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 theory of first order functional differential equations with deviating arguments

Summary New oscillation criteria are established for the first order functional differential equation (*) y'(t)+p(t)y(g(t))=0and its nonlinear analogue. The results are presented so that a remarkable duality existing between the case where (*) is retarded (g(t)t) is apparent. Possible extension of the results for (*) to equations with several deviating arguments is attempted. Finally, it is shown that there exists a class of autonomous equations for which the oscillation situation can be completely characterized.     

Oscillation of first-order differential equations with retarded arguments

In this paper, we provide a test under which every solution of a first-order delay differential equation oscillates. An example is given to illustrate the significance of the result.     

On analytic solutions of nonlinear differential functional equations with nonlinear deviations of arguments

We obtain the conditions for existence and uniqueness of an analytic solution of a nonlinear differential functional equation with nonlinear deviations of the argument Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 234–240, February, 1999.     

Oscillation of even order nonlinear functional differential equations with damping

This paper is concerned with a class of even order nonlinear damped differential equations

Oscillation of third order functional dynamic equations with mixed arguments on time scales

In this paper we investigate the oscillation of third order nonlinear functional dynamic equation with mixed arguments. Our results extend and improve many known results for oscillation of third order dynamic equations. Some examples are given to illustrate the main results.     

An oscillation criterion for second order nonlinear differential equations with functional arguments

Best possible conditions are given here, under which all solutions of the equation y″(t) + p(t)f(y(t), y(g(t))) = 0 are oscillatory.     

Asymptotic properties of third-order half-linear neutral dynamic equations with mixed arguments

This work is concerned with the asymptotic behavior of a class of third-order half-linear neutral dynamic equations with mixed arguments on a time scale. Some new criteria and an example are presented.     

Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities

where τ0, p(t)C¹[0,∞), q(t),q_i(t),e(t)C[0,∞), p(t)>0, . Without assuming that q(t), q_i(t) and e(t) are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.     

Oscillation of second-order nonlinear differential equations with nonlinear damping

This paper is concerned with the oscillation of a class of general type second-order differential equations with nonlinear damping terms. Several new oscillation criteria are established for such a class of differential equations under quite general assumptions. Examples are also given to illustrate the results.     

Oscillation of nonlinear vector differential equations

Summary Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈E^m, where G is an exterior domain in E_n, and b is a continuous nonnegative valued function in G × E^m. A solution v: G→E^m is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in Eⁿ satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G. Entrata in Redazione il 26 giugno 1975.     

二阶非线性泛函微分方程解的振动性质

研究了一类二阶非线性泛函微分方程解的振动性以及非振动解的有界性.在一定条件下,建立了几个新的振动性和有界性定理,其结果推广和改进了已有的一些结果.     

Oscillation of second-order nonlinear differential equations with damping

We study oscillatory properties of solutions to a class of nonlinear second-order differential equations with a nonlinear damping. New oscillation criteria extend those reported in [ROGOVCHENKO, Yu. V.—TUNCAY, F.: Oscillation criteria for second-order nonlinear differential equations with damping, Nonlinear Anal. 69 (2008), 208–221] and improve a number of related results.