Topics in fractional differential equations
type: Item
instance of: scholarly article
title: Topics in fractional differential equations
zbMATH Open document ID: 1273.35001
DOI: 10.1007/978-1-4614-4036-9
author: Said Abbas
author: Mouffak Benchohra
author: G. M. N'Guérékata
published in: Developments in Mathematics
publication date: 2012-05-14T00:00:00Z
review text: The book under review starts with a historical review of the topic in Chapter 1 and proceeds in the second chapter with introductory material such as necessary notations, definitions and some preliminary notions such as the Riemann-Liouville and Caputo fractional derivatives, partial fractional derivatives of order \(\alpha\), properties of set-valued maps; also, fixed point theorems and generalizations of Gronwoll's lemmas for two independent variables and singular kernel are given there. The main aims of the book are initial value problems (IVP) for partial fractional order hyperbolic differential equations and inclusions with Caputo derivative and implicitly given hyperbolic PDEs. Chapter 3 is devoted to IVPs and nonlocal IVPs for hyperbolic PDEs of fractional order. These are the results of solution existence for perturbed and neutral functional PDEs, extremal solutions existence for discontinuous PDEs in Banach algebras with proofs based on various fixed-point theorems. For the proof of solution existence for discontinuous PDEs and for the Darboux problem also the upper and lower solutions technique is used. Further, existence results for the Darboux problem for fractional order IVPs with infinite delay and existence and uniqueness solutions for some classes of fractional order IVPs of hyperbolic and neutral hyperbolic PDEs with state-dependent finite and infinite delays are proved at the usage of the Banach contraction principle and a nonlinear alternative of Leray-Schauder type. In the conclusive section sufficient conditions for the global existence and uniqueness of solutions to four classes of fractional order hyperbolic PDEs are provided. Chapter 4 contains four existence results for some classes of IVPs for partial hyperbolic differential inclusions with Caputo fractional order derivative when the right-hand side is convex and nonconvex valued. Here the nonlinear alternative of Leray-Schauder type and various fixed point theorems for contractive multival…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 6034365
zbMATH Keywords: partial hyperbolic fractional order differential equations and inclusions
zbMATH Keywords: Caputo fractional derivative
zbMATH Keywords: initial value problems
zbMATH Keywords: impulsive equations with and without delays
zbMATH Keywords: implicit differential equations
zbMATH Keywords: fractional order Riemann-Liouville integral equation
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4614-4036-9
OpenAlex ID: W2099721887
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description: scientific article; zbMATH DE number 6034365