The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type ∆x(n) + ∑ q k=−p ak(n)x(n + k) = 0, n > n0, where ∆x(n) = x(n + 1) − x(n) is the difference operator and {ak(n)} are sequences of real numbers for k = −p, . . . , q, and p > 0, q > 0. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.
In this paper we present some new oscillatory criteria for the $n$-th order neutral differential equations of the form \[ (x(t)\pm p(t)x[\tau (t)])^{(n)} +q(t)x[\sigma (t)] =0. \] The results obtained extend and improve a number of existing criteria.
Sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations \[ (-1)^{m+1}\frac{d^my_i(t)}{dt^m} + \sum ^n_{j=1} q_{ij} y_j(t-h_{jj})=0, \quad m \ge 1, \ i=1,2,\ldots ,n, \] to be oscillatory, where $q_{ij} \varepsilon (-\infty ,\infty )$, $h_{jj} \in (0,\infty )$, $i,j = 1,2,\ldots ,n$. Also, we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations \[ (-1)^{m+1} \frac{d^m}{dt^m} (y_i(t)+cy_i(t-g)) + \sum ^n_{j=1} q_{ij} y_j (t-h)=0,
\] where $c$, $g$ and $h$ are real constants and $i=1,2,\ldots ,n$.
In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of $n$th order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.
This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.
The asymptotic and oscillatory behavior of solutions of mth order damped nonlinear difference equation of the form \[ \Delta (a_n \Delta ^{m-1} y_n) + p_n \Delta ^{m-1} y_n + q_n f(y_{\sigma (n+m-1)}) = 0 \] where $m$ is even, is studied. Examples are included to illustrate the results.
In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form yn+3 + rnyn+2 + qnyn+1 + pnyn = 0, n ≥ 0. These results are generalization of the results concerning difference equations with constant coefficients yn+3 + ryn+2 + qyn+1 + pyn = 0, n ≥ 0. Oscillation, nonoscillation and disconjugacy of a certain class of linear third order difference equations are discussed with help of a class of linear second order difference equations.
In this paper we are concerned with the oscillation of solutions of a certain more general higher order nonlinear neutral type functional differential equation with oscillating coefficients. We obtain two sufficient criteria for oscillatory behaviour of its solutions.
Necessary and sufficient conditions are obtained for oscillation of all bounded solutions of (∗) [y(t) − y(t − τ )](n) + Q(t)G(y(t − σ)) = 0, t ≥ 0, where n ≥ 3 is odd. Sufficient conditions are obtained for all solutions of (∗) to oscillate. Further, sufficient conditions are given for all solutions of the forced equation associated with (∗) to oscillate or tend to zero as t → ∞. In this case, there is no restriction on n.
We study oscillatory properties of the second order half-linear difference equation \[ \Delta (r_k|\Delta y_k|^{\alpha -2}\Delta y_k)-p_k|y_{k+1}|^{\alpha -2}y_{k+1}=0, \quad \alpha >1. \qquad \mathrm{(HL)}\] It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation \[ \Delta (r_k\Delta y_k)-p_ky_{k+1}=0. \] We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.