Given a Young function $\Phi$, we study the existence of copies of $c_0$ and $\ell _{\infty }$ in $\mathop {\mathrm cabv}\nolimits _{\Phi} (\mu ,X)$ and in $\mathop {\mathrm cabsv}\nolimits _{\Phi } (\mu ,X)$, the countably additive, $\mu $-continuous, and $X$-valued measure spaces of bounded $\Phi $-variation and bounded
$\Phi$-semivariation, respectively.
Let $E$ be a real linear space. A vectorial inner product is a mapping from $E\times E$ into a real ordered vector space $Y$ with the properties of a usual inner product. Here we consider $Y$ to be a $\mathcal B$-regular Yosida space, that is a Dedekind complete Yosida space such that $\bigcap _{J\in {\mathcal B}}J=\lbrace 0 \rbrace $, where $\mathcal B$ is the set of all hypermaximal bands in $Y$. In Theorem 2.1.1 we assert that any $\mathcal B$-regular Yosida space is Riesz isomorphic to the space $B(A)$ of all bounded real-valued mappings on a certain set $A$. Next we prove Bessel Inequality and Parseval Identity for a vectorial inner product with range in the $\mathcal B$-regular and norm complete Yosida algebra $(B(A),\sup _{\alpha \in A}|x(\alpha )|)$.
Some new criteria for the oscillation of difference equations of the form \[ \Delta ^2 x_n - p_n \Delta x_{n-h} + q_n |x_{g_n}|^c \mathop {\mathrm sgn}x_{g_n} = 0 \] and \[ \Delta ^i x_n + p_n \Delta ^{i-1} x_{n-h} + q_n |x_{g_n}|^c \mathop {\mathrm sgn}x_{g_n} = 0, \ i = 2,3, \] are established.
Sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations \[ (-1)^{m+1}\frac{d^my_i(t)}{dt^m} + \sum ^n_{j=1} q_{ij} y_j(t-h_{jj})=0, \quad m \ge 1, \ i=1,2,\ldots ,n, \] to be oscillatory, where $q_{ij} \varepsilon (-\infty ,\infty )$, $h_{jj} \in (0,\infty )$, $i,j = 1,2,\ldots ,n$. Also, we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations \[ (-1)^{m+1} \frac{d^m}{dt^m} (y_i(t)+cy_i(t-g)) + \sum ^n_{j=1} q_{ij} y_j (t-h)=0,
\] where $c$, $g$ and $h$ are real constants and $i=1,2,\ldots ,n$.
The asymptotic and oscillatory behavior of solutions of mth order damped nonlinear difference equation of the form \[ \Delta (a_n \Delta ^{m-1} y_n) + p_n \Delta ^{m-1} y_n + q_n f(y_{\sigma (n+m-1)}) = 0 \] where $m$ is even, is studied. Examples are included to illustrate the results.
By a paramedial groupoid we mean a groupoid satisfying the equation ax·yb=bx·ya. This equation is, in certain sense, symmetric to the equation of mediality xa·by=xb·ay and, in fact, the theories of both varieties of groupoids are parallel. The present paper, initiating the study of paramedial groupoids, is meant as a modest contribution to the enormously difficult task of describing algebraic properties of varieties determined by strong linear identities (and, especially,of the corresponding simple algebras).
In this paper it is proved that every $3$-connected planar graph contains a path on $3$ vertices each of which is of degree at most $15$ and a path on $4$ vertices each of which has degree at most $23$. Analogous results are stated for $3$-connected planar graphs of minimum degree $4$ and $5$. Moreover, for every pair of integers $n\ge 3$, $ k\ge 4$ there is a $2$-connected planar graph such that every path on $n$ vertices in it has a vertex of degree $k$.
It is known that the ring $B(\mathbb R)$ of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring $C(\mathbb R)$ of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of $C(\mathbb R)$. In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of $C(\mathbb R)$ which differs from $B(\mathbb R)$.
In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation $(r(t)u^{\prime }(t))^{\prime }+p(t)u(t)=0$ are derived. One of them deals with the case $\int ^{\infty }\frac{{\mathrm d}t}{r(t)}=\infty $, and the other with the case $\int ^{\infty }\frac{{\mathrm d}t}{r(t)}<\infty $.
We discuss here the question whether there exists an invariant subspace M for a contraction T in the class $A$ such that the spectrum of the restriction of T on M is the whole closed unit disc.