We are concerned with the problem of approximating a locally unique solution of a generalized equation using a multipoint method in a Banach spaces. In [9]-[11] the authors showed that the previous method is superquadratically (or cubically) convergent when the second Fréchet derivative satisfies the usual Hölder continuity condition (or center--Hölder continuity condition). Here, we weaken these conditions by using $\omega$--condition (or $\sigma$--condition) on the second derivative introduced by us [1]-[4],[22] (for nonlinear equations), with $\omega$ and $\sigma $ a non--decreasing continuous real functions. We provide also an improvement of the ratio of our algorithm under some $\omega$--center--condition (or $\sigma$--center--condition) and less computational cost.

Let $\mathcal{E}$ be a set of primes with density $\tau > 0$ in the set of primes. Write $\mathcal{A}$ for the set of positive integers composed solely of primes from $\mathcal{E}$. We discuss the distribution of the integers from $\mathcal{A}$ in short intervals, and whether for fixed $k \in \mathbb{Z}$ there are solutions to $m+k = p$ with $m \in \mathcal{A}$, where $p$ denotes a prime, or $m+k=n$ where $n$ has a large prime factor ($>n^{\xi}$ for $\xi > \tfrac{1}{2}$

We give estimates for the error term of the Mertens product over primes in arithmetic progressions of the Bombieri-Vinogradov and Barban-Davenport-Halberstam type.

In this paper, we prove that almost all integers $N$ satisfying $N\equiv 3(mod 24)$ and $5\nmid N$ or $N\equiv 4(mod 24)$ are the sum of three or four almost equal prime squares, respectively.

The class of (PLS)-spaces covers most of the natural spaces of analysis, e. g. the space of real analytic functions, spaces of distributions. We characterize those (PLS)-spaces for which there exists a 'reasonable' (LFS)-topology, i. e. a topology of the inductive limit of a sequence of Fréchet-Schwartz spaces. Then we characterize - in terms of the defining sequence - power series (PLS)-type spaces which satisfy the same condition. It is known that power series (PLS)-type spaces appear naturally as kernels of convolution operators.

In a recent paper, Bugeaud and Dubickas have given an explicit characterisation of a rather remarkable class of transcendental numbers which are exceptional from the perspective of distribution of exponential sequences modulo $1$. Muchbefore, Helson and Kahane, from a completely different point-of-view hadexistentially exhibited another class of exceptional real numbers which conjecturally are either rational or transcendental. Wondering whether these two rather large class of real numbers overlap, we study their distribution functions and our investigation gives the first indication that these two interesting class of real numbers originating from different contexts are most likely different. We also frame a natural conjecture in this set up which would establish the above assertion. Our results can be regarded as the first step towards this conjecture.

In the present paper we characterize the Besov-Morrey spaces and the Triebel-Lizorkin-Morrey spaces in terms of wavelet. As an application we investigate some embedding properties and diversity of the function spaces.

It is known that the isometry equation is stable in Banach spaces. In this paper we investigate stability of isometries in real $p$-Banach spaces, that is Fréchet spaces with $p$-homogenous norms, where $p \in (0,1]$. Let $X,Y$ be $p$-Banach spaces and let $f:X \to Y$ be an {\it $\varepsilon$-isometry}, that is a function such that $|||f(x)-f(y)||-||x-y|| |\leq \varepsilon$ for all $x,y \in X$. We show that if $f$ is a surjective then there exists an affine surjective isometry $U: X \to Y$ and a constant $C_p$ such that $$||f(x)-U(x)||\leq C_p (\varepsilon+\varepsilon^p ||x||^{(1-p)}) for x \in X.$$ We also show that in general the above estimation cannot be improved.