In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for $$\left\}\begin{array}{c}{(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0,\text{in}(\alpha ,\infty ),u^{\left(i\right)}\left(\xi \right)=0,i=0,1,\cdots ,n-1,\text{and}\xi \in (\alpha ,\infty ),\hfill \end{array}\right.$$ must be unbounded, provided $f(t,z)z\ge 0$, in $E\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E\times I$. (B) Every bounded solution for ${(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0$, in $ℝ$, must be constant, provided $f(t,z)z\ge 0$ in $ℝ\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $ℝ\times I$.

The neutral differential equation (1.1) $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+x(t-\tau )]+\sigma F(t,x\left(g\left(t\right)\right))=0,$$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma =\pm 1$, $F(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and is nondecreasing in $u\in (0,\infty )$, and $limg\left(t\right)=\infty$ as $t\to \infty$. It is shown that equation (1.1) has a solution $x\left(t\right)$ such that (1.2) $$\underset{t\to \infty }{lim}\frac{x\left(t\right)}{t^k}\text{exists}\text{and}\text{is}\text{a}\text{positive}\text{finite}\text{value}\text{if}\text{and}\text{only}\text{if}{\int }_{t_0}^{\infty }{t}^{n-k-1}F(t,c{\left[g\left(t\right)\right]}^k)\mathrm{d}t<\infty \text{for}\text{some}c>0.$$ Here, $k$ is an integer with $0\le k\le n-1$. To prove the existence of a solution $x\left(t\right)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.

A function $f:ℝ\to ℝ$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers ${\alpha }_0,...,{\alpha }_{n-1}$ such that $s^{n+1}{\int }_0^{\delta }{e}^{-st}[f(x+t)-{\sum }_{i=0}^{n-1}{\alpha }_i{t}^i/i!]dt$ converges as $s\to +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }\left(x\right)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than...

Let $\mathbb{P}:={\left(P_t\right)}_{t>0}$ be a measurable semigroup and $m$ a $\sigma$-finite positive measure on a Lusin space $X$. An $m$-exit law for $\mathbb{P}$ is a family ${\left(f_t\right)}_{t>0}$ of nonnegative measurable functions on $X$ which are finite $m$-a.e. and satisfy for each $s,t>0$ $P_s{f}_t=f_{s+t}$ $m$-a.e. An excessive function $u$ is said to be in $ℛ$ if there exits an $m$-exit law ${\left(f_t\right)}_{t>0}$ for $\mathbb{P}$ such that $u={\int }_0^{\infty }{f}_{t}dt$, $m$-a.e. Let $𝒫$ be the cone of $m$-purely excessive functions with respect to $\mathbb{P}$ and $ℐmV$ be the cone of $m$-potential functions. It is clear that $ℐmV\subseteq ℛ\subseteq 𝒫$. In this...