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 we investigate the existence of solutions of a class of partial impulsive hyperbolic differential inclusions involving the Caputo fractional derivative. Our main tools are appropriate fixed point theorems from multivalued analysis.

Impulsive partial hyperbolic differential inclusions of fractional order

This paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

/pdf/existence-and-ulam-stability-for-implicit-fractional-q-462qrhay07.pdf

Existence and Ulam stability for implicit fractional q-difference equations

/pdf/nonlocal-initial-value-problem-for-hybrid-generalized-hilfer-5983d7i0hy.pdf

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

Mouffak BenchohraLaboratory of Mathematics,University of Sidi Bel Abbes,PO Box 89, 22000 Sidi BelAbbes, Algeria.benchohra@yahoo.comOmar BennihiD´epartement demath´ematiques etinformatique,Universit´e de Saida,20000, Saida, Alg´erie.obennihidz@yahoo.frKhalilEzzinbiD´epartement deMath´ematiques,Facult´e des Sciences,Semlalia, B.P.2390,Marrakech, Morocco.ezzinbi@ucam.ac.maABSTRACTIn this paper we provide suﬃcient conditions for the existence and uniqueness of mildsolutions for a class of neutral partial functional diﬀerential equations of fractionalorder with state-dependent delay. The nonlinear alternative of Frigon-Granas type forcontractions maps in Fr´echet spaces combined with α-resolvent family is the main toolin our analysis.RESUMENEn este art´iculo entregamos condiciones suﬁcientes para la existencia y unicidad desolucionesmild parauna clase de ecuaciones diferencialesfuncionales parcialesneutralesde orden fraccionario con retraso dependiente del estado. La alternativa no lineal detipo Frigon-Granas para contracciones en espacios de F´echet combinados con familiasα-resolvente es la herramienta principal en nuestro an´alisis.Keywords and Phrases: Neutral functional diﬀerential equations; Riemman-Liouville fractionalderivative; mild solution; delay; ﬁxed point; α-resolvent; Fr´echet space.2010 AMS Mathematics Subject Classiﬁcation: 26A33; 34A08; 34K37.

/pdf/existence-results-for-some-neutral-partial-functional-2a6j19xuix.pdf

Mouffak Benchohra

Papers

Impulsive partial hyperbolic differential inclusions of fractional order

Existence and Ulam stability for implicit fractional q-difference equations

Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type Fractional Implicit Differential Equations

Existence Results for Some Neutral Partial Functional Differential Equations of Fractional order with State-Dependent Delay

Dynamics and stability of Fredholm type fractional order Hadamard integral equations