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9783110521382

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Table of Content: Introduction1 Fractional difference equations1.1 Note on fractional difference Gronwall inequalities1.1.1 Introduction1.1.2 Caputo like fractional difference1.1.3 Linear fractional difference equation1.1.4 Fractional difference inequalities1.1.5 Conclusion1.2 S-asymptotically periodic solution1.2.1 Introduction1.2.2 Preliminaries1.2.3 Nonexistence of periodic solutions1.2.4 Existence and uniqueness resultsReference2 Fractional integral equations2.1 Abel-type integral equations with weakly singular kernels2.1.1 Introduction2.1.2 Preliminary2.1.3 Existence and uniqueness of nontrivial solution2.1.4 General solutions of Erdelyi-Kober type integral equations2.1.5 Illustrative examples2.2 Quadratic Erdelyi-Kober type integral equations of fractional order2.2.1 Introduction2.2.2 Preliminaries2.2.3 Existence and limit property of solutions2.2.4 Uniqueness and another existence results2.2.5 Applications2.2.6 Conclusion2.3 Functional equations involving Erdelyi-Kober fractional integrals2.3.1 Introduction2.3.2 Preliminary2.3.3 Main results2.3.4 An example2.4 Quadratic Weyl fractional integral equations2.4.1 Introduction2.4.2 Preliminaries2.4.3Some basic properties on Weyl kernel2.4.4 Existence and uniform local attractivity of periodic solutions2.4.5 Example2.4.6 ConclusionReference3 Fractional differential equations3.1 Asymptotically periodic solutions3.1.1 Introduction3.1.2 Preliminaries3.1.3 Nonexistence results for periodic solutions3.1.4 Existence results for asymptotically periodic solutions3.1.5 Further extensions3.1.6 Conclusions3.2 Modified fractional iterative functional differential equations3.2.1 Introduction3.2.2 Notation, definitions and auxiliary facts3.2.3 Existence3.2.4 Data dependence3.2.5 Examples3.3 Ulam-Hyers-Rassias stability for semilinear equation3.3.1 Introduction3.3.2 Surjective linear equations3.3.3 Linear equations with closed ranges3.3.Surjective semilinear equations3.4 Practical Ulam-Hyers-Rassias stability for nonlinear equations3.4.1 Introduction3.4.2 Main results3.4.3 Examples3.5 Ulam-Hyers-Mittag-Leffler stability of fractional delay differential equations3.5.1 Introduction3.5.2 Preliminary3.5.3 Main results3.5.4 Examples3.6 Impulsive problems for fractional differential equations3.6.1 Introduction3.6.2 Preliminaries3.6.3 Existence results for impulsive Cauchy problems3.6.4 Ulam stability results3.6.5 Existence results for impulsive boundary value problems3.6.6 Application3.7 Fractional differential switched systems with coupled nonlocal initialand impulsive conditions3.7.1 Introduction3.7.2 Preliminaries3.7.3 Existence and uniqueness results3.7.4 Existence result via Krasnoselski fixed point theorem3.7.5 Existence result via Leray-Schauderfixed point theorem3.7.6 Existence result for resonant case: Landesman-Lazer conditions3.7.7 Ulam type stability results3. 8 New classes of impulsive fractional differential equations3.8.1 Introduction3.8.2 Linear impulsive fractional Cauchy problem3.8.3 Generalized Ulam-Hyers-Rassias stability concept3.8.4 Main results via fixed point methods3.9 Center stable manifold for planar fractional damped equations3.9.1 Introduction3.9.2 Asymptotic behavior of Mittag-Leffler functions3.9.3 Planar fractional Cauchy problems3.9.4 Center stable manifolds resultReference4 Fractional evolution equations4.1 Alternative results and robustness for periodic problems4.1.1 Introduction4.1.2 Preliminaries4.1.3 Homogeneous periodic boundary value problem4.1.4 Nonhomogeneous periodic boundary value problem4.1.5 Parameter perturbation methods for robustness4.1.6 Example4.2 Abstract Cauchy problem of fractional differential equations4.2.1 Introduction4.2.2 Preliminaries4.2.3 Existence and uniqueness theorems of solutions for Problem (I)4.2.4 Existence and uniqueness theorems of solutions for Problem (II)4.2.5 Existence and uniqueness theorems of solutions for Problem (III)4.3 Nonlocal Cauchy problems involving Volterra-Fredholm type integral operators4.3.1 Introduction4.3.2 Preliminaries4.3.3 Existence of mild solutions4.3.4 An example4.4 Controllability of fractional functional evolution equations of Sobolev type4.4.1 Introduction4.4.2 Preliminaries4.4.3 Characteristic solution operators and their properties4.4.4 Main results4.4.5 An example4.4.6 Conclusions4.5 Relaxed controls for nonlinear fractional impulsive evolution equations4.5.1 Introduction4.5.2 Problem statement4.5.3 Preliminaries4.5.4 Original and relaxed fractional impulsive control systems4.5.5 Properties of relaxed trajectories4.5.6 Example4.5.7 ConclusionsReference5 Fractional inclusions problems5.1 Fractional differential inclusions with anti-periodic conditions5.1.1 Introduction5.1.2 Preliminaries and notations5.1.3 Existence results for (5.1)5.1.4 Existence results for (5.2)5.1.5 Applications to fractional lattice inclusions5.2 Nonlocal Cauchy problems for semilinear fractional differential inclusions5.2.1 Introduction5.2.2 Preliminaries and notations5.2.3 Existence results5.2.4 Examples5.2.5 Conclusions5.3 Nonlocal impulsive fractional differential inclusions with sectorial operators5.3.1 Introduction5.3.2 Preliminaries and notation5.3.3 Existence results for convex case5.3.4 Existence results for nonconvex caseReferenceBibliographyIndex

Table of Content: Introduction 1 Fractional Difference Equations 1.1 Fractional difference Gronwall inequalities 1.1.1 Introduction 1.1.2 Caputo like fractional difference 1.1.3 Linear fractional difference equation 1.1.4 Fractional difference inequalities 1.2 S-asymptotically periodic solutions 1.2.1 Introduction 1.2.2 Preliminaries 1.2.3 Non-existence of periodic solutions 1.2.4 Existence and uniqueness results 2 Fractional Integral Equations 2.1 Abel-type nonlinear integral equations 2.1.1 Introduction 2.1.2 Preliminaries 2.1.3 Existence and uniqueness of non-trivial solution in an order interval 2.1.4 General solutions of Erdelyi-Kober type integral equations 2.1.5 Illustrative examples 2.2 Quadratic Erdelyi-Kober type integral equations of fractional order 2.2.1 Introduction 2.2.2 Preliminaries 2.2.3 Existence and limit property of solutions 2.2.4 Uniqueness and another existence results 2.2.5 Applications 2.3 Fully nonlinear Erdelyi-Kober fractional integral equations 2.3.1 Introduction 2.3.2 Main result 2.3.3 Example 2.4 Quadratic Weyl fractional integral equations 2.4.1 Introduction 2.4.2 Preliminaries 2.4.3 Some basic properties of Weyl kernel 2.4.4 Existence and uniform local attractivity of 2 -periodic solutions 2.4.5 Example 3 Fractional Differential Equations 3.1 Asymptotically periodic solutions 3.1.1 Introduction 3.1.2 Preliminaries 3.1.3 Non-existence results for periodic solutions 3.1.4 Existence results for asymptotically periodic solutions 3.1.5 Further extensions 3.2 Modified fractional iterative functional differential equations 3.2.1 Introduction 3.2.2 Notation, definitions and auxiliary facts 3.2.3 Existence 3.2.4 Data dependence 3.2.5 Examples 3.3 Ulam-Hyers-Rassias stability for semilinear equations 3.3.1 Introduction 3.3.2 Ulam-Hyers-Rassias stability for surjective linear equations on Banach spaces 3.3.3 Ulam-Hyers-Rassias stability for linear equations on Banach spaces with closed ranges 3.3.4 Ulam-Hyers-Rassias stability for surjective semilinear equations between Banach spaces 3.4 Practical Ulam-Hyers-Rassias stability for nonlinear equations 3.4.1 Introduction 3.4.2 Main results 3.4.3 Examples 3.5 Ulam-Hyers-Mittag-Leffler stability of fractional delay differential equations 3.5.1 Introduction 3.5.2 Preliminaries 3.5.3 Main results 3.5.4 Examples 3.6 Nonlinear impulsive fractional differential equations 3.6.1 Introduction 3.6.2 Preliminaries 3.6.3 Existence results for impulsive Cauchy problems 3.6.4 Ulam stability results for impulsive fractional differential equations 3.6.5 Existence results for impulsive boundary value problems 3.6.6 Applications 3.7 Fractional differential switched systems with coupled nonlocal initial and impulsive conditions 3.7.1 Introduction 3.7.2 Preliminaries 3.7.3 Existence and uniqueness result via Banach fixed point theorem 3.7.4 Existence result via Krasnoselskii fixed point theorem 3.7.5 Existence result via Leray-Schauder fixed point theorem 3.7.6 Existence result for the resonant case: Landesman-Lazer conditions 3.7.7 Ulam type stability results 3.8 Not instantaneous impulsive fractional differential equations 3.8.1 Introduction 3.8.2 Framework of linear impulsive fractional Cauchy problem 3.8.3 Generalized Ulam-Hyers-Rassias stability concept 3.8.4 Main results via fixed point methods 3.9 Center stable manifold result for planar fractional damped equations 3.9.1 Introduction 3.9.2 Asymptotic behavior of Mittag-Leffler functions E , 3.9.3 Planar fractional Cauchy problems 3.9.4 Center stable manifold result 3.10 Periodic fractional differential equations with impulses 3.10.1 Introduction 3.10.2 FDE with Caputo derivatives with varying lower limits 3.10.3 FDE with Caputo derivatives with fixed lower limits 3.10.4 Conclusions 4 Fractional Evolution Equations: Continued 4.1 Fractional evolution equations with periodic boundary conditions 4.1.1 Introduction 4.1.2 Homogeneous periodic boundary value problem 4.1.3 Non-homogeneous periodic boundary value problem 4.1.4 Parameter perturbation methods for robustness 4.1.5 Example 4.2 Abstract Cauchy problems for fractional evolution equations 4.2.1 Introduction 4.2.2 Preliminaries 4.2.3 Existence and uniqueness theorems of solutions for Problem (I) 4.2.4 Existence and uniqueness theorems of solutions for Problem (II) 4.2.5 Existence and uniqueness theorems of solutions for Problem (III) 4.3 Nonlocal Cauchy problems for Volterra-Fredholm type fractional evolution equations 4.3.1 Introduction 4.3.2 Preliminaries 4.3.3 Existence of mild solutions 4.3.4 Example 4.4 Controllability of Sobolev type fractional functional evolution equations 4.4.1 Introduction 4.4.2 Preliminaries 4.4.3 Characteristic solution operators and their properties 4.4.4 Main results 4.4.5 Example 4.5 Relaxed controls for nonlinear impulsive fractional evolution equations 4.5.1 Introduction 4.5.2 Problem statement 4.5.3 Original and relaxed fractional impulsive control systems 4.5.4 Properties of relaxed trajectories for fractional impulsive control systems 4.5.5 Example 5 Fractional Differential Inclusions 5.1 Fractional differential inclusions with anti-periodic conditions 5.1.1 Introduction 5.1.2 Preliminaries 5.1.3 Existence results for (5.1) 5.1.4 Existence results for (5.2) 5.1.5 Applications to fractional lattice inclusions 5.2 Nonlocal Cauchy problems for semilinear fractional differential inclusions 5.2.1 Introduction 5.2.2 Preliminaries and notation 5.2.3 Existence results 5.2.4 Examples 5.3 Nonlocal impulsive fractional differential inclusions 5.3.1 Introduction 5.3.2 Preliminaries and notation 5.3.3 Existence results for convex case 5.3.4 Existence results for non-convex case A Appendix A.1 Functional analysis A.1.1 Basic notation and results A.1.2 Banach functional spaces A.1.3 Linear operators A.1.4 Semigroup of linear operators A.1.5 Metric spaces A.2 Fractional differential calculus A.3 Henry-Gronwall's inequality A.4 Measures of noncompactness A.5 Multifunctions Bibliography Index

Author Information

J. Wang, Guizhou University, China; M. Fečkan, Comenius University, Slovakia; M. Pospíšil, Slovak Academy of Sciences, Slovakia.

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