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.

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 ${\int }_a^{\infty }1/\left(t\omega \left(t\right)\right)dt<\infty$.

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form $$\Delta \left[\begin{array}{c}x\left(n\right)+p\left(n\right)x(n-m)\\ y\left(n\right)+p\left(n\right)y(n-m)\end{array}\right]=\left[\begin{array}{cc}a\left(n\right)& b\left(n\right)\\ c\left(n\right)& d\left(n\right)\end{array}\right]\left[\begin{array}{c}x(n-\alpha )\\ y(n-\beta )\end{array}\right]$$ are established, where $m>0$, $\alpha \ge 0$, $\beta \ge 0$ are integers and $a\left(n\right)$, $b\left(n\right)$, $c\left(n\right)$, $d\left(n\right)$, $p\left(n\right)$ are sequences of real numbers.

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 $$y^{\left(n\right)}={\left|y\right|}^k\mathrm{sgn}y,k>1,n=12,13,14.$$

The paper considers a scalar differential equation of an advance-delay type $$\dot{y}\left(t\right)=-\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma ),$$ where constants $a_0$, $b_0$, $\tau$ and $\sigma$ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\to \infty$ is analyzed provided that the transcendental equation $$\lambda =-a_0{\mathrm{e}}^{-\lambda \tau }+b_0{\mathrm{e}}^{\lambda \sigma }$$ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.

This paper discusses the existence and multiplicity of solutions for a class of $p\left(x\right)$-Kirchhoff type problems with Dirichlet boundary data of the following form $$\left\}\begin{array}{cc}-\left(a+b{\int }_{\Omega }\frac{1}{p\left(x\right)}{\left|\nabla u\right|}^{p\left(x\right)}dx\right){\mathrm{div}\left(\right|\nabla u|}^{p\left(x\right)-2}\nabla u)=f(x,u),\hfill & in\Omega \hfill \\ u=0\hfill & on\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a smooth open subset of ${ℝ}^N$ and $p\in C\left(\overline{\Omega }\right)$ with $N<p^{-}={inf}_{x\in \Omega }p\left(x\right)\le p^{+}={sup}_{x\in \Omega }p\left(x\right)<+\infty$, $a$, $b$ are positive constants and $f:\overline{\Omega }\times ℝ\to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

The half-linear differential equation $$\left(\right|u^{\text{'}}{|}^{\alpha }\mathrm{sgn}{u}^{\text{'}}{)}^{\text{'}}=\alpha ({\lambda }^{\alpha +1}+b\left(t\right)){\left|u\right|}^{\alpha }\mathrm{sgn}u,t\ge t_0,$$ is considered, where $\alpha$ and $\lambda$ are positive constants and $b\left(t\right)$ is a real-valued continuous function on $[t_0,\infty )$. It is proved that, under a mild integral smallness condition of $b\left(t\right)$ which is weaker than the absolutely integrable condition of $b\left(t\right)$, the above equation has a nonoscillatory solution $u_0\left(t\right)$ such that $u_0\left(t\right)\sim {\mathrm{e}}^{-\lambda t}$ and $u_0^{\text{'}}\left(t\right)\sim -\lambda {\mathrm{e}}^{-\lambda t}$ ($t\to \infty$), and a nonoscillatory solution $u_1\left(t\right)$ such that $u_1\left(t\right)\sim {\mathrm{e}}^{\lambda t}$ and $u_1^{\text{'}}\left(t\right)\sim \lambda {\mathrm{e}}^{\lambda t}$ ($t\to \infty$).