On montre que toute fonction positive de classe $C^{2m}$ définie sur un intervalle de $ℝ$ est somme de deux carrés de fonctions de classe $C^m$. En dimension 2, toute fonction positive $f$ de classe $C^4$ est somme d’un nombre fini de carrés de fonctions de classe $C^2$, pourvu que ses dérivées d’ordre 4 s’annulent aux points où $f$ et ${\nabla }^{2}f$ s’annulent.

Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of $\mathbf{K}$-analytic spaces, called disjoint analytic spaces, and on all pseudocompact spaces. The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either $0,1$, or $\Omega$ (the first uncountable ordinal). In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces with index...

We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.

On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $su{p}_{n\in \mathbb{N},z\in }\left|\right|{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))k^{-1}{z}^k{U}^k\left|\right|<\infty$. (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with Young-Stieltjes...

In this paper, stability of linear neutral systems with distributed delay is investigated. A bounded half circular region which includes all unstable characteristic roots, is obtained. Using the argument principle, stability criteria are derived which are necessary and sufficient conditions for asymptotic stability of the neutral systems. The stability criteria need only to evaluate the characteristic function on a straight segment on the imaginary axis and the argument on the boundary of a bounded...

We study the relation between standard ideals of the convolution Sobolev algebra ${₊}^{\left(n\right)}\left(tⁿ\right)$ and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ${₊}^{\left(n\right)}\left(tⁿ\right)$ with compact and countable hull are standard.

We are concerned with first order set-valued problems with very general boundary value conditions $$\left\{\begin{array}{c}{u}_g^{\text{'}}\left(t\right)\in F(t,u\left(t\right)),{\mu }_g\text{-a.e.}\in [0,T],\hfill \\ L\left(u\right(0),T))=0\hfill \end{array}\right.$$ involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g:[0,T]\to ℝ$, a Carathéodory multifunction $F:[0,T]\times ℝ\to 𝒫\left(ℝ\right)$ and a continuous $L:{ℝ}^2\to ℝ$. Using appropriate notions of lower and upper solutions, we prove the existence of solutions via a fixed point result for condensing mappings. In the periodic single-valued case, combining an existence theory for the linear case with a recent result involving...

We study a stochastic fractional partial differential equations of order $\alpha \&gt;1$ driven by a compensated Poisson measure. We prove existence and uniqueness of the solution and we study the regularity of its trajectories.

Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes.

MSC 2010: 26A33, 34A08, 34K37

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory....