Abstract
In this paper, we state and prove a new discrete fractional version of the generalized Gronwall inequality. Based on this, a particular version expressed by means of discrete Mittag-Leer functions is provided. As an application, we prove the uniqueness and obtain an estimate for the solutions of nonlinear delay Caputo fractional difference system. Numerical example is presented to demonstrate the applicability of the main results.
Highlights
The theory of fractional differential equations has been extensively investigated over the last years due to widespread applications in various fields of science and engineering
It has been realized that the discrete analogue of ordinary differential equations has tremendous applications in computational analysis and computer simulations [4]
Few mathematicians have taken the lead to develop the theory of fractional difference equations which is the discrete analogue of fractional differential equations
Summary
The theory of fractional differential equations has been extensively investigated over the last years due to widespread applications in various fields of science and engineering. It has been realized that the discrete analogue of ordinary differential equations has tremendous applications in computational analysis and computer simulations [4]. Gronwall inequality, which is our main concern has been studied for fractional differential equations [18– 22]. The existence and uniqueness of solutions, which is a main application of Gronwall inequality, has been the object of many researchers prior to the study of the qualitative properties for different types of differential or difference equations. There have appeared many results about the existence and uniqueness of solutions for fractional differential equations [26–32]. The authors claim that there is few literature on the existence and uniqueness of solutions [33–38]. In parallel to the development of fractional difference equations in the recent years, we state and prove a new discrete fractional version of the generalized Gronwall inequality. Our result is different and generalize some existing results in the literature
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.
