Fractional Difference, Differential Equations, and Inclusions: Analysis and Stability

Saïd Abbas · Bashir Ahmad · Mouffak Benchohra · Abdelkrim Salim

jan 2024 · Elsevier

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The field of fractional calculus (FC) is more than 300 years old, and it presumably stemmed from a question about a fractional-order derivative raised in communication between L'Hopital and Leibniz in the year 1695. This branch of mathematical analysis is regarded as the generalization of classical calculus, as it deals with the derivative and integral operators of fractional order. 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. 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. Some equations include delay effects of finite, infinite, or state-dependent nature. Others are subject to impulsive effect which may be fixed or non-instantaneous. The 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. All the abstract results are illustrated by examples in applied mathematics, engineering, biomedical, and other applied sciences. - Introduces notation, definitions, and foundational concepts of fractional q-calculus - Presents existence and attractivity results for a class of implicit fractional q-difference equations in Banach and Fréchet spaces - Focuses on the study of a class of coupled systems of Hilfer and Hilfer-Hadamard fractional differential equations

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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.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.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.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.