Fractional Difference, Differential Equations, and Inclusions

Analysis and Stability

1st Edition - January 11, 2024
Authors: Saïd Abbas, Bashir Ahmad, Mouffak Benchohra, Abdelkrim Salim
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 2 3 6 0 1 - 3
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 3 6 0 2 - 0

Fractional Difference, Differential Equations, and Inclusions: Analysis and Stability is devoted to the existence and stability (Ulam-Hyers-Rassias stability and asymptotic stabil… Read more

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Fractional Difference, Differential Equations, and Inclusions: Analysis and Stability is devoted to the existence and stability (Ulam-Hyers-Rassias stability and asymptotic stability) of solutions for several classes of functional fractional difference equations and inclusions. Covered equations include delay effects of finite, infinite, or state-dependent nature, and tools used to establish the existence results for the proposed problems include fixed point theorems, densifiability techniques, monotone iterative technique, notions of Ulam stability, attractivity and the measure of non-compactness, as well as the measure of weak noncompactness. The tools of fractional calculus are found to be of great utility in improving the mathematical modeling of many natural phenomena and processes occurring in the areas of engineering, social, natural, and biomedical sciences. All abstract results in the book are illustrated by examples in applied mathematics, engineering, biomedical, and other applied sciences.

Cover image
Title page
Table of Contents
Copyright
Dedication
Biography
Saïd Abbas
Bashir Ahmad
Mouffak Benchohra
Abdelkrim Salim
Preface
1: Introduction
Abstract
References
2: Preliminary background
Abstract
2.1. Notations and definitions
2.2. Elements from fractional calculus theory
2.3. Fractional q-calculus
2.4. Multi-valued analysis
2.5. Measure of noncompactness
2.6. Measure of weak noncompactness
2.7. Degree of nondensifiability
2.8. Some attractivity concepts
2.9. Some Ulam stability concepts
2.10. Some fixed-point theorems
2.11. Auxiliary lemmas
References
3: Caputo fractional difference equations in Banach spaces
Abstract
3.1. Implicit fractional q-difference equations in Banach spaces
3.2. Fractional q-difference equations on the half line
3.3. On the solution of Caputo fractional q-difference equations in Banach spaces
3.4. Notes and remarks
References
4: Caputo fractional difference inclusions
Abstract
4.1. Fractional q-difference inclusions in Banach spaces
4.2. Weak solutions for Pettis fractional q-difference inclusions
4.3. Upper and lower solutions for fractional q-difference inclusions
4.4. Notes and remarks
References
5: Ulam stability for fractional difference equations
Abstract
5.1. Existence and Ulam stability for implicit fractional q-difference equations
5.2. Ulam stability results
5.3. Implicit fractional q-difference equations: analysis and stability
5.4. Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory
5.5. Notes and remarks
References
6: Impulsive fractional difference equations
Abstract
6.1. Impulsive implicit Caputo fractional q-difference equations
6.2. Implicit Caputo fractional q-difference equations with noninstantaneous impulses
6.3. Instantaneous and noninstantaneous impulsive integro-differential equations in Banach spaces
6.4. Notes and remarks
References
7: Coupled fractional difference systems
Abstract
7.1. Implicit coupled Caputo fractional q-difference systems
7.2. Existence and oscillation for coupled fractional q-difference systems
7.3. Notes and remarks
References
8: Coupled Caputo–Hadamard fractional differential systems in generalized Banach spaces
Abstract
8.1. Coupled Caputo–Hadamard fractional differential systems with multipoint boundary conditions
8.2. Random coupled Caputo–Hadamard fractional differential systems with four-point boundary conditions
8.3. Random coupled systems of implicit Caputo–Hadamard fractional differential equations with multi-point boundary conditions
8.4. Notes and remarks
References
9: Coupled Hilfer–Hadamard fractional differential systems in generalized Banach spaces
Abstract
9.1. Coupled Hilfer and Hadamard fractional differential systems
9.2. Coupled Hilfer and Hadamard random fractional differential systems with finite delay
9.3. Random coupled Hilfer and Hadamard fractional differential systems
9.4. Notes and remarks
References
10: Oscillation and nonoscillation results for fractional q-difference equations and inclusions
Abstract
10.1. Oscillation and nonoscillation results for Caputo fractional q-difference equations and inclusions
10.2. Existence and oscillation for coupled fractional q-difference systems
10.3. Notes and remarks
References
11: A Filippov's theorem and topological structure of solution sets for fractional q-difference inclusions
Abstract
11.1. Existence and topological structure of solution sets
11.2. Filippov theorem
11.3. Notes and remarks
References
12: On ψ-Caputo fractional differential equations in Banach spaces
Abstract
12.1. Boundary value problem for fractional differential equations via densifiability techniques
12.2. Application of Meir–Keeler condensing operators
12.3. Notes and remarks
References
13: Ulam stability for ψ-Caputo fractional differential equations and systems
Abstract
13.1. Existence and Mittag–Leffler–Ulam stability of fractional partial differential equations
13.2. Coupled system of fractional differential equations without and with delay in generalized Banach spaces
13.3. Coupled system of nonlinear hyperbolic partial fractional differential equations in generalized Banach spaces
13.4. Notes and remarks
References
14: Monotone iterative technique for ψ-Caputo fractional differential equations
Abstract
14.1. Initial value problem for nonlinear ψ-Caputo fractional differential equations
14.2. Sequential ψ-Caputo fractional differential equations with nonlinear boundary conditions
14.3. Hyperbolic fractional partial differential equation
14.4. Notes and remarks
References
References
References
Index

