Advanced Topics in Fractional Differential Equations: A Fixed Point Approach

Description

This book explores fractional differential equations with a fixed point approach. The authors highlight the existence, uniqueness, and stability results for various classes of fractional differential equations. All of the problems in the book also deal with some form of of the well-known Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. Classical and new fixed point theorems, associated with the measure of noncompactness in Banach spaces as well as several generalizations of the Gronwall's lemma, are employed as tools. The book is based on many years of research in this area, and provides suggestions for further study as well. The authors have included illustrations in order to support the readers’ understanding of the concepts presented.  Includes illustrations in order to support readers understanding of the presented concepts ·         Approaches the topic of fractional differential equations while employing fixed point theorems as tools ·         Presents novel results, which build upon previous literature and many years of research by the authors Preface Contents 1 Introduction [DELETE] 2 Preliminary Background [DELETE] 2.1 Notations and Functional Spaces 2.1.1 Space of Continuous Functions 2.1.2 Spaces of Integrable Functions 2.1.3 Spaces of Continuous Functions with Weight 2.2 Special Functions of the Fractional Calculus 2.2.1 Gamma Function 2.2.2 kk-Gamma and kk-Beta Functions 2.2.3 Mittag-Leffler Function 2.3 Elements from Fractional Calculus Theory 2.3.1 Fractional Integrals 2.3.2 Fractional Derivatives 2.3.3 Necessary Lemmas, Theorems, and Properties 2.4 Grönwall's Lemma 2.5 Kuratowski Measure of Noncompactness 2.6 Fixed Point Theorems 3 Implicit Fractional Differential Equations 3.1 Introduction and Motivations 3.2 Existence and Ulam Stability Results for Generalized Hilfer-Type Boundary Value Problem 3.2.1 Existence Results 3.2.2 Ulam-Hyers-Rassias Stability 3.2.3 An Example 3.3 Existence and Ulam Stability Results for kk-Generalized psiψ-Hilfer Initial Value Problem 3.3.1 Existence Results 3.3.2 Ulam-Hyers-Rassias Stability 3.3.3 Examples 3.4 Existence and Ulam Stability Results for kk-Generalized psiψ-Hilfer Initial Value Problem in Banach Spaces 3.4.1 Existence Results 3.4.2 Ulam-Hyers-Rassias Stability 3.4.3 Examples 3.5 Existence and kk-Mittag-Leffler-Ulam-Hyers Stability Results of kk-Generalized psiψ-Hilfer Boundary Valued Problem 3.5.1 Existence Results 3.5.2 kk-Mittag-Leffler-Ulam-Hyers Stability 3.5.3 Examples 3.6 Notes and Remarks 4 Fractional Differential Equations with Instantaneous Impulses 4.1 Introduction and Motivations 4.2 Existence and Ulam Stability Results for Generalized Hilfer … 4.2.1 Existence Results 4.2.2 Ulam-Hyers-Rassias Stability 4.2.3 Examples 4.3 Existence and Ulam Stability Results … 4.3.1 Existence Results 4.3.2 Ulam-Type Stability 4.3.3 Examples 4.4 Notes and Remarks 5 Fractional Differential Equations with Non-Instantaneous Impulses 5.1 Introduction and Motivations 5.2 Initial Value Problem for Nonlinear Implicit Generalized … 5.2.1 Existence Results 5.2.2 Nonlocal Impulsive Differential Equations 5.2.3 Ulam-Hyers-Rassias Stability 5.2.4 Examples 5.3 Initial Value Problem for Nonlinear Implicit Generalized … 5.3.1 Existence Results 5.3.2 Ulam-Hyers-Rassias Stability 5.3.3 An Example 5.4 Boundary Value Problem for Fractional Order Generalized … 5.4.1 Existence Results 5.4.2 Ulam-Hyers-Rassias Stability 5.4.3 An Example 5.5 Notes and Remarks References Index