There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

In this article, we give sucient conditions for the existence of solutions for an integral equation of fractional order with multiple time delays in Banach spaces. Our main tool is a fixed point theorem of Monch type associated with measures of noncompactness. Our results are illustrated by an example.

/pdf/integral-equations-of-fractional-order-with-multiple-time-1jn4k2miq4.pdf

Integral equations of fractional order with multiple time delays in banach spaces

The idea of fractional calculus and fractional order differential equations and inclusions has been a subject of interest not only among mathematicians, but also among physicists and engineers. Fractional calculus techniques are widely used in rheology, control, porous media, viscoelasticity, electrochemistry, electromagnetism, etc. [18, 21, 26, 28, 30]. There has been a significant development in ordinary and partial fractional differential equations in recent years; see the monographs of Kilbas et al. [22], Miller and Ross [29], Podlubny [31], Samko et al. [33], the papers of Abbas and Benchohra [2, 3, 4], Abbas et al. [1, 5, 6], Belarbi et al. [7], Benchohra et al. [8, 9, 10], Diethelm [16], Kilbas and Marzan [23], Mainardi [26], Podlubny et al [32], Vityuk and Golushkov [34], Zhang [35] and the references therein.

On the set of solutions of fractional order Riemann–Liouville integral inclusions

We study the existence and the stability of solutions for Riemann-Liouville, Volterra-Stieltjes quadratic delay integral equations of fractional order. Our results are obtained by applying some fixed point theorems.

https://projecteuclid.org/download/pdf_1/euclid.jiea/1383573820

Existence and stability of nonlinear, fractional order Riemann-Liouville Volterra-Stieltjes multi-delay integral equations

In this paper, the concept of upper and lower solutions method combined with the fixed point theorem is used to investigate the existence of oscillatory and nonoscillatory solutions for a class of initial value problem for Caputo–Hadamard impulsive fractional differential inclusions.

/pdf/oscillation-and-nonoscillation-for-caputo-hadamard-impulsive-569qiencre.pdf

Oscillation and nonoscillation for Caputo–Hadamard impulsive fractional differential inclusions

The subjuct of this paper is the existence of solutions for a class of Caputo-Fabrizio fractional differential equations with instantaneous impulses. Our results are based on Schauder's and Monch's fixed point theorems and the technique of the measure of noncompactness. Two illustrative examples are the subject of the last section.

https://www.aimspress.com/data/article/export-pdf?id=5ff832caba35de13a108a874

Mouffak Benchohra

Papers

Integral equations of fractional order with multiple time delays in banach spaces

On the set of solutions of fractional order Riemann–Liouville integral inclusions

Existence and stability of nonlinear, fractional order Riemann-Liouville Volterra-Stieltjes multi-delay integral equations

Oscillation and nonoscillation for Caputo–Hadamard impulsive fractional differential inclusions

Caputo-Fabrizio fractional differential equations with instantaneous impulses