Oscillation theorems for nth order nonlinear functional differential equations

In the modern geometric approach partial differential equations are cast into equivalent ideals of differential forms. The invariance of forms under transformation groups is used for constructing invariant solutions by geometric methods. In the present paper the concept of partially invariant solutions introduced earlier by Ovsjannikov is studied in order to obtain geometric methods for partially invariant solutions too.     

Boundedness theorems for some fourth order differential equations

Summary In this paper a new approach involving the use of two signum functions together with a suitably chosen Lyapunov function is employed to investigate the boundedness property of solutions of two special cases of(1.3). This approach makes for considerable reduction in the conditions imposed on f, g in an earlier paper[1]. Entrata in Redazione il 25 ottobre 1970.     

Oscillation theorems for second-order functional differential equations

This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ₁, τ₂ be nonnegative continuous functions on (0, ∞) such that τ₁ ? τ₂ is a bounded function on [1, ∞) and t ? τ₁(t) → ∞ if t → ∞. Then $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_1\text{(t)) = 0}$ is oscillatory if and only if $\text{y}\text{?}\text{(t) + q(t) y(t ? τ}{}_2\text{(t)) = 0}$ is oscillatory.     

Oscillation theorems for second order nonlinear differential equations

Some oscillation criteria are established for certain second order nonlinear differential equations of the form (a(t)ψ(x(t)) x^. (t))^. + p(t) x^. (t) + q(t)f(x(t)) = 0. These criteria improve upon some of the known results by Kura, Kamenev and Philos.     

Oscillation of fourth order sub-linear differential equations

In this paper, we deal with oscillatory and asymptotic properties of solutions of a fourth order sub-linear differential equation with the oscillatory operator. We establish conditions for the nonexistence of positive and bounded solutions and an oscillation criterion.     

Oscillation properties of fourth order differential equations

We define various ways in which linear-quadratic control problems can be robust and show that there are conditions which are simultaneously necessary and sufficient for robust problems to have a solution when such conditions do not exist for nonrobust problems.     

Oscillation theorems for second order quasilinear perturbed differential equations

Abstract. New oscillation criteria for the second order perturbed differential equation are pre-sented. The special case of the results includes the corresponding results in previous papers,extends and unifies a number of known results.     

Nonlinear oscillation of fourth order functional differential equations

Summary In the oscillation theory of nonlinear differential equations one of the important problems is to find necessary and sufficient conditions for the equations under consideration to be oscillatory. Beginning with the pionearing work of F. V. Atkinson, there have been a number of papers. Recently, Kusano and Naito proved the interesting results to the jourth order nonlinear ordinary differential equations of the from [r(t)y″(t)]″+y(t)F(y(t) ₂ ,t)=0. In the present paper, we will extend them to the more general functional differential equations and improve the not clear parts of them. Also, we will propose a new simple definition of nonlinearity of the functional differential equations. Entrata in Redazione il 5 settembre 1977.     

Oscillation criteria of a class of fourth order differential equations

In the present paper, we consider the oscillation of a class of self‐adjoint fourth order differential equation. Some oscillation and non‐oscillation criteria on a perturbation equation of Euler differential equation are given. In particular, we find the exact value of a constant in some oscillation criteria, which has answered the open problem presented by Do?lý in 2004. Copyright © 2011 John Wiley & Sons, Ltd.     

Oscillation criteria for a class of nonlinear fourth order neutral differential equations

In this paper, sufficient conditions are obtained for oscillation of a class of nonlinear fourth order mixed neutral differential equations of the form (E) $$\left( {\frac{1} {{a\left( t \right)}}\left( {\left( {y\left( t \right) + p\left( t \right)y\left( {t - \tau } \right)} \right)^{\prime \prime } } \right)^\alpha } \right)^{\prime \prime } = q\left( t \right)f\left( {y\left( {t - \sigma _1 } \right)} \right) + r\left( t \right)g\left( {y\left( {t + \sigma _2 } \right)} \right)$$ under the assumption $$\int\limits_0^\infty {\left( {a\left( t \right)} \right)^{\tfrac{1} {\alpha }} dt} = \infty .$$ where α is a ratio of odd positive integers. (E) is studied for various ranges of p(t).     

二阶非线性椭圆型微分方程解的振动比较定理

建立几个微分不等式,讨论了一类二阶非线性椭圆型微分方程解的振动性,得到几个新的振动比较定理．     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.