Oscillation criteria for third-order nonlinear differential equations

In this paper, we are concerned with the oscillation properties of the third order differential equation

Oscillation criteria for second order nonlinear differential equations

Summary Various sufficient conditions are obtained which guarantee that all continuable solutions of (1.1) y″+q(t)y′+p(t)f(y)=0 are oscillatory. No explicit sign assumptions are made on p(t) although certain integral conditions are assumed to hold with regard to f(y), p(t) and q(t). Examples are given of the form p(t) = λ/t^μ + (βsint)/t^α, λ, β, μ, α>0. This research was supported by NRC Grant A-7673 and CMC Summer Research Grant. Entrata in Redazione il 4 settembre 1971.     

Nonoscillation criteria for second order nonlinear differential equations

A number of known nonoscillation criteria for the second order nonlinear differential equation y″ + q(x) y^γ = 0, γ > 0, where q is positive and locally of bounded variation, are improved by energy function techniques.     

A posteriori estimators for nonlinear elliptic partial differential equations

Many works have reported results concerning the mathematical analysis of the performance of a posteriori error estimators for the approximation error of finite element discrete solutions to linear elliptic partial differential equations. For each estimator there is a set of restrictions defined in such a way that the analysis of its performance is made possible. Usually, the available estimators may be classified into two types, i.e., the implicit estimators (based on the solution of a local problem) and the explicit estimators (based on some suitable norm of the residual in a dual space). Regarding the performance, an estimator is called asymptotically exact if it is a higher-order perturbation of a norm of the exact error. Nowadays, one may say that there is a larger understanding about the behavior of estimators for linear problems than for nonlinear problems. The situation is even worse when the nonlinearities involve the highest derivatives occurring in the PDE being considered (strongly nonlinear PDEs). In this work we establish conditions under which those estimators, originally developed for linear problems, may be used for strongly nonlinear problems, and how that could be done. We also show that, under some suitable hypothesis, the estimators will be asymptotically exact, whenever they are asymptotically exact for linear problems. Those results allow anyone to use the knowledge about estimators developed for linear problems in order to build new reliable and robust estimators for nonlinear problems.     

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x″+ p ( t ) x′+ q ( t ) f ( x ) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 694–700, May, 2008.     

二阶非线性矩阵微分方程的振动性

Some new oscillation criteria are established for the second-order matrix differential system(r(t)Z′(t))′ p(t)Z′(t) Q(t)F(Z′(t))G(Z(t)) = 0, t ≥ to ＞ 0,are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. The results weaken the condition of Q(t) and generalize some well-known results of Wong (1999) to nonlinear matrix differential equation.     

Quasi-periodic solutions of nonlinear elliptic partial differential equations

In a recent paper [9] the KAM theory has been extended to non-linear partial differential equations, to construct quasi-periodic solutions. In this article this theory is illustrated with three typical examples: an elliptic partial differential equation, an ordinary differential equation and a difference equation related to monotone twist mappings.     

Interval criteria for oscillation of second-order nonlinear differential equations

By employing the generalized Riccati technique and the integral averaging technique, new interval oscillation criteria are established for second-order nonlinear differential equations.     

Oscillation criteria for second order nonlinear neutral differential equations

By refining the standard integral averaging technique, we obtain new oscillation criteria for a class of second order nonlinear neutral differential equations of the form

(r(t)(x(t)+p(t)x(t-τ))^′)^′+q(t)f(x(t),x(σ(t)))=0.     

On elliptic solutions of nonlinear ordinary differential equations

The problem of constructing and classifying exact elliptic solutions of autonomous nonlinear ordinary differential equations is studied. An algorithm for finding elliptic solutions in explicit form is presented.     

Oscillation criteria for nonlinear second order damped differential equations

For all solutions of a class of second order nonlinear damped differential equations, new oscillation criteria are established. Asymptotic behavior for forced equations is also discussed.     

Oscillation criteria for second-order nonlinear differential equations with damping

For a second-order nonlinear differential equation with a damping term, we obtain new oscillation criteria without an assumption that has been required for related results reported in the literature over the last two decades. Two approaches refining the standard integral averaging technique suggested in the paper can be used to derive a variety of simpler oscillation theorems for different classes of nonlinear differential equations, and both sets of oscillation criteria established in the paper are of independent interest. Several examples are provided to illustrate the relevance of the new theorems.     

New oscillation criteria of second-order nonlinear differential equations

By employing a class of new functions Φ=Φ(t,s,l) and a generalized Riccati technique, some new oscillation and interval oscillation criteria are established for the second-order nonlinear differential equation

(r(t)y^′(t))^′+Q(t,y(t),y^′(t))=0.     

Oscillation criteria for nonlinear second order differential equations with damping

Summary Using the integral average method, we give some new oscillation criteria for the second order differential equation with damped term (a(t)Ψ(x(t))K(x'(t)))'+p(t)K(x'(t))+q(t)f(x(t))=0, t<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>≧t₀. These results improve and generalize the oscillation criteria in<span lang=EN-US style='font-size:10.0pt;mso-ansi-language:EN-US'>[1], because they eliminate both the differentiability of p(t) and the sign of p(t), q(t). As a consequence, improvements of Sobol's type oscillation criteria are obtained.     

Oscillation criteria for forced second-order nonlinear differential equations with damping

In this paper, we are concerned with oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using a classical variational principle and an averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. An example is also given to illustrate the results.     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

In this paper, we consider the second-order nonlinear differential equation

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.