The Newton-Mysovskikh theorem provides sufficient conditions for the semilocal convergence of Newton's method to a locally unique solution of an equation in a Banach space setting. It turns out that under weaker hypotheses and a more precise error analysis than before, weaker sufficient conditions can be obtained for the local as well as semilocal convergence of Newton's method. Error bounds on the distances involved as well as a larger radius of convergence are obtained. Some numerical examples...

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(·)$-Harmonic and $p(·)$-biharmonic operators $$\begin{array}{c}{\Delta }_{p\left(x\right)}^{2}u-{\Delta }_{p\left(x\right)}u=\lambda w\left(x\right){\left|u\right|}^{q\left(x\right)-2}u\text{in}\Omega ,\\ u\in W^{2,p(·)}\left(\Omega \right)\cap W_0^{1,p(·)}\left(\Omega \right),\end{array}$$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(·)}\left(\Omega \right)$ and $W^{m,p(·)}\left(\Omega \right)$.

Let $u$ be a weight on $(0,\infty )$. Assume that $u$ is continuous on $(0,\infty )$. Let the operator $S_u$ be given at measurable non-negative function $\varphi$ on $(0,\infty )$ by $$S_u\varphi \left(t\right)=\underset{0<\tau \le t}{sup}u\left(\tau \right)\varphi \left(\tau \right).$$ We characterize weights $v,w$ on $(0,\infty )$ for which there exists a positive constant $C$ such that the inequality $${\left({\int }_0^{\infty }{\left[S_u\varphi \left(t\right)\right]}^{q}w\left(t\right)dt\right)}^{\frac{1}{q}}\lesssim {\left({\int }_0^{\infty }{\left[\varphi \left(t\right)\right]}^{p}v\left(t\right)dt\right)}^{\frac{1}{p}}$$ holds for every $0<p,q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

In the theory of elliptic equations, the technique of Schwarz symmetrization is one of the tools used to obtain a priori bounds for classical and weak solutions in terms of general information on the data. A basic result says that, in the absence of lower-order terms, the symmetric rearrangement of the solution $u$ of an elliptic equation, that we write $u^{*}$, can be compared pointwise with the solution of the symmetrized problem. The main question we address here is the modification of the method to...

We consider a discrete Schrödinger operator 𝒥 with Wigner-von Neumann potential not belonging to l². We find the asymptotics of orthonormal polynomials associated to 𝒥. We prove a Weyl-Titchmarsh type formula, which relates the spectral density of 𝒥 to a coefficient in the asymptotics of the orthonormal polynomials.

The Coifman-Fefferman inequality implies quite easily that a Calderón-Zygmund operator T acts boundedly in a Banach lattice X on ℝⁿ if the Hardy-Littlewood maximal operator M is bounded in both X and X'. We establish a converse result under the assumption that X has the Fatou property and X is p-convex and q-concave with some 1 < p, q < ∞: if a linear operator T is bounded in X and T is nondegenerate in a certain sense (for example, if T is a Riesz transform) then M is bounded in both X and...

Let $\left(X_j\right)$ be a sequence of independent identically distributed random operators on a Banach space. We obtain necessary and sufficient conditions for the Abel means of $X_n...X_2{X}_1$ to belong to Hardy and Lipschitz spaces a.s. We also obtain necessary and sufficient conditions on the Fourier coefficients of random Taylor series with bounded martingale coefficients to belong to Lipschitz and Bergman spaces.

We give a brief survey of recent results of order limited operators related to some properties on Banach lattices.