In this paper, necessary and sufficient conditions for the existence of nonoscillatory solutions of the forced nonlinear difference equation ∆(xn = pnxτ(n)) + f(n, xσ(n)) = qn are obtained. Examples are included to illustrate the results.
In this paper, necessary and sufficient conditions are obtained for every bounded solution of \[ [y (t) - p (t) y (t - \tau )]^{(n)} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \ge 0, \qquad \mathrm{(*)}\] to oscillate or tend to zero as $t \rightarrow \infty $ for different ranges of $p (t)$. It is shown, under some stronger conditions, that every solution of $(*)$ oscillates or tends to zero as $t \rightarrow \infty $. Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by Ladas and Sficas, Austral. Math. Soc. Ser. B 27 (1986), 502–511, and generalize some known results.
In this paper we consider the third-order nonlinear delay differential equation (∗) (a(t) x ′′(t) ) γ ) ′ + q(t)x γ (τ (t)) = 0, t ≥ t0, where a(t), q(t) are positive functions, γ > 0 is a quotient of odd positive integers and the delay function τ (t) 6 t satisfies lim t→∞ τ (t) = ∞. We establish some sufficient conditions which ensure that (∗) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.
Necessary and sufficient conditions are obtained for every solution of
\[ \Delta (y_{n}+p_{n}y_{n-m})\pm q_{n}G(y_{n-k})=f_{n} \] to oscillate or tend to zero as $n\rightarrow \infty $, where $p_{n}$, $q_{n}$ and $f_{n}$ are sequences of real numbers such that $q_{n}\ge 0$. Different ranges for $p_{n}$ are considered.
In the paper we offer criteria for oscillation of the third order Euler differential equation with delay y ′′′(t) + k 2 ⁄ t 3 y(ct) = 0. We provide detail analysis of the properties of this equation, we fill the gap in the oscillation theory and provide necessary and sufficient conditions for oscillation of equation considered.
Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form ∆(an(∆2 (xn + bnxn−δ))α ) + qnx α n+1−τ = 0 and ∆(an(∆2 (xn − bnxn−δ))α ) + qnx α n+1−τ = 0 are established. Some examples are presented to illustrate the main results.
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.
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.
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.
The second order linear difference equation (1) ∆(rk∆xk) + ckxk+1 = 0, where rk ≠ 0 and k ∈ ℤ , is considered as a special type of symplectic systems. The concept of the phase for symplectic systems is introduced as the discrete analogy of the Borůvka concept of the phase for second order linear differential equations. Oscillation and nonoscillation of (1) and of symplectic systems are investigated in connection with phases and trigonometric systems. Some applications to summation of number series are given, too.