This paper deals with the oscillation problems on the nonlinear differential equation $\left(a\left(t\right)\right|x^{\text{'}}{|}^{p\left(t\right)-2}{x}^{\text{'}}{)}^{\text{'}}+b\left(t\right){\left|x\right|}^{\lambda -2}x=0$ involving $p\left(t\right)$-Laplacian. Sufficient conditions are given under which all proper solutions are oscillatory. In addition, we give a-priori estimates for nonoscillatory solutions and propose an open problem.

Recently there has been an increasing interest in studying $p\left(t\right)$-Laplacian equations, an example of which is given in the following form $$\left(\right|u^{\text{'}}{\left(t\right)|}^{p\left(t\right)-2}{u}^{\text{'}}{\left(t\right))}^{\text{'}}+c\left(t\right){\left|u\left(t\right)\right|}^{q\left(t\right)-2}u\left(t\right)=0,t>0.$$ In particular, the first study of sufficient conditions for oscillatory solution of $p\left(t\right)$-Laplacian equations was made by Zhang (2007), but to our knowledge, there has not been a paper which gives the oscillatory conditions by utilizing Riccati inequality. Therefore, we establish sufficient conditions for oscillatory solution of nonlinear differential equations...

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...

We consider the half-linear differential equation of the form $$\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha }\mathrm{sgn}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\alpha }\mathrm{sgn}x=0,t\ge t_0,$$ under the assumption that $p{\left(t\right)}^{-1/\alpha }$ is integrable on $[t_0,\infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t\to \infty$.

We obtain conditions for existence and (almost) non-oscillation of solutions of a second order linear homogeneous functional differential equations $$u^{\text{'}\text{'}}\left(x\right)+\sum _i{p}_i\left(x\right)u^{\text{'}}\left(h_i\left(x\right)\right)+\sum _i{q}_i\left(x\right)u\left(g_i\left(x\right)\right)=0$$ without the delay conditions $h_i\left(x\right),g_i\left(x\right)\le x$, $i=1,2,...$, and $$u^{\text{'}\text{'}}\left(x\right)+{\int }_0^{\infty }{u}^{\text{'}}\left(s\right){\mathrm{d}}_s{r}_1(x,s)+{\int }_0^{\infty }u\left(s\right){\mathrm{d}}_s{r}_0(x,s)=0.$$

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

Consider the third order nonlinear dynamic equation $x^{\Delta \Delta \Delta }\left(t\right)+p\left(t\right)f\left(x\right)=g\left(t\right)$, (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $\Delta ³x\left(n\right)+n^{\alpha }{\left|x\right|}^{\gamma }sgn\left(n\right)=(-1)ⁿn^c$, where α ≥ -1, γ > 0, c > 3, is oscillatory.

Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi $h(x)\text{sgn}x\ge 0$ $for|x|>R$, si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

We consider the second order self-adjoint differential equation (1) (r(t)y’(t))’ + p(t)y(t) = 0 on an interval I, where r, p are continuous functions and r is positive on I. The aim of this paper is to show one possibility to write equation (1) in the same form but with positive coefficients, say r₁, p₁ and to derive a sufficient condition for equation (1) to be oscillatory in the case p is nonnegative and ${\int }^{\infty }[1/r\left(t\right)]dt$ converges.

We obtain a necessary and sufficient condition for $L^p$ boundedness of commutators of certain oscillatory integral operators and Lipschitz functions.

Sufficient oscillation conditions involving $lim\; sup$ and $lim\; inf$ for first-order differential equations with non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples, numerically solved in MATLAB, are also given to illustrate the applicability and strength of the obtained conditions over known ones.

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.

We study the existence of positive solutions to the fourth-order two-point boundary value problem $$\left\{\begin{array}{cc}{u}^{\text{'}\text{'}\text{'}\text{'}}\left(t\right)+f(t,u\left(t\right))=0,\hfill & 0<t<1,\hfill \\ u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=0,\hfill & u\left(0\right)=\alpha \left[u\right],\hfill \end{array}\right.$$ where $\alpha \left[u\right]={\int }_0^{1}u\left(t\right)\mathrm{d}A\left(t\right)$ is a Riemann-Stieltjes integral with $A\ge 0$ being a nondecreasing function of bounded variation and $f\in 𝒞([0,1]\times {ℝ}_{+},{ℝ}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

Oscillation criteria are obtained for nonlinear homogeneous third order differential equations of the form $y^{\text{'}\text{'}\text{'}}+q\left(t\right)y^{\text{'}}+p\left(t\right)y^{\alpha }=0$ and y”’ + q(t)y’ + p(t)f(y) = 0, where p and q are real-valued continuous functions on [a,∞), f is a real-valued continuous function on (-∞, ∞) and α > 0 is a quotient of odd integers. Sign restrictions are imposed on p(t) and q(t). These results generalize some of the results obtained earlier in this direction.