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 present existence results for some classes of initial value problems for fractional order partial hyperbolic differential equations with impulses at fixed or variable times impulses.

https://link.springer.com/content/pdf/10.1007%2F978-1-4614-4036-9_7.pdf

Impulsive Partial Hyperbolic Functional Differential Equations

In the present article, we prove some results concerning the existence of solutions for some classes of Hadamard fractional differential equations with instantaneous and noninstantaneous impulses in Banach spaces. The technique relies on the concept of measure of noncompactness and the fixed point theory.

Measure of noncompactness and impulsive Hadamard fractional implicit differential equations in Banach spaces

In this paper, we study a class of Caputo fractional q-difference inclusions in Banach spaces. We obtain some existence results by using the set-valued analysis, the measure of noncompactness, and the fixed point theory (Darbo and Monch’s fixed point theorems). Finally we give an illustrative example in the last section. We initiate the study of fractional q-difference inclusions on infinite dimensional Banach spaces.

https://www.mdpi.com/2227-7390/8/1/91/pdf

Fractional q-Difference Inclusions in Banach Spaces

We first present several existence results and compactness of solutions set for the following Volterra type integral inclusions of the form: , where , is the infinitesimal generator of an integral resolvent family on a separable Banach space , and is a set-valued map. Then the Filippov's theorem and a Filippov-Wazewski result are proved.

/pdf/some-results-for-integral-inclusions-of-volterra-type-in-joyt496vh6.pdf

Some Results for Integral Inclusions of Volterra Type in Banach Spaces

In this paper we investigate the existence of multiple solutions for first and second order impulsive functional dierential equations with bound- ary conditions. Our main tool is the Leggett and Williams fixed point theorem.

Mouffak Benchohra

Papers

Impulsive Partial Hyperbolic Functional Differential Equations

Measure of noncompactness and impulsive Hadamard fractional implicit differential equations in Banach spaces

Fractional q-Difference Inclusions in Banach Spaces

Some Results for Integral Inclusions of Volterra Type in Banach Spaces

Multiple solutions for nonresonance impulsive functional differential equations.