Vol 13 - À paraître

Avancées en Mathématiques Pures et Appliquées

Articles parus

This paper presents a new generalization of the Genocchi numbers and the Genocchi theorem. As consequences, we obtain some important families of integer-valued polynomials those are closely related to the Bernoulli polynomials. Denoting by $$${(B_n)}_{n \in \mathbb{N}}$$$ the sequence of the Bernoulli numbers and by $$${(B_n(X))}_{n \in \mathbb{N}}$$$ the sequence of the Bernoulli polynomials, we especially obtain that for any natural number $$$n$$$, the reciprocal polynomial of the polynomial $$$\big(B_n(X) - B_n\big)$$$ is integer-valued.

We characterize the weights for which we have the boundedness of standard weighted integral operators induced by the Bergman-Besov kernels acting between two weighted Lebesgue classes on the unit ball of ℂ^N in terms of Békollè - Bonami type condition on the weights. To accomplish this we employ the proof strategy originated by Békollè.

We generalize the well-known transforms such as short-time Fourier transform, wavelet transform and shearlet transform and refer it as Continuous Modulated Shearlet Transform. Important properties like Plancherel formula and inversion formula have been investigated. Uncertainty inequalities associated with this transform are presented.

In this paper, we study a discrete anisotropic Kirchho type problem using variational methods and critical point theory, and we discuss the existence of two solutions for the problem. A example is presented to demonstrate the application of our main results.

This paper is concerned with the existence of solutions for the following class of singular fourth order elliptic equations
$$$ \left\{ \begin{array}{ll} \Delta\Big(｜x｜^{p(x)}｜\Delta u｜^{p(x)-2}\Delta u\Big)=a(x)u^{-\gamma (x)}+\lambda f(x,u),\quad \mbox{in }\Omega, \\ u=\Delta u=0, \quad \mbox{on }\partial\Omega. \end{array} \right.$$$
where $$$\Omega$$$ is a smooth bounded domain in $$$\mathbb{R}^N, \gamma :\overline{\Omega}\rightarrow (0,1)$$$ be a continuous function, $$$f\in C^{1}( \overline{\Omega}\times \mathbb{R}), p:\; \overline{\Omega}\longrightarrow \;(1,\infty)$$$ and $$$a$$$ is a function that is almost everywhere positive in $$$\Omega$$$. Using variational techniques combined with the theory of the generalized Lebesgue-Sobolev spaces, we prove the existence at least one nontrivial weak solution.

This paper mainly concerns about establishing the Bruck conjecture for differential-difference polynomial generated by an entire function. The polynomial considered is of finite order and involves the entire function $$$f(z)$$$ and its shift $$$f(z + c)$$$ where $$$c \in ℂ$$$. Suitable examples are given to prove the sharpness of sharing exceptional values of Borel and Nevanlinna.

The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and $$${V}$$$ is a divergence-free $$$C^{1}$$$-vector field with finitely many singularities on M then every orbit L of $$$\mathcal{V}$$$ is one of the following types : (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M* = M - Sing($$$\mathcal{V}$$$), (iv) a locally dense orbit, where Sing($$$\mathcal{V}$$$) denotes the set of singular points of $$$\mathcal{V}$$$. On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M*. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431-445] established when the divergence-free vector field $$$\mathcal{V}$$$ is regular that is all its singular points are non-degenerate.