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

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

In this chapter, we are concerned by the existence of mild solutions of functional differential inclusions with delay and multi-valued jumps in a Banach space.

Functional Differential Inclusions with Multi-valued Jumps

This paper deals with the existence of periodic mild solutions for a class of functional evolution inclusions. We use a multivalued fixed point theorem in Banach spaces combined with the technique of measure of noncompactness. We show that the Poincare operator is a condensing operator with respect to Kuratowski’s measure of noncompactness in a determined phase space, and then derive periodic solutions from bounded solutions by using Sadovskii’s fixed point theorem.

Periodic Mild Solutions of Infinite Delay not Instantaneous Impulsive Evolution Inclusions

In this chapter, we study some first order classes of partial functional, neutral functional, integro-differential, and neutral integro-differential evolution equations on a positive line \(\mathbb{R}_{+}\) with local and nonlocal conditions when the historical interval H is bounded, i.e., when the delay is finite. In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on H for r > 0, endowed with the uniform norm topology. Using a recent nonlinear alternative of Leray–Schauder type for contractions in Frechet spaces due to Frigon and Granas combined with the semigroup theory, the existence and uniqueness of the mild solution will be obtained. The method we are going to use is to reduce the existence of the unique mild solution to the search for the existence of the unique fixed point of an appropriate contraction operator in a Frechet space.

Partial Functional Evolution Equations with Infinite Delay

/pdf/caputo-type-modification-of-the-erdelyi-kober-coupled-35l2g1girp.pdf

Mouffak Benchohra

Papers

Uniqueness and Ulam stability for implicit fractional q-difference equations via Picard operators theory

Functional Differential Inclusions with Multi-valued Jumps

Periodic Mild Solutions of Infinite Delay not Instantaneous Impulsive Evolution Inclusions

Partial Functional Evolution Equations with Infinite Delay

Caputo type modification of the Erdélyi-Kober coupled implicit fractional differential systems with retardation and anticipation