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

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.

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

Consider the first-order linear delay (advanced) differential equation$$x^{\text{'}}\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)=0(x^{\text{'}}\left(t\right)-q\left(t\right)x\left(\sigma \left(t\right)\right)=0),t\ge t_0,$$ where $p$ $\left(q\right)$ is a continuous function of nonnegative real numbers and the argument $\tau \left(t\right)$ $\left(\sigma \right(t\left)\right)$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions$$\underset{t\to \infty }{lim\; sup}{\int }_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s>1\left(\underset{t\to \infty }{lim\; sup}{\int }_t^{\sigma \left(t\right)}q\left(s\right)\mathrm{d}s>1\right)$$ and $$\underset{t\to \infty }{lim\; inf}{\int }_{\tau \left(t\right)}^{t}p\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}}\left(\underset{t\to \infty }{lim\; inf}{\int }_t^{\sigma \left(t\right)}q\left(s\right)\mathrm{d}s>\frac{1}{\mathrm{e}}\right)$$ are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known...

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator ${\left(a\left(t\right){\phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right){\phi }_p\left(x^{\text{'}}\right)+c\left(t\right){\phi }_p\left(x\right)=0$, where ${\phi }_p\left(x\right)={\left|x\right|}^{p-1}\mathrm{sgn}x$ for $x\in ℝ$ and $p>1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial...

A couple ($\sigma ,\tau$) of lower and upper slopes for the resonant second order boundary value problem $$x^{\text{'}\text{'}}=f(t,x,x^{\text{'}}),x\left(0\right)=0,x^{\text{'}}\left(1\right)={\int }_0^1{x}^{\text{'}}\left(s\right)\mathrm{d}g\left(s\right),$$ with $g$ increasing on $[0,1]$ such that ${\int }_0^{1}dg=1$, is a couple of functions $\sigma ,\tau \in C^1\left([0,1]\right)$ such that $\sigma \left(t\right)\le \tau \left(t\right)$ for all $t\in [0,1]$, $$\begin{array}{c}{\sigma }^{\text{'}}\left(t\right)\ge f(t,x,\sigma \left(t\right)),\sigma \left(1\right)\le {\int }_0^1\sigma \left(s\right)\mathrm{d}g\left(s\right),\\ {\tau }^{\text{'}}\left(t\right)\le f(t,x,\tau \left(t\right)),\tau \left(1\right)\ge {\int }_0^1\tau \left(s\right)\mathrm{d}g\left(s\right),\end{array}$$ in the stripe ${\int }_0^t\sigma \left(s\right)\mathrm{d}s\le x\le {\int }_0^t\tau \left(s\right)\mathrm{d}s$ and $t\in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

The fixed infinitely differentiable function $\rho \left(x\right)$ is such that $\left\{n\rho \right(nx\left)\right\}$ is a regular sequence converging to the Dirac delta function $\delta$. The function ${\delta }_{𝐧}\left(𝐱\right)$, with $𝐱=(x_1,\cdots ,x_m)$ is defined by $${\delta }_{𝐧}\left(𝐱\right)=n_1\rho \left(n_1{x}_1\right)\cdots n_m\rho \left(n_m{x}_m\right).$$ The product $f\circ g$ of two distributions $f$ and $g$ in ${𝒟}_m^{\text{'}}$ is the distribution $h$ defined by $$\underset{n_1\to \infty }{error}\cdots \underset{n_m\to \infty }{error}\langle f_{𝐧}{g}_{𝐧},\phi \rangle =\langle h,\phi \rangle ,$$ provided this neutrix limit exists for all $\phi \left(𝐱\right)={\phi }_1\left(x_1\right)\cdots {\phi }_m\left(x_m\right)$, where $f_{𝐧}=f*{\delta }_{𝐧}$ and $g_{𝐧}=g*{\delta }_{𝐧}$.

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 $u_1^{\text{'}}\left(t\right)\sim \lambda {\mathrm{e}}^{\lambda t}$ ($t\to \infty$).

We consider the equation $$-r\left(x\right)y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ$$ where $f\in L_p\left(ℝ\right)$, $p\in [1,\infty ]$ ($L_{\infty }\left(ℝ\right):=C\left(ℝ\right)$) and $$0<r\in C^{}\left(ℝ\right),0\le q\in L_1^{}\left(ℝ\right).$$ We obtain minimal requirements to the functions $r$ and $q$, in addition to (), under which equation () is correctly solvable in $L_p\left(ℝ\right)$, $p\in [1,\infty ]$.