Research Article Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations Saif Ur Rehman 1 and Hassen Aydi 2,3,4 1 Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan 2 Institut Supérieur d’Informatique et des Techniques de Communication, Université de Sousse, H. Sousse, Tunisia 3 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa 4 China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan Correspondence should be addressed to Hassen Aydi; hassen.aydi@isima.rnu.tn Received 28 February 2021; Accepted 22 April 2021; Published 10 May 2021 Academic Editor: Liliana Guran Copyright © 2021 Saif Ur Rehman and Hassen Aydi. 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. This paper is aimed at proving some common fixed point theorems for mappings involving generalized rational-type fuzzy cone- contraction conditions in fuzzy cone metric spaces. Some illustrative examples are presented to support our work. Moreover, as an application, we ensure the existence of a common solution of the Fredholm integral equations: μðτÞ = Ð τ 0 Γðτ, v, μðvÞÞdv and νðτÞ = Ð τ 0 Γðτ, v, νðvÞÞdv, for all μ ∈ U , v ∈ ½0, η, and 0< η ∈ ℝ, where U = Cð½0, η, ℝÞ is the space of all ℝ-valued continuous functions on the interval ½0, η and Γ : ½0, η × ½0, η × ℝ ⟶ ℝ. 1. Introduction In 1922, Banach [1] proved a “Banach contraction principle,” which is stated as follows: “A self-mapping on a complete metric space verifying the contraction condition has a unique fixed point.” This principle plays a very important role in the fixed point theory. A number of researches have generalized it in many directions for single-valued and multivalued map- pings in the context of metric spaces. Some of the findings can be found in [2–13] and the references therein. Currently, the fixed point theory is one of the most interested research areas in the field of mathematics. In the last decades, it has been investigated in many fields, such as game theory, graph theory, economics, computer sciences, and engineering. The theory of fuzzy sets was introduced by Zadeh [14], while the concept of a fuzzy metric space (FM space) was given by Kramosil and Michalek [15]. After that, the stronger form of the metric fuzziness was presented by George and Veeramani in [16]. Later on, in [17], Gregori and Sapena proved some contractive-type fixed point results in complete FM spaces. Some more fixed point results in FM spaces can be found in [18–27] and the references therein. Initially, in 2007, the concept of a cone metric space was reintroduced by Huang and Zhang [28]. They proved some nonlinear contractive-type fixed point results in cone metric spaces. After the publication of this article, a number of researchers have contributed their ideas in cone metric spaces. Some of such works can be found in [29–34] and the references therein. In 2015, the basic concept of a fuzzy cone metric space (FCM space) was given by Öner et al. [35]. They presented some key attributes and a “fuzzy cone Banach contraction theorem” in FCM spaces. Later, Rehman and Li [36] extended and improved a “fuzzy cone Banach contraction theorem” and proved some generalized fixed point theorems in FCM spaces. Some more properties and related fixed point results can be found in [37–47]. The aim of this research work is to establish some rational-type fuzzy cone-contraction results in FCM spaces. We use the concept of [36, 39] and prove some common fixed theorems under generalized rational-type fuzzy cone- contraction conditions in FCM spaces. Some illustrative examples are presented. In the last section, we give an appli- cation of two Fredholm integral equations (FIEs). Hindawi Journal of Function Spaces Volume 2021, Article ID 5527864, 13 pages https://doi.org/10.1155/2021/5527864