By means of Picone's type identities and inequalities some comparison results for problems related to the equation $$\nabla .\{ A(u) \nabla u \} + c(x)u + f(u) = g(x) \text{ in } \mathbb R^n$$ are established.Because of its principal part,this type of equation finds its applications in various physical phenomena like in nonlinear diffusion problems,flows through porous media, plasma physics,...etc. In this paper we show how versatile can the use of Picone-type formulae be for these type of quasilinear equations. Our main focus is to establish some oscillation criteria for classical non trivial and bounded solutions of some of these types of equations.We will display here some criteria conditions for some model equations.The ultimate aim is that by means of comparison methods and some Picone-type formulae,this could lead to getting the oscillation criteria of some more general equations.

In this paper, the authors obtain the $(L^1,L^{1,\infty})$ type boundedness for the parametrized Littlewood-Paley operator $\mu_\lambda^{*,\rho}$ with kernel satisfying the logarithmic type Lipschitz condition. Moreover, the $L^p(\mathbb{R}^{n})$ boundedness of the operator $\mu_\lambda^{*,\rho}$ can be deduced, where $1<p<\infty$.

Let $\gamma$ be a non-rectifiable closed Jordan curve in $\mathbb{C}$, which is merely assumed to be $d$-summable ($1<d<2$) in the sense of Harrison and Norton [7]. We are interested in the so-called jump problem over $\gamma$, which is that of finding an analytic function in $\mathbb{C}$ having a prescribed jump across the curve. The goal of this note is to show that the sufficient solvability condition of the jump problem given by $\displaystyle \nu > \frac{d}{2}$, being the jump function defined in $\gamma$ and satisfying a Hölder condition with exponent $\nu$, $0<\nu\leq 1$, cannot be weakened on the whole class of $d$-summable curves.

In this paper, we obtain an extensions of majorization type results and extensions of weighted Favard's and Berwald's inequality. We prove positive semi-definiteness of matrices generated by differences deduced from majorization type results and differences deduced from weighted Favard's and Berwald's inequality. This implies a surprising property of exponentially convexity and $\log$-convexity of this differences which allows us to deduce Lyapunov's inequalities for the differences, which are improvements of majorization type results and weighted Favard's and Berwald's inequalities. Analogous Cauchy's type means, as equivalent forms of exponentially convexity and log-convexity, are also studied and the monotonicity properties are proved.

A relation between double Dirichlet averages and multivariate complex B-splines is presented. Based on this relationship, a formula for the computation of certain moments of multivariate complex B-splines is derived. In addition, an infinite-dimensional analogue of the Lauricella function $F_B$ is defined and a relation to the moments of multivariate complex B-splines is presented.

In this paper we define a vertical Liouville distribution in the vertical foliated distribution on a complex Finsler bundle and we prove that it is an integrable one. Next, some new operators on foliated forms along the vertical Liouville distribution are defined, a Dolbeault type lemma is proved and new cohomology groups are studied.

In this paper we study the existence of solutions for differential equations with state dependent delay on an unbounded domain. Our results are based on the properties of the Kuratowski measure of noncompactness and Darbo's fixed point theorem.

In this paper we consider the discrete dynamic evolution of a connected network related to an average distance functional minimization problem, with associated dissipation term, under irreversibility assumption. We will analyze its geometric and topological properties, and study whether it inherits regularity properties verified by the static or quasi static case.

We consider a class of $p$-Laplacian equations in $\mathbb{R}^{N}$. By carefully analyzing the compactness of the Palais-Smale sequences and constructing Nehari manifolds, we prove that for every positive integer $m\geq 2$, there exists a nodal solution with at least $2m$ nodal domains.