Using families of curves to generalize vector fields, the Lie bracket is defined on a metric space, $M$. For $M$ complete, versions of the local and global Frobenius theorems hold, and flows are shown to commute if and only if their bracket is zero. An example is given showing $L^{2}\left( \mathbb{R}\right) $ is controllable by two elementary flows.

We study the action of each subgroup $A$, $N$ and $K$ of the group $SL_2(\mathbb{R})$ for the Clifford algebra $\mathcal{C}\ell(a)$ and calculate their vector fields, using the derived representation of the Lie algebra $sl_{2}$.

In this paper, we prove the generalized Hyers -Ulam -Rassias stability of double derivations on Banach algebras.

Reiterated homogenization of linear elliptic Neuman eigenvalue problems in multiscale perforated domains is considered beyond the periodic setting. The classical periodicity hypothesis on the coefficients of the operator is here substituted on each microscale by an abstract hypothesis covering a large set of concrete behaviors such as the periodicity, the almost periodicity, the weakly almost periodicity and many more besides. Furthermore, the usual double periodicity is generalized by considering a type of structure where the perforations on each scale follow not only the periodic distribution but also more complicated but realistic ones. Our main tool is Nguetseng's Sigma convergence.

The main goal of the paper is to define new type of integrals for vector-valued functions on time scales. This allows to make possible the advantages of dynamic equations also for vector-valued functions i.e. for dynamic modeling in Banach spaces. To do it we define some appropriate integrals for vector-valued functions on time scales and we prove their properties. We emphasize on the particular ones, which are useful for solving dynamic equations in Banach spaces.

We introduce a new property that we call the property (W.T). This property generalizes and unifies the concept of noncompatible mappings due to Jungck (1986), the property (E.A) of Aamri and Moutawakil (2002) and the concept of asymptotically regular maps due to Browder and Petryshyn (1966). We use this concept to prove a general common fixed point for two pairs of compatible maps under a contractive condition of Lipschitz type. We study also the well-posedness of the fixed point problem for these maps. Our main result extends, unifies and improves several recent results involving this Lipschitz type condition which is is not a contractive condition of the classical type. So our work provides some new contributions to the field of metric fixed point theory.

This paper addresses new results on the factorization of the General Heun's operator, extending the investigations performed in previous works [ Applied Mathematics and Computation 141 (2003), 177-184 and 189 (2007), 816-820]. Both generalized $k-$Lamé and $k-$Heun's second order differential equations are considered.

In this paper, using the Leray-Schauder degree theory, the new results on the existence and uniqueness of anti-periodic solutions are established for a kind of nonlinear high order differential equations with multiple deviating arguments of the form \begin{eqnarray*} x^{(n)}(t)+f(t,x^{(n-1)}(t))+\sum_{i=1}^n g_i(t,x(t-\tau_i(t)))=e(t) \end{eqnarray*} Finally, an example is also given to demonstrate the obtaining results.