It is well known that if φ(t) ≡ t, then the system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${\phi ⁿ\left(t\right)}_{n=0}^{\infty }$ is a basis in some Lebesgue space $L_p$. The aim of this short note is to show that the answer to this question is negative.

In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)${}_{\epsilon }$ with a nonnegative constant $\epsilon$, and for any $\epsilon \ge 0$, (S)${}_{\epsilon }$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma :=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method,...

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨$\alpha u\left(0\right)-\beta u^{\text{'}}\left(0\right)={\sum }_{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)$ ⎩$\gamma u\left(1\right)+\delta u^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\xi }_i\right)$, where ${\xi }_i\in (0,1)$, $a_i,b_i\in (0,\infty )$ (for i ∈ 1,…,m-2) are given constants satisfying $\varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)>0$, $\varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)>0$ and $\Delta :=\left|\begin{array}{cc}-{\sum }_{i=1}^{m-2}{a}_i\psi \left({\xi }_i\right)& \varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)\\ \varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)& -{\sum }_{i=1}^{m-2}{b}_i\varphi \left({\xi }_i\right)\end{array}\right|<0$. We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and...

We consider functionals of a potential energy ${\mathbb{P}}_{\psi }\left(u\right)$ corresponding to $\mathrm{𝑎𝑛}\mathrm{𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}-\mathrm{𝑣𝑎𝑙𝑢𝑒}\mathrm{𝑝𝑟𝑜𝑏𝑙𝑒𝑚}$. We are dealing with $𝑎\mathrm{𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛}\mathrm{𝑜𝑓}𝑎\mathrm{𝑡ℎ𝑖𝑛}\mathrm{𝑎𝑛𝑛𝑢𝑙𝑎𝑟}\mathrm{𝑝𝑙𝑎𝑡𝑒}$ with $\mathrm{𝑁𝑒𝑢𝑚𝑎𝑛𝑛}\mathrm{𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦}\mathrm{𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠}$. Various types of the subsoil of the plate are described by various types of the $\mathrm{𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒}$ nonlinear term $\psi \left(u\right)$. The aim of the paper is to find a suitable computational algorithm.

A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_p$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_p$ (1 ≤ p < +∞).

Let ${Rₙ}_{n=1}^{\infty }$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an ${ℒ}_p$-space, then both X and A have bases. We apply these results to show that the spaces $C_{\Lambda }=\overline{span}{z}^k:k\in \Lambda \subset C\left(\right)$ and $L_{\Lambda }=\overline{span}{z}^k:k\in \Lambda \subset L₁\left(\right)$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

We consider the equation $$-{\left(r\left(x\right)y^{\text{'}}\left(x\right)\right)}^{\text{'}}+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p\left(ℝ\right)$, $p\in (1,\infty )$ and $$r>0,\frac{1}{r}\in L_1^{\mathrm{loc}}\left(ℝ\right),q\in L_1^{\mathrm{loc}}\left(ℝ\right).$$ For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions $r$ and $q$ under which the above equation is correctly solvable in the space $L_p\left(ℝ\right),$ $p\in (1,\infty ).$

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

We study boundary values of functions in Cegrell’s class ${}_{\psi }$.

Let ${}_{\infty }\left(B_X\right)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let ${}_u\left(B_X\right)$ be the subspace of ${}_{\infty }\left(B_X\right)$ of those functions which are uniformly continuous on $B_X$. A subset $B\subset B_X$ is a boundary for ${}_{\infty }\left(B_X\right)$ if $\parallel f\parallel =su{p}_{x\in B}\left|f\left(x\right)\right|$ for every $f{\in }_{\infty }\left(B_X\right)$. We prove that for X = d(w,1) (the Lorentz sequence space) and X = C₁(H), the trace class operators, there is a minimal closed boundary for ${}_{\infty }\left(B_X\right)$. On the other hand, for X = , the Schreier...

Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^1$-base and data in $h_c^1$, a subspace of $L$1. We derive our results, considering the action of an adjoint operator on $B_{T}MOC$, a predual of $h_c^1$, and using known properties of this last space.

Let $Q={\left(q_n\right)}_{n=1}^{\infty }$ be a sequence of bases with $q_i\ge 2$. In the case when the $q_i$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$-Cantor series expansion is both $Q$-normal and $Q$-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$, and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p,q}\left(B\right)$........ 235. Examples in $L_{p,q}$ theory...................................................................................

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $$\mathrm{d}x=\mathrm{d}A\left(t\right)·f(t,x),h\left(x\right)=0$$ is established, where $f:[a,b]\times {ℝ}^n\to {ℝ}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A:[a,b]\to {ℝ}^{n\times n}$ with bounded total variation components, and $h:{BV}_s([a,b],{ℝ}^n)\to {ℝ}^n$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x\left(t_1\left(x\right)\right)=ℬ\left(x\right)·x\left(t_2\left(x\right)\right)+c_0,$ where $t_i:{BV}_s([a,b],{ℝ}^n)\to [a,b]$ $(i=1,2)$ and $ℬ:{BV}_s([a,b],{ℝ}^n)\to {ℝ}^n$ are continuous...

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.