The present article is a survey of known results on Schur and operator multipliers. It starts with the classical description of Schur multipliers due to Grothendieck, followed by a discussion of measurable Schur multipliers and a generalisation of Grothendieck's Theorem due to Peller. Thereafter, a non-commutative version of Schur multipliers, called operator multipliers and introduced by Kissin and Schulman, is discussed, and a characterisation extending the description in the commutative case...

Let $B_w\left({\ell }^p\right)$ denote the space of infinite matrices $A$ for which $A\left(x\right)\in {\ell }^p$ for all $x={\left\{x_k\right\}}_{k=1}^{\infty }\in {\ell }^p$ with $|x_k|\searrow 0$. We characterize the upper triangular positive matrices from $B_w\left({\ell }^p\right)$, $1<p<\infty$, by using a special kind of Schur multipliers and the G. Bennett factorization technique. Also some related results are stated and discussed.

A new approach to the generalization of Schwartz’s kernel theorem to Colombeau algebras of generalized functions is given. It is based on linear maps from algebras of classical functions to algebras of generalized ones. In particular, this approach enables one to give a meaning to certain hypotheses in preceding similar work on this theorem. Results based on the properties of $G^{\infty }$-generalized functions class are given. A straightforward relationship between the classical and the generalized versions...

The aim of this contribution is to propose and analyze some computational means to approximate solving mathematical problems appearing in some recent studies devoted to biological and chemical networks.

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition $$\begin{array}{c}{u}^{\text{'}\text{'}}+g\left(t\right)f(t,u)=0,t\in (0,1),\\ u\left(0\right)=\alpha u\left(\xi \right)+\lambda ,u\left(1\right)=\beta u\left(\eta \right)+\mu .Criteriafortheexistenceofnontrivialsolutionsoftheproblemareestablished.Thenonlineartermf(t,x)maytakenegativevaluesandmaybeunboundedfrombelow.Conditionsaredeterminedbytherelationshipbetweenthebehavioroff(t,x)/xforxnear0and\pm \infty ,andthesmallestpositivecharacteristicvalueofanassociatedlinearintegraloperator.Theanalysismainlyreliesontopologicaldegreetheory.Thisworkcomplementssomerecentresultsintheliterature.Theresultsareillustratedwithexamples.\end{array}$$

The paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.

The existence of anti-periodic solutions is studied for a second order difference inclusion associated with a maximal monotone operator in Hilbert spaces. It is the discrete analogue of a well-studied class of differential equations.

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the...

We give some theorems on continuity and differentiability with respect to (h,t) of the solution of a second order evolution problem with parameter $h\in \Omega \subset {ℝ}^m$. Our main tool is the theory of strongly continuous cosine families of linear operators in Banach spaces.

In this study, we establish existence and uniqueness theorems for solutions of second order nonlinear differential equations on a finite interval subject to linear impulse conditions and periodic boundary conditions. The results obtained yield periodic solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.