##
**Topics in fractional differential equations.**
*(English)*
Zbl 1273.35001

Developments in Mathematics 27. Berlin: Springer (ISBN 978-1-4614-4035-2/hbk; 978-1-4614-4036-9/ebook). xiii, 396 p. (2012).

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 multivalued mappings are applied. Further, the solution existence for functional differential inclusions is deduced using lower and upper solutions technique in combination with the nonlinear alternative of Leray-Schauder type and also the relevant results for IVPs for fractional order hyperbolic and neutral hyperbolic functional differential inclusions with infinite delay. In the conclusive section, the solutions existence for a system of fractional order Riemann-Liouville integral inclusions with two independent variables and multiple time delay is investigated when their right-hand sides are convex and nonconvex valued.

In Chapter 5 existence results are presented for some classes of initial value problems for fractional order partial hyperbolic PDEs with impulses at fixed or variable times impulses. These are two results on existence and uniqueness of solutions at fixed times impulses with extension to nonlocal problems with the usage of the Banach contraction principle and a nonlinear alternative of Leray-Schauder type; two existence and uniqueness results at variable type impulses using the Schaefer fixed-point and the Banach contraction principle; existence of solutions and extremal solutions to fractional order hyperbolic discontinuous PDEs with impulses in Banach algebras under Lipschitz, Carathéodory and certain monotonicity conditions. Then impulsive hyperbolic PDEs with state-dependent finite and infinite delays both with fixed and variable times of impulses follow also based on various fixed point principles. In the last section the solution existence for a class of IVPs for impulsive hyperbolic PDEs is investigated using lower and upper solution technique and the Schauder fixed-point theorem. Chapter 6 is devoted to differential inclusions of fractional order at first when the right-hand-side is compact and convex valued and then nonconvex valued. Further, impulsive hyperbolic differential inclusions with variable times of impulses and the method of upper and lower solutions for hyperbolic differential inclusions with fixed times of impulses follow. Every section of Chapters 5 and 6 contains illustrative examples.

In Chapter 7 implicit functional partial hyperbolic PDEs are presented: two results on existence and uniqueness of solutions based on the Banach contraction principle and nonlinear alternative of Leray-Schauder type; a global uniqueness result using nonlinear Leray-Schauder type alternative for contraction maps on Fréchet spaces; implicit equations with finite and infinite delays and state-dependent delay; existence results for the Darboux problem to fractional order IVPs for implicit impulsive hyperbolic PDEs. In the conclusive section, existence results for implicit impulsive hyperbolic PDEs with state-dependent finite and infinite delays are obtained.

Chapter 8 Is devoted to existence results for some classes of Riemann-Liouville integral equations (IEs) in two variables based on some fixed-point theorems. These are Fredholm type fractional order equations and IEs with multiple time delay, nonlinear quadratic Volterra IEs with asymptotic stability investigation of their solutions, local asymptotic attractivity results for nonlinear quadratic Volterra type Riemann-Liouville integral equations in Banach algebras. Every result in the Chapters 7 and 8 is equipped with illustrative examples.

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 multivalued mappings are applied. Further, the solution existence for functional differential inclusions is deduced using lower and upper solutions technique in combination with the nonlinear alternative of Leray-Schauder type and also the relevant results for IVPs for fractional order hyperbolic and neutral hyperbolic functional differential inclusions with infinite delay. In the conclusive section, the solutions existence for a system of fractional order Riemann-Liouville integral inclusions with two independent variables and multiple time delay is investigated when their right-hand sides are convex and nonconvex valued.

In Chapter 5 existence results are presented for some classes of initial value problems for fractional order partial hyperbolic PDEs with impulses at fixed or variable times impulses. These are two results on existence and uniqueness of solutions at fixed times impulses with extension to nonlocal problems with the usage of the Banach contraction principle and a nonlinear alternative of Leray-Schauder type; two existence and uniqueness results at variable type impulses using the Schaefer fixed-point and the Banach contraction principle; existence of solutions and extremal solutions to fractional order hyperbolic discontinuous PDEs with impulses in Banach algebras under Lipschitz, Carathéodory and certain monotonicity conditions. Then impulsive hyperbolic PDEs with state-dependent finite and infinite delays both with fixed and variable times of impulses follow also based on various fixed point principles. In the last section the solution existence for a class of IVPs for impulsive hyperbolic PDEs is investigated using lower and upper solution technique and the Schauder fixed-point theorem. Chapter 6 is devoted to differential inclusions of fractional order at first when the right-hand-side is compact and convex valued and then nonconvex valued. Further, impulsive hyperbolic differential inclusions with variable times of impulses and the method of upper and lower solutions for hyperbolic differential inclusions with fixed times of impulses follow. Every section of Chapters 5 and 6 contains illustrative examples.

In Chapter 7 implicit functional partial hyperbolic PDEs are presented: two results on existence and uniqueness of solutions based on the Banach contraction principle and nonlinear alternative of Leray-Schauder type; a global uniqueness result using nonlinear Leray-Schauder type alternative for contraction maps on Fréchet spaces; implicit equations with finite and infinite delays and state-dependent delay; existence results for the Darboux problem to fractional order IVPs for implicit impulsive hyperbolic PDEs. In the conclusive section, existence results for implicit impulsive hyperbolic PDEs with state-dependent finite and infinite delays are obtained.

Chapter 8 Is devoted to existence results for some classes of Riemann-Liouville integral equations (IEs) in two variables based on some fixed-point theorems. These are Fredholm type fractional order equations and IEs with multiple time delay, nonlinear quadratic Volterra IEs with asymptotic stability investigation of their solutions, local asymptotic attractivity results for nonlinear quadratic Volterra type Riemann-Liouville integral equations in Banach algebras. Every result in the Chapters 7 and 8 is equipped with illustrative examples.

Reviewer: Boris V. Loginov (Ul’yanovsk)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35R11 | Fractional partial differential equations |

45K05 | Integro-partial differential equations |