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 chapter, we shall establish sufficient conditions for the existence of mild, extremal mild, integral, and extremal integral solutions for some impulsive semi-linear neutral functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators.

Impulsive Functional Differential Inclusions with Unbounded Delay

Existence of Concave Positive Solutions for Nonlinear Fractional Differential Equation with p-Laplacian Operator

In this paper we prove the existence of a mild solution for a impulsive semilinear evolution differential inclusions with infinite delay in a separable Banach space.

Impulsive semilinear evolution differential inclusions with nonconvex right hand side

This paper deals with some existence results for a class of k-generalized ψ-Hilfer implicit fractional differential equations in b-metric spaces. The results are based on the α-φ-Geraghty type contraction and the fixed point theory. We illustrate our results by an example in the last section.

Functional k-Generalized Ψ-Hilfer Fractional Differential Equations in b-Metric Spaces

In this chapter, we shall prove the existence of solutions of some classes of functional differential equations and inclusions. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis \(\mathbb{R}.\) We will use some fixed point theorems combined with the semigroup theory.

Mouffak Benchohra

Papers

Impulsive Functional Differential Inclusions with Unbounded Delay

Existence of Concave Positive Solutions for Nonlinear Fractional Differential Equation with p-Laplacian Operator

Impulsive semilinear evolution differential inclusions with nonconvex right hand side

Functional k-Generalized Ψ-Hilfer Fractional Differential Equations in b-Metric Spaces

Functional Differential Equations and Inclusions with Delay