Citation: Nallaselli, G.; Gnanaprakasam, A.J.; Mani, G.; Ege, O.; Santina, D.; Mlaiki, N. A Study on Fixed-Point Techniques under the α--Convex Contraction with an Application. Axioms 2023, 12, 139. https://doi.org/10.3390/ axioms12020139 Academic Editors: Shengda Zeng, Stanislaw Migórski, Yongjian Liu and Hsien-Chung Wu Received: 27 December 2022 Revised: 23 January 2023 Accepted: 28 January 2023 Published: 29 January 2023 Copyright: © 2023 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/). axioms Article A Study on Fixed-Point Techniques under the α--Convex Contraction with an Application Gunasekaran Nallaselli 1 , Arul Joseph Gnanaprakasam 1 , Gunaseelan Mani 2 , Ozgur Ege 3 , Dania Santina 4 and Nabil Mlaiki 4,* 1 Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, Tamil Nadu, India 2 Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India 3 Departmentof Mathematics, Ege University, Bornova, 35100 Izmir, Turkey 4 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia * Correspondence: nmlaiki2012@gmail.com or nmlaiki@psu.edu.sa Abstract: In this paper, we consider several classes of mappings related to the class of α--contraction mappings by introducing a convexity condition and establish some fixed-point theorems for such mappings in complete metric spaces. The present result extends and generalizes the well-known results of Singh, Khan, and Kang (Mathematics, 2018, 6(6), 105), Istr ˘ atescu (Liberta Math., 1981, 1, 151–163), and many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results. Finally, we derive the existence and uniqueness of a solution to an integral equation to support our main result and give a numerical example to validate the application of our obtained results. Keywords: α-admissible; -contraction; fixed point; α ∗ -admissible; α--convex contraction; integral equation 1. Introduction and Preliminaries In recent years, a great number of papers have presented generalizations of the well- known Banach–Picard contraction principle. ˇ Ciri ´ c [1] investigated the generalized contrac- tion and extension of Banach’s contraction to combine x, y, Tx, and Ty of all six possible values for all x, y ∈ X and T a self-mapping on a metric space. In 1982, Istr ˘ atescu [2] pro- posed a generalization of seven contraction principle values by introducing a “convexity” condition for the mapping iterates. He deduced that these conditions might be adapted for other classes of mappings to obtain some extensions of known fixed-point results. Al- ghamdi et al. [3] proved a generalization of the Banach contraction principle to the class of convex contractions in non-normal cone metric spaces. In 2015, Miculescu et al. [4] obtained a generalization of Istr˘ a¸ tescu’s fixed-point theorem concerning convex contractions. In 2017, Miculescu et al. [5] obtained a generalization of Matkowski’s fixed-point theorem and Istr˘ a¸ tescu’s fixed-point theorem of convex contraction of a comparison function φ such that d( f [m] ( x), f [m] (y)) ≤ φ(max d( x, y), d( f ( x), f (y)),..., d( f [m−1] ( x), f [m−1] (y))) for all x, y ∈ X. Latif et al. [6] introduced the new concepts of partial generalized convex contrac- tions and partial generalized convex contractions of order two. Moreover, they established some approximate fixed-point results in a metric space endowed with an arbitrary binary relation and some approximate fixed-point results in a metric space endowed with a graph. In 2022, Latif et al. [7] established fixed points in the setting of metric spaces for generalized multivalued contractive mappings with respect to the w b -distance . In 2013, Miandaragh et al. [8] expanded the concept of convex contractions to generalized convex contractions and generalized convex contractions of order two. In the same year, they proved some approximate fixed-point results in the setting of generalized α-convex contractive mapping Axioms 2023, 12, 139. https://doi.org/10.3390/axioms12020139 https://www.mdpi.com/journal/axioms