Abstract
We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.
Highlights
Introduction and Some PreliminariesModeling real-life problems with fractional differential equations (FDE) has a significant role in recent years
In the last decades, new fractional derivative operators have been defined by using an exponential kernel called Caputo–Fabrizio (CF) [10] and the Mittag–Leffler kernel called Atangana–Baleanu (AB) [11]
Caputo and Fabrizio have given a different perspective to fractional operators by introducing a new fractional operator without a singular kernel
Summary
Modeling real-life problems with fractional differential equations (FDE) has a significant role in recent years. Dieudonne [35] provided the first example of a continuous map from an infinitely dimensional Banach space c0 for which there is no solution to the related Cauchy problem in Equation (1). Benchohra and Seba [39] studied the existence of solutions in Banach spaces for a class of initial value problems In these mentioned studies, the Caputo derivative is considered. Lv et al [41] employed about a new existence and uniqueness theorem for solutions of a special equation by using a Caputo fractional derivative in a Banach space. After giving preliminary material, we obtained EUC of solutions of the following Cauchy problem with nonlocal initial conditions (nonlocal Cauchy problem) which includes the Caputo–Fabrizio operator in a Banach space E.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
