We show that the half-linear differential equation $${\left[r\left(t\right)\Phi \left(x^{\text{'}}\right)\right]}^{\text{'}}+\frac{s\left(t\right)}{t^p}\Phi \left(x\right)=0*$$ with $\alpha$-periodic positive functions $r,s$ is conditionally oscillatory, i.e., there exists a constant $K>0$ such that () with $\frac{\gamma s\left(t\right)}{t^p}$ instead of $\frac{s\left(t\right)}{t^p}$ is oscillatory for $\gamma >K$ and nonoscillatory for $\gamma <K$.

We obtain monotonicity results concerning the oscillatory solutions of the differential equation $\left(a\left(t\right)\right|y^{\text{'}}{{|}^{p-1}{y}^{\text{'}})}^{\text{'}}+f(t,y,y^{\text{'}})=0$. The obtained results generalize the results given by the first author in [1] (1976). We also give some results concerning a special case of the above differential equation.

We consider the Robin eigenvalue problem $\Delta u+\lambda u=0$ in $\Omega$, $\partial u/\partial \nu +\alpha u=0$ on $\partial \Omega$ where $\Omega \subset {ℝ}^n$, $n\ge 2$ is a bounded domain and $\alpha$ is a real parameter. We investigate the behavior of the eigenvalues ${\lambda }_k\left(\alpha \right)$ of this problem as functions of the parameter $\alpha$. We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative ${\lambda }_1^{\text{'}}\left(\alpha \right)$. Assuming that the boundary $\partial \Omega$ is of class $C^2$ we obtain estimates to the difference ${\lambda }_k^D-{\lambda }_k\left(\alpha \right)$ between the $k$-th eigenvalue of the Laplace operator with...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0...