Oscillatory solutions of Emden-Fowler type differential equation

The paper deals with the coexistence between the oscillatory dynamics and the nonoscillatory one for a generalized super-linear Emden–Fowler differential equation. In particular, the coexistence of infinitely many oscillatory solutions with unbounded positive solutions are proved. The asymptotics of the unbounded positive solutions are described as well

Nonoscillatory solutions for super-linear Emden-Fowler type dynamic equations on time scales

In this paper, we consider the following Emden-Fowler type dynamic equations on time scales \begin{equation*} \big(a(t)|x^\Delta(t)|^\alpha \operatorname{sgn} x^\Delta(t)\big)^\Delta+b(t)|x(t)|^\beta \operatorname{sgn}x(t)=0, \end{equation*} when $\alpha<\beta$. The classification of the nonoscillatory solutions are investigated and some necessary and sufficient conditions of the existence of oscillatory and nonoscillatory solutions are given by using the Schauder-Tychonoff fixed point theorem. Three possibilities of two classes of double integrals which are not only related to the coefficients of the equation but also linked with the classification of the nonoscillatory solutions and oscillation of solutions are put forward. Moreover, an important property of the intermediate solutions on time scales is indicated. At last, an example is given to illustrate our main results

Oscillation criteria for fourth order half-linear differential equations

summary:Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form \begin{equation*} \big (|y^{\prime \prime }|^\alpha {\rm sgn\ } y^{\prime \prime }\big )^{\prime \prime } + q(t)|y|^\alpha {\rm sgn}\ y = 0, \quad t \ge a > 0, A \end{equation*} where $\alpha > 0$ is a constant and $q(t)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega (t)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies $\int _a^\infty 1/(t\omega (t))\,dt < \infty$

Existence and classification of nonoscillatory solutions of two dimensional time scale systems

During the past years, there has been an increasing interest in studying oscillation and nonoscillation criteria for dynamic equations and systems on time scales that harmonize the oscillation and nonoscillation theory for the continuous and discrete cases in order to combine them in one comprehensive theory and eliminate obscurity from both. We not only classify nonoscillatory solutions of dynamic equations and systems on time scales but also guarantee the (non)existence of such solutions by using the Knaster fixed point theorem, Schauder - Tychonoff fixed point theorem, and Schauder fixed point theorem. The approach is based on the sign of nonoscillatory solutions. A short introduction to the time scale calculus is given as well. Examples are significant in order to see if nonoscillatory solutions exist or not. Therefore, we give several examples in order to highlight our main results for the set of real numbers R, the set of integers Z, and qN0 = {1, q, q2, q3, ...}, q \u3e 1, which are the most well-known time scales --Abstract, page iv

Asymptotic proximity to higher order nonlinear differential equations

Abstract The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation

Oscillation of third order differential equation with damping term

summary:We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x'''(t)+q(t)x'(t)+r(t)|x|^{\lambda }(t)\mathop {\rm sgn} x(t)=0 ,\quad t\geq 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case $\lambda \leq 1$ and if the corresponding second order differential equation $h''+q(t)h=0$ is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions

Asymptotic and oscillatory behavior of higher order quasilinear delay differential equations

In the paper, we offer such generalization of a lemma due to Philos (and partially Staikos), that yields many applications in the oscillation theory. We present its disposal in the comparison theory and we establish new oscillation criteria for $n-$th order delay differential equation \begin{equation*} \left(r(t)\left[x'(t)\right]^{\gamma}\right)^{(n-1)}+q(t)x^{\gamma}(\tau(t))=0.\tag{$E$} \end{equation*} The presented technique essentially simplifies the examination of the higher order differential equations