The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x\left(n\right)+\sum _{k=-p}^q{a}_k\left(n\right)x(n+k)=0,n>n_0,$$ where $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ is the difference operator and $\left\{a_k\left(n\right)\right\}$ are sequences of real numbers for $k=-p,...,q$, and $p>0$, $q\ge 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

This paper is concerned with the delay partial difference equation (1) $A_{m+1,n}+A_{m,n+1}-A_{m,n}+{\Sigma }_{i=1}^u{p}_i{A}_{m-k_i,n-l_i}=0$ where $p_i$ are real numbers, $k_i$ and $l_i$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form $$\Delta \left(a_n\left(\Delta \left(b_n{\left(\Delta y_n\right)}^{\alpha }\right)\right)\right)+q_{n}f\left(y_{\sigma \left(n\right)}\right)=0$$ to have property $\left(\mathrm{A}\right)$ or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results.

We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $$\Delta \left(p\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha }\right)+\eta \left(n\right)y^{\beta }(n-k)=0$$ under the condition ${\sum }_{n=n_0}^{\infty }{p}^{-\frac{1}{\alpha }}\left(n\right)<\infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

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 → ∞.

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type $${\left(r\left(t\right){\left(z^{\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+\sum _{i=1}^m{q}_i\left(t\right)x^{{\alpha }_i}\left({\sigma }_i\left(t\right)\right)=0,t\ge t_0,$$ where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \right(t\left)\right)$. Under the assumption ${\int }^{\infty }{\left(r\left(\eta \right)\right)}^{-1/\gamma }\mathrm{d}\eta =\infty$, we consider two cases when $\gamma >{\alpha }_i$ and $\gamma <{\alpha }_i$. Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

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.

The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation ${\left[p\left(t\right){\left[{\left(r\left(t\right)x^{\Delta }\left(t\right)\right)}^{\Delta }\right]}^{\gamma }\right]}^{\Delta }+q\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right)=0$, t ≥ t₀, on a time scale , where γ > 0 is a quotient of odd positive integers, and p, q, r and τ are positive right-dense continuous functions defined on . We classify the nonoscillatory solutions into certain classes $C_i$, i = 0,1,2,3, according to the sign of the Δ-quasi-derivatives and obtain sufficient conditions in order that $C_i=\varnothing$. Also,...

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.

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...