In this paper by using the measure of noncompactness concept, we present new fixed point theorems for multivalued maps. In further we introduce a new class of mappings which are general than Meir-Keeler mappings. Finally, we use these results to investigate the existence of weak solutions to an Evolution differential inclusion with lack of compactness.

A fondamental theorem due to Gindikin [Russian Math. Surveys, 29 (1964), 1-89] says that the‎ ‎generalized power $\Delta_{s}(-\theta^{-1})$ defined on a symmetric‎ ‎cone is the Laplace transform of a positive measure $R_{s}$ if and ‎only if $s$ is in a given subset $\Xi$ of $\mathbb{R}^{r}$‎, ‎where $r$‎ ‎is the rank of the cone‎. ‎When $s$ is in a well defined part of‎ ‎$\Xi$‎, ‎the measure $R_{s}$ is absolutely continuous with respect to‎ ‎Lebesgue measure and has a known expression‎. ‎For the other elements‎ ‎$s$ of $\Xi$‎, ‎the measure $R_{s}$ is concentrated on the boundary of‎ ‎the cone and it has never been explicitly determined‎. ‎The aim of the‎ ‎present paper is to give an explicit description of the measure‎ ‎$R_{s}$ for all $s$ in $\Xi$‎. ‎The work is motivated by the‎ ‎importance of these measures in probability theory and in statistics‎ ‎since they represent a generalization of the class of measures‎ ‎generating the famous Wishart probability distributions‎.

‎Let $X$ be a completely regular Hausdorff space‎, ‎and let $\mathcal{D}$ be a‎ ‎cover of $X$ by $C_{b}$-embedded sets‎. ‎Let $\pi‎: ‎\mathcal{E} \rightarrow X$‎ ‎be a bundle of Banach spaces (algebras)‎, ‎and let $\Gamma(\pi)$ be the‎ ‎section space of the bundle $\pi‎.‎$ Denote by $\Gamma _{b}(\pi‎,‎\mathcal{D})$‎ ‎the subspace of $\Gamma (\pi )$ consisting of sections which are bounded on‎ ‎each $D \in \mathcal{D}$. We construct a bundle $\rho ^{\prime }: \mathcal{F}‎^{\prime}\rightarrow \beta X$ such that $\Gamma _{b}(\pi‎ ,‎ \mathcal{D}) ‎$ is topologically and algebraically isomorphic to $\Gamma(\rho^\prime‎‎)‎$, ‎and use this to study the subspaces (ideals) and quotients resulting‎ ‎from endowing $\Gamma _{b}(\pi‎,‎\mathcal{D})$ with the cover topology‎ ‎determined by $\mathcal{D}$‎.

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases.

The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.

We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.

We study the truncated two-dimensional moment problem (with rectangular data) to find a non-negative measure $\mu(\delta)$, $\delta\in\mathfrak{B}(\mathbb{R}^2)$, such that $\int_{\mathbb{R}^2} x_1^m x_2^n d \mu = s_{m,n}$, $0\leq m\leq M,\quad 0\leq n\leq N$, where $\{ s_{m,n} \}_{0\leq m\leq M, 0\leq n\leq N}$ is a prescribed sequence of real numbers; $M,N\in\mathbb{Z}_+$. For the cases $M=N=1$ and $M=1, N=2$ explicit numerical necessary and sufficient conditions for the solvability of the moment problem are given. In the cases $M=N=2$; $M=2, N=3$; $M=3, N=2$; $M=3, N=3$ some explicit numerical sufficient conditions for the solvability are obtained. In all the cases some solutions (not necessarily atomic) of the moment problem can be constructed.

In this paper, we study some connection between the compactness of radial operators and the boundary behavior of the corresponding Berezin transform on weighted Bergman spaces. More precisely, we prove that, under some mild condition, the vanishing of the Berezin transform on the unit circle is equivalent to the compactness of a class of radial operators on weighted Bergman spaces. Moreover, we also study the radial essential commutant of the Toeplitz operator $T_z$.

Matrix splittings are useful in finding a solution of linear systems of equations, iteratively. In this note, we present some more convergence and comparison results for recently introduced matrix splittings called index-proper regular and index-proper weak regular splittings. We then apply to theory of double index-proper splittings.

We define two new types of spectrum, called the $\varepsilon$-left (or right) pseudospectrum and the $\varepsilon$-left (or right) condition spectrum, of an element $a$ in a complex unital Banach algebra $A$. We prove some basic properties among them the property that the $\varepsilon$-left (or right) condition spectrum is a particular case of Ransford spectrum. We study also the linear preserver problem for our defined functions and we establish the following:

(1) Let $A$ and $B$ be complex unital Banach algebras and $\varepsilon>0$. Let $\phi : A \longrightarrow B$ be an $\varepsilon$-left (or right) pseudospectrum preserving onto linear map. Then $\phi$ preserves certain standard spectral functions.

(2) Let $A$ and $B$ be complex unital Banach algebras and $0 \lt \varepsilon \lt 1$. Let $\phi : A \longrightarrow B $ be a unital linear map. Then

(a) If $\phi$ is an $\varepsilon$-almost multiplicative map, then $\sigma^{l}(\phi(a))\subseteq \sigma^{l}_\varepsilon(a)$ and $\sigma^{r}(\phi(a))\subseteq \sigma^{r}_\varepsilon(a)$, for all $a \in A$.

(b) If $\phi$ is an $\varepsilon$-left (or right) condition spectrum preserving, then (i) if $A$ is semi-simple, then $\phi$ is injective; (ii) if B is spectrally normed, then $\phi$ is continuous.

The $\Lambda$-sequence spaces $\Lambda_p$ for $1 \lt p\leq\infty$ and their generalized forms $\Lambda_{\hat{p}}$ for $1 \lt \hat{p} \lt \infty$, $\hat{p}=(p_n)$, $n\in \mathbb{N}_0$ are introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1 \lt p \leq \infty$ are determined. It is proved that the generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is a closed subspace of the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$, $n\in \mathbb{N}_0$. Hence it follows that sequence spaces $\Lambda_p$ and $\Lambda_{\hat{p}}$ possess the uniform Opial property, ($\beta$)-property of Rolewicz, and weak uniform normal structure. Moreover, it is established that $\Lambda_{\hat{p}}$ possesses the coordinate wise uniform Kadec–Klee property. Further, necessary and sufficient conditions for element $x\in S(\Lambda_{\hat{p}})$ to be an extreme point of $B(\Lambda_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $\Lambda$-sequence space $\Lambda_2^{(2)}$ are carried out. Upper bound for the Hausdorff matrix operator norm on the non-absolute type $\Lambda$-sequence spaces is also obtained.