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On the oscillation of a class of linear homogeneous third order differential equations

N. Parhi, P. Das (1998)

Archivum Mathematicum

In this paper we have considered completely the equation y ' ' ' + a ( t ) y ' ' + b ( t ) y ' + c ( t ) y = 0 , ( * ) where a ∈ C 2 ( [ σ , ∞ ) , R ) , b ∈ C 1 ( [ σ , ∞ ) , R ) , c ∈ C ( [ σ , ∞ ) , R ) and σ ∈ R such that a ( t ) ≤ 0 , b ( t ) ≤ 0 and c ( t ) ≤ 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.

On the oscillation of certain class of third-order nonlinear delay differential equations

S. H. Saker, J. Džurina (2010)

Mathematica Bohemica

In this paper we consider the third-order nonlinear delay differential equation (*) ( a ( t ) x ' ' ( t ) γ ) ' + q ( t ) x γ ( τ ( t ) ) = 0 , t ≥ t 0 , where a ( t ) , q ( t ) are positive functions, γ > 0 is a quotient of odd positive integers and the delay function τ ( t ) ≤ t satisfies lim t → i n f t y τ ( t ) = i n f t y . We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria....

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