This paper discusses oscillatory and asymptotic properties of solutions of a class of third-order nonlinear neutral differential equations. Some new sufficient conditions for a solution of the equation to be either oscillatory or to converges to zero are presented. The results obtained can easily be extended to more general neutral differential equations as well as to neutral dynamic equations on time scales. Two examples are provided to illustrate the results.
This note is concerned with the oscillation of third order nonlinear delay differential equations of the form (r2(t) (r1(t)y'(t))')' +p(t)y'(t) + q(t)ƒ(y(g(t))) = 0. (*) In the papers [A.Tiryaki, M.F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M.F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear-functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (*) oscillates or converges to zero, provided that the second order equation (r2(t)z'(t))' + (p(t)/r1(t))z(t) =0 (**) is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (*) oscillates if equation (**) is nonoscillatory. We also establish results for the oscillation of equation (*) when equation (**) is oscillatory.
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Some new criteria for the oscillation of advanced functional differential equations of the form are presented in this paper. A discussion of neutral equations will also be included.
In this paper we investigate the oscillatory character of the second order nonlinear difference equations of the forms (wzór) n = 1,2, ... and the corresponding nonhomogeneous equation (wzór) n= 1,2,... via comparison with certain second order linear difference equations where the function f is not necessarily monotonic. The results of this paper are essentially new and can be extended to more general equations.
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