Research Article Qualitative Analysis of Implicit Dirichlet Boundary Value Problem for Caputo-Fabrizio Fractional Differential Equations Rozi Gul, 1 Muhammad Sarwar , 1 Kamal Shah, 1 Thabet Abdeljawad , 2,3,4 and Fahd Jarad 5 1 Department of Mathematics, University of Malakand, Chakdara Dir(L), Pakistan 2 Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia 3 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 4 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan 5 Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Ankara 06790, Turkey Correspondence should be addressed to Muhammad Sarwar; sarwarswati@gmail.com and Thabet Abdeljawad; tabdeljawad@psu.edu.sa Received 16 June 2020; Accepted 23 September 2020; Published 23 November 2020 Academic Editor: Adrian Petrusel Copyright © 2020 Rozi Gul 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. This article studies a class of implicit fractional differential equations involving a Caputo-Fabrizio fractional derivative under Dirichlet boundary conditions (DBCs). Using classical fixed-point theory techniques due to Banch’s and Krasnoselskii, a qualitative analysis of the concerned problem for the existence of solutions is established. Furthermore, some results about the stability of the Ulam type are also studied for the proposed problem. Some pertinent examples are given to justify the results. 1. Introduction and Preliminaries The concerned area of fractional order differential equations (FODEs) have many concentrations in real-world problems and have paid close attention to numerous researchers in the past few decades [1–5]. The mentioned area has been studied from several aspects, such as the existence and uniqueness of solutions via using the classical fixed-point theory, the numerical analysis, the optimization theory, and also the theory of stability corresponding to various frac- tional differential operators like Caputo, Hamdard, and Riemann-Liouville (we refer few as [6–9]). In the aforemen- tioned operators, there exists a singular kernel. Therefore, recently some authors introduced some new types of frac- tional derivative operators in which they have replaced a sin- gular kernel by a nonsingular kernel. The nonsingular kernel derivative has been proved as a good tool to model real-world problems in different fields of science and engineering [10, 11]. In fractional, it is called nonsingular exponential type or Caputo-Fabrizio fractional differential (CFFD) operator. The CFFD operator introduced two researchers, Caputo and Fabrizio for the first time in 2015 [12]. They replaced the singular kernel in the usual Caputo and Riemann- Liouville derivative by an exponential nonsingular kernel. The new operator of this type was found to be more practical than the usual Caputo and Riemann-Liouville fractional differential operators in some problems (see some detailed references such as [13–15]). Recently, many researchers have studied the existence and uniqueness of the solutions at the initial value problems for FODEs under the said operator. But the investigation has been limited to initial value prob- lems only. On the other hand, boundary value problems have significant applications in engineering and other physical sci- ences during modeling numerous phenomena (we refer to see [16–19]). Furthermore, during optimization and numer- ical analysis of the mentioned problems, researchers need stable results from theoretical as well as practical sides. A sta- ble result may lead us to a stable process. Therefore, the stability theory has also got proper attention during the last many decades. It is well known fact that stability analysis plays an important role. Various stability concepts such as exponential stability, Mittag-Lefler stability and Hayers- Ulam’s stability have been adopted in literature to study the stability of different systems of FODEs. The analysis of Hindawi Journal of Function Spaces Volume 2020, Article ID 4714032, 9 pages https://doi.org/10.1155/2020/4714032