In this paper, we give a sufficient condition for a linear composition operator to map the space of functions of bounded $\textit{Koremblum}$ variation, $\kappa BV[a,b]$, into itself. We present several results concerning quasi monotonic properties of the functionals of $\kappa$-variation and prove a Burenkov's type result for functions belonging to $\kappa BV[a,b]$.

In this paper, we establish the $M^k$-type sharp maximal function inequalities for the Toeplitz type operator associated to the pseudo-differential operator. As an application, we obtain the boundedness of the operator on Lebesgue and Morrey spaces.

In this paper we study properties of complete sets and of completions of sets in Banach spaces, with relation to balls: in particular, we study conditions implying that a set is a ball, or it has a completion which is a ball. We also discuss some related simple properties considered in a few recent papers, a new similar property, and relations among classes of sets defined by them. Results are illustrated by several examples.

We give an existence result for strongly nonlinear elliptic equations of the form \[ -{\rm div}(a(x,u,\nabla u))+g(x,u,\nabla u)+H(x,\nabla u) = \mu\ \text{in}\ \Omega, \] where the right hand side belongs to $L^1(\Omega)+W^{-1,p'}(\Omega)$ and $- {\rm div}(a(x,u,\nabla u))$ is a Leray--Lions type operator with growth $|\nabla u|^{p-1}$ in $\nabla u$. The critical growth condition on $g$ is with respect to $\nabla u$ and no growth condition with respect to $u$, while the function $H(x,\nabla u)$ grows as $|\nabla u|^{p-1}$.

The transcendental radius of the Volterra integration operator $V$ acting in the space $L^2 \left( {0;1} \right)$ is calculated. The latter is compared with the norm of the self-commutator of $V.$

n this paper, we define positive functionals by using the Jensen's inequality, converse of Jensen's inequality, and Jensen-Mercer's inequality on time scales for superquadratic functions. We give mean-value theorems and introduce related Cauchy-type means by using the functionals mentioned above and show the monotonicity of these means. We also show that these functionals are exponentially convex and give some applications of them by using the log-convexity and exponential convexity.

We prove that $\frac{1}{n!}\sum_{\pi\in\mathfrak{S}_n} \left( \sum\limits_{i=1}^n | {x_ia_{i,\pi(i)}}^2 \right)^{\frac{1}{2}}$ is equivalent to a Musielak--Orlicz norm $\|{x}\|_{\Sigma M_i}$. We also provide the converse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak--Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak--Orlicz spaces into $L_1$.

In recent years, the problem of learning and methods for learning functions have received increasing attention in Machine Learning. This problem is motivated by several applications in which it is required to estimate a function representation from available data. Recently, an extension of hypercircle inequality to data error $(Hide)$ was proposed by Kannika Khompurngson and Charles A. Micchelli and the results on this subject have constructed a new learning method. Unfortunately, the material on Hide only applies to circumstances for which all data are known within error. In this paper, our purpose is to extend the hypercircle inequality to circumstances for which data set contains both accurate and inaccurate data.

An operator $T$ on a Hilbert space $H$ is called $p$-quasiposinormal operator if $c^2T^*(T^*T)^pT\ge T^*(TT^*)^pT$ where $p \in (0, 1]$ and for some $c\in (0, \infty)$. In this paper, we have obtained conditions for composition and weighted composition operators to be $p$-quasiposinormal operators.

In this paper, we introduce the concept of Stepanov-like square-mean pseudo almost periodic functions and establish the existence and uniqueness of square-mean almost periodic mild solutions for several neutral partial stochastic differential equations with Stepanov-like almost periodic coefficients in a real separable Hilbert space. Moreover, two examples are given to illustrate the general theorems.

Recently, many recent studies have focused on the James type constant introduced by Takahashi. In this study, we have presented two inequalities on the James type constant, and shown that the inequalities can be used to estimate the values of James type constant for some concrete spaces. In particular, for any $\tau\geq0$, we have calculated the exact values of James type constant, $J_{X,t}(\tau)$ of $l_1-l_2$, for $t\geq1.$.

In this paper we use Banach algebra techniques to study the spectrum, pseudospectrum and condition spectrum of a block Laurent operator with continuous symbol and a lower triangular block Toeplitz operator with continuous symbol.

In this paper, new sequence spaces $X(r, s, t ;\Delta^{(m)})$ for $X\in \{l_\infty, c,$ $c_0\}$ defined by using generalized means and difference operator of order $m$ are introduced. It is shown that these spaces are complete normed linear spaces and the spaces $c_0(r, s, t ;\Delta^{(m)})$, $c(r, s, t ;\Delta^{(m)})$ have Schauder basis. Furthermore, the $\alpha$-, $\beta$-, $\gamma$- duals of these spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t ;\Delta^{(m)})$ to $X$. Finally, some classes of compact operators on the spaces $c_0(r, s, t ;\Delta^{(m)})$ and $l_{\infty}(r, s, t ;\Delta^{(m)})$ are characterized by using the Hausdorff measure of .

In this paper, using the projection operator, we introduce two new dynamical systems for extended general quasi variational inequalities. These dynamical systems are called extended general implicit projected dynamical system and extended general implicit Wiener-Hopf dynamical system. We prove that these new dynamical systems converge globally exponentially to a unique solution of the extended general quasi variational inequalities under some suitable conditions. Some special cases are also discussed. The ideas and technique of this paper may stimulate further research.

In this paper, we prove the following assertions: (1) If $T$ is a $k$-quasi-$*$-$A(n)$ operator, then $T$ is polaroid. (2) If $T$ is a $k$-quasi-$*$-$A(n)$ operator, then the spectrum $\sigma$ is continuous. (3) If $T$ or $T^{*}$ is a $k$-quasi-$*$-$A(n)$ operator, then Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. (4) If $T^{*}$ is a $k$-quasi-$*$-$A(n)$ operator, then generalized $a$-Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. Finally, the finiteness of a quasi-$*$-$A(n)$ operator is also studied.

The distance from the nonconstant function $\varphi$ in $L^{\infty}% (\mathbb{T})$ to the set $\mathcal{F}_{\text{const}}$ of all constant functions is estimated in terms of Hankel operators on the Hardy space $H^{2}(\mathbb{D})$ over the unit disk $\mathbb{D}=\left\{ z\in \mathbb{C}:\left\vert z\right\vert \in [0, 1)\right\} $. We give a sufficient condition ensuring the equality $dist(\varphi,\mathcal{F}_{\text{const}})=\left\Vert \varphi\right\Vert_{L^{\infty}}$. Some other $dist$-formulas are also discussed.

In this note we express the localized numerical range of elements in locally multiplicative convex (l.m.c.) algebras as the intersection of closed discs. We characterize the spectral states on l.m.c. algebras in terms of their radicals.

In this paper, we prove some weak and strong convergence results for a generalized three-step-three-mappings iteration scheme using a more satisfactory definition of nonself mappings. Our results approximate common fixed points of three nonself total asymptotically nonexpansive mappings in a uniformly convex Banach space.

We consider in four space dimensions, the initial value problem for a fourth-order semi-linear Schröodinger equation with exponential type nonlinearity. In the defocusing sign, we obtain unconditional global well-posedness in the energy space. In the focusing case, global well-posedness via existence of ground state holds for small radial data.

In this paper, we present generalizations of Beckner's inequality by using symmetric absolute normalized norms on $\mathbb{R}^2$.