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 paper, a fixed-point theorem for condensing multivalued maps due to Martelli combined with the concept of upper and lower solutions is used to investigate the existence of solutions for first-order impulsive differential inclusions with nonlinear boundary conditions.

Upper and lower solutions method for first-order impulsive differential inclusions with nonlinear boundary conditions

The well-known Krasnoselskii twin fixed point theorem is used to investigate the existence of mild solutions for first- and second-order impulsive semilinear functional and neutral functional differential equations in Hilbert spaces.

/pdf/multiple-solutions-for-impulsive-semilinear-functional-and-3t5qu9y5mf.pdf

Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space

In this paper, sufficient conditions are provided for the controllability of impulsive fractional evolution inclusions with state-dependent delay in Banach spaces. We used  a fixed-point theorem for condensing maps due to Bohnenblust--Karlin and the theory of semigroup for the achievement of the results. An Illustrative example is presented.

/pdf/controllability-for-impulsive-fractional-evolution-4zete2122g.pdf

Controllability for Impulsive Fractional Evolution Inclusions with State-Dependent Delay

. This paper deals with the existence of solutions for hybrid fractional differential equa- tions involving Caputo fractional derivative of order 2 < ζ (cid:2) 3. We base our arguments on a generalization of Darbo’s ﬁ xed point theorem combined with the approaches related with mea- sures of noncompactness in Banach algebras. To demonstrate the argument, an illustration is provided.

Measure of noncompactness and fractional hybrid differential equations with hybrid conditions

We give some existence results and Ulam stability results for a class of Hadamard-Stieltjes integral equations. We present two results: the first one is an existence result based on Schauder’s fixed point theorem and the second one is about the generalized Ulam-Hyers-Rassias stability.

/pdf/existence-and-stability-of-solutions-for-hadamard-stieltjes-3dsyacjyfx.pdf

Mouffak Benchohra

Papers

Upper and lower solutions method for first-order impulsive differential inclusions with nonlinear boundary conditions

Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space

Controllability for Impulsive Fractional Evolution Inclusions with State-Dependent Delay

Measure of noncompactness and fractional hybrid differential equations with hybrid conditions

Existence and Stability of Solutions for Hadamard-Stieltjes Fractional Integral Equations