functions ϑ, θ : (0, ∞) → R. A mapping T : X → X is said to be a Proinov type contraction if ϑ(d(T x, T y)) ≤ θ (d(x, y)), (1) for all x, y ∈ X with d(T x, T y) > 0. Theorem 1 ([5]). Let (X, d) be a complete metric space and T : X → X be a Proinov type contrac- tion, where the functions ϑ, θ : (0, ∞) → R are such that the following conditions are satisfied: (1) ϑ is non-decreasing; (2) θ (s) < ϑ(s) for any s > 0; (3) lim sup s→s 0 + θ (s) < ϑ(s 0 +) for any s 0 > 0, then T admits a unique fixed point. Definition 2. Let (X, d) be a metric space and the functions ϑ, θ : (0, ∞) → R. A mapping T : X → X is said to be a generalized Proinov type contraction if Symmetry 2021, 13, 962. https://doi.org/10.3390/sym13060962 https://www.mdpi.com/journal/symmetry