Research Article Solvability of Implicit Fractional Order Integral Equation in ℓ p ð1 ≤ p<∞Þ Space via Generalized Darbo’s Fixed Point Theorem Inzamamul Haque , 1 Javid Ali , 1 and M. Mursaleen 1,2 1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India 2 Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan Correspondence should be addressed to M. Mursaleen; mursaleenm@gmail.com Received 8 March 2022; Revised 8 April 2022; Accepted 18 April 2022; Published 20 May 2022 Academic Editor: Reny George Copyright © 2022 Inzamamul Haque et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in ℓ p ð1 ≤ p<∞Þ space. The fundamental tool in the presentation of our proofs is the measure of noncompactness ðmncÞ approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations. 1. Introduction The integral equations have a variety of practical applications in defining specific real-world problems and situations, such as the law of physics, the theory of radioactive transmission, statistical mechanics, and cytotoxic activity (see [1–3]). Fixed point theory and mnc have several applications in solving var- ious types of differential and integral equations [4, 5]. The existence of solutions for the system of integral equations is studied by Aghajani et al. [6]. Mursaleen and Mohiuddin [7] proved the existence theorem for the infinite system of differ- ential equations in the space ℓ p . Banas [8] studied the solution of nonlinear differential and integral equations. Arab et al. [9] proved the existence of functional integral equations using mnc. Also, Çakan [10] proved the existence of nonlinear inte- gral equations in Banach spaces by using the technique of mnc . The notions of α-ψ and β-ψ condensing operators were recently developed in [11], and they were used to show some new fixed point results using the technique of measure of noncompactness. Fractional calculus theory and applications grew rapidly in the nineteenth and early twentieth century, and many contrib- utors provided interpretations for fractional derivatives and integrals. Rezapour et al. [12] studied a fractional-order model for anthrax disease among animals based on the Caputo- Fabrizio derivative. The Erdélyl-Kober fractional integral is utilized in various areas of mathematics, including porous media and viscoelasticity [13–15]. The study of fractional order integral equations has become necessary due to their importance. The Erdélyl-Kober fractional operator was stud- ied by various researchers for differential and integral equa- tions. Darwish [16] studied the existence of a solution for Erdélyl-Kober fractional Urysohn-Volterra quadratic integral equations. Mollapouras and Ostadi [17] investigated the exis- tence and stability of the solution of the functional integral equations of fractional order arising in physics, mechanics, and chemical reactions. Mohammadi et al. [18] and Jleli et al. [19] have been generalized Darbo’s fixed point theorem with the help of a new type of contraction operator. Motivated by these, we have generalized Darbo’s fixed point theorem by Hindawi Journal of Function Spaces Volume 2022, Article ID 1674243, 8 pages https://doi.org/10.1155/2022/1674243