In this article, we show how the concept of Hilbert $C^*$-module can be used to investigate completely positive linear maps. We show when two unital pure completely positive linear maps of a $C^*$-algebra into $M_n$ are unitarily equivalent. We also develop and characterize a concept of weak containment between two completely positive linear maps of a $C^*$-algebra into a von Neumann algebra. In preparation, we exhibit some basic known properties of Hilbert $C^*$-modules. In addition, we explore the norm of the standard Hilbert column $C^*$-modules and show it is the Haagerup tensor norm of two operator spaces.

It is shown that $iB$ generates a strongly continuous group of exponential type $\omega$ on a Hilbert space if and only if for all $\alpha \gt \omega$, $B$ is similar to an operator with spectrum and numerical range contained in the horizontal strip $\{z \in {\bf C} \, | \, |Im(z)| \lt \alpha \}$.

In this paper global existence results for certain higher order differential equations are established. Our analysis is based on a simple and classical application of the Leray-Schauder alternative.

In this paper, we point out an error in [3, Main theorem] and obtain some sufficient conditions for multivalently starlikeness.

In this paper we study the asymptotic behavior of the solution $v(x)$ of initial value problem (1.1) which arises from a mathematical model describing the large deformations of a nonumiform cantilever.

Let $M$ be a $\Gamma$-ring in the sense of Nobusawa. The ring $M_2=\left (\begin{array}{ll} R~~& \Gamma\\ M& \Gamma\\ \end{array}\right )$ was defined by Kyuno. Let $\cal{P}$ be a class of prime rings such that for every prime ring $R$ and any $0\neq e^2=e\in R,~R\in \cal{P}$ if and only if $eRe\in \cal{P}$. In this paper, the $\cal{P}$-Jacobson $\Gamma$-rings which include the Jacobson property and Brown-McCoy property as special case are defined. Relationships between $\cal{P}$-Jacobson properties of $\Gamma$-ring $M$ and the corresponding properties of $\Gamma_{n,m}$-ring $M_{m,n}$, the right operator ring $R$ of $\Gamma$-ring $M,~M$-ring $\Gamma$ and the ring $M_2$ are established.

A $CVM(k)$ game is a game of the form $f\circ \lambda $, where $\lambda $ is a $k$-dimensional non-atomic measure and $f$ is a continuously differentiable function on $R^k$. For a convex $CVM(1)$ game, we characterize the ``least upper bound'' and ``greatest lower bound'' of the core elements in terms of the distribution function. We also show that the core of a convex $CVM(1)$ game expands as the underlying measure $\lambda$ changes in a ``convex manner''. These results provide a partial geometric picture for the core and its variations of a convex $CVM(1)$ game.

It is well known that solutions of linear difference equations with constant coefficients are exponentially bounded. Similar statements, however, are not valid for solutions of partial difference equations. In this paper, we consider two linear partial difference equations and obtain sufficient conditions for their solutions to be exponentially bounded.

In this paper we consider the quasilinear elliptic equation \def\theequation{1} \begin{equation} {\rm div}(|\nabla u|^{m-2}\nabla u)+f(u)=0 \end{equation} where $n\gt m\gt 1$. We obtain a necessary and sufficient condition for the existence of positive radial solutions $u=u(r)$ on $[r_0, \infty)$, where $r_0 \gt 0$. If $f$ satisfies a further condition, then Eq. (1) possesses infinitely many singular ground state solutions $u(r)$ satisfying $u(r)\sim r^{-{(n-m)}\over {m-1}}$ at $\infty $ and $u(r)\to \infty \hbox{\ as\ }r\to 0^+$. We also obtain some important conclusions via our main results.

In this article we prove the existence of a minimizer of semilinear elliptic equations in axial symmetric domains.

Let $G$ be a compact affine exponentially Nash group and let $\eta$ be a $C^\infty G$ vector bundle over a compact affine exponentially Nash $G$ manifold $X$. We prove that $\eta$ admits a unique strongly exponentially Nash $G$ vector bundle structure $\zeta$, and that $\eta$ admits a non-strongly exponentially Nash $G$ vector bundle structure if dim $X\geq 1$, rank $\eta\geq 1$ and $X$ has a 0-dimensional orbit. Moreover we show that every exponentially Nash $G$ vector bundle structure of $\eta$ which is not necessarily strongly exponentially Nash is exponentially Nash $G$ vector bundle isomorphic to $\zeta$ if the action on $X$ is transitive.