axioms Article θ * -Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations Atiya Perveen 1 , Waleed M. Alfaqih 2 , Salvatore Sessa 3, * and Mohammad Imdad 1   Citation: Perveen, A.; Alfaqih, W.M.; Sessa, S.; Imdad, M. θ*-Weak Contractions and Discontinuity at the Fixed Point with Applications to Matrix and Integral Equations. Axioms 2021, 10, 209. https:// doi.org/10.3390/axioms10030209 Academic Editor: Chiming Chen Received: 30 July 2021 Accepted: 25 August 2021 Published: 31 August 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, Aligarh Muslim University, Aligarh 202002, India; atiyaperveen2@gmail.com (A.P.); mhimdad@gmail.com (M.I.) 2 Department of Mathematics, Hajjah University, Hajjah 1729, Yemen; wmnaalfaqih@myamu.ac.in 3 Dipartimento di Architettura, Università degli Studi di Napoli Federico II, Via Toledo 402, 80134 Napoli, Italy * Correspondence: sessa@unina.it Abstract: In this paper, the notion of θ ∗ -weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan and Rhoades on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well. Keywords: θ*-weak contraction; fixed point; discontinuity at the fixed point; property P; matrix equation; integral equation MSC: 47H10; 54H25 1. Introduction and Preliminaries In order to study the existence of fixed point for discontinuous mappings, Kannan [1] introduced a weaker contraction condition and proved the following theorem: Every self-mapping S defined on a complete metric space ( M, d) satisfying the condi- tion d(Sz, Sw) ≤ β[d(z, Sz)+ d(w, Sw)], where β ∈  0, 1 2  , (1) ∀z, w ∈ M, has a unique fixed point. We refer such a mappings as Kannan type mappings. Reader can find a lot of literature in this conntext. One such type of result can be seen in [2]. In his paper, [3], Rhoades presented 250 contractive definitions (including (1)) and compared them. He found that though most of them do not force the mapping to be continuous in the entire domain but under these definitions, all the mapping are continuous at the fixed point. Rhoades [4] constructed a very fascinating open problem: Open Question 1. Does there exist a contractive definition which is strong enough to ensure the existence and uniqueness of a fixed point but does not force the mapping to be continuous at the underlying fixed point? After more than a decade, Pant [5] was the first to give an answer to this Open Question 1. In other direction, Jleli and Samet [6] introduced another class of mappings and by using it, they defined θ-contractions. Definition 1 ([6–8]). Let θ : (0, ∞) → (1, ∞) be a mapping satisfying the following conditions: Θ1 : θ is non-decreasing; Θ2 : for each sequence {β k }⊂ (0, ∞), lim k→∞ θ ( β k )= 1 ⇐⇒ lim k→∞ β k = 0; Axioms 2021, 10, 209. https://doi.org/10.3390/axioms10030209 https://www.mdpi.com/journal/axioms