Functional and impulsive differential equations of fractional order. Qualitative analysis and applications (Q2825463)

This book presents an overview of qualitative analysis of various types of non-integer order systems. In particular, the authors focus on the topics such as stability, boundedness and almost periodicity of solutions of functional and impulsive systems with fractional order derivatives. The authors formulate the introduction part with more care and exactly present what they intend to do. Further, the motivation and aim of the investigated research topic is clearly demonstrated with necessary basic information. In my knowledge, the topic under investigation has rich literature and wide open problems. The contents in the book are arranged into four chapters which include an introduction and applications of proposed results. All the analysis and studies are made with the help of Lyapunov theory. Chapter 1 presents some basic ideas about fractional calculus and fractional functional and impulsive differential equations. The fractional derivatives in the sense of Caputo and Riemann-Liouville are clearly explained and the results are derived using the former one.NEWLINENEWLINEChapter 2 is devoted to stability results and boundedness theorems. The Lyapunov stability conditions for functional and impulsive fractional order systems are derived using Mittag-Leffler functions. The practical stability for impulsive control fractional systems and Lipschitz stability of fractional functional differential systems are examined. The main results for fractional order impulsive systems are expressed by using piece-wise continuous vector Lyapunov functions. Further, the stability results for fractional order impulsive functional differential equations are discussed using the method of integral manifolds. In Chapter 3, the almost periodicity results are derived by applying the fractional powers of an infinitesimal generator of an analytic semigroup. The existence theorems are proved for impulsive functional and impulsive integrodifferential equations of fractional order by using fractional Lyapunov direct method.NEWLINENEWLINERobustness conditions are established by taking the uncertainty factors like modeling inaccuracies, unpredictability and other factors. The authors chose to use the Lyapunov-Razumikhin method because of the reason that this technique allows to verify the results with some initial data rather than all the initial data required in other methods. Also due to the Lyapunov-Razumikhin method, the Lyapunov functions not necessarily have any information about the time delays that occurred in the systems, which helps the authors to propose some delay-independent conditions for stability, boundedness and almost periodicity of solutions. Chapter 4 presents the applications of the proposed results in the earlier chapters which include the fractional version of neural network models, some biological models and population dynamics models. The proposed sufficient conditions are examined on the above models and the applicability of the results is extensively illustrated. Finally, references and index are provided. As mentioned earlier the proposed topic has rich literature and the authors tried their level best to cover them from the origin.NEWLINENEWLINEIt seems that the authors documented their recent research problems and results, which are acceptable and have some new contributions. In view of my overall reading and consideration, the topic of the book is an interesting one. Also these topics need much more attention among the researchers to get more significant results in future. This book can be recommended to researchers for further development of fractional order impulsive and functional differential equations.