The zeros $c_k\left(\nu \right)$ of the solution $z(t,\nu )$ of the differential equation $z^{\text{'}\text{'}}+q(t,\nu )z=0$ are investigated when $\underset{t\to \infty }{lim}q(t,\nu )=1$, ${\int }^{\infty }|q(t,\nu )-1|dt<\infty$ and $q(t,\nu )$ has some monotonicity properties as $t\to \infty$. The notion $c_{\kappa }\left(\nu \right)$ is introduced also for $\kappa$ real, too. We are particularly interested in solutions $z(t,\nu )$ which are “close" to the functions $sint$, $cost$ when $t$ is large. We derive a formula for $d{c}_{\kappa }\left(\nu \right)/d\nu$ and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair $J_{\nu }\left(t\right)$, $Y_{\nu }\left(t\right)$. We show the concavity of $c_{\kappa }\left(\nu \right)$ for...

In this paper we show that from an estimate of the form $su{p}_{t\ge 0}\left|\right|C\left(t\right)-cos\left(at\right)I\left|\right|<1$, we can conclude that C(t) equals cos(at)I. Here ${\left(C\left(t\right)\right)}_{t\ge 0}$ is a strongly continuous cosine family on a Banach space.

We study extension of $p$-trigonometric functions ${sin}_p$ and ${cos}_p$ to complex domain. For $p=4,6,8,\cdots$, the function ${sin}_p$ satisfies the initial value problem which is equivalent to (*) $$-{\left(u^{\text{'}}\right)}^{p-2}{u}^{\text{'}\text{'}}-u^{p-1}=0,u\left(0\right)=0,u^{\text{'}}\left(0\right)=1$$ in $ℝ$. In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_p\left(x\right)$ is a real analytic function for $p=4,6,8,\cdots$ on $(-{\pi }_p/2,{\pi }_p/2)$, where ${\pi }_p/2={\int }_0^1{(1-s^p)}^{-1/p}$. This allows us to extend ${sin}_p$ to complex domain by its Maclaurin series convergent on the disc $\{z\in ℂ:|z|<{\pi }_p/2\}$. The question is whether this extensions ${sin}_p\left(z\right)$ satisfies (*) in the sense of differential equations in complex domain. This...

Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R\left(A\right)$ of $A$ as follows. Let $0\ne {\lambda }_n\in \rho \left(A\right)$ for all $1\le n<\infty$, where $\rho \left(A\right)$ denotes the resolvent set of $A$, and assume that ${lim}_{n\to \infty }{\lambda }_n=0$ and ${sup}_{n\ge 1}\parallel {\lambda }_n{({\lambda }_n-A)}^{-1}\parallel <\infty$. Furthermore, assume that there exists ${\lambda }_{\infty }\in \rho \left(A\right)$ such that $\parallel {\lambda }_{\infty }{({\lambda }_{\infty }-A)}^{-1}\parallel \le 1$. Then $f\in R\left(A\right)$ is equivalent to ${sup}_{n\ge 1}{\parallel {({\lambda }_n-A)}^{-1}f\parallel }_1<\infty$. This generalizes Shaw’s result for scalar-valued functions.