We consider the existence of at least one positive solution to the dynamic boundary value problem $$\begin{array}{cccc}\hfill -y^{\Delta \Delta }\left(t\right)& =\lambda f(t,y\left(t\right))\text{,}t\in {[0,T]}_{𝕋}y\left(0\right)\hfill & \hfill ={\int }_{{\tau }_1}^{{\tau }_2}{F}_1(s,y\left(s\right))\Delta sy\left({\sigma }^2\left(T\right)\right)& ={\int }_{{\tau }_3}^{{\tau }_4}{F}_2(s,y\left(s\right))\Delta s,\hfill \end{array}$$ where $𝕋$ is an arbitrary time scale with $0<{\tau }_1<{\tau }_2<{\sigma }^2\left(T\right)$ and $0<{\tau }_3<{\tau }_4<{\sigma }^2\left(T\right)$ satisfying ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, ${\tau }_4\in 𝕋$, and where the boundary conditions at $t=0$ and $t={\sigma }^2\left(T\right)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.

This article presents an alternative approach to statistical moments within non-standard models and by the help of these moments some limit theorems are reformulated and proved.

We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ${}_{\left(\right)}\left({[-1,1]}^r\right)$; (b) there is no continuous linear extension map from ${\Lambda }_{\left(\right)}^{\left(r\right)}$ into ${}_{\left(\right)}\left({ℝ}^r\right)$; (c) under some additional assumption on , there is an explicit extension map from ${}_{\left(\right)}\left({[-1,1]}^r\right)$ into ${}_{\left(\right)}\left({[-2,2]}^r\right)$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.

For a large class of Hardy fields their extensions containing non-${}^{\infty }$-germs are constructed. Hardy fields composed of only non-${}^{\infty }$-germs, apart from constants, are also considered.

We extend a result of M. Tamm as follows:Let $f:A\to ℝ,A\subseteq {ℝ}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\to ℝ,r\in ℝ$. Then there exists $N\in \mathbb{N}$ such that for all $(a,b)\in A$, if $y\mapsto f(a,y)$ is $C^N$ in a neighborhood of $b$, then $y\mapsto f(a,y)$ is real analytic in a neighborhood of $b$.

In [3], J. Chaumat and A.-M. Chollet prove, among other things, a Whitney extension theorem, for jets on a compact subset E of ℝⁿ, in the case of intersections of non-quasi-analytic classes with moderate growth and a Łojasiewicz theorem in the regular situation. These intersections are included in the intersection of Gevrey classes. Here we prove an extension theorem in the case of more general intersections such that every $C^{\infty }$-Whitney jet belongs to one of them. We also prove a linear extension theorem...