Saïd Abbas

Dr. Saïd Abbas is a Full Professor at the Department of Mathematics at Tahar Moulay University of Saida, Algeria. Dr. Abbas received his MSc in Functional Analysis from Mostaganem University, Algeria, and his PhD in Differential Equations from Djillali Liabes University of Sidi Bel Abbes, Algeria. His research fields include fractional differential equations and inclusions, evolution equations and inclusions, control theory and applications, and other topics in applied mathematics. Dr. Abbas is the author of Topics in Fractional Differential Equations, Springer; Advanced Functional Evolution Equations and Inclusions, Springer; Fractional Differential Equations and Inclusions: Classical and Advanced Topics, World Scientific; and Implicit Fractional Differential and Integral Equations, DeGruyter.

Affiliations and expertise

Professor, Department of Mathematics, Tahar Moulay University of Saida, Algeria

Bashir Ahmad

Dr. Bashir Ahmad is a Full Professor of Mathematics at King Abdulaziz University, Jeddah, Saudi Arabia. He received his Ph.D. from Quaid-i-Azam University, Islamabad, Pakistan. His research interest includes approximate/numerical methods for nonlinear problems involving a variety of differential equations, existence theory of fractional differential equations, stability and instability properties of dynamical systems, impulsive systems, control theory, mathematical biology, and fluid mechanics. He was honored with “Best Researcher of King Abdulaziz University” award in 2009. He is the Managing Editor of Bulletin of Mathematical Sciences, Editor-in-Chief of the journal Fractional Differential Calculus, and member of editorial boards of several journals. He has been “Highly-Cited Researcher” in the category of Mathematics (Web of Science/Clarivate Analytics) from 2014 to 2019.

Affiliations and expertise

Professor, King Abdulaziz University, Jeddah, Saudi Arabia

Mouffak Benchohra

Dr. Mouffak Benchohra is a Full Professor in the Department of Mathematics, Djillali Liabes University of Sidi Bel Abbes. Dr. Benchohra received his MSc in Nonlinear Analysis from Tlemcen University, Algeria, and his PhD in Mathematics from Djillali Liabes University, Sidi Bel Abbes, Algeria. His research fields include fractional differential equations, evolution equations and inclusions, control theory and applications, and other topics in applied mathematics. Dr. Benchohra is the author of Topics in Fractional Differential Equations, Springer; Advanced Functional Evolution Equations and Inclusions, Springer; Fractional Differential Equations and Inclusions: Classical and Advanced Topics, World Scientific; and Implicit Fractional Differential and Integral Equations, DeGruyter. He is a Highly Cited Researcher in Mathematics from Thompson Reuters (2014) and Clarivate Analytics (2017 and 2018). He is also among the top 2% researchers in the world (2020, 2021, 2022) released by Stanford University. Dr. Benchohra has also been the chair of the Department of Mathematics at Djillali Liabes University, Sidi Bel Abbes, and he serves on the Editorial Board of 10 international journals.

Affiliations and expertise

Professor, Department of Mathematics, Djillali Liabes University of Sidi Bel Abbes, Algeria

Abdelkrim Salim

Dr. Abdelrkim Salim is an Associate Professor at the Faculty of Technology, Hassiba Benbouali University of Chlef, Algeria. Salim received his MSc in functional analysis and differential equations, and his PhD in mathematical analysis and applications from Djillali Liabes University of Sidi Bel Abbes, Algeria. His research fields include fractional differential equations and inclusions, control theory and applications, and other topics in applied mathematics.