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

The purpose of this paper is to establish some  types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for a class of implicit fractional-order differential equation.

/pdf/on-stability-for-nonlinear-implicit-fractional-differential-3q35xcosv3.pdf

On stability for nonlinear implicit fractional differential equations

/pdf/impulsive-caputo-fabrizio-fractional-differential-equations-2jzse6j19i.pdf

Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces

In this article, the authors establish sucient conditions for the existence of solutions for a class of boundary value problem for fractional dierential inclusions involving the Caputo fractional derivative and nonlinear integral conditions. Both cases of convex and nonconvex valued right hand sides are considered. The topological structure of the set of solutions also examined.

Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions.

In this paper, we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the Riemann–Liouville derivative. Our approach is based on integrated semigroup theory, the Banach contraction principle, and the nonlinear alternative of Leray–Schauder type.

Existence results for fractional order semilinear functional differential equations with nondense domain

In this paper, we shall establish sufficient conditions for the existence of solutions for a boundary value problem for fractional differential inclusions. Both cases of convex valued and nonconvex valued right hand sides are considered.

Mouffak Benchohra

Papers

On stability for nonlinear implicit fractional differential equations

Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces

Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions.

Existence results for fractional order semilinear functional differential equations with nondense domain

Boundary value problems for differential inclusions with fractional order