mathematics Article An Implicit Hybrid Delay Functional Integral Equation: Existence of Integrable Solutions and Continuous Dependence Ahmed M. A. El-Sayed 1,† , Hind H. G. Hashem 1,† and Shorouk M. Al-Issa 2, * ,†,‡   Citation: El-Sayed, A.M.A.; Hashem, H.H.G.; Al-Issa, S.M. An Implicit Hybrid Delay Functional Integral Equation: Existence of Integrable Solutions and Continuous Dependence. Mathematics 2021, 9, 3234. https://doi.org/10.3390/ math9243234 Academic Editor: Jan Awrejcewicz Received: 1 October 2021 Accepted: 26 November 2021 Published: 14 December 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 Faculty of Science, Alexandria University, Alexandria 21544, Egypt; masayed@alexu.edu.eg (A.M.A.E.-S.); hindhghashem@gmail.com (H.H.G.H.) 2 Department of Mathematics, Faculty of Arts and Sciences, The International University of Beirut, Beirut 1107, Lebanon * Correspondence: shorouk.alissa@liu.edu.lb † These authors contributed equally to this work. ‡ Current address: Department of Mathematics, Faculty of Arts and Sciences, Lebanese International University, Saida 1600, Lebanon. Abstract: In this work, we are discussing the solvability of an implicit hybrid delay nonlinear functional integral equation. We prove the existence of integrable solutions by using the well known technique of measure of noncompactnes. Next, we give the sufficient conditions of the uniqueness of the solution and continuous dependence of the solution on the delay function and on some functions. Finally, we present some examples to illustrate our results. Keywords: measure of noncompactness; darbo fixed point theorem; monotonic integrable solutions; uniqueness of the solution; continuous dependence of solution 1. Introduction The study of implicit differential and integral equations has received much atten- tion over the last 30 years or so. For instance, Nieto et al. [1] have studied IFDE via the Liouville–Caputo Derivative. The integrable solutions of IFDEs has been studied in [2]. Moreover, IFDEs have recently been studied by several researchers; Dhage and Laksh- mikantham [3] have proposed and studied hybrid differential equations. Zhao et al. [4] have worked at hybrid fractional differential equations and expanded Dhage’s approach to fractional order. A fractional hybrid two-point boundary value problem had been studied by Sun et al. [5]. The technique of measure of noncompactness is found to be a fruitful one to obtain the existence results for a variety of differential and integral equations, for example, see [6–14]. Srivastava et al. [15] have studied the existence of monotonic integrable a.e. solution of nonlinear hybrid implicit functional differential inclusions of arbitrary fractional orders by using the measure of noncompactness technique. Here, we investigate the existence of integrable solutions of an implicit hybrid delay functional integral equation x(t) − h(t, x(t)) g(t, x(t)) = f 1  t, x(t) − h(t, x(t)) g(t, x(t)) ,  ϕ(t) 0 f 2 ( t, s, x(s) − h(s, x(s)) g(s, x(s)) ) ds  , t ∈ [0, 1]. (1) where ϕ : [0, 1] → [0, 1], ϕ(t) ≤ t is nondecreasing continuous function. The main tool of our study is the technique of measure noncompactness. Furthermore, we studied the continuous dependence on the delay function ϕ and on the two functions f 1 and f 2 . Our article is organized as follows: In Section 2 we introduce some preliminaries. Existence results are presented in Section 3. Section 4 contains the continuous dependence of the unique solution on the delay function ϕ and of the two functions f 1 and f 2 . Section 5 presents two examples to verify our theorems. Lastly, conclusions are stated. Mathematics 2021, 9, 3234. https://doi.org/10.3390/math9243234 https://www.mdpi.com/journal/mathematics