Sufficient conditions are obtained so that every solution of ${[y\left(t\right)-p\left(t\right)y(t-\tau )]}^{\left(n\right)}+Q\left(t\right)G\left(y(t-\sigma )\right)=f\left(t\right)$ where n ≥ 2, p,f ∈ C([0,∞),ℝ), Q ∈ C([0,∞),[0,∞)), G ∈ C(ℝ,ℝ), τ > 0 and σ ≥ 0, oscillates or tends to zero as $t\to \infty$. Various ranges of p(t) are considered. In order to accommodate sublinear cases, it is assumed that ${\int }_0^{\infty }Q\left(t\right)dt=\infty$. Through examples it is shown that if the condition on Q is weakened, then there are sublinear equations whose solutions tend to ±∞ as t → ∞.

For the equation $$y^{\left(n\right)}+{\left|y\right|}^k\mathrm{sgn}y=0,k>1,n=3,4,$$ existence of oscillatory solutions $$y={(x^{*}-x)}^{-\alpha }h(log(x^{*}-x)),\alpha =\frac{n}{k-1},x<x^{*},$$ is proved, where $x^{*}$ is an arbitrary point and $h$ is a periodic non-constant function on $ℝ$. The result on existence of such solutions with a positive periodic non-constant function $h$ on $ℝ$ is formulated for the equation $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{\varphi }:f\left(x\right)\mapsto {\int }_0^{\varphi \left(x\right)}f\left(t\right)dt$ be defined on L₂[0,1]. We prove that $V_{\varphi }$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{\varphi }$ always equals 1.

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form $$\left(\right|y^{\text{'}\text{'}}{|}^{\alpha }\mathrm{sgn}{y}^{\text{'}\text{'}}{)}^{\text{'}\text{'}}+q\left(t\right){\left|y\right|}^{\alpha }\mathrm{sgn}y=0,t\ge a>0,A$$ where $\alpha >0$ is a constant and $q\left(t\right)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega \left(t\right)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies ${\int }_a^{\infty }1/\left(t\omega \left(t\right)\right)dt<\infty$.

Let $u$ be a weight on $(0,\infty )$. Assume that $u$ is continuous on $(0,\infty )$. Let the operator $S_u$ be given at measurable non-negative function $\varphi$ on $(0,\infty )$ by $$S_u\varphi \left(t\right)=\underset{0<\tau \le t}{sup}u\left(\tau \right)\varphi \left(\tau \right).$$ We characterize weights $v,w$ on $(0,\infty )$ for which there exists a positive constant $C$ such that the inequality $${\left({\int }_0^{\infty }{\left[S_u\varphi \left(t\right)\right]}^{q}w\left(t\right)dt\right)}^{\frac{1}{q}}\lesssim {\left({\int }_0^{\infty }{\left[\varphi \left(t\right)\right]}^{p}v\left(t\right)dt\right)}^{\frac{1}{p}}$$ holds for every $0<p,q<\infty$. Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.

Let $I=[a,b]\subset ℝ$, let , let $u$ and $v$ be positive functions with $u\in L_{p^{\prime }}(I)$ e $v\in L_q(I)$ and let $T:L_p(I)\to L_q(I)$be the Hardy-type operator given by $$\left(Tf\right)\left(x\right)=v\left(x\right){\int }_a^{x}f\left(t\right)u\left(t\right)dt,x\in I.$$ We show that the asymptotic behavior of the eigenvalues$\lambda$of the non-linear integral system $$g\left(x\right)=\left(TF\right)\left(x\right){\left(f\left(x\right)\right)}_{\left(p\right)}=\lambda \left(T^{*}{g}_{\left(p\right)}\right))\left(x\right)$$ (where, for example,$t_{(p)}={|t|}^{p-1}\mathrm{sgn}(t)$is given by $$\begin{array}{cc}& \underset{n\to \infty }{lim}n{\widehat{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^r{)}^{1/r}dt\right)}^{1/r},\text{for}1<p<q<\infty \hfill \\ & \underset{n\to \infty }{lim}n{\check{\lambda }}_n\left(T\right)=c_{p,q}{\left({\int }_I{\left(uv\right)}^{r}dt\right)}^{1/r}\text{for}1<q<p<\infty \hfill \end{array}$$ Here$r=\frac{1}{p^{\prime }}+\frac{1}{p}$, $c_{p,q}$ is an explicit constant depending only on $p$ and $q$, $\widehat{\lambda }(T)=\max (s{p}_n(T,p,q))$, ${\check{\lambda }}_n(T)=\min (s{p}_n(T,p,q))$ where $s{p}_n(T,p,q)$ stands for the set of all eigenvalues $\lambda$ corresponding to eigenfunctions $g$ with $n$ zeros.

The second order linear differential equation $${\left(p\left(x\right)y^{\text{'}}\right)}^{\text{'}}+q\left(x\right)y=0,x\in (0,x_0]$$ is considered, where $p$, $q\in C^1(0,x_0]$, $p\left(x\right)>0$, $q\left(x\right)>0$ for $x\in (0,x_0]$. Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near $x=0$ without the Hartman–Wintner condition.