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

Elementary Fixed Point Theorems * Theorem of Borsuk and Topological Transversality * Homology and Fixed Points * Leray-Schauder Degree and Fixed Point Index * The Lefschetz-Hopf Theory * Selected Topics * Index

http://ieeexplore.ieee.org/iel5/5/31196/01451889.pdf

Random differential equations in science and engineering

In this work, we use a new random fixed point theorem in vector metric spaces due to Sinacer et al. [M.L. Sinacer et al., Random Oper. Stoch. Equ. 24, 93 (2016)] to prove the existence of solutions and the compactness of solution sets of a random system of fractional differential equations via the Hadamard-type derivative. The existence, modification and stochastically continuity of an M2-solution are also proved.

Random solutions to a system of fractional differential equations via the Hadamard fractional derivative

In this paper, we study the existence of solutions for systems of random semilinear impulsive differential equations. The existence results are established by means of a new version of Perov’s, a nonlinear alternative of Leray-Schauder’s fixed point principles combined with a technique based on vector-valued metrics and convergent to zero matrices. Also, we give a random abstract formulation to Sadovskii’s fixed point theorem in a vector-valued Banach space. Examples illustrating the results are included.

/pdf/random-semilinear-system-of-differential-equations-with-48vfx5i0w7.pdf

Random semilinear system of differential equations with impulses

The authors study the existence of mild solutions to a functional differential equation with delay and random effects. They use a random fixed point theorem with a stochastic domain. An example is included to illustrate their results.

Functional Differential Equations With Delay and Random Effects

We study the existence of mild solutions of a functional evolution equation with infinite delay and random effects. We use a random fixed point theorem with stochastic domain. AMS (MOS) Subject Classification. 34G20, 34K20, 34K30.

Amel Benaissa

Papers

Functional Differential Equations With Delay and Random Effects

Global existence results for functional evolution equations with delay and random effects

Functional delay random semilinear differential equations