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Si dimostra un teorema sul comportamento asintotico delle soluzioni di un'equazione non lineare del secondo ordine.

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 the paper, conditions are obtained, in terms of coefficient functions, which are necessary as well as sufficient for non-oscillation/oscillation of all solutions of self-adjoint linear homogeneous equations of the form $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q{y}_n=0,n\ge 1,$$ where $q$ is a constant. Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of all solutions of nonlinear non-homogeneous equations of the type $$\Delta \left(p_{n-1}\Delta y_{n-1}\right)+q_{n}g\left(y_n\right)=f_{n-1},n\ge 1,$$ where, unlike earlier works, $f_n\ge 0$ or $\le 0$ (but $\neg \equiv 0)$ for large $n$. Further, these results are used to obtain...

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.

Some results concerning oscillation of second order self-adjoint matrix differential equations are obtained. These may be regarded as a generalization of results for the corresponding scalar equations.

The lower bounds of the spacings b-a or a’-a of two consecutive zeros or three consecutive zeros of solutions of third order differential equations of the form y”’ + q(t)y’ + p(t)y = 0 (*) are derived under very general assumptions on p and q. These results are then used to show that $t_{n+1}-tₙ\to \infty$ or $t_{n+2}-tₙ\to \infty$ as n → ∞ under suitable assumptions on p and q, where ⟨tₙ⟩ is a sequence of zeros of an oscillatory solution of (*). The Opial-type inequalities are used to derive lower bounds of the spacings d-a or b-d for...

In questa Nota si danno condizioni sufficienti perchè tutte le soluzioni delle equazioni ${(r(t)y^{\prime })}^{\prime }-p(t)y^{\nu }=f(t)$ ed $y^{\mathrm{\prime \prime }}-g(t,y)=f(t)$ siano non oscillatorie.

A Liapunov-type inequality for a class of third order delay-differential equations is derived.

In this paper we have considered completely the equation $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0,(*)$$ where $a\in C^2([\sigma ,\infty ),R)$, $b\in C^1([\sigma ,\infty ),R)$, $c\in C\left(\right[\sigma ,\infty ),R)$ and $\sigma \in R$ such that $a\left(t\right)\le 0$, $b\left(t\right)\le 0$ and $c\left(t\right)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.

This paper deals with property A and B of a class of canonical linear homogeneous delay differential equations of $n$-th order.

Sufficient conditions are obtained in terms of coefficient functions such that a linear homogeneous third order differential equation is strongly oscillatory.

In this paper, oscillation and asymptotic behaviour of solutions of $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0$$ have been studied under suitable assumptions on the coefficient functions $a,b,c\in C\left(\right[\sigma ,\infty ),R)$, $\sigma \in R$, such that $a\left(t\right)\ge 0$, $b\left(t\right)\le 0$ and $c\left(t\right)<0$.

In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form $$y_{n+3}+r_n{y}_{n+2}+q_n{y}_{n+1}+p_n{y}_n=0,n\ge 0.$$ These results are generalization of the results concerning difference equations with constant coefficients $$y_{n+3}+r{y}_{n+2}+q{y}_{n+1}+p{y}_n=0,n\ge 0.$$ 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.

Sufficient conditions are obtained for oscillation of all solutions of a class of forced nth order linear and nonlinear neutral delay differential equations. Also, asymptotic behaviour of nonoscillatory solutions of a class of forced first order neutral equations is studied.

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 → ∞